Planar tilings based on Goursat tetrahedra

Higher-dimensional geometry (previously "Polyshapes").

Planar tilings based on Goursat tetrahedra

Postby Marek14 » Sun May 01, 2016 9:27 am

I was wondering about this: Goursat tetrahedra can, obviously, tile their respective spaces. And since all their dihedral angles divide the straight angle, it means that there are planes tiled by the faces of the tetrahedra (though, they don't necessarily have to be tiled by ONE type of faces).

But how do those tilings actually look? Part of the problem is to derive the face angles of Goursat tetrahedra from their dihedral angles. So far I've managed to understand the reason for weird dihedral angle of tetrahedron (it's double the arcsin of sqrt(3)/3, and I now understand why it must be so). But is there a general way to determine triangles of tetrahedron from its dihedral angles?
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Re: Planar tilings based on Goursat tetrahedra

Postby wendy » Sun May 01, 2016 12:14 pm

If you know the dihedral angles of a tetrahedron, you can immediately populate a dynkins matrix for it, and from that a stott matrix. If the thing is a tiling, it still works, though.

Some points to note. If you imagine that there is a 'drop of paint' on a mirror, it will walk to every mirror to which it is connected by an odd branch, but not step over an even branch, so the mirrors in o---o---o-5-o are all the same, but o---o---o-4-x the o mirrors can't reach the x mirror. There is a mirror-group comprised entirely of the o and another comprised entirely of the x mirrors.

The dual of o---o---x---o has faces entirely comprised of the cell-wall marked x, and this is true for each kind of mirror.
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Re: Planar tilings based on Goursat tetrahedra

Postby Marek14 » Sun May 01, 2016 1:02 pm

Hm, ok, let's try an example with pentachoric group {3,3,3}.

Dynkin matrix should be 1/2 *

Code: Select all
( 2 -1  0  0)
(-1  2 -1  0)
( 0 -1  2 -1)
( 0  0 -1  2)


This is then inverted to Stott matrix, which comes out as 1/5 *

Code: Select all
a ( 8  6  4  2)
b ( 6 12  8  4)
c ( 4  8 12  6)
d ( 2  4  6  8)


Now, I suppose I can take the rows as vectors and compute their angles?

This gives me:

a&b or c&d: acos(4*sqrt(2/39)) = 25.066°
a&c or b&d: acos(7/sqrt(78)] = 37.571°
a&d: acos[2/3] = 48.190°
b&c: acos[12/13] = 22.620°

What's the next step? How can I use these angles to find actual face angles of this tetrahedron?
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Re: Planar tilings based on Goursat tetrahedra

Postby Klitzing » Sun May 01, 2016 7:22 pm

Hi Marek,
not too clear what you are truely after here. But did you already read that http://bendwavy.org/klitzing/explain/dihedral.htm? Might be that helps.
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Re: Planar tilings based on Goursat tetrahedra

Postby Marek14 » Sun May 01, 2016 8:10 pm

Klitzing wrote:Hi Marek,
not too clear what you are truely after here. But did you already read that http://bendwavy.org/klitzing/explain/dihedral.htm? Might be that helps.
--- rk


Well, what I am after is to start with a face plane of a Goursat tetrahedron and see how the rest of that plane looks. The whole plane should be formed by faces of that tetrahedron in certain configuration. That's what I'm going after.
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Re: Planar tilings based on Goursat tetrahedra

Postby wendy » Sun May 01, 2016 11:48 pm

It should not be too hard. A tiling of Goursat simplexes, in any dimension, is a pennant tiling, because a reflection in any face will cause all but one vertex to stay still. This means that if you number the vertices of a simplex 0 to n, the whole tiling consists of verticies of type 0 to n.

In the Conway Hart system, 0-n represents the centres of the surtopes of 0 to n dimensions. In a symmetry group, which is in essence, a goursat simplex, 0 to n represent the nodes of the graph. When the two groups intersect, you get the regular figures, with Coxeter's 'transitive on the flags' as the definition of the intersection here.

The resulting tiling can be constructed as follows.

If the margin is odd (eg, 3, 5) the next cell is found by rolling the simplex over the margin. If the margin is even, then the next cell is found by a reflection through the wall. What lies in the plane contains a symmetry group, but does not need to be completely one itself. It does need to be made of cell walls, though.
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Re: Planar tilings based on Goursat tetrahedra

Postby Marek14 » Mon May 02, 2016 5:54 am

wendy wrote:It should not be too hard. A tiling of Goursat simplexes, in any dimension, is a pennant tiling, because a reflection in any face will cause all but one vertex to stay still. This means that if you number the vertices of a simplex 0 to n, the whole tiling consists of verticies of type 0 to n.

In the Conway Hart system, 0-n represents the centres of the surtopes of 0 to n dimensions. In a symmetry group, which is in essence, a goursat simplex, 0 to n represent the nodes of the graph. When the two groups intersect, you get the regular figures, with Coxeter's 'transitive on the flags' as the definition of the intersection here.

The resulting tiling can be constructed as follows.

If the margin is odd (eg, 3, 5) the next cell is found by rolling the simplex over the margin. If the margin is even, then the next cell is found by a reflection through the wall. What lies in the plane contains a symmetry group, but does not need to be completely one itself. It does need to be made of cell walls, though.


I got that far, yes. I explain how I got there, that would be the best.

It started as a musing about directions. How many "special directions" there is in a polytope or a tiling and what will you go through when you follow them? It's best seen in omnitruncates -- for example when you are on a truncated octahedron (omnitruncated tetrahedron), there's only one kind of circuits to follow. If you start in a square, you go to a hexagon, then through opposite face to another hexagon, and another square, where it repeats. Twice, in this case.
But truncated cuboctahedron (omnitruncated cube) has two fundamentally different circuits: 4-8-4 and 4-6-8-6-4.

In 4D, this is very similar. If we take, for example, omnitruncated {4,3,5}, there are up to six possible fundamental directions (one for each combination of cells), but some of them might merge together. In this case, we have:
A 5-fold direction (the line has 5-fold symmetry around it) passing through alternating truncated icosidodecahedra and decagonal prisms.
A 4-fold direction passing through alternating truncated cuboctahedra and octagonal prisms.
A 3-fold direction passing through alternating truncated cuboctahedra and truncated icosidodecahedra.
A 2-fold direction passing through sequence truncated cuboctahedron - decagonal prism - octagonal prism - decagonal prism - truncated cuboctahedron
A 2-fold direction passing through alternating truncated icosidodecahedra and octagonal prisms.

If you had, for example, a chess variant on this tiling -- or on any tiling with this symmetry, these would be the basic five types of movement, comparable to orthogonal and diagonal movement in normal chess.

And then I became interested in how a plane through the tiling would look. In 3D, there are 4 basic kinds planes (corresponding to 4 sides of the Goursat tetrahedron), but thanks to odd branches, some of them might merge into one. So the question was how many kinds of fundamental planes there are for various tilings and how do they look?

One practical effect would be that it would allow to more easily draw a thin "slice" of a hyperbolic tiling which should be more easily pondered than full 3D rendering.
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Re: Planar tilings based on Goursat tetrahedra

Postby wendy » Mon May 02, 2016 9:18 am

The number of different planes is not hard to find.

A goursat tiling, for being a crossing of planes, is an alternation of black and white cells. You see this, for example, in presentations of the symmetries as alternating black and white triangles.

An odd margin, will open up that margin flat, and have white cells on each side.

An even margin, will reflect the same wall, and will have different cells above and below the plane.

If you take a goursat tetrahedra made of triangles, the outside would be black and the inside white, the mirror image would have the inside black and the outside white: that is the cells support a kind of 'out-vector' that reverses only on even margins.

You get as many different kinds of wall, as there are sets of cells that are separated from each other entirely by even walls. For example, in o3o4o3o, it is not possible to reach mirrors 3 or 4 from either 1 or 2, but you can reach mirror 3 from 4, and 1 from 2. If you remove the even branches, it reduces to two separate things o3o.o3o, and these are the separate mirrors.

Note that the space enclosed by even mirrors only form a reflective region. Such a region is a subgroup of the whole group.

For example, in the icosahedral group, the triangles have vertices 2,3,5, meaning that 4, 6, 10 meet at a corner. If you start at a 2-corner, the straight line continues to a 3 vertex, and then from the 3 to a 5 vertex, all the time with white on the right. Then you cross a 2 vertex, and white is on the left, so you then pass through a 5, 3, and 2 vertex. The total space bounded by these lines form a face of a {3,4}, of alternating black and white mirrors. All mirrors are the same, and you end up with the five octahedra.

In 2,3,4, you have two mirrors, the edge 2-4 can not be reached from 2-3 or 3-4 mirrors. The 2-4 mirrors form loops of four mirrors 2-4-2-4-2-4-2-4 while the 2-3 and 3-4 mirrors form a loop 2-3-4-3-2-3-4-3... through the vertex and midedge of an octahedron. What we get here is that the tetrahedral group is the intersection of three mirror groups formed by pairs of crossing lines: the tetrahedron is in three ways xo2ox&#xt disphenoid tetrahedron.

In {4,3,5}, for example, you can see that the rule of even branches apply, so o4o3o5o gives o.o=o=o, two different mirrors. One set of mirrors is represented by the first wall only, leads to dodecahedral cells, while the remaining three mirrors give the symmetry o5o3oAo, formed by the icosahedral reflection applied to a rhombic tricontahedron, three to an edge, or icosahedra, five to an edge. This is what o5o3mAo, o5o3oAm, and m5o3oAo mean. Since both are mirror-walls, we now look at the symmetry point at the centre of the face of a tricontahedron. This is the figure o5m3oAo, a flattened tetrahedron. The four sharp edges belong to angles of c/5, the eight blunt edges are c/3 (c=2pi).
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Re: Planar tilings based on Goursat tetrahedra

Postby Marek14 » Sun May 29, 2016 3:24 pm

OK, thought about this a bit more and I found an algorithm that should work:

As an example, let's consider the octahedral prismatic group with Dynkin diagram o o3o4o. This is the simplest group where each of the four nodes is distinct.

Step 1: Construct an omnitruncated polytope/honeycomb. In this case, this will be x x3x4x -- truncated cuboctahedral prism.

Step 2: For each node, consider three cells and three faces that include this node.

Node 1: x .3.4. -- Faces: squares x x3.4., x .3x4. and x .3.4x. Cells: hexagonal prism x x3x4., cube x x3.4x, and octagonal prism x .3x4x
Node 2: . x3.4. -- Faces: squares x x3.4. and . x3.4x, and hexagon . x3x4. Cells: hexagonal prism x x3x4, cube x x3.4x, and truncated cuboctahedron . x3x4x
Node 3: . .3x4. -- Faces: square x .3x4., hexagon . x3x4., and octagon . .3x4x Cells: hexagonal prism x x3x4, octagonal prism x .3x4x, and truncated cuboctahedron . x3x4x
Node 4: . .3.4x -- Faces: squares x .3.4x and . x3.4x, and octagon . .3x4x Cells: cube x x3.4x, octagonal prism x .3x4x, and truncated cuboctahedron . x3x4x

Now, each of these can be identified with a particular triangle:

Node 1: inner angle between normal vectors to adjacent square faces of hexagonal prism is 60 degrees (180 - dihedral angle between the faces for Euclidean version of polyhedron). For cube, this is 90 degrees, for octagonal prism 45 degrees. So we have a spherical triangle with angles 60, 90, 45.

Node 2: Here we need dihedral angles of truncated cuboctahedron. These are 144.736 for square-hexagon (central angle 35.264), 125.264 for hexagon-octagon (central angle 54.736) and 135 for square-octagon (central angle 45). Inner angle square-n-gon in any n-gonal prism is naturally 90. So Node 2 corresponds to a triangle with angles 90, 90 and 35.264 degrees.

Node 3: Similarly, here it is 90, 90 and 54.736 degrees.

Node 4: Here it is 90, 90 and 45 degrees.

Now, nodes 2 and 3 are joined by an odd branch (3). This means that these two triangles will be combined for a plane tiling. Generally, any group of nodes that is connected through odd branches will be combined.

So we have 3 possible planes:

Plane 1: tiled with 90-60-45 triangles.
Plane 23: tiled with combination of Node 2 and Node 3 triangles; the weird central angles add to 90 degrees.
Plane 4: tiled with 90-90-45 triangles.

Other fundamental domains should yield planes in the same way.

So, for example, {4,3,5}.

If we mark the nodes A, B, C, D, there are two planes, one tiled solely by A triangles, and one tiled by a combination of B, C and D.

Triangle A has angles 36, 45, 90 degrees. It's the Schwarz triangle of {4,5}, which makes sense since {4,5} and {4,3,5} are related.

Triangle B has angles 20.905, 54.736, 90 degrees. Side across from right angle leads to triangle C, other two sides are reflective.
Triangle C has angles 35.264, 37.377, 90 degrees. Side across from 37.377 is reflective, side across from right angle leads to triangle B, side across from 35.264 leads to triangle D.
Triangle D has angles 31.717, 45, 90 degrees. Side across from 45 leads to triangle C, other two sides are reflective.

Vertices of this tiling:
90-degree vertex of B-triangle. Has 4 B-triangles around it, forming a rhombus with angles 41.81 and 109.472.
54.736-degree vertex of B-triangle is combined with 35.264-degree vertex of C-triangle; these two angles add to 90 degrees, so there are 4 B-triangles and 4 C-triangles around this vertex, in pattern BCCBBCCB. These eight triangles together form a square with inner angle 58.282 degrees.
20.905-degree vertex of B-triangle is combined with 37.377-degree vertex of C-triangle and 31.717-degree vertex of D-triangle; these three angles add to 90 degrees, so there are 4 of each around the vertex, in pattern BCDDCBBCDDCB. These twelve triangles together form a right-angled hexagon, though it's not necessarily a regular one.
90-degree vertex of C-triangle is combined with 90-degree vertex of D-triangle; there are 4 triangles around the vertex in pattern CCDD. Together they form a deltoid with angles 70.529, 69.094, 90, 69.094.
45-degree vertex of D-triangle. Has 8 D-triangles around it, forming a square with inner angle 63.434 degrees.
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Re: Planar tilings based on Goursat tetrahedra

Postby Marek14 » Wed Jun 01, 2016 8:28 am

I believe I have cracked this, I have successfully managed to find plane tilings with triangles for all 4-dimensional groups and most of the 5-dimensional ones, and if I find the dihedral angles in polytera and higher, there is no reason why it couldn't continue higher.

Code: Select all
{3,3,3}
A and D triangles identical.
B and C triangles identical.
One plane tiled by combination of A, B, C, and D.

Triangle A: 54.736, 60,     90; ABA
Triangle B: 54.736, 70.529, 90; ABC
Triangle C: 54.736, 70.529, 90; DCB
Triangle D: 54.736, 60,     90; DCD

Vertices:
A-54.736 & B-70.529 & C-54.736
Pattern: AABCCB
Triangle with angles 120, 125.265, 125.265

A-60
Pattern: AAAAAA
Triangle with angles 109.472, 109.472, 109.472

A-90 & B-90
Pattern: AABB
Quadrangle with angles 109.472, 125.265, 120, 125.265

B-54.736 & C-70.529 & D-54.736
Pattern: BBCDDC
Triangle with angles 120, 125.265, 125.265

C-90 & D-90
Pattern: CCDD
Quadrangle with angles 109.472, 125.265, 120, 125.265

D-60
Pattern: DDDDDD
Equilateral triangle with angle 109.472

Repeating unit: Digonal strip of angle 60. Composed of 1 A, 1 B, 1 C and 1 D.

{3,3,4}
Plane 1: tiled by combination of A, B, and C.
Plane 2: tiled solely by D.

Triangle A: 45,     54.736, 90; BAA
Triangle B: 35.264, 70.529, 90; ABC
Triangle C: 54.736, 54.736, 90; CCB - double of triangle A. Note that each of its 54.736 angles belongs to a different type of vertex.
Triangle D: 45,     60,     90; DDD

Vertices:
A-45
Pattern: AAAAAAAA
Square with angle 109.472

A-54.736 & B-70.529 & C-54.736
Pattern: AABCCB
Triangle with angles 90, 90, 109.472

A-90 & B-90
Pattern: AABB
Quadrangle with angles 90, 125.265, 109.472, 125.265

B-35.264 & C-54.736
Pattern: BBCCBBCC
Square with angle 125.265

C-90
Pattern: CCCC
Square with angle 109.472

D-45
Pattern: DDDDDDDD
Square with angle 120

D-60
Pattern: DDDDDD
Equilateral triangle with angle 90

D-90
Pattern: DDDD
Rhombus with angles 90, 120, 90, 120

Repeating unit 1: Triangle with angles 45, 90, 90. Composed of 1 A, 1 B, 1 C.
Repeating unit 2: Triangle D.

{3,3,5}
One plane tiled by combination of A, B, C, and D.

Triangle A: 36,     54.736, 90; BAA
Triangle B: 20.905, 70.529, 90; ABC
Triangle C: 37.377, 54.736, 90; CDB
Triangle D: 31.717, 60,     90; DCD

Vertices:
A-36
Pattern: AAAAAAAAAA
Regular pentagon with angle 109.472

A-54.736 & B-70.529 & C-54.736
Pattern: AABCCB
Triangle with angles 58.282, 58.282, 72

A-90 & B-90
Pattern: AABB
Quadrangle with angles 41.810, 125.265, 72, 125.265

B-20.905 & C-37.377 & D-31.717
Pattern: BBCDDCBBCDDC
Hexagon with angles 120, 125.265, 125.265, 120, 125.265, 125.265

C-90 & D-90
Pattern: CCDD
Quadrangle with angles 69.094, 109.472, 69.094, 120

D-60
Pattern: DDDDDD
Equilateral triangle with angle 63.434

Repeating unit: Triangle with angles 36, 60, 90. Composed of 1 A, 1 B, 1 C, 1 D.

{3,4,3}
A and D triangles identical.
B and C triangles identical.
Plane 1: tiled by combination of A and B
Plane 2: tiled by combination of C and D

Triangle A: 35.264, 60,     90; ABA
Triangle B: 45,     54.736, 90; ABB
Triangle C: 45,     54.736, 90; DCC
Triangle D: 35.264, 60,     90; DCD

Vertices:
A-35.264 & B-54.736
Pattern: AABBAABB
Rhombus with angles 90, 120, 90, 120

A-60
Pattern: AAAAAA
Equilateral triangle with angle 70.528

A-90 & B-90
Pattern: AABB
Quadrangle with angles 90, 90, 90, 120

B-45
Pattern: BBBBBBBB
Square with angle 109.472

C-45
Pattern: CCCCCCCC
Square with angle 109.472

C-54.736 & D-35.264
Pattern: CCDDCCDD
Rhombus with angles 90, 120, 90, 120

C-90 & D-90
Pattern: CCDD
Quadrangle with angles 90, 90, 90, 120

D-60
Pattern: DDDDDD
Equilateral triangle with angle 70.529

Repeating unit 1: Triangle with angles 45, 60, 90. Composed of 1 A, 1 B.
Repeating unit 2: Triangle with angles 45, 60, 90. Composed of 1 C, 1 D.

Branched 333 (demitesseractic)
A, C, and D triangles identical.
One plane tiled by combination of A, B, C, and D.

Triangle A: 54.736, 54.736, 90    ; AAB
Triangle B: 70.529, 70.529, 70.529; ABC
Triangle C: 54.736, 54.736, 90    ; CCB
Triangle D: 54.736, 54.736, 90    ; DDB

Vertices:
A-54.736 & B-70.529 & C-54.736
Pattern: AABCCB
Quadrangle with angles 125.265, 125.265, 125.265, 125.265 (not square, apparently)

A-54.736 & B-70.529 & D-54.736
Pattern: AABDDB
Quadrangle with angles 125.265, 125.265, 125.265, 125.265 (not square, apparently)

A-90
Pattern: AAAA
Square with angle 109.472

B-70.529 & C-54.736 & D-54.736
Pattern: BCCBDD
Quadrangle with angles 125.265, 125.265, 125.265, 125.265 (not square, apparently)

C-90
Pattern: CCCC
Square with angle 109.472

D-90
Pattern: DDDD
Square with angle 109.472

Repeating unit: Equilateral triangle of angle 90. Composed of 1 A, 1 B, 1 C and 1 D.

{4,3,4}
A and D triangles identical.
B and C triangles identical.
Plane 1: tiled solely by A
Plane 2: tiled by combination of B and C
Plane 3: tiled solely by D

Triangle A: 45,     45,     90; AAA
Triangle B: 35.264, 54.736, 90; BBC
Triangle C: 35.264, 54.736, 90; CCB
Triangle D: 45,     45,     90; DDD

Vertices:
A-45
Pattern: AAAAAAAA
Square with angle 90

A-45
Pattern: AAAAAAAA
Square with angle 90
(The two 45 angles at A look identical within the plane, but differ in how other tetrahedron faces are connected to them.)

A-90
Pattern: AAAA
Square with angle 90

B-35.264 & C-54.736
Pattern: BBCC
Rectangle with angle 90

B-54.736 & C-35.264
Pattern: BBCC
Rectangle with angle 90

B-90
Pattern: BBBB
Rhombus with angles 70.529, 109.472, 70.529, 109.472

C-90
Pattern: CCCC
Rhombus with angles 70.529, 109.472, 70.529, 109.472

D-45
Pattern: DDDDDDDD
Square with angle 90

D-45
Pattern: DDDDDDDD
Square with angle 90

D-90
Pattern: DDDD
Square with angle 90

Repeating unit 1: Triangle A.
Repeating unit 2: Rectangle of angle 90. Composed of 1 B, 1 C.
Repeating unit 3: Triangle D.

Branched 334 (tetrahedral/octahedral honeycomb)
A and C triangles identical.
Plane 1: tiled by combination of A, B, and C
Plane 2: tiled solely by D

Triangle A: 35.264, 54.736, 90    ; AAB
Triangle B: 54.736, 54.736, 70.529; ACB (double of A or C triangle)
Triangle C: 35.264, 54.736, 90    ; CCB
Triangle D: 45,     45,     90    ; DDD

Vertices:
A-35.264 & B-54.736
Pattern: AABBAABB
Hexagon with angles 109.472, 125.565, 125.565, 109.472, 125.565, 125.565

A-54.736 & B-70.529 & C-54.736
Pattern: AABCCB
Square with angle 90

A-90
Pattern: AAAA
Rhombus with angles 70.529, 109.472, 70.529, 109.472

B-54.736 & C-35.264
Pattern: BBCCBBCC
Hexagon with angles 109.472, 125.565, 125.565, 109.472, 125.565, 125.565

C-90
Pattern: CCCC
Rhombus with angles 70.529, 109.472, 70.529, 109.472

D-45
Pattern: DDDDDDDD
Square with angle 90

D-45
Pattern: DDDDDDDD
Square with angle 90

D-90
Pattern: DDDD
Square with angle 90

Repeating unit 1: Rectangle of angle 90. Composed of 1 A, 1 B, 1 C.
Repeating unit 2: Triangle D.

Cyclical 3333
Triangles A, B, C, and D all identical.
One plane tiled by combination of A, B, C, and D.

Triangle A: 54.736, 54.736, 70.529; BDA
Triangle B: 54.736, 54.736, 70.529; ACB
Triangle C: 54.736, 54.736, 70.529; BDC
Triangle D: 54.736, 54.736, 70.529; ACD

Vertices:
A-54.736 & B-70.529 & C-54.736
Pattern: AABCCB
Hexagon with angles 109.472, 125.565, 125.565, 109.472, 125.565, 125.565

A-54.736 & C-54.736 & D-70.529
Pattern: AADCCD
Hexagon with angles 109.472, 125.565, 125.565, 109.472, 125.565, 125.565

A-70.529 & B-54.736 & D-54.736
Pattern: ABBADD
Hexagon with angles 109.472, 125.565, 125.565, 109.472, 125.565, 125.565

B-54.736 & C-70.529 & D-54.736
Pattern: BBCDDC
Hexagon with angles 109.472, 125.565, 125.565, 109.472, 125.565, 125.565

Repeating unit: An infinite strip formed by repeating triangles A, B, C, D.

{4,3,5}
Plane 1: tiled solely by A
Plane 2: tiled by combination of B, C, and D

Triangle A: 36,     45,     90; AAA
Triangle B: 20.905, 54.736, 90; BBC
Triangle C: 35.264, 37.377, 90; DCB
Triangle D: 31.717, 45,     90; DCD

Vertices:
A-36
Pattern: AAAAAAAAAA
Regular pentagon with angle 90

A-45
Pattern: AAAAAAAA
Square with angle 72

A-90
Pattern: AAAA
Rhombus with angles 72, 90, 72, 90

B-20.905 & C-37.377 & D-31.717
Pattern: BBCDDCBBCDDC
Right-angled hexagon, not regular

B-54.736 & C-35.264
Pattern: BBCCBBCC
Rectangle with angle 58.282

B-90
Pattern: BBBB
Rhombus with angles 41.810, 109.472, 41.810, 109.472

C-90 & D-90
Pattern: CCDD
Quadrangle with angles 69.095, 70.529, 90, 69.095

D-45
Pattern: DDDDDDDD
Square with angle 63.434

Repeating unit: Quadrangle with angles 45, 90, 90, 90. Composed of 1 B, 1 C, 1 D.

{5,3,5}
A and D triangles identical.
B and C triangles identical.
One plane tiled by combination of A, B, C, and D.

Triangle A: 31.717, 36,     90; ABA
Triangle B: 20.905, 37.377, 90; ABC
Triangle C: 20.905, 37.377, 90; DCB
Triangle D: 31.717, 36,     90; DCC

Vertices:
A-31.717 & B-37.377 & C-20.905
Pattern: AABCCBAABCCB
Hexagon with angles 58.282, 58.282, 72, 58.282, 58.282, 72

A-36
Pattern: AAAAAAAAAA
Regular pentagon with angle 63.434

A-90 & B-90
Pattern: AABB
Quadrangle with angles 41.810, 69.095, 72, 69.095

B-20.905 & C-37.377 & D-31.717
Pattern: BBCDDCBBCDDC
Hexagon with angles 58.282, 58.282, 72, 58.282, 58.282, 72

C-90 & D-90
Pattern: CCDD
Quadrangle with angles 41.810, 69.095, 72, 69.095

D-36
Pattern: DDDDDDDDDD
Regular pentagon with angle 63.434

Repeating unit: Quadrangle with angles 36, 90, 36, 90. Composed of 1 A, 1 B, 1 C, 1 D.

{3,5,3}
A and D triangles identical.
B and C triangles identical.
One plane tiled by combination of A, B, C, and D.

Triangle A: 20.905,     60, 90; ABA
Triangle B: 31.717, 37.377, 90; ABC
Triangle C: 31.717, 37.377, 90; DCB
Triangle D: 20.905,     60, 90; DCD

Vertices:
A-20.905 & B-37.377 & C-31.717
Pattern: AABCCBAABCCB
Hexagon with angles 69.095, 69.095, 120, 69.095, 69.095, 120

A-60
Pattern: AAAAAA
Equilateral triangle with angle 41.810

A-90 & B-90
Pattern: AABB
Quadrangle with angles 58.282, 63.434, 58.282 and 120

B-31.717 & C-37.377 & D-20.905
Pattern: BBCDDCBBCDDC
Hexagon with angles 69.095, 69.095, 120, 69.095, 69.095, 120

C-90 & D-90
Pattern: CCDD
Quadrangle with angles 58.282, 63.434, 58.282 and 120

D-60
Pattern: AAAAAA
Equilateral triangle with angle 41.810

Repeating unit: Quadrangle with angles 60, 90, 60, 90. Composed of 1 A, 1 B, 1 C, 1 D.

Branched 335 (tetrahedral/icosahedral honeycomb)
A and C triangles identical.
One plane tiled by combination of A, B, C, and D.

Triangle A: 20.905, 54.736,     90; AAB
Triangle B: 37.377, 37.377, 70.529; ACD
Triangle C: 20.905, 54.736,     90; CCB
Triangle D: 31.717, 31.717,     90; DDB

Vertices:
A-20.905 & B-37.377 & D-31.717
Pattern: AABDDBAABDDB
Octagon with alternating angles of 69.095 and 125.565

A-54.736 & B-70.529 & C-54.736
Pattern: AABCCB
Rectangle with angle 58.282

A-90
Pattern: AAAA
Rhombus with angles 41.810, 109.472, 41.810, 109.472

B-37.377 & C-20.905 & D-31.717
Pattern: BCCBDDBCCBDD
Octagon with alternating angles of 69.095 and 125.565

C-90
Pattern: CCCC
Rhombus with angles 41.810, 109.472, 41.810, 109.472

D-90
Pattern: DDDD
Square with angle 63.434

Repeating unit: Right-angled pentagon; not regular. Composed of 1 A, 1 B, 1 C, 1 D.

Cyclical 3334
A and D triangles identical.
B and C triangles identical.
One plane tiled by combination of A, B, C, and D.

Triangle A: 45,     54.736, 54.736; BAA
Triangle B: 35.264, 54.736, 70.529; CAB
Triangle C: 35.264, 54.736, 70.529; BDC
Triangle D: 45,     54.736, 54.736; CDD

Vertices:
A-45
Pattern: AAAAAAAA
Regular octagon with angle 109.472

A-54.736 & B-35.264
Pattern: AABBAABB
Octagon with angles 90, 125.565, 109.472, 125.565, 90, 125.565, 109.472, 125.565

A-54.736 & B-70.529 & C-54.736
Pattern: AABCCB
Hexagon with angles 70.529, 125.565, 90, 90, 90, 125.565

B-54.736 & C-70.529 & D-54.736
Pattern: BBCDDC
Hexagon with angles 70.529, 125.565, 90, 90, 90, 125.565

C-35.264 & D-54.736
Pattern: CCDDCCDD
Octagon with angles 90, 125.565, 109.472, 125.565, 90, 125.565, 109.472, 125.565

D-45
Pattern: DDDDDDDD
Regular octagon with angle 109.472

Repeating unit: Quadrangle with angles 45, 90, 45, 90. Composed of 1 A, 1 B, 1 C, 1 D.

Cyclical 3335
A and D triangles identical.
B and C triangles identical.
One plane tiled by combination of A, B, C, and D.

Triangle A: 31.717, 37.377, 54.736; BAD
Triangle B: 20.905, 54.736, 70.529; CAB
Triangle C: 20.905, 54.736, 70.529; BDC
Triangle D: 31.717, 37.377, 54.736; CDA

Vertices:
A-31.717 & C-20.905 & D-37.377
Pattern: AADCCDAADCCD
Dodecagon with angles 69.095, 109.472, 69.095, 125.565, 109.472, 125.565, 69.095, 109.472, 69.095, 125.565, 109.472, 125.565

A-37.377 & B-20.905 & D-31.717
Pattern: ABBADDABBADD
Dodecagon with angles 69.095, 109.472, 69.095, 125.565, 109.472, 125.565, 69.095, 109.472, 69.095, 125.565, 109.472, 125.565

A-54.736 & B-70.529 & C-54.736
Pattern: AABCCD
Hexagon with angles 41.810, 125.565, 58.282, 63.434, 58.282, 125.565

B-54.736 & C-70.529 & D-54.736
Pattern: BBCDDC
Hexagon with angles 41.810, 125.565, 58.282, 63.434, 58.282, 125.565

Repeating unit: An infinite strip formed by repeating triangles A, B, C, D. Its two edges form right-angled pseudogons.

Cyclical 3434
Triangles A, B, C, and D all identical.
Plane 1: tiled by combination of A and B
Plane 2: tiled by combination of C and D

Triangle A: 35.264, 45, 54.736; ABA
Triangle B: 35.264, 45, 54.736; BAB
Triangle C: 35.264, 45, 54.736; CDC
Triangle D: 35.264, 45, 54.736; DCD

Vertices:
A-35.264 & B-54.736
Pattern: AABBAABB
Right-angled octagon, not regular

A-45
Pattern: AAAAAAAA
Octagon with alternating angles 70.529 and 125.565

A-54.736 & B-35.264
Pattern: AABBAABB
Right-angled octagon, not regular

B-45
Pattern: BBBBBBBB
Octagon with alternating angles 70.529 and 125.565

C-35.264 & D-54.736
Pattern: CCDDCCDD
Right-angled octagon, not regular

C-45
Pattern: CCCCCCCC
Octagon with alternating angles 70.529 and 125.565

C-54.736 & D-35.264
Pattern: CCDDCCDD
Right-angled octagon, not regular

D-45
Pattern: DDDDDDDD
Octagon with alternating angles 70.529 and 125.565

Repeating unit 1: Quadrangle with angles 45, 90, 45, 90. Composed of 1 A, 1 B.
Repeating unit 2: Quadrangle with angles 45, 90, 45, 90. Composed of 1 C, 1 D.

Cyclical 3435
A and D triangles identical.
B and C triangles identical.
One plane tiled by combination of A, B, C, and D.

Triangle A: 31.717, 35.264, 37.377; BDA
Triangle B: 20.905, 45,     54.736; BAB
Triangle C: 20.905, 45,     54.736; CDC
Triangle D: 31.717, 35.264, 37.377; CAD

Vertices:
A-31.717 & C-20.905 & D-37.377
Pattern: AADCCDAADCCD
Dodecagon with angles 69.095, 70.529, 69.095, 90, 90, 90, 69.095, 70.529, 69.095, 90, 90, 90

A-35.264 & B-54.736
Pattern: AABBAABB
Octagon with angles 58.282, 63.434, 58.282, 90, 58.282, 63.434, 58.282, 90

A-37.377 & B-20,905 & D-31.717
Pattern: ABBADDABBADD
Dodecagon with angles 69.095, 70.529, 69.095, 90, 90, 90, 69.095, 70.529, 69.095, 90, 90, 90

B-45
Pattern: BBBBBBBB
Octagon with alternating angles 41.810 and 109.472

C-45
Pattern: CCCCCCCC
Octagon with alternating angles 41.810 and 109.472

C-54.736 & D-35.264
Pattern: CCDDCCDD
Octagon with angles 58.282, 63.434, 58.282, 90, 58.282, 63.434, 58.282, 90

Repeating unit: Hexagon with angles 45, 90, 90, 45, 90, 90. Composed of 1 A, 1 B, 1 C, 1 D.

Cyclical 3535
Triangles A, B, C, and D all identical.
One plane tiled by combination of A, B, C, and D.

Triangle A: 20.905, 31.717, 37.377; DBA
Triangle B: 20.905, 31.717, 37.377; CAB
Triangle C: 20.905, 31.717, 37.377; BDC
Triangle D: 20.905, 31.717, 37.377; ACD

Vertices:
A-20.905 & C-31.717 & D-37.377
Pattern: AADCCDAADCCD
Dodecagon with angles 41.810, 58.282, 69.095, 63.434, 69.095, 58.282, 41.810, 58.282, 69.095, 63.434, 69.095, 58.282

A-31.717 & B-37.377 & C-20.905
Pattern: AABCCBAABCCB
Dodecagon with angles 41.810, 58.282, 69.095, 63.434, 69.095, 58.282, 41.810, 58.282, 69.095, 63.434, 69.095, 58.282

A-37.377 & B-20.905 & D-31.717
Pattern: ABBADDABBADD
Dodecagon with angles 41.810, 58.282, 69.095, 63.434, 69.095, 58.282, 41.810, 58.282, 69.095, 63.434, 69.095, 58.282

B-31.717 & C-37.377 & D-20.905
Pattern: BBCDDCBBCDDC
Dodecagon with angles 41.810, 58.282, 69.095, 63.434, 69.095, 58.282, 41.810, 58.282, 69.095, 63.434, 69.095, 58.282

Repeating unit: An infinite strip formed by repeating triangles A, B, C, D. Its two edges form right-angled pseudogons.

{3,3,6}
Plane 1: tiled by combination of A, B, and C.
Plane 2: tiled solely by D.

Triangle A: 30, 54.736, 90; BAA
Triangle B: 0,  70.529, 90; ABC
Triangle C: 0,  54.736, 90; CCB
Triangle D: 0,  60,     90; DDD

Vertices:
A-30
Pattern: AAAAAAAAAAAA
Regular hexagon with angle 109.472

A-54.736 & B-70.529 & C-54.736
Pattern: AABCCB
Triangle with angles 0, 0, 60

A-90 & B-90
Pattern: AABB
Quadrangle with angles 0, 125.565, 60, 125.565

B-0 & C-0
Pattern: BBCC...
Apeirogon with angle 125.565, not regular

C-90
Pattern: CCCC
Rhombus with angles 0, 109.472, 0, 109.472

D-0
Pattern: D...
Regular apeirogon with angle 120

D-60
Pattern: DDDDDD
Equilateral triangle with angle 0

D-90
Pattern: DDDD
Rhombus with angles 0, 120, 0, 120

Repeating unit 1: Triangle with angles 0, 30, 90. Composed of 1 A, 1 B, 1 C.
Repeating unit 2: Triangle D.

{4,3,6}
Plane 1: tiled solely by A.
Plane 2: tiled by combination of B and C.
Plane 3: tiled solely by D.

Triangle A: 30, 45,     90; AAA
Triangle B: 0,  54.736, 90; BBC
Triangle C: 0,  35.264, 90; CCB
Triangle D: 0,  45,     90; DDD

Vertices:
A-30
Pattern: AAAAAAAAAAAA
Right-angled dodecagon

A-45
Pattern: AAAAAAAA
Regular octagon with angle 60

A-90
Pattern: AAAA
Rhombus with angles 60, 90, 60, 90

B-0
Pattern: B...
Regular apeirogon with angle 109.472

B-54.736 & C-35.264
Pattern: BBCCBBCC
Rectangle with angle 0

B-90 & C-90
Pattern: BBCC
Quadrangle with angles 0, 90, 0, 90

C-0
Pattern: C...
Regular apeirogon with angle 70.529

D-0
Pattern: D...
Right-angled apeirogon

D-45
Pattern: DDDDDDDD
Square with angle 0

D-90
Pattern: DDDD
Rhombus with angles 0, 90, 0, 90

Repeating unit 1: Triangle A.
Repeating unit 2: Triangle with angles 0, 0, 90. Composed of 1 B and 1 C.
Repeating unit 3: Triangle D.

{5,3,6}
Plane 1: tiled by combination of A, B, and C.
Plane 2: tiled solely by D.

Triangle A: 30, 31.717, 90; BAA
Triangle B: 0,  37.377, 90; ABC
Triangle C: 0,  20.905, 90; CCB
Triangle D: 0,  36,     90; DDD

Vertices:
A-30
Pattern: AAAAAAAA
Regular hexagon with angle 63.434

A-31.717 & B-37.377 & C-20.905
Pattern: AABCCBAABCCB
Hexagon with angles 0, 0, 60, 0, 0, 60

A-90 & B-90
Pattern: AABB
Quadrangle with angles 0, 69.095, 60, 69.095

B-0 & C-0
Pattern: BBCC...
Apeirogon with angle 58.282, not regular

C-90
Pattern: CCCC
Rhombus with angles 0, 41.810, 0, 41.810

D-0
Pattern: D...
Regular apeirogon with angle 72

D-36
Pattern: DDDDDDDDDD
Regular pentagon with angle 0

D-90
Pattern: DDDD
Rhombus with angles 0, 72, 0, 72

Repeating unit 1: Quadrangle with angles 0, 30, 90, 90. Composed of 1 A, 1 B, 1 C.
Repeating unit 2: Triangle D.

{6,3,6}
A and D triangles identical.
B and C triangles identical.
Plane 1: tiled solely by A
Plane 2: tiled by combination of B and C
Plane 3: tiled solely by D

Triangle A: 0, 30, 90; AAA
Triangle B: 0, 0,  90; BBC
Triangle C: 0, 0,  90; CCB
Triangle D: 0, 30, 90; DDD

Vertices:
A-0
Pattern: A...
Regular apeirogon with angle 60

A-30
Pattern: AAAAAAAAAAAA
Regular hexagon with angle 0

A-90
Pattern: AAAA
Rhombus with angles 0, 60, 0, 60

B-0 & C-0
Pattern: BBCC...
Regular apeirogon with angle 0

B-0 & C-0
Pattern: BBCC...
Regular apeirogon with angle 0

B-90
Pattern: BBBB
Square with angle 0

C-90
Pattern: CCCC
Square with angle 0

D-0
Pattern: D...
Regular apeirogon with angle 60

D-30
Pattern: DDDDDDDDDDDD
Regular hexagon with angle 0

D-90
Pattern: DDDD
Rhombus with angles 0, 60, 0, 60

Repeating unit 1: Triangle A.
Repeating unit 2: Rhombus of angles 0, 90, 0, 90. Composed of 1 B, 1 C.
Repeating unit 3: Triangle D.

{3,4,4}
Plane 1: tiled by combination of A and B.
Plane 2: tiled solely by C.
Plane 3: tiled solely by D.

Triangle A: 35.264, 45,     90; ABA
Triangle B: 0,      54.736, 90; ABB
Triangle C: 0,      45,     90; CCC
Triangle D: 0,      60,     90; CCC

Vertices:
A-35.264 & B-54.736
Rhombus with angles 0, 90, 0, 90

A-45
Pattern: AAAAAAAA
Square with angle 70.529

A-90 & B-90
Pattern: AABB
Quadrangle with angles 0, 90, 90, 90

B-0
Pattern: B...
Regular apeirogon with angle 109.472

C-0
Pattern: C...
Right-angled apeirogon

C-45
Pattern: CCCCCCCC
Square with angle 0

C-90
Pattern: CCCC
Rhombus with angles 0, 90, 0, 90

D-0
Pattern: D...
Regular apeirogon with angle 120

D-60
Pattern: DDDDDD
Equilateral triangle with angle 0

D-90
Pattern: DDDD
Rhombus with angles 0, 120, 0, 120

Repeating unit 1: Triangle with angles 0, 45, 90. Composed of 1 A and 1 B.
Repeating unit 2: Triangle C.
Repeating unit 3: Triangle D.

{4,4,4}
A and D triangles identical.
B and C triangles identical.
Plane 1: tiled solely by A.
Plane 2: tiled solely by B.
Plane 3: tiled solely by C.
Plane 4: tiled solely by D.

Triangle A: 0, 45, 90; AAA
Triangle B: 0, 0,  90; BBB
Triangle C: 0, 0,  90; CCC
Triangle D: 0, 45, 90; DDD

Vertices:
A-0
Pattern: A...
Right-angled apeirogon

A-45
Pattern: AAAAAAAA
Square with angle 0

A-90
Pattern: AAAA
Rhombus with angles 0, 90, 0, 90

B-0
Pattern: B...
Regular apeirogon with angle 0

B-0
Pattern: B...
Regular apeirogon with angle 0

B-90
Pattern: BBBB
Square with angle 0

C-0
Pattern: C...
Regular apeirogon with angle 0

C-0
Pattern: C...
Regular apeirogon with angle 0

C-90
Pattern: CCCC
Square with angle 0

D-0
Pattern: D...
Right-angled apeirogon

D-45
Pattern: DDDDDDDD
Square with angle 0

D-90
Pattern: DDDD
Rhombus with angles 0, 90, 0, 90

Repeating unit 1: Triangle A.
Repeating unit 2: Triangle B.
Repeating unit 3: Triangle C.
Repeating unit 4: Triangle D.

{3,6,3}
A and D triangles identical.
B and C triangles identical.
Plane 1: tiled by combination of A and B
Plane 2: tiled by combination of C and D

Triangle A: 0, 60, 90; ABA
Triangle B: 0, 0,  90; ABB
Triangle C: 0, 0,  90; CDC
Triangle D: 0, 60, 90; DCD

Vertices:
A-0 & B-0
Pattern: AABB...
Apeirogon with alternating angles 0 and 120

A-60
Pattern: AAAAAA
Equilateral triangle with angle 0

A-90 & B-90
Pattern: AABB
Quadrangle with angles 0, 0, 0, 120

B-0
Pattern: B...
Regular apeirogon with angle 0

C-0
Pattern: C...
Regular apeirogon with angle 0

C-0 & D-0
Pattern: CCDD...
Apeirogon with alternating angles 0 and 120

C-90 & D-90
Pattern: CCDD
Quadrangle with angles 0, 0, 0, 120

D-60
Pattern: DDDDDD
Equilateral triangle with angle 0

Repeating unit 1: Triangle with angles 0, 0, 60. Composed of 1 A, 1 B.
Repeating unit 2: Triangle with angles 0, 0, 60. Composed of 1 C, 1 D.

Branched 336
A and C triangles identical.
Plane 1: tiled by combination of A, B, and C
Plane 2: tiled solely by D

Triangle A: 0, 54.736, 90    ; AAB
Triangle B: 0, 0,      70.529; ACB
Triangle C: 0, 54.736, 90    ; CCB
Triangle D: 0, 0,      90    ; DDD

Vertices:
A-0 & B-0
Pattern: AABB...
Apeirogon with alternating angles 0 and 109.472

A-54.736 & B-70.529 & C-54.736
Pattern: AABCCB
Hexagon with all angles 0, not regular

A-90
Pattern: AAAA
Rhombus with angles 0, 109.472, 0, 109.472

B-0 & C-0
Pattern: CCDD...
Apeirogon with alternating angles 0 and 109.472

C-90
Pattern: CCCC
Rhombus with angles 0, 109.472, 0, 109.472

D-0
Pattern: D...
Regular apeirogon with angle 0

D-0
Pattern: D...
Regular apeirogon with angle 0

D-90
Pattern: DDDD
Square with angle 0

Repeating unit 1: Quadrangle with angles 0, 0, 90, 90. Composed of 1 A, 1 B, 1 C.
Repeating unit 2: Triangle D.

Branched 344
C and D triangles identical.
Plane 1: tiled by combination of A and B
Plane 2: tiled solely by C
Plane 3: tiled solely by D

Triangle A: 35.264, 35.264, 90    ;
Triangle B: 0,      54.736, 54.736;
Triangle C: 0,      45,     90    ;
Triangle D: 0,      45,     90    ;

Vertices:
A-35.264 & B-54.736
Pattern: AABB
Hexagon with angles 0, 90, 90, 0, 90, 90

A-35.264 & B-54.736
Pattern: AABB
Hexagon with angles 0, 90, 90, 0, 90, 90

A-90
Pattern: AAAA
Square with angle 70.529

B-0
Pattern: B...
Regular apeirogon with angle 109.472

C-0
Pattern: C...
Right-angled apeirogon

C-45
Pattern: CCCCCCCC
Square with angle 0

C-90
Pattern: CCCC
Rhombus with angles 0, 90, 0, 90

D-0
Pattern: D...
Right-angled apeirogon

D-45
Pattern: DDDDDDDD
Square with angle 0

D-90
Pattern: DDDD
Rhombus with angles 0, 90, 0, 90

Repeating unit 1: Quadrangle with angles 0, 90, 90, 90. Composed of 1 A and 1 B.
Repeating unit 2: Triangle C.
Repeating unit 3: Triangle D.

Branched 444
A, C and D triangles identical.
Plane 1: tiled solely by A
Plane 2: tiled solely by B
Plane 3: tiled solely by C
Plane 4: tiled solely by D

Triangle A: 0, 0, 90; AAA
Triangle B: 0, 0, 0 ; BBB
Triangle C: 0, 0, 90; CCC
Triangle D: 0, 0, 90; DDD

Vertices:
A-0
Pattern: A...
Regular apeirogon with angle 0

A-0
Pattern: A...
Regular apeirogon with angle 0

A-90
Pattern: AAAA
Square with angle 0

B-0
Pattern: B...
Regular apeirogon with angle 0

B-0
Pattern: B...
Regular apeirogon with angle 0

B-0
Pattern: B...
Regular apeirogon with angle 0

C-0
Pattern: C...
Regular apeirogon with angle 0

C-0
Pattern: C...
Regular apeirogon with angle 0

C-90
Pattern: CCCC
Square with angle 0

D-0
Pattern: D...
Regular apeirogon with angle 0

D-0
Pattern: D...
Regular apeirogon with angle 0

D-90
Pattern: CCCC
Square with angle 0

Repeating unit 1: Triangle A.
Repeating unit 2: Triangle B.
Repeating unit 3: Triangle C.
Repeating unit 4: Triangle D.

Cyclical 3336
A and D triangles identical.
B and C triangles identical.
One plane tiled by combination of A, B, C, and D.

Triangle A: 0, 0,      54.736; BAA
Triangle B: 0, 54.736, 70.529; CAB
Triangle C: 0, 54.736, 70.529; BDC
Triangle D: 0, 0,      54.736; DCD

Vertices:
A-0
Pattern: A...
Regular apeirogon with angle 0

A-0 & B-0
Pattern: AABB...
Apeirogon with angles 0, 125.565, 109.472, 125.565, 0...

A-54.736 & B-70.529 & C-54.736
Pattern: AABCCB
Hexagon with angles 0, 0, 0, 125.565, 0, 125.565

B-54.736 & C-70.529 & D-54.736
Pattern: AABCCB
Hexagon with angles 0, 0, 0, 125.565, 0, 125.565

C-0 & D-0
Pattern: CCDD...
Apeirogon with angles 0, 125.565, 109.472, 125.565, 0...

D-0
Pattern: D...
Regular apeirogon with angle 0

Repeating unit: Rectangle with angle 0. Composed of 1 A, 1 B, 1 C, 1 D.

Cyclical 3436
A and D triangles identical.
B and C triangles identical.
Plane 1: tiled by combination of A and B
Plane 2: tiled by combination of C and D

Triangle A: 0, 0,  35.264; BAA
Triangle B: 0, 45, 54.736; ABA
Triangle C: 0, 45, 54.736; CDC
Triangle D: 0, 0,  35.264; DCD

Vertices:
A-0
Pattern: A...
Apeirogon with alternating angles 0 and 70.529

A-0 & B-0
Pattern: AABB...
Apeirogon with angles 0, 90, 90, 90...

A-35.264 & B-54.736
Pattern: AABBAABB
Octagon with angles 0, 0, 0, 90, 0, 0, 0, 90

B-45
Pattern: BBBBBBBB
Octagon with alternating angles 0 and 109.472

C-0 & D-0
Pattern: CCDD...
Apeirogon with angles 0, 90, 90, 90...

C-45
Pattern: CCCCCCCC
Octagon with alternating angles 0 and 109.472

C-54.736 & D-35.264
Pattern: CCDDCCDD
Octagon with angles 0, 0, 0, 90, 0, 0, 0, 90

D-0
Pattern: D...
Apeirogon with alternating angles 0 and 70.529

Repeating unit 1: Quadrangle with angles 0, 0, 45, 90. Composed of 1 A and 1 B.
Repeating unit 2: Quadrangle with angles 0, 0, 45, 90. Composed of 1 A and 1 B.

Cyclical 3536
A and D triangles identical.
B and C triangles identical.
One plane tiled by combination of A, B, C, and D.

Triangle A: 0, 0,      20.905; BAA
Triangle B: 0, 31.717, 37.377; CAB
Triangle C: 0, 31.717, 37.377; BDC
Triangle D: 0, 0,      20.905; DCD

Vertices:
A-0
Pattern: A...
Apeirogon with alternating angles 0 and 41.810

A-0 & B-0
Pattern: AABB...
Apeirogon with angles 0, 58.282, 63.434, 58.282, 0...

A-20.905 & B-37.377 & C-31.717
Pattern: AABCCBAABCCB
Dodecagon with angles 0, 0, 0, 69.095, 0, 69.095, 0, 0, 0, 69.095, 0, 69.095

B-31.717 & C-37.377 & D-20.905
Pattern: BBCDDCBBCDDC
Dodecagon with angles 0, 0, 0, 69.095, 0, 69.095, 0, 0, 0, 69.095, 0, 69.095

C-0 & D-0
Pattern: CCDD...
Apeirogon with angles 0, 58.282, 63.434, 58.282, 0...

D-0
Pattern: D...
Apeirogon with alternating angles 0 and 41.810

Repeating unit: Hexagon with angles 0, 0, 90, 0, 0, 90. Composed of 1 A, 1 B, 1 C, 1 D.

Cyclical 3636
Triangles A, B, C, and D all identical.
Plane 1: tiled by combination of A and B
Plane 2: tiled by combination of C and D

Triangle A: 0, 0, 0; BAA
Triangle B: 0, 0, 0; ABB
Triangle C: 0, 0, 0; CCD
Triangle D: 0, 0, 0; DDC

Vertices:
A-0
Pattern: A...
Regular apeirogon with angle 0

A-0 & B-0
Pattern: AABB...
Regular apeirogon with angle 0

A-0 & B-0
Pattern: AABB...
Regular apeirogon with angle 0

B-0
Pattern: B...
Regular apeirogon with angle 0

C-0
Pattern: C...
Regular apeirogon with angle 0

C-0 & D-0
Pattern: CCDD...
Regular apeirogon with angle 0

C-0 & D-0
Pattern: CCDD...
Regular apeirogon with angle 0

D-0
Pattern: D...
Regular apeirogon with angle 0

Repeating unit 1: Square with angle 0. Composed of 1 A, 1 B.
Repeating unit 2: Square with angle 0. Composed of 1 C, 1 D.

Cyclical 3344
A and C triangles identical
Plane 1: tiled by combination of A, B, and C.
Plane 2: tiled solely by D.

Triangle A: 0,      54.736, 54.736; BAA
Triangle B: 35.264, 35.264, 70.529; ACB
Triangle C: 0,      54.736, 54.736; BCC
Triangle D: 0,      45,     45    ; DDD

Vertices:
A-0
Pattern: A...
Regular apeirogon with angle 109.472

A-54.736 & B-35.264
Pattern: AABBAABB
Octagon with angles 0, 125.565, 70.529, 125.565, 0, 125.565, 70.529, 125.565

A-54.736 & B-70.529 & C-54.736
Pattern: AABCCB
Hexagon with angles 0, 90, 90, 0, 90, 90

B-35.264 & C-54.736
Pattern: BBCCBBCC
Octagon with angles 0, 125.565, 70.529, 125.565, 0, 125.565, 70.529, 125.565

C-0
Pattern: C...
Regular apeirogon with angle 109.472

D-0
Pattern: D...
Right-angled apeirogon

D-45
Pattern: DDDDDDDD
Octagon with alternating angles 0 and 90

D-45
Pattern: DDDDDDDD
Octagon with alternating angles 0 and 90

Repeating unit 1: Quadrangle with angles 0, 0, 90, 90. Composed of 1 A, 1 B, 1 C.
Repeating unit 2: Triangle D.

Cyclical 3444
A and B triangles identical
C and D triangles identical
Plane 1: tiled by combination of A and B.
Plane 2: tiled solely by C.
Plane 3: tiled solely by D.

Triangle A: 0, 35.264, 54.736; BAA
Triangle B: 0, 35.264, 54.736; ABB
Triangle C: 0, 0,      45    ; CCC
Triangle D: 0, 0,      45    ; DDD

Vertices:
A-0
Pattern: A...
Apeirogon with alternating angles 70.529 and 109.472

A-35.264 & B-54.736
Pattern: AABBAABB
Octagon with angles 0, 90, 0, 90, 0, 90, 0, 90

A-54.736 & B-35.264
Pattern: AABBAABB
Octagon with angles 0, 90, 0, 90, 0, 90, 0, 90

B-0
Pattern: B...
Apeirogon with alternating angles 70.529 and 109.472

C-0
Pattern: C...
Apeirogon with alternating angles 0 and 90

C-0
Pattern: C...
Apeirogon with alternating angles 0 and 90

C-45
Patttern: CCCCCCCC
Regular octagon with angle 0

D-0
Pattern: D...
Apeirogon with alternating angles 0 and 90

D-0
Pattern: D...
Apeirogon with alternating angles 0 and 90

D-45
Patttern: DDDDDDDD
Regular octagon with angle 0

Repeating unit 1: Quadrangle with angles 0, 90, 0, 90. Composed of 1 A and 1 B.
Repeating unit 2: Triangle C.
Repeating unit 3: Triangle D.

Cyclical 4444
Triangles A, B, C, and D all identical.
Plane 1: tiled solely by A.
Plane 2: tiled solely by B.
Plane 3: tiled solely by C.
Plane 4: tiled solely by D.

Triangle A: 0, 0, 0; AAA
Triangle B: 0, 0, 0; BBB
Triangle C: 0, 0, 0; CCC
Triangle D: 0, 0, 0; DDD

Vertices:
A-0
Pattern: A...
Regular apeirogon with angle 0

A-0
Pattern: A...
Regular apeirogon with angle 0

A-0
Pattern: A...
Regular apeirogon with angle 0

B-0
Pattern: B...
Regular apeirogon with angle 0

B-0
Pattern: B...
Regular apeirogon with angle 0

B-0
Pattern: B...
Regular apeirogon with angle 0

C-0
Pattern: C...
Regular apeirogon with angle 0

C-0
Pattern: C...
Regular apeirogon with angle 0

C-0
Pattern: C...
Regular apeirogon with angle 0

D-0
Pattern: D...
Regular apeirogon with angle 0

D-0
Pattern: D...
Regular apeirogon with angle 0

D-0
Pattern: D...
Regular apeirogon with angle 0

Repeating unit 1: Triangle A.
Repeating unit 2: Triangle B.
Repeating unit 3: Triangle C.
Repeating unit 4: Triangle D.

Triangle with added 3-branch
A and B triangles identical.
One plane tiled by combination of A, B, C, and D.

Triangle A: 0,      54.736,     90; ABC
Triangle B: 0,      54.736,     90; BAC
Triangle C: 0,      70.529, 70.529; DAB
Triangle D: 54.736, 54.736,     60; DDC

Vertices:
A-0 & B-0 & C-0
Pattern: ABC...
Apeirogon with angle 125.565; not regular

A-54.736 & C-70.529 & D-54.736
Pattern: AACDDC
Pentagon with angles 0, 0, 125.565, 120, 125.565

A-90 & B-90
Pattern: AABB
Rhombus with angles 0, 109.472, 0, 109.472

B-54.736 & C-70.529 & D-54.736
Pattern: BBCDDC
Pentagon with angles 0, 0, 125.565, 120, 125.565

D-60
Pattern: DDDDDD
Regular hexagon with angle 109.472

Repeating unit: Regular apeirogon with angle 60. Basically the full A-0 B-0 C-0 vertex with triangle D capping the finite sides of triangles C. A quadrangle with angles 0, 90, 60, 90 can also be considered a repeating unit, but when reflecting through a 0-90 side, triangles A and B will switch.

Triangle with added 4-branch
A and B triangles identical.
Plane 1: tiled by combination of A, B, and C.
Plane 2: tiled solely by D.

Triangle A: 0,  35.264, 90    ; ABC
Triangle B: 0,  35.264, 90    ; BAC
Triangle C: 0,  54.736, 54.736; CAB
Triangle D: 45, 45,     60    ; DDD

Vertices:
A-0 & B-0 & C-0
Pattern: ABC...
Non-regular right-angled apeirogon

A-35.264 & C-54.736
Pattern: AACCAACC
Hexagon with angles 0, 0, 109.472, 0, 0, 109.472

A-90 & B-90
Pattern: AABB
Rhombus with angles 0, 70.529, 0, 70.529

B-35.264 & C-54.736
Pattern: BBCCBBCC
Hexagon with angles 0, 0, 109.472, 0, 0, 109.472

D-45
Pattern: DDDDDDDD
Octagon with alternating angles 90 and 120

D-45
Pattern: DDDDDDDD
Octagon with alternating angles 90 and 120

D-60
Pattern: DDDDDD
Right-angled hexagon.

Repeating unit 1: The basic one is a pentagon with angles 0, 90, 90, 90, 90, but reflecting through a 0-90 side will switch A and B. Full repeating unit is a non-regular right-angled apeirogon.
Repeating unit 2: Triangle D.

Triangle with added 5-branch
A and B triangles identical.
One plane tiled by combination of A, B, C, and D.

Triangle A: 0,      20.905, 90    ; ABC
Triangle B: 0,      20.905, 90    ; BAC
Triangle C: 0,      37.377, 37.377; DAB
Triangle D: 31.717, 31.717, 60    ; DDC

Vertices:
A-0 & B-0 & C-0
Pattern: ABC...
Apeirogon with angle 58.282, not regular

A-20.905 & C-37,377 & D-31.317
Pattern: AACDDCAACDDC
Decagon with angles 0, 0, 69.094, 120, 69.094, 0, 0, 69.094, 120, 69.094

A-90 & B-90
Pattern: AABB
Rhombus with angles 0, 41.810, 0, 41.810

B-20.905 & C-37.377 & 31.717
Pattern: BBCDDCBBCDDC
Decagon with angles 0, 0, 69.094, 120, 69.094, 0, 0, 69.094, 120, 69.094

D-60
Pattern: DDDDDD
Regular hexagon with angle 63.434

Repeating unit: The basic one is a hexagon with angles 0, 90, 90, 60, 90, 90, but reflecting through a 0-90 side will switch A and B. Full repeating unit is an apeirogon with repeating angle sequence 60, 90, 90.

Triangle with added 6-branch
A and B triangles identical.
Plane 1: tiled by combination of A, B, and C.
Plane 2: tiled solely by D.

Triangle A: 0b, 0d, 90c; BAC
Triangle B: 0a, 0d, 90c; ABC
Triangle C: 0a, 0b, 0d ; ABC
Triangle D: 0a, 0b, 60c; DDD

Vertices:
A-0 & B-0 & C-0
Pattern: ABC...
Apeirogon with angle 0

A-0 & C-0
Pattern: AACC...
Apeirogon with angle 0

A-90 & B-90
Pattern: AABB
Square with angle 0

B-0 & C-0
Pattern: BBCC...
Apeirogon with angle 0

D-0
Pattern: D...
Apeirogon with alternating angles 0 and 120

D-0
Pattern: D...
Apeirogon with alternating angles 0 and 120

D-60
Pattern: DDDDDD
Regular hexagon with angle 0

Repeating unit 1: The basic one is a pentagon with angles 0, 0, 90, 0, 90, but reflecting through a 0-90 side will switch A and B. Full repeating unit is an apeirogon with angle 0.
Repeating unit 2: Triangle D.

Two fused triangles
A and D triangles identical.
B and C triangles identical.
One plane tiled by combination of A, B, C, and D.

Triangle A: 0, 54.736, 54.736; ABC
Triangle B: 0, 0,      70.529; ADC
Triangle C: 0, 0,      70.529; ADB
Triangle D: 0, 54.736, 54.736; DCB

Vertices:
A-0 & B-0 & C-0
Pattern: ABC...
Apeirogon with angles 0, 125.565, 125.565...

A-54.736 & B-70.529 & D-54.736
Pattern: AABDDB
Hexagon with angles 0, 0, 109.472, 0, 0, 109.472

A-54.736 & C-70.529 & D-54.736
Pattern: AACDDC
Hexagon with angles 0, 0, 109.472, 0, 0, 109.472

B-0 & C-0 & D-0
Pattern: BCD...
Apeirogon with angles 0, 125.565, 125.565...

Repeating unit: The notion of repeating unit starts breaking down a bit here. The ABD or ACD vertices work, with some swaps caused by reflection.

Tetrahedron of 3-edges
Triangles A, B, C, and D all identical.
One plane tiled by combination of A, B, C, and D.

Triangle A: 0, 0, 0; BCD
Triangle B: 0, 0, 0; ACD
Triangle C: 0, 0, 0; ABD
Triangle D: 0, 0, 0; ABC

Vertices:
A-0 & B-0 & C-0
Pattern: ABC...
Regular apeirogon with angle 0

A-0 & B-0 & D-0
Pattern: ABC...
Regular apeirogon with angle 0

A-0 & C-0 & D-0
Pattern: ABC...
Regular apeirogon with angle 0

B-0 & C-0 & D-0
Pattern: ABC...
Regular apeirogon with angle 0

Repeating unit: Each of the ideal triangles can be considered a repeating unit; they are just differently labeled.

{3,3,3,3}
AB and DE triangles identical.
AC and CE triangles identical.
AD and BE triangles identical.
BC and CD triangles identical.
Plane 1: tiled by combination of AB, BC, CD, and DE.
Plane 2: tiled by combination of AC, AD, AE, BD, BE, and CE.

Triangle AB: 52.239, 60,     90; AB BC AB
Triangle AC: 54.736, 65.905, 90; AC AC AD
Triangle AD: 48.190, 70.529, 90; AE BD AC
Triangle AE: 54.736, 54.736, 90; AD BE AE
Triangle BC: 52.239, 75.522, 90; AB BC CD
Triangle BD: 65.905, 65.905, 90; AD BE BD
Triangle BE: 48.190, 70.529, 90; BD AE CE
Triangle CD: 52.239, 75.522, 90; DE CD BC
Triangle CE: 54.736, 65.905, 90; CE CE BE
Triangle DE: 52.239, 60,     90; DE CD DE

Repeating unit 1: Digonal strip of angle 60, made from AB, BC, CD, DE.
Repeating unit 2: Digonal strip of angle 90, made from AC, AD, AE, BD, BE, CE.

{3,3,3,4}
Plane 1: tiled by combination of AB, BC, and CD.
Plane 2: tiled by combination of AC, AD, and BD.
Plane 3: tiled by combination of AE, BE, and CE.
Plane 4: tiled solely by DE.

Triangle AB: 45,     52.239, 90; BC AB AB
Triangle AC: 35.264, 65.905, 90; AC AC AD
Triangle AD: 48.190, 54.736, 90; AD BD AC
Triangle AE: 45,     54.736, 90; BE AE AE
Triangle BC: 30,     75.522, 90; AB BC CD
Triangle BD: 45,     65.905, 90; AD BD BD
Triangle BE: 35.264, 70.529, 90; AE BE CE
Triangle CD: 52.239, 60,     90; CD CD BC
Triangle CE: 54.736, 54.736, 90; CE CE BE
Triangle DE: 45,     60,     90; DE DE DE

Repeating unit 1: Equilateral triangle with angle 90, made from AB, BC, and CD.
Repeating unit 2: Triangle with angles 45, 90, 90, made from AC, AD, and BD.
Repeating unit 3: Triangle with angles 45, 90, 90, made from AE, BE, and CE.
Repeating unit 4: Triangle DE.

Demipenteractic
A, B are one branch, C is the center, D and E are ends of short branches.
AD and AE triangles identical.
BD and BE triangles identical.
CD and CE triangles identical.
Plane 1: tiled by combination of AB, BC, CD, and CE.
Plane 2: tiled by combination of AC, AD, AE, BD, and BE.
Plane 3: tiled solely by DE.

Triangle AB: 52.239, 52.239, 90    ; AB AB BC
Triangle AC: 65.905, 65.905, 70.529; AD AE AC
Triangle AD: 48.190, 54.736, 90    ; AD BD AC
Triangle AE: 48.190, 54.736, 90    ; AE BE AC
Triangle BC: 75.522, 75.522, 90    ; CD CE AB
Triangle BD: 45,     65.905, 90    ; AD BD BD
Triangle BE: 45,     65.905, 90    ; AE BE BE
Triangle CD: 52.239, 60,     90    ; CE CD BC
Triangle CE: 52.239, 60,     90    ; CD CE BC
Triangle DE: 45,     60,     90    ; DE DE DE

Repeating unit 1: Digonal strip of angle 90, made from AB, BC, CD, and CE (two of each). Half of it, equilateral triangle of angle 90, works, but swaps some labels when reflecting.
Repeating unit 1: Digonal strip of angle 45, made from AC, AD, AE, BD, and BE.
Repeating unit 3: Triangle DE.

C~4 {4,3,3,4}
AB and DE triangles identical.
AC and CE triangles identical.
AD and BE triangles identical.
BC and CD triangles identical.
Plane 1: tiled solely by AB.
Plane 2: tiled by combination of AC and AD.
Plane 3: tiled solely by AE.
Plane 4: tiled by combination of BC and CD.
Plane 5: tiled solely by BD.
Plane 6: tiled by combination of BE and CE.
Plane 7: tiled solely by DE.

Triangle AB: 45,     45,     90; AB AB AB
Triangle AC: 35.264, 54.736, 90; AC AC AD
Triangle AD: 35.264, 54.736, 90; AD AD AC
Triangle AE: 45,     45,     90; AE AE AE
Triangle BC: 30,     60,     90; BC BC CD
Triangle BD: 45,     45,     90; BD BD BD
Triangle BE: 35.264, 54.736, 90; BE BE CE
Triangle CD: 30,     60,     90; CD CD BC
Triangle CE: 35.264, 54.736, 90; CE CE BE
Triangle DE: 45,     45,     90; DE DE DE

Repeating unit 1: Triangle AB
Repeating unit 2: Rectangle made from AC and AD.
Repeating unit 3: Triangle AE
Repeating unit 4: Rectangle made from BC and CD.
Repeating unit 5: Triangle BD
Repeating unit 6: Rectangle made from BE and CE.
Repeating unit 7: Triangle DE

F~4 {3,3,4,3}
Plane 1: tiled by combination of AB and BC.
Plane 2: tiled solely by AC.
Plane 3: tiled by combination of AD, AE, BD, BE, and CE.
Plane 4: tiled solely by CD.
Plane 5: tiled solely by DE.

Triangle AB: 30,     60,     90; AB BC AB
Triangle AC: 45,     45,     90; AC AC AC
Triangle AD: 35.264, 54.736, 90; AE BD AD
Triangle AE: 35.264, 54.736, 90; BE AD AE
Triangle BC: 30,     60,     90; BC AB BC
Triangle BD: 35.264, 54.736, 90; AD BE BD
Triangle BE: 19.471, 70.529, 90; AE BD CE
Triangle CD: 45,     45,     90; CD CD CD
Triangle CE: 35.264, 54.736, 90; CE CE BE
Triangle DE: 30,     60,     90; DE DE DE

Repeating unit 1: Equilateral triangle made from AB and BC.
Repeating unit 2: Triangle AC.
Repeating unit 3: This is an interesting one. The five triangles AD, AE, BD, BE, and CE form a rectangle. Four of these five triangles are similar; only the "central" one, BE, has a different shape.
Repeating unit 4: Triangle CD.
Repeating unit 5: Triangle DE.

B~4 Branched Euclidean group (half of tesseractic honeycomb)
Same marking as demipenteractic, AB branch is 4, BC, CD and DE are 3.
AD and AE triangles identical.
BD and BE triangles identical.
CD and CE triangles identical.
Plane 1: tiled solely by AB.
Plane 2: tiled by combination of AC, AD, and AE.
Plane 3: tiled by combination of BC, CD, and CE.
Plane 4: tiled solely by BD.
Plane 5: tiled solely by BE.
Plane 6: tiled solely by DE.

Triangle AB: 45,     45,     90    ; AB AB AB
Triangle AC: 54.736, 54.736, 70.529; AD AE AC
Triangle AD: 35.264, 54.736, 90    ; AD AD AC
Triangle AE: 35.264, 54.736, 90    ; AE AE AC
Triangle BC: 60,     60,     60    ; BC CD CE
Triangle BD: 45,     45,     90    ; BD BD BD
Triangle BE: 45,     45,     90    ; BE BE BE
Triangle CD: 30,     60,     90    ; CD CD BC
Triangle CE: 30,     60,     90    ; CE CE BC
Triangle DE: 45,     45,     90    ; DE DE DE

Repeating unit 1: Triangle AB.
Repeating unit 2: Rectangle made from triangles AC, AD, and AE.
Repeating unit 3: Rectangle made from triangles BC, CD, and CE, two of each. Half of it works as well, but swaps some labels when reflecting.
Repeating unit 4: Triangle BE.
Repeating unit 5: Triangle BE.
Repeating unit 6: Triangle DE.

D~4 Cross group (quarter of tesseractic honeycomb)
AB, BC, BD, BE branches are 3.
AB, BC, BC, BE triangles identical.
AC, AD, AE, CD, CE, DE triangles identical.
Plane 1: tiled by combination of AB, BC, BD, and BE.
Plane 2: tiled solely by AC.
Plane 3: tiled solely by AD.
Plane 4: tiled solely by AE.
Plane 5: tiled solely by CD.
Plane 6: tiled solely by CE.
Plane 7: tiled solely by DE.

Triangle AB: 60, 60, 60; BC BD BE
Triangle AC: 45, 45, 90; AC AC AC
Triangle AD: 45, 45, 90; AD AD AD
Triangle AE: 45, 45, 90; AE AE AE
Triangle BC: 60, 60, 60; AB BD BE
Triangle BD: 60, 60, 60; AB BC BE
Triangle BE: 60, 60, 60; AB BC BD
Triangle CD: 45, 45, 90; CD CD CD
Triangle CE: 45, 45, 90; CE CE CE
Triangle DE: 45, 45, 90; DE DE DE

Repeating unit 1: Plane 1 is tiled by equilateral triangles with a particular 4-coloring such as that each vertex has triangles of three colors around it, with opposite pairs colored alike, and each triangle has three vertices with different color combinations.
Repeating unit 2: Triangle AC.
Repeating unit 3: Triangle AD.
Repeating unit 4: Triangle AE.
Repeating unit 5: Triangle CD.
Repeating unit 6: Triangle CE.
Repeating unit 7: Triangle DE.

A~4 Cyclical 33333
AB, AE, BC, CD, DE triangles identical.
AC, AD, BD, BE, CE triangles identical.
Plane 1: tiled by combination of AB, AE, BC, CD, and DE.
Plane 2: tiled by combination of AC, AD, BD, BE, and CE.

Triangle AB: 52.239, 52.239, 75.522; AE BC AB
Triangle AC: 48.190, 65.905, 65.905; AC AD CE
Triangle AD: 48.190, 65.905, 65.905; AD AC BD
Triangle AE: 52.239, 52.239, 75.522; AB DE AE
Triangle BC: 52.239, 52.239, 75.522; AB CD BC
Triangle BD: 48.190, 65.905, 65.905; BD AD BE
Triangle BE: 48.190, 65.905, 65.905; BE BD CE
Triangle CD: 52.239, 52.239, 75.522; BC DE CD
Triangle CE: 48.190, 65.905, 65.905; CE AC BE
Triangle DE: 52.239, 52.239, 75.522; AE CD DE

Repeating unit 1: Triangles form strips where five "colors" repeat endlessly.
Repeating unit 2: Triangles form strips where five "colors" repeat endlessly.

H4 {3,3,3,5}
Plane 1: tiled by combination of AB, BC, and CD.
Plane 2: tiled by combination of AC, AD, AE, BD, BE, and CE.
Plane 3: tiled solely by DE.

Triangle AB: 36,     52.239, 90; BC AB AB
Triangle AC: 20.905, 65.905, 90; AC AC AD
Triangle AD: 37.377, 48.190, 90; BD AE AC
Triangle AE: 31.717, 54.736, 90; BE AD AE
Triangle BC: 7.761,  75.522, 90; AB BC CD
Triangle BD: 13.283, 65.905, 90; AD BE BD
Triangle BE: 10.812, 70.529, 90; AE BD CE
Triangle CD: 22.239, 52.239, 90; CD CD BC
Triangle CE: 20.905, 54.736, 90; CE CE BE
Triangle DE: 18,     60,     90; DE DE DE

Repeating unit 1: Triangle with angles 30, 36, 90, made from AC, BC, and CD.
Repeating unit 2: Quadrangle with angles 45, 90, 90, 90, made from AC, AD, AE, BD, BE, and CE.
Repeating unit 3: Triangle DE.

BH4 {4,3,3,5}
Plane 1: tiled solely by AB.
Plane 2: tiled by combination of AC, AD, and AE.
Plane 3: tiled by combination of BC and CD.
Plane 4: tiled by combination of BD, BE, and CE.
Plane 5: tiled solely by DE.

Triangle AB: 36,     45,     90; AB AB AB
Triangle AC: 20.905, 54.736, 90; AC AC AD
Triangle AD: 35.264, 37.377, 90; AE AD AC
Triangle AE: 31.717, 45,     90; AE AD AE
Triangle BC: 7.761,  60,     90; BC BC CD
Triangle BD: 13.283, 45,     90; BD BE BD
Triangle BE: 10.812, 54.736, 90; BE BD CE
Triangle CD: 22.239, 30,     90; CD CD BC
Triangle CE: 20.905, 35.264, 90; CE CE BE
Triangle DE: 18,     45,     90; DE DE DE

Repeating unit 1: Triangle AB.
Repeating unit 2: Quadrangle with angles 45, 90, 90, 90, made from AC, AD, and AE.
Repeating unit 3: Quadrangle with angles 30, 90, 90, 90, made from BC and CD.
Repeating unit 4: Quadrangle with angles 45, 45, 90, 90, made from BD, BE, and CE.
Repeating unit 5: Triangle DE.

K4 {5,3,3,5}
AB and DE triangles identical.
AC and CE triangles identical.
AD and BE triangles identical.
BC and CD triangles identical.
Plane 1: tiled solely by AB.
Plane 2: tiled by combination of AC, AD, AE, BD, BE, and CE.
Plane 3: tiled by combination of BC and CD.
Plane 4: tiled solely by DE.

Triangle AB: 18,     36,     90; AB AB AB
Triangle AC: 20.905, 20.905, 90; AC AC AD
Triangle AD: 10.812, 37.377, 90; AE BD AC
Triangle AE: 31.717, 31.717, 90; AD BE AE
Triangle BC: 7.761,  22.239, 90; BC CD BC
Triangle BD: 13.283, 13.283, 90; AD BE BD
Triangle BE: 10.812, 37.377, 90; AE BD CE
Triangle CD: 7.761,  22.239, 90; CD BC CD
Triangle CE: 20.905, 20.905, 90; CE CE BE
Triangle DE: 18,     36,     90; DE DE DE

Repeating unit 1: Triangle AB.
Repeating unit 3: Hexagon with angles 45, 45, 90, 90, 90, 90, made from AC, AD, AE, BD, BE, and CE.
Repeating unit 3: Quadrangle with angles 30, 90, 30, 90, made from BC and CD.
Repeating unit 4: Triangle DE.

DH4 Branched half of {4,3,3,5}
AB is 5, BC, CD and CE are 3
AD and AE triangles identical.
BD and BE triangles identical.
CD and CE triangles identical.
Plane 1: tiled solely by AB.
Plane 2: tiled by combination of AC, AD, AE, BD, and BE.
Plane 3: tiled combination of BC, CD, and CE.
Plane 4: tiled solely by DE.

Triangle AB: 18,     18,     90    ; AB AB AB
Triangle AC: 20.905, 20.905, 70.529; AD AE AC
Triangle AD: 10.812, 54.736, 90    ; AD BD AC
Triangle AE: 10.812, 54.736, 90    ; AE BE AC
Triangle BC: 22.239, 22.239, 60    ; CD CE BC
Triangle BD: 13.283, 45,     90    ; BD AD BD
Triangle BE: 13.283, 45,     90    ; BE AE BE
Triangle CD: 7.761,  60,     90    ; CE CD BC
Triangle CE: 7.761,  60,     90    ; CD CE BC
Triangle DE: 36,     45,     90    ; DE DE DE

Repeating unit 1: Triangle AB.
Repeating unit 2: Quadrangle with all angles 45, made from AC, AD, AE, BD, and BE.
Repeating unit 3: Quadrangle with all angles 30, made from BC, CD, and CE, two of each. Half of it works as well, but swaps some labels when reflecting.
Repeating unit 4: Triangle DE.

AF4 Cyclical 33334
AE is 4, AB, BC, CD, and DE are 3
AB and DE triangles identical.
AC and CE triangles identical.
AD and BE triangles identical.
BC and CD triangles identical.
Plane 1: tiled by combination of AB, BC, CD, and DE.
Plane 2: tiled by combination of AC, AD, BD, BE, and CE.
Plane 3: tiled solely by AE.

Triangle AB: 30,     52.239, 60    ; BC AB AB
Triangle AC: 35.264, 45,     65.905; AC AD AC
Triangle AD: 35.264, 48.190, 54.736; AC BD AD
Triangle AE: 45,     45,     45    ; AE AE AE
Triangle BC: 30,     52.239, 70.529; CD AB BC
Triangle BD: 19.471, 65.905, 65.905; BD AD BE
Triangle BE: 35.264, 48.190, 54.736; CE BD BE
Triangle CD: 30,     52.239, 70.529; BC DE CD
Triangle CE: 35.264, 45,     65.905; CE BE CE
Triangle DE: 30,     52.239, 60    ; CD DE DE

Repeating unit 1: Quadrangle with angles 30, 90, 30, 90, made from AB, BC, CD, and DE.
Repeating unit 1: Quadrangle with angles 45, 45, 90, 90, made from AC, AD, BD, BE, and CE.
Repeating unit 3: Triangle AE.


For 4-dimensional groups, there are 4 triangles to check, corresponding to faces of the Goursat tetrahedron. For 5-dimensional groups, there are 10 triangles, corresponding to triangles in the fundamental pentachoron, however it's called.
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Re: Planar tilings based on Goursat tetrahedra

Postby wendy » Wed Jun 01, 2016 12:51 pm

{4,3,5}, apart from the usual suspects, has ; {3,5,A} of order 2 ; {3,5/2,5,5/2:} of order 4.
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Re: Planar tilings based on Goursat tetrahedra

Postby Marek14 » Wed Jun 01, 2016 2:09 pm

wendy wrote:{4,3,5}, apart from the usual suspects, has ; {3,5,A} of order 2 ; {3,5/2,5,5/2:} of order 4.


Well, I'm not interested in star groups at this point. But I finally cracked the rule af adjacency. In 4D, it's simply a matter of odd/even branches, but in higher dimensions it's like this:

If you have two triangles AB and BC, they will be adjacent in the plane is the omnitruncated polytope formed by all three nodes, ABC, has AB and BC as opposite faces.

This resolves all. Odd/even branches in 4D work because truncated n-gon, for odd n, has two different types of edges in opposite positions. In 5D, this generalizes for n-gonal prisms with odd n and to tetrahedral symmetry. In 6D, the analogical rule would add pentachoric symmetry, and generally all further simplex symmetries will fall here.
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Re: Planar tilings based on Goursat tetrahedra

Postby wendy » Wed Jun 01, 2016 2:22 pm

On the other hand the groups that descend from 8,4,A and 8,3,4 do not have simplex groups to relate to.

Someone calculated the space of each simplex, and they exactly matched the subgroups i found. On the other hand, the unrelated groups are unrelated numbers. With the 4D tilings, like all even numbers, Euler's characteristic suffices to find the volume, and if i recall correctly, the group descending from {8,3,4,3} can not be constructed from any of the other symmetries.

What makes things even more interesting, is that there are common subgroups, that divide in different ways, but there is no over-group that they are all subgroups of. For example, the common subgroup of 7,3 and 7,4 is the group with the orbifold 7 * 2, and {3,9} and {3,18} share in a fairly ordinary way, the subgroup {9,3,3:} as {18/2,3,3} d2 and {9,9/3,3} d4.

The goursat path from {3,3,5} will find {3,4,3} but not {3,3,3}, because the latter mirrors do not cut the lunes or gores of {3,3} in {3,3,5} at the edges of the second group.
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Re: Planar tilings based on Goursat tetrahedra

Postby Marek14 » Thu Jun 02, 2016 1:20 pm

Hm, I'm now done with 5D groups. Here are my results so far:

Dihedral angles treatise:
Code: Select all
truncated octahedron:
4-6: 125.264390 - 54.735610 - arccos(1/sqrt(3))
6-6: 109.471221 - 70.528779 - arccos(1/3)
cycle 4-6-6-4-6-6
opposites: 4/4, 6a/6b

truncated cuboctahedron:
4-6: 144.735610 - 35.264390 - arccos(sqrt(2/3))
4-8: 135 - 45
6-8: 125.264390 - 54.735610 - arccos(1/sqrt(3))
cycle 1: 4-6-8-6-4-6-8-6
cycle 2: 4-8-4-8-4-8-4-8
opposites: 4/4, 6/6, 8/8

truncated icosidodecahedron:
4-6: 159.094843 - 20.905157 - arccos(sqrt[(3+sqrt(5))/6])
4-10: 148.282526 - 31.717474 - arccos(sqrt[(5+sqrt(5))/10])
6-10: 142.622632 - 37.377368 - arccos(sqrt[(5+2 sqrt(5))/15])
cycle: 4-6-10-4-10-6-4-6-10-4-10-6
opposites: 4/4, 6/6, 10/10

o{3,3,3}
truncated octahedron 1/hexagonal prism 1
127.761244 - 52.238756 - arccos(sqrt[3/8])
truncated octahedron 1/hexagonal prism 2
114.094843 - 65.905157 - arccos(sqrt[1/6])
truncated octahedron 1/truncated octahedron 2
104.477512 - 75.522488 - arccos(1/4)
hexagonal prism 1/hexagonal prism 2
131.810315 - 48.189685 - arccos(2/3)
hexagonal prism 1/truncated octahedron 2
114.094843 - 65.905157 - arccos(sqrt[1/6])
hexagonal prism 2/truncated octahedron 2
127.761244 - 52.238756 - arccos(sqrt[3/8])
cycle 1: |to1-4-hp2-4-hp1-4-to2| - 65.905 + 48.190 + 65.905 = 180
cycle 2: |hp1-6-to1-6-to2-6-hp2| - 52.239 + 75.522 + 52.239 = 180
opposites: to1/to2, hp1/hp2

o{3,3,4}
truncated octahedron/hexagonal prism
150 - 30
truncated octahedron/octagonal prism
135 - 45
truncated octahedron/truncated cuboctahedron
120 - 60
hexagonal prism/octagonal prism
144.735610 - 35.264390 - arccos(sqrt[2/3])
hexagonal prism/truncated cuboctahedron
125.264390 - 54.735610 - arccos(1/sqrt(3))
octagonal prism/truncated cuboctahedron
135 - 45
cycle 1: |to-4-op| - 45
cycle 2: |op-4-hp-4-tco| - 35.264 + 54.736 = 90
cycle 3: |hp-6-to-6-tco| - 30 + 60 = 90
cycle 4: |op-8-tco| - 45
opposites: to/to, hp/hp/ op/op, tco/tco

o{3,3,5}
truncated octahedron/hexagonal prism
172.238756 - 7.761244 - arccos(sqrt[9+3*sqrt(5)]/4)
truncated octahedron/decagonal prism
166.717474 - 13.282526 - arccos(sqrt[(5+2 sqrt(5))/10])
truncated octahedron/truncated icosidodecahedron
157.761244 - 22.238756 - arccos(sqrt[7+3sqrt(5)]/4)
hexagonal prism/decagonal prism
169.187683 - 10.812317 - arccos(sqrt[(10+2*sqrt(5))/15])
hexagonal prism/truncated icosidodecahedron
159.094843 - 20.905157 - arccos(sqrt[(3+sqrt(5))/6])
decagonal prism/truncated icosidodecahedron
162 - 18
cycle 1: |to-4-dp-4-hp-4-tid| - 13.283 + 10.812 + 20.905 = 45
cycle 2: |hp-6-to-6-tid| - 7.761 + 22.239 = 30
cycle 3: |dp-10-tid| - 18
opposites: to/to, hp/hp, dp/dp, tid/tid

o{3,4,3}
truncated cuboctahedron 1/hexagonal prism 1
150 - 30
truncated cuboctahedron 1/hexagonal prism 2
144.735610 - 35.264390 - arccos(sqrt[2/3])
truncated cuboctahedron 1/truncated cuboctahedron 2
135 - 45
hexagonal prism 1/hexagonal prism 2
160.528779 - 19.471221 - arccos(sqrt[8/9])
hexagonal prism 1/truncated cuboctahedron 2
144.735610 - 35.264390 - arccos(sqrt[2/3])
hexagonal prism 2/truncated cuboctahedron 2
150 - 30
cycle 1: |tco1-4-hp2-4-hp1-4-tco2| - 35.264 + 19.471 + 35.264 = 90
cycle 2: |tco1-6-hp1| - 30
cycle 3: |hp2-6-tco2| - 30
cycle 4: |tco1-8-tco2| - 45
opposites: tco1/tco1, tco2/tco2, hp1/hp1, hp2/hp2

o(b333)
truncated octahedron 1/truncated octahedron 2
120 - 60
truncated octahedron 1/cube
135 - 45
truncated octahedron 1/truncated octahedron 3
120 - 60
truncated octahedron 2/cube
135 - 45
truncated octahedron 2/truncated octahedron 3
120 - 60
cube/truncated octahedron 3
135 - 45
cycle 1: |to1-4-c| - 45
cycle 2: |to2-4-c| - 45
cycle 3: |c-4-to3| - 45
cycle 4: (to1-6-to2-6-to3-6-) - 60 + 60 + 60 = 180
opposites: to1/to1, to2/to2, to3/to3, c/c


Plane data:
Code: Select all
{3,3,3}
A and D triangles identical.
B and C triangles identical.
One plane tiled by combination of A, B, C, and D.

Triangle A: 54.736, 60,     90; ABA
Triangle B: 54.736, 70.529, 90; ABC
Triangle C: 54.736, 70.529, 90; DCB
Triangle D: 54.736, 60,     90; DCD

Vertices:
A-54.736 & B-70.529 & C-54.736
Pattern: AABCCB
Triangle with angles 120, 125.265, 125.265

A-60
Pattern: AAAAAA
Triangle with angles 109.472, 109.472, 109.472

A-90 & B-90
Pattern: AABB
Quadrangle with angles 109.472, 125.265, 120, 125.265

B-54.736 & C-70.529 & D-54.736
Pattern: BBCDDC
Triangle with angles 120, 125.265, 125.265

C-90 & D-90
Pattern: CCDD
Quadrangle with angles 109.472, 125.265, 120, 125.265

D-60
Pattern: DDDDDD
Equilateral triangle with angle 109.472

Repeating unit: Digonal strip of angle 60. Composed of 1 A, 1 B, 1 C and 1 D.

{3,3,4}
Plane 1: tiled by combination of A, B, and C.
Plane 2: tiled solely by D.

Triangle A: 45,     54.736, 90; BAA
Triangle B: 35.264, 70.529, 90; ABC
Triangle C: 54.736, 54.736, 90; CCB - double of triangle A. Note that each of its 54.736 angles belongs to a different type of vertex.
Triangle D: 45,     60,     90; DDD

Vertices:
A-45
Pattern: AAAAAAAA
Square with angle 109.472

A-54.736 & B-70.529 & C-54.736
Pattern: AABCCB
Triangle with angles 90, 90, 109.472

A-90 & B-90
Pattern: AABB
Quadrangle with angles 90, 125.265, 109.472, 125.265

B-35.264 & C-54.736
Pattern: BBCCBBCC
Square with angle 125.265

C-90
Pattern: CCCC
Square with angle 109.472

D-45
Pattern: DDDDDDDD
Square with angle 120

D-60
Pattern: DDDDDD
Equilateral triangle with angle 90

D-90
Pattern: DDDD
Rhombus with angles 90, 120, 90, 120

Repeating unit 1: Triangle with angles 45, 90, 90. Composed of 1 A, 1 B, 1 C.
Repeating unit 2: Triangle D.

{3,3,5}
One plane tiled by combination of A, B, C, and D.

Triangle A: 36,     54.736, 90; BAA
Triangle B: 20.905, 70.529, 90; ABC
Triangle C: 37.377, 54.736, 90; CDB
Triangle D: 31.717, 60,     90; DCD

Vertices:
A-36
Pattern: AAAAAAAAAA
Regular pentagon with angle 109.472

A-54.736 & B-70.529 & C-54.736
Pattern: AABCCB
Triangle with angles 58.282, 58.282, 72

A-90 & B-90
Pattern: AABB
Quadrangle with angles 41.810, 125.265, 72, 125.265

B-20.905 & C-37.377 & D-31.717
Pattern: BBCDDCBBCDDC
Hexagon with angles 120, 125.265, 125.265, 120, 125.265, 125.265

C-90 & D-90
Pattern: CCDD
Quadrangle with angles 69.094, 109.472, 69.094, 120

D-60
Pattern: DDDDDD
Equilateral triangle with angle 63.434

Repeating unit: Triangle with angles 36, 60, 90. Composed of 1 A, 1 B, 1 C, 1 D.

{3,4,3}
A and D triangles identical.
B and C triangles identical.
Plane 1: tiled by combination of A and B
Plane 2: tiled by combination of C and D

Triangle A: 35.264, 60,     90; ABA
Triangle B: 45,     54.736, 90; ABB
Triangle C: 45,     54.736, 90; DCC
Triangle D: 35.264, 60,     90; DCD

Vertices:
A-35.264 & B-54.736
Pattern: AABBAABB
Rhombus with angles 90, 120, 90, 120

A-60
Pattern: AAAAAA
Equilateral triangle with angle 70.528

A-90 & B-90
Pattern: AABB
Quadrangle with angles 90, 90, 90, 120

B-45
Pattern: BBBBBBBB
Square with angle 109.472

C-45
Pattern: CCCCCCCC
Square with angle 109.472

C-54.736 & D-35.264
Pattern: CCDDCCDD
Rhombus with angles 90, 120, 90, 120

C-90 & D-90
Pattern: CCDD
Quadrangle with angles 90, 90, 90, 120

D-60
Pattern: DDDDDD
Equilateral triangle with angle 70.529

Repeating unit 1: Triangle with angles 45, 60, 90. Composed of 1 A, 1 B.
Repeating unit 2: Triangle with angles 45, 60, 90. Composed of 1 C, 1 D.

Branched 333 (demitesseractic)
A, C, and D triangles identical.
One plane tiled by combination of A, B, C, and D.

Triangle A: 54.736, 54.736, 90    ; AAB
Triangle B: 70.529, 70.529, 70.529; ABC
Triangle C: 54.736, 54.736, 90    ; CCB
Triangle D: 54.736, 54.736, 90    ; DDB

Vertices:
A-54.736 & B-70.529 & C-54.736
Pattern: AABCCB
Quadrangle with angles 125.265, 125.265, 125.265, 125.265 (not square, apparently)

A-54.736 & B-70.529 & D-54.736
Pattern: AABDDB
Quadrangle with angles 125.265, 125.265, 125.265, 125.265 (not square, apparently)

A-90
Pattern: AAAA
Square with angle 109.472

B-70.529 & C-54.736 & D-54.736
Pattern: BCCBDD
Quadrangle with angles 125.265, 125.265, 125.265, 125.265 (not square, apparently)

C-90
Pattern: CCCC
Square with angle 109.472

D-90
Pattern: DDDD
Square with angle 109.472

Repeating unit: Equilateral triangle of angle 90. Composed of 1 A, 1 B, 1 C and 1 D.

{4,3,4}
A and D triangles identical.
B and C triangles identical.
Plane 1: tiled solely by A
Plane 2: tiled by combination of B and C
Plane 3: tiled solely by D

Triangle A: 45,     45,     90; AAA
Triangle B: 35.264, 54.736, 90; BBC
Triangle C: 35.264, 54.736, 90; CCB
Triangle D: 45,     45,     90; DDD

Vertices:
A-45
Pattern: AAAAAAAA
Square with angle 90

A-45
Pattern: AAAAAAAA
Square with angle 90
(The two 45 angles at A look identical within the plane, but differ in how other tetrahedron faces are connected to them.)

A-90
Pattern: AAAA
Square with angle 90

B-35.264 & C-54.736
Pattern: BBCC
Rectangle with angle 90

B-54.736 & C-35.264
Pattern: BBCC
Rectangle with angle 90

B-90
Pattern: BBBB
Rhombus with angles 70.529, 109.472, 70.529, 109.472

C-90
Pattern: CCCC
Rhombus with angles 70.529, 109.472, 70.529, 109.472

D-45
Pattern: DDDDDDDD
Square with angle 90

D-45
Pattern: DDDDDDDD
Square with angle 90

D-90
Pattern: DDDD
Square with angle 90

Repeating unit 1: Triangle A.
Repeating unit 2: Rectangle of angle 90. Composed of 1 B, 1 C.
Repeating unit 3: Triangle D.

Branched 334 (tetrahedral/octahedral honeycomb)
A and C triangles identical.
Plane 1: tiled by combination of A, B, and C
Plane 2: tiled solely by D

Triangle A: 35.264, 54.736, 90    ; AAB
Triangle B: 54.736, 54.736, 70.529; ACB (double of A or C triangle)
Triangle C: 35.264, 54.736, 90    ; CCB
Triangle D: 45,     45,     90    ; DDD

Vertices:
A-35.264 & B-54.736
Pattern: AABBAABB
Hexagon with angles 109.472, 125.565, 125.565, 109.472, 125.565, 125.565

A-54.736 & B-70.529 & C-54.736
Pattern: AABCCB
Square with angle 90

A-90
Pattern: AAAA
Rhombus with angles 70.529, 109.472, 70.529, 109.472

B-54.736 & C-35.264
Pattern: BBCCBBCC
Hexagon with angles 109.472, 125.565, 125.565, 109.472, 125.565, 125.565

C-90
Pattern: CCCC
Rhombus with angles 70.529, 109.472, 70.529, 109.472

D-45
Pattern: DDDDDDDD
Square with angle 90

D-45
Pattern: DDDDDDDD
Square with angle 90

D-90
Pattern: DDDD
Square with angle 90

Repeating unit 1: Rectangle of angle 90. Composed of 1 A, 1 B, 1 C.
Repeating unit 2: Triangle D.

Cyclical 3333
Triangles A, B, C, and D all identical.
One plane tiled by combination of A, B, C, and D.

Triangle A: 54.736, 54.736, 70.529; BDA
Triangle B: 54.736, 54.736, 70.529; ACB
Triangle C: 54.736, 54.736, 70.529; BDC
Triangle D: 54.736, 54.736, 70.529; ACD

Vertices:
A-54.736 & B-70.529 & C-54.736
Pattern: AABCCB
Hexagon with angles 109.472, 125.565, 125.565, 109.472, 125.565, 125.565

A-54.736 & C-54.736 & D-70.529
Pattern: AADCCD
Hexagon with angles 109.472, 125.565, 125.565, 109.472, 125.565, 125.565

A-70.529 & B-54.736 & D-54.736
Pattern: ABBADD
Hexagon with angles 109.472, 125.565, 125.565, 109.472, 125.565, 125.565

B-54.736 & C-70.529 & D-54.736
Pattern: BBCDDC
Hexagon with angles 109.472, 125.565, 125.565, 109.472, 125.565, 125.565

Repeating unit: An infinite strip formed by repeating triangles A, B, C, D.

{4,3,5}
Plane 1: tiled solely by A
Plane 2: tiled by combination of B, C, and D

Triangle A: 36,     45,     90; AAA
Triangle B: 20.905, 54.736, 90; BBC
Triangle C: 35.264, 37.377, 90; DCB
Triangle D: 31.717, 45,     90; DCD

Vertices:
A-36
Pattern: AAAAAAAAAA
Regular pentagon with angle 90

A-45
Pattern: AAAAAAAA
Square with angle 72

A-90
Pattern: AAAA
Rhombus with angles 72, 90, 72, 90

B-20.905 & C-37.377 & D-31.717
Pattern: BBCDDCBBCDDC
Right-angled hexagon, not regular

B-54.736 & C-35.264
Pattern: BBCCBBCC
Rectangle with angle 58.282

B-90
Pattern: BBBB
Rhombus with angles 41.810, 109.472, 41.810, 109.472

C-90 & D-90
Pattern: CCDD
Quadrangle with angles 69.095, 70.529, 90, 69.095

D-45
Pattern: DDDDDDDD
Square with angle 63.434

Repeating unit: Quadrangle with angles 45, 90, 90, 90. Composed of 1 B, 1 C, 1 D.

{5,3,5}
A and D triangles identical.
B and C triangles identical.
One plane tiled by combination of A, B, C, and D.

Triangle A: 31.717, 36,     90; ABA
Triangle B: 20.905, 37.377, 90; ABC
Triangle C: 20.905, 37.377, 90; DCB
Triangle D: 31.717, 36,     90; DCC

Vertices:
A-31.717 & B-37.377 & C-20.905
Pattern: AABCCBAABCCB
Hexagon with angles 58.282, 58.282, 72, 58.282, 58.282, 72

A-36
Pattern: AAAAAAAAAA
Regular pentagon with angle 63.434

A-90 & B-90
Pattern: AABB
Quadrangle with angles 41.810, 69.095, 72, 69.095

B-20.905 & C-37.377 & D-31.717
Pattern: BBCDDCBBCDDC
Hexagon with angles 58.282, 58.282, 72, 58.282, 58.282, 72

C-90 & D-90
Pattern: CCDD
Quadrangle with angles 41.810, 69.095, 72, 69.095

D-36
Pattern: DDDDDDDDDD
Regular pentagon with angle 63.434

Repeating unit: Quadrangle with angles 36, 90, 36, 90. Composed of 1 A, 1 B, 1 C, 1 D.

{3,5,3}
A and D triangles identical.
B and C triangles identical.
One plane tiled by combination of A, B, C, and D.

Triangle A: 20.905,     60, 90; ABA
Triangle B: 31.717, 37.377, 90; ABC
Triangle C: 31.717, 37.377, 90; DCB
Triangle D: 20.905,     60, 90; DCD

Vertices:
A-20.905 & B-37.377 & C-31.717
Pattern: AABCCBAABCCB
Hexagon with angles 69.095, 69.095, 120, 69.095, 69.095, 120

A-60
Pattern: AAAAAA
Equilateral triangle with angle 41.810

A-90 & B-90
Pattern: AABB
Quadrangle with angles 58.282, 63.434, 58.282 and 120

B-31.717 & C-37.377 & D-20.905
Pattern: BBCDDCBBCDDC
Hexagon with angles 69.095, 69.095, 120, 69.095, 69.095, 120

C-90 & D-90
Pattern: CCDD
Quadrangle with angles 58.282, 63.434, 58.282 and 120

D-60
Pattern: AAAAAA
Equilateral triangle with angle 41.810

Repeating unit: Quadrangle with angles 60, 90, 60, 90. Composed of 1 A, 1 B, 1 C, 1 D.

Branched 335 (tetrahedral/icosahedral honeycomb)
A and C triangles identical.
One plane tiled by combination of A, B, C, and D.

Triangle A: 20.905, 54.736,     90; AAB
Triangle B: 37.377, 37.377, 70.529; ACD
Triangle C: 20.905, 54.736,     90; CCB
Triangle D: 31.717, 31.717,     90; DDB

Vertices:
A-20.905 & B-37.377 & D-31.717
Pattern: AABDDBAABDDB
Octagon with alternating angles of 69.095 and 125.565

A-54.736 & B-70.529 & C-54.736
Pattern: AABCCB
Rectangle with angle 58.282

A-90
Pattern: AAAA
Rhombus with angles 41.810, 109.472, 41.810, 109.472

B-37.377 & C-20.905 & D-31.717
Pattern: BCCBDDBCCBDD
Octagon with alternating angles of 69.095 and 125.565

C-90
Pattern: CCCC
Rhombus with angles 41.810, 109.472, 41.810, 109.472

D-90
Pattern: DDDD
Square with angle 63.434

Repeating unit: Right-angled pentagon; not regular. Composed of 1 A, 1 B, 1 C, 1 D.

Cyclical 3334
A and D triangles identical.
B and C triangles identical.
One plane tiled by combination of A, B, C, and D.

Triangle A: 45,     54.736, 54.736; BAA
Triangle B: 35.264, 54.736, 70.529; CAB
Triangle C: 35.264, 54.736, 70.529; BDC
Triangle D: 45,     54.736, 54.736; CDD

Vertices:
A-45
Pattern: AAAAAAAA
Regular octagon with angle 109.472

A-54.736 & B-35.264
Pattern: AABBAABB
Octagon with angles 90, 125.565, 109.472, 125.565, 90, 125.565, 109.472, 125.565

A-54.736 & B-70.529 & C-54.736
Pattern: AABCCB
Hexagon with angles 70.529, 125.565, 90, 90, 90, 125.565

B-54.736 & C-70.529 & D-54.736
Pattern: BBCDDC
Hexagon with angles 70.529, 125.565, 90, 90, 90, 125.565

C-35.264 & D-54.736
Pattern: CCDDCCDD
Octagon with angles 90, 125.565, 109.472, 125.565, 90, 125.565, 109.472, 125.565

D-45
Pattern: DDDDDDDD
Regular octagon with angle 109.472

Repeating unit: Quadrangle with angles 45, 90, 45, 90. Composed of 1 A, 1 B, 1 C, 1 D.

Cyclical 3335
A and D triangles identical.
B and C triangles identical.
One plane tiled by combination of A, B, C, and D.

Triangle A: 31.717, 37.377, 54.736; BAD
Triangle B: 20.905, 54.736, 70.529; CAB
Triangle C: 20.905, 54.736, 70.529; BDC
Triangle D: 31.717, 37.377, 54.736; CDA

Vertices:
A-31.717 & C-20.905 & D-37.377
Pattern: AADCCDAADCCD
Dodecagon with angles 69.095, 109.472, 69.095, 125.565, 109.472, 125.565, 69.095, 109.472, 69.095, 125.565, 109.472, 125.565

A-37.377 & B-20.905 & D-31.717
Pattern: ABBADDABBADD
Dodecagon with angles 69.095, 109.472, 69.095, 125.565, 109.472, 125.565, 69.095, 109.472, 69.095, 125.565, 109.472, 125.565

A-54.736 & B-70.529 & C-54.736
Pattern: AABCCD
Hexagon with angles 41.810, 125.565, 58.282, 63.434, 58.282, 125.565

B-54.736 & C-70.529 & D-54.736
Pattern: BBCDDC
Hexagon with angles 41.810, 125.565, 58.282, 63.434, 58.282, 125.565

Repeating unit: An infinite strip formed by repeating triangles A, B, C, D. Its two edges form right-angled pseudogons.

Cyclical 3434
Triangles A, B, C, and D all identical.
Plane 1: tiled by combination of A and B
Plane 2: tiled by combination of C and D

Triangle A: 35.264, 45, 54.736; ABA
Triangle B: 35.264, 45, 54.736; BAB
Triangle C: 35.264, 45, 54.736; CDC
Triangle D: 35.264, 45, 54.736; DCD

Vertices:
A-35.264 & B-54.736
Pattern: AABBAABB
Right-angled octagon, not regular

A-45
Pattern: AAAAAAAA
Octagon with alternating angles 70.529 and 125.565

A-54.736 & B-35.264
Pattern: AABBAABB
Right-angled octagon, not regular

B-45
Pattern: BBBBBBBB
Octagon with alternating angles 70.529 and 125.565

C-35.264 & D-54.736
Pattern: CCDDCCDD
Right-angled octagon, not regular

C-45
Pattern: CCCCCCCC
Octagon with alternating angles 70.529 and 125.565

C-54.736 & D-35.264
Pattern: CCDDCCDD
Right-angled octagon, not regular

D-45
Pattern: DDDDDDDD
Octagon with alternating angles 70.529 and 125.565

Repeating unit 1: Quadrangle with angles 45, 90, 45, 90. Composed of 1 A, 1 B.
Repeating unit 2: Quadrangle with angles 45, 90, 45, 90. Composed of 1 C, 1 D.

Cyclical 3435
A and D triangles identical.
B and C triangles identical.
One plane tiled by combination of A, B, C, and D.

Triangle A: 31.717, 35.264, 37.377; BDA
Triangle B: 20.905, 45,     54.736; BAB
Triangle C: 20.905, 45,     54.736; CDC
Triangle D: 31.717, 35.264, 37.377; CAD

Vertices:
A-31.717 & C-20.905 & D-37.377
Pattern: AADCCDAADCCD
Dodecagon with angles 69.095, 70.529, 69.095, 90, 90, 90, 69.095, 70.529, 69.095, 90, 90, 90

A-35.264 & B-54.736
Pattern: AABBAABB
Octagon with angles 58.282, 63.434, 58.282, 90, 58.282, 63.434, 58.282, 90

A-37.377 & B-20,905 & D-31.717
Pattern: ABBADDABBADD
Dodecagon with angles 69.095, 70.529, 69.095, 90, 90, 90, 69.095, 70.529, 69.095, 90, 90, 90

B-45
Pattern: BBBBBBBB
Octagon with alternating angles 41.810 and 109.472

C-45
Pattern: CCCCCCCC
Octagon with alternating angles 41.810 and 109.472

C-54.736 & D-35.264
Pattern: CCDDCCDD
Octagon with angles 58.282, 63.434, 58.282, 90, 58.282, 63.434, 58.282, 90

Repeating unit: Hexagon with angles 45, 90, 90, 45, 90, 90. Composed of 1 A, 1 B, 1 C, 1 D.

Cyclical 3535
Triangles A, B, C, and D all identical.
One plane tiled by combination of A, B, C, and D.

Triangle A: 20.905, 31.717, 37.377; DBA
Triangle B: 20.905, 31.717, 37.377; CAB
Triangle C: 20.905, 31.717, 37.377; BDC
Triangle D: 20.905, 31.717, 37.377; ACD

Vertices:
A-20.905 & C-31.717 & D-37.377
Pattern: AADCCDAADCCD
Dodecagon with angles 41.810, 58.282, 69.095, 63.434, 69.095, 58.282, 41.810, 58.282, 69.095, 63.434, 69.095, 58.282

A-31.717 & B-37.377 & C-20.905
Pattern: AABCCBAABCCB
Dodecagon with angles 41.810, 58.282, 69.095, 63.434, 69.095, 58.282, 41.810, 58.282, 69.095, 63.434, 69.095, 58.282

A-37.377 & B-20.905 & D-31.717
Pattern: ABBADDABBADD
Dodecagon with angles 41.810, 58.282, 69.095, 63.434, 69.095, 58.282, 41.810, 58.282, 69.095, 63.434, 69.095, 58.282

B-31.717 & C-37.377 & D-20.905
Pattern: BBCDDCBBCDDC
Dodecagon with angles 41.810, 58.282, 69.095, 63.434, 69.095, 58.282, 41.810, 58.282, 69.095, 63.434, 69.095, 58.282

Repeating unit: An infinite strip formed by repeating triangles A, B, C, D. Its two edges form right-angled pseudogons.

{3,3,6}
Plane 1: tiled by combination of A, B, and C.
Plane 2: tiled solely by D.

Triangle A: 30, 54.736, 90; BAA
Triangle B: 0,  70.529, 90; ABC
Triangle C: 0,  54.736, 90; CCB
Triangle D: 0,  60,     90; DDD

Vertices:
A-30
Pattern: AAAAAAAAAAAA
Regular hexagon with angle 109.472

A-54.736 & B-70.529 & C-54.736
Pattern: AABCCB
Triangle with angles 0, 0, 60

A-90 & B-90
Pattern: AABB
Quadrangle with angles 0, 125.565, 60, 125.565

B-0 & C-0
Pattern: BBCC...
Apeirogon with angle 125.565, not regular

C-90
Pattern: CCCC
Rhombus with angles 0, 109.472, 0, 109.472

D-0
Pattern: D...
Regular apeirogon with angle 120

D-60
Pattern: DDDDDD
Equilateral triangle with angle 0

D-90
Pattern: DDDD
Rhombus with angles 0, 120, 0, 120

Repeating unit 1: Triangle with angles 0, 30, 90. Composed of 1 A, 1 B, 1 C.
Repeating unit 2: Triangle D.

{4,3,6}
Plane 1: tiled solely by A.
Plane 2: tiled by combination of B and C.
Plane 3: tiled solely by D.

Triangle A: 30, 45,     90; AAA
Triangle B: 0,  54.736, 90; BBC
Triangle C: 0,  35.264, 90; CCB
Triangle D: 0,  45,     90; DDD

Vertices:
A-30
Pattern: AAAAAAAAAAAA
Right-angled dodecagon

A-45
Pattern: AAAAAAAA
Regular octagon with angle 60

A-90
Pattern: AAAA
Rhombus with angles 60, 90, 60, 90

B-0
Pattern: B...
Regular apeirogon with angle 109.472

B-54.736 & C-35.264
Pattern: BBCCBBCC
Rectangle with angle 0

B-90 & C-90
Pattern: BBCC
Quadrangle with angles 0, 90, 0, 90

C-0
Pattern: C...
Regular apeirogon with angle 70.529

D-0
Pattern: D...
Right-angled apeirogon

D-45
Pattern: DDDDDDDD
Square with angle 0

D-90
Pattern: DDDD
Rhombus with angles 0, 90, 0, 90

Repeating unit 1: Triangle A.
Repeating unit 2: Triangle with angles 0, 0, 90. Composed of 1 B and 1 C.
Repeating unit 3: Triangle D.

{5,3,6}
Plane 1: tiled by combination of A, B, and C.
Plane 2: tiled solely by D.

Triangle A: 30, 31.717, 90; BAA
Triangle B: 0,  37.377, 90; ABC
Triangle C: 0,  20.905, 90; CCB
Triangle D: 0,  36,     90; DDD

Vertices:
A-30
Pattern: AAAAAAAA
Regular hexagon with angle 63.434

A-31.717 & B-37.377 & C-20.905
Pattern: AABCCBAABCCB
Hexagon with angles 0, 0, 60, 0, 0, 60

A-90 & B-90
Pattern: AABB
Quadrangle with angles 0, 69.095, 60, 69.095

B-0 & C-0
Pattern: BBCC...
Apeirogon with angle 58.282, not regular

C-90
Pattern: CCCC
Rhombus with angles 0, 41.810, 0, 41.810

D-0
Pattern: D...
Regular apeirogon with angle 72

D-36
Pattern: DDDDDDDDDD
Regular pentagon with angle 0

D-90
Pattern: DDDD
Rhombus with angles 0, 72, 0, 72

Repeating unit 1: Quadrangle with angles 0, 30, 90, 90. Composed of 1 A, 1 B, 1 C.
Repeating unit 2: Triangle D.

{6,3,6}
A and D triangles identical.
B and C triangles identical.
Plane 1: tiled solely by A
Plane 2: tiled by combination of B and C
Plane 3: tiled solely by D

Triangle A: 0, 30, 90; AAA
Triangle B: 0, 0,  90; BBC
Triangle C: 0, 0,  90; CCB
Triangle D: 0, 30, 90; DDD

Vertices:
A-0
Pattern: A...
Regular apeirogon with angle 60

A-30
Pattern: AAAAAAAAAAAA
Regular hexagon with angle 0

A-90
Pattern: AAAA
Rhombus with angles 0, 60, 0, 60

B-0 & C-0
Pattern: BBCC...
Regular apeirogon with angle 0

B-0 & C-0
Pattern: BBCC...
Regular apeirogon with angle 0

B-90
Pattern: BBBB
Square with angle 0

C-90
Pattern: CCCC
Square with angle 0

D-0
Pattern: D...
Regular apeirogon with angle 60

D-30
Pattern: DDDDDDDDDDDD
Regular hexagon with angle 0

D-90
Pattern: DDDD
Rhombus with angles 0, 60, 0, 60

Repeating unit 1: Triangle A.
Repeating unit 2: Rhombus of angles 0, 90, 0, 90. Composed of 1 B, 1 C.
Repeating unit 3: Triangle D.

{3,4,4}
Plane 1: tiled by combination of A and B.
Plane 2: tiled solely by C.
Plane 3: tiled solely by D.

Triangle A: 35.264, 45,     90; ABA
Triangle B: 0,      54.736, 90; ABB
Triangle C: 0,      45,     90; CCC
Triangle D: 0,      60,     90; CCC

Vertices:
A-35.264 & B-54.736
Rhombus with angles 0, 90, 0, 90

A-45
Pattern: AAAAAAAA
Square with angle 70.529

A-90 & B-90
Pattern: AABB
Quadrangle with angles 0, 90, 90, 90

B-0
Pattern: B...
Regular apeirogon with angle 109.472

C-0
Pattern: C...
Right-angled apeirogon

C-45
Pattern: CCCCCCCC
Square with angle 0

C-90
Pattern: CCCC
Rhombus with angles 0, 90, 0, 90

D-0
Pattern: D...
Regular apeirogon with angle 120

D-60
Pattern: DDDDDD
Equilateral triangle with angle 0

D-90
Pattern: DDDD
Rhombus with angles 0, 120, 0, 120

Repeating unit 1: Triangle with angles 0, 45, 90. Composed of 1 A and 1 B.
Repeating unit 2: Triangle C.
Repeating unit 3: Triangle D.

{4,4,4}
A and D triangles identical.
B and C triangles identical.
Plane 1: tiled solely by A.
Plane 2: tiled solely by B.
Plane 3: tiled solely by C.
Plane 4: tiled solely by D.

Triangle A: 0, 45, 90; AAA
Triangle B: 0, 0,  90; BBB
Triangle C: 0, 0,  90; CCC
Triangle D: 0, 45, 90; DDD

Vertices:
A-0
Pattern: A...
Right-angled apeirogon

A-45
Pattern: AAAAAAAA
Square with angle 0

A-90
Pattern: AAAA
Rhombus with angles 0, 90, 0, 90

B-0
Pattern: B...
Regular apeirogon with angle 0

B-0
Pattern: B...
Regular apeirogon with angle 0

B-90
Pattern: BBBB
Square with angle 0

C-0
Pattern: C...
Regular apeirogon with angle 0

C-0
Pattern: C...
Regular apeirogon with angle 0

C-90
Pattern: CCCC
Square with angle 0

D-0
Pattern: D...
Right-angled apeirogon

D-45
Pattern: DDDDDDDD
Square with angle 0

D-90
Pattern: DDDD
Rhombus with angles 0, 90, 0, 90

Repeating unit 1: Triangle A.
Repeating unit 2: Triangle B.
Repeating unit 3: Triangle C.
Repeating unit 4: Triangle D.

{3,6,3}
A and D triangles identical.
B and C triangles identical.
Plane 1: tiled by combination of A and B
Plane 2: tiled by combination of C and D

Triangle A: 0, 60, 90; ABA
Triangle B: 0, 0,  90; ABB
Triangle C: 0, 0,  90; CDC
Triangle D: 0, 60, 90; DCD

Vertices:
A-0 & B-0
Pattern: AABB...
Apeirogon with alternating angles 0 and 120

A-60
Pattern: AAAAAA
Equilateral triangle with angle 0

A-90 & B-90
Pattern: AABB
Quadrangle with angles 0, 0, 0, 120

B-0
Pattern: B...
Regular apeirogon with angle 0

C-0
Pattern: C...
Regular apeirogon with angle 0

C-0 & D-0
Pattern: CCDD...
Apeirogon with alternating angles 0 and 120

C-90 & D-90
Pattern: CCDD
Quadrangle with angles 0, 0, 0, 120

D-60
Pattern: DDDDDD
Equilateral triangle with angle 0

Repeating unit 1: Triangle with angles 0, 0, 60. Composed of 1 A, 1 B.
Repeating unit 2: Triangle with angles 0, 0, 60. Composed of 1 C, 1 D.

Branched 336
A and C triangles identical.
Plane 1: tiled by combination of A, B, and C
Plane 2: tiled solely by D

Triangle A: 0, 54.736, 90    ; AAB
Triangle B: 0, 0,      70.529; ACB
Triangle C: 0, 54.736, 90    ; CCB
Triangle D: 0, 0,      90    ; DDD

Vertices:
A-0 & B-0
Pattern: AABB...
Apeirogon with alternating angles 0 and 109.472

A-54.736 & B-70.529 & C-54.736
Pattern: AABCCB
Hexagon with all angles 0, not regular

A-90
Pattern: AAAA
Rhombus with angles 0, 109.472, 0, 109.472

B-0 & C-0
Pattern: CCDD...
Apeirogon with alternating angles 0 and 109.472

C-90
Pattern: CCCC
Rhombus with angles 0, 109.472, 0, 109.472

D-0
Pattern: D...
Regular apeirogon with angle 0

D-0
Pattern: D...
Regular apeirogon with angle 0

D-90
Pattern: DDDD
Square with angle 0

Repeating unit 1: Quadrangle with angles 0, 0, 90, 90. Composed of 1 A, 1 B, 1 C.
Repeating unit 2: Triangle D.

Branched 344
C and D triangles identical.
Plane 1: tiled by combination of A and B
Plane 2: tiled solely by C
Plane 3: tiled solely by D

Triangle A: 35.264, 35.264, 90    ;
Triangle B: 0,      54.736, 54.736;
Triangle C: 0,      45,     90    ;
Triangle D: 0,      45,     90    ;

Vertices:
A-35.264 & B-54.736
Pattern: AABB
Hexagon with angles 0, 90, 90, 0, 90, 90

A-35.264 & B-54.736
Pattern: AABB
Hexagon with angles 0, 90, 90, 0, 90, 90

A-90
Pattern: AAAA
Square with angle 70.529

B-0
Pattern: B...
Regular apeirogon with angle 109.472

C-0
Pattern: C...
Right-angled apeirogon

C-45
Pattern: CCCCCCCC
Square with angle 0

C-90
Pattern: CCCC
Rhombus with angles 0, 90, 0, 90

D-0
Pattern: D...
Right-angled apeirogon

D-45
Pattern: DDDDDDDD
Square with angle 0

D-90
Pattern: DDDD
Rhombus with angles 0, 90, 0, 90

Repeating unit 1: Quadrangle with angles 0, 90, 90, 90. Composed of 1 A and 1 B.
Repeating unit 2: Triangle C.
Repeating unit 3: Triangle D.

Branched 444
A, C and D triangles identical.
Plane 1: tiled solely by A
Plane 2: tiled solely by B
Plane 3: tiled solely by C
Plane 4: tiled solely by D

Triangle A: 0, 0, 90; AAA
Triangle B: 0, 0, 0 ; BBB
Triangle C: 0, 0, 90; CCC
Triangle D: 0, 0, 90; DDD

Vertices:
A-0
Pattern: A...
Regular apeirogon with angle 0

A-0
Pattern: A...
Regular apeirogon with angle 0

A-90
Pattern: AAAA
Square with angle 0

B-0
Pattern: B...
Regular apeirogon with angle 0

B-0
Pattern: B...
Regular apeirogon with angle 0

B-0
Pattern: B...
Regular apeirogon with angle 0

C-0
Pattern: C...
Regular apeirogon with angle 0

C-0
Pattern: C...
Regular apeirogon with angle 0

C-90
Pattern: CCCC
Square with angle 0

D-0
Pattern: D...
Regular apeirogon with angle 0

D-0
Pattern: D...
Regular apeirogon with angle 0

D-90
Pattern: CCCC
Square with angle 0

Repeating unit 1: Triangle A.
Repeating unit 2: Triangle B.
Repeating unit 3: Triangle C.
Repeating unit 4: Triangle D.

Cyclical 3336
A and D triangles identical.
B and C triangles identical.
One plane tiled by combination of A, B, C, and D.

Triangle A: 0, 0,      54.736; BAA
Triangle B: 0, 54.736, 70.529; CAB
Triangle C: 0, 54.736, 70.529; BDC
Triangle D: 0, 0,      54.736; DCD

Vertices:
A-0
Pattern: A...
Regular apeirogon with angle 0

A-0 & B-0
Pattern: AABB...
Apeirogon with angles 0, 125.565, 109.472, 125.565, 0...

A-54.736 & B-70.529 & C-54.736
Pattern: AABCCB
Hexagon with angles 0, 0, 0, 125.565, 0, 125.565

B-54.736 & C-70.529 & D-54.736
Pattern: AABCCB
Hexagon with angles 0, 0, 0, 125.565, 0, 125.565

C-0 & D-0
Pattern: CCDD...
Apeirogon with angles 0, 125.565, 109.472, 125.565, 0...

D-0
Pattern: D...
Regular apeirogon with angle 0

Repeating unit: Rectangle with angle 0. Composed of 1 A, 1 B, 1 C, 1 D.

Cyclical 3436
A and D triangles identical.
B and C triangles identical.
Plane 1: tiled by combination of A and B
Plane 2: tiled by combination of C and D

Triangle A: 0, 0,  35.264; BAA
Triangle B: 0, 45, 54.736; ABA
Triangle C: 0, 45, 54.736; CDC
Triangle D: 0, 0,  35.264; DCD

Vertices:
A-0
Pattern: A...
Apeirogon with alternating angles 0 and 70.529

A-0 & B-0
Pattern: AABB...
Apeirogon with angles 0, 90, 90, 90...

A-35.264 & B-54.736
Pattern: AABBAABB
Octagon with angles 0, 0, 0, 90, 0, 0, 0, 90

B-45
Pattern: BBBBBBBB
Octagon with alternating angles 0 and 109.472

C-0 & D-0
Pattern: CCDD...
Apeirogon with angles 0, 90, 90, 90...

C-45
Pattern: CCCCCCCC
Octagon with alternating angles 0 and 109.472

C-54.736 & D-35.264
Pattern: CCDDCCDD
Octagon with angles 0, 0, 0, 90, 0, 0, 0, 90

D-0
Pattern: D...
Apeirogon with alternating angles 0 and 70.529

Repeating unit 1: Quadrangle with angles 0, 0, 45, 90. Composed of 1 A and 1 B.
Repeating unit 2: Quadrangle with angles 0, 0, 45, 90. Composed of 1 A and 1 B.

Cyclical 3536
A and D triangles identical.
B and C triangles identical.
One plane tiled by combination of A, B, C, and D.

Triangle A: 0, 0,      20.905; BAA
Triangle B: 0, 31.717, 37.377; CAB
Triangle C: 0, 31.717, 37.377; BDC
Triangle D: 0, 0,      20.905; DCD

Vertices:
A-0
Pattern: A...
Apeirogon with alternating angles 0 and 41.810

A-0 & B-0
Pattern: AABB...
Apeirogon with angles 0, 58.282, 63.434, 58.282, 0...

A-20.905 & B-37.377 & C-31.717
Pattern: AABCCBAABCCB
Dodecagon with angles 0, 0, 0, 69.095, 0, 69.095, 0, 0, 0, 69.095, 0, 69.095

B-31.717 & C-37.377 & D-20.905
Pattern: BBCDDCBBCDDC
Dodecagon with angles 0, 0, 0, 69.095, 0, 69.095, 0, 0, 0, 69.095, 0, 69.095

C-0 & D-0
Pattern: CCDD...
Apeirogon with angles 0, 58.282, 63.434, 58.282, 0...

D-0
Pattern: D...
Apeirogon with alternating angles 0 and 41.810

Repeating unit: Hexagon with angles 0, 0, 90, 0, 0, 90. Composed of 1 A, 1 B, 1 C, 1 D.

Cyclical 3636
Triangles A, B, C, and D all identical.
Plane 1: tiled by combination of A and B
Plane 2: tiled by combination of C and D

Triangle A: 0, 0, 0; BAA
Triangle B: 0, 0, 0; ABB
Triangle C: 0, 0, 0; CCD
Triangle D: 0, 0, 0; DDC

Vertices:
A-0
Pattern: A...
Regular apeirogon with angle 0

A-0 & B-0
Pattern: AABB...
Regular apeirogon with angle 0

A-0 & B-0
Pattern: AABB...
Regular apeirogon with angle 0

B-0
Pattern: B...
Regular apeirogon with angle 0

C-0
Pattern: C...
Regular apeirogon with angle 0

C-0 & D-0
Pattern: CCDD...
Regular apeirogon with angle 0

C-0 & D-0
Pattern: CCDD...
Regular apeirogon with angle 0

D-0
Pattern: D...
Regular apeirogon with angle 0

Repeating unit 1: Square with angle 0. Composed of 1 A, 1 B.
Repeating unit 2: Square with angle 0. Composed of 1 C, 1 D.

Cyclical 3344
A and C triangles identical
Plane 1: tiled by combination of A, B, and C.
Plane 2: tiled solely by D.

Triangle A: 0,      54.736, 54.736; BAA
Triangle B: 35.264, 35.264, 70.529; ACB
Triangle C: 0,      54.736, 54.736; BCC
Triangle D: 0,      45,     45    ; DDD

Vertices:
A-0
Pattern: A...
Regular apeirogon with angle 109.472

A-54.736 & B-35.264
Pattern: AABBAABB
Octagon with angles 0, 125.565, 70.529, 125.565, 0, 125.565, 70.529, 125.565

A-54.736 & B-70.529 & C-54.736
Pattern: AABCCB
Hexagon with angles 0, 90, 90, 0, 90, 90

B-35.264 & C-54.736
Pattern: BBCCBBCC
Octagon with angles 0, 125.565, 70.529, 125.565, 0, 125.565, 70.529, 125.565

C-0
Pattern: C...
Regular apeirogon with angle 109.472

D-0
Pattern: D...
Right-angled apeirogon

D-45
Pattern: DDDDDDDD
Octagon with alternating angles 0 and 90

D-45
Pattern: DDDDDDDD
Octagon with alternating angles 0 and 90

Repeating unit 1: Quadrangle with angles 0, 0, 90, 90. Composed of 1 A, 1 B, 1 C.
Repeating unit 2: Triangle D.

Cyclical 3444
A and B triangles identical
C and D triangles identical
Plane 1: tiled by combination of A and B.
Plane 2: tiled solely by C.
Plane 3: tiled solely by D.

Triangle A: 0, 35.264, 54.736; BAA
Triangle B: 0, 35.264, 54.736; ABB
Triangle C: 0, 0,      45    ; CCC
Triangle D: 0, 0,      45    ; DDD

Vertices:
A-0
Pattern: A...
Apeirogon with alternating angles 70.529 and 109.472

A-35.264 & B-54.736
Pattern: AABBAABB
Octagon with angles 0, 90, 0, 90, 0, 90, 0, 90

A-54.736 & B-35.264
Pattern: AABBAABB
Octagon with angles 0, 90, 0, 90, 0, 90, 0, 90

B-0
Pattern: B...
Apeirogon with alternating angles 70.529 and 109.472

C-0
Pattern: C...
Apeirogon with alternating angles 0 and 90

C-0
Pattern: C...
Apeirogon with alternating angles 0 and 90

C-45
Patttern: CCCCCCCC
Regular octagon with angle 0

D-0
Pattern: D...
Apeirogon with alternating angles 0 and 90

D-0
Pattern: D...
Apeirogon with alternating angles 0 and 90

D-45
Patttern: DDDDDDDD
Regular octagon with angle 0

Repeating unit 1: Quadrangle with angles 0, 90, 0, 90. Composed of 1 A and 1 B.
Repeating unit 2: Triangle C.
Repeating unit 3: Triangle D.

Cyclical 4444
Triangles A, B, C, and D all identical.
Plane 1: tiled solely by A.
Plane 2: tiled solely by B.
Plane 3: tiled solely by C.
Plane 4: tiled solely by D.

Triangle A: 0, 0, 0; AAA
Triangle B: 0, 0, 0; BBB
Triangle C: 0, 0, 0; CCC
Triangle D: 0, 0, 0; DDD

Vertices:
A-0
Pattern: A...
Regular apeirogon with angle 0

A-0
Pattern: A...
Regular apeirogon with angle 0

A-0
Pattern: A...
Regular apeirogon with angle 0

B-0
Pattern: B...
Regular apeirogon with angle 0

B-0
Pattern: B...
Regular apeirogon with angle 0

B-0
Pattern: B...
Regular apeirogon with angle 0

C-0
Pattern: C...
Regular apeirogon with angle 0

C-0
Pattern: C...
Regular apeirogon with angle 0

C-0
Pattern: C...
Regular apeirogon with angle 0

D-0
Pattern: D...
Regular apeirogon with angle 0

D-0
Pattern: D...
Regular apeirogon with angle 0

D-0
Pattern: D...
Regular apeirogon with angle 0

Repeating unit 1: Triangle A.
Repeating unit 2: Triangle B.
Repeating unit 3: Triangle C.
Repeating unit 4: Triangle D.

Triangle with added 3-branch
A and B triangles identical.
One plane tiled by combination of A, B, C, and D.

Triangle A: 0,      54.736,     90; ABC
Triangle B: 0,      54.736,     90; BAC
Triangle C: 0,      70.529, 70.529; DAB
Triangle D: 54.736, 54.736,     60; DDC

Vertices:
A-0 & B-0 & C-0
Pattern: ABC...
Apeirogon with angle 125.565; not regular

A-54.736 & C-70.529 & D-54.736
Pattern: AACDDC
Pentagon with angles 0, 0, 125.565, 120, 125.565

A-90 & B-90
Pattern: AABB
Rhombus with angles 0, 109.472, 0, 109.472

B-54.736 & C-70.529 & D-54.736
Pattern: BBCDDC
Pentagon with angles 0, 0, 125.565, 120, 125.565

D-60
Pattern: DDDDDD
Regular hexagon with angle 109.472

Repeating unit: Regular apeirogon with angle 60. Basically the full A-0 B-0 C-0 vertex with triangle D capping the finite sides of triangles C. A quadrangle with angles 0, 90, 60, 90 can also be considered a repeating unit, but when reflecting through a 0-90 side, triangles A and B will switch.

Triangle with added 4-branch
A and B triangles identical.
Plane 1: tiled by combination of A, B, and C.
Plane 2: tiled solely by D.

Triangle A: 0,  35.264, 90    ; ABC
Triangle B: 0,  35.264, 90    ; BAC
Triangle C: 0,  54.736, 54.736; CAB
Triangle D: 45, 45,     60    ; DDD

Vertices:
A-0 & B-0 & C-0
Pattern: ABC...
Non-regular right-angled apeirogon

A-35.264 & C-54.736
Pattern: AACCAACC
Hexagon with angles 0, 0, 109.472, 0, 0, 109.472

A-90 & B-90
Pattern: AABB
Rhombus with angles 0, 70.529, 0, 70.529

B-35.264 & C-54.736
Pattern: BBCCBBCC
Hexagon with angles 0, 0, 109.472, 0, 0, 109.472

D-45
Pattern: DDDDDDDD
Octagon with alternating angles 90 and 120

D-45
Pattern: DDDDDDDD
Octagon with alternating angles 90 and 120

D-60
Pattern: DDDDDD
Right-angled hexagon.

Repeating unit 1: The basic one is a pentagon with angles 0, 90, 90, 90, 90, but reflecting through a 0-90 side will switch A and B. Full repeating unit is a non-regular right-angled apeirogon.
Repeating unit 2: Triangle D.

Triangle with added 5-branch
A and B triangles identical.
One plane tiled by combination of A, B, C, and D.

Triangle A: 0,      20.905, 90    ; ABC
Triangle B: 0,      20.905, 90    ; BAC
Triangle C: 0,      37.377, 37.377; DAB
Triangle D: 31.717, 31.717, 60    ; DDC

Vertices:
A-0 & B-0 & C-0
Pattern: ABC...
Apeirogon with angle 58.282, not regular

A-20.905 & C-37,377 & D-31.317
Pattern: AACDDCAACDDC
Decagon with angles 0, 0, 69.094, 120, 69.094, 0, 0, 69.094, 120, 69.094

A-90 & B-90
Pattern: AABB
Rhombus with angles 0, 41.810, 0, 41.810

B-20.905 & C-37.377 & 31.717
Pattern: BBCDDCBBCDDC
Decagon with angles 0, 0, 69.094, 120, 69.094, 0, 0, 69.094, 120, 69.094

D-60
Pattern: DDDDDD
Regular hexagon with angle 63.434

Repeating unit: The basic one is a hexagon with angles 0, 90, 90, 60, 90, 90, but reflecting through a 0-90 side will switch A and B. Full repeating unit is an apeirogon with repeating angle sequence 60, 90, 90.

Triangle with added 6-branch
A and B triangles identical.
Plane 1: tiled by combination of A, B, and C.
Plane 2: tiled solely by D.

Triangle A: 0b, 0d, 90c; BAC
Triangle B: 0a, 0d, 90c; ABC
Triangle C: 0a, 0b, 0d ; ABC
Triangle D: 0a, 0b, 60c; DDD

Vertices:
A-0 & B-0 & C-0
Pattern: ABC...
Apeirogon with angle 0

A-0 & C-0
Pattern: AACC...
Apeirogon with angle 0

A-90 & B-90
Pattern: AABB
Square with angle 0

B-0 & C-0
Pattern: BBCC...
Apeirogon with angle 0

D-0
Pattern: D...
Apeirogon with alternating angles 0 and 120

D-0
Pattern: D...
Apeirogon with alternating angles 0 and 120

D-60
Pattern: DDDDDD
Regular hexagon with angle 0

Repeating unit 1: The basic one is a pentagon with angles 0, 0, 90, 0, 90, but reflecting through a 0-90 side will switch A and B. Full repeating unit is an apeirogon with angle 0.
Repeating unit 2: Triangle D.

Two fused triangles
A and D triangles identical.
B and C triangles identical.
One plane tiled by combination of A, B, C, and D.

Triangle A: 0, 54.736, 54.736; ABC
Triangle B: 0, 0,      70.529; ADC
Triangle C: 0, 0,      70.529; ADB
Triangle D: 0, 54.736, 54.736; DCB

Vertices:
A-0 & B-0 & C-0
Pattern: ABC...
Apeirogon with angles 0, 125.565, 125.565...

A-54.736 & B-70.529 & D-54.736
Pattern: AABDDB
Hexagon with angles 0, 0, 109.472, 0, 0, 109.472

A-54.736 & C-70.529 & D-54.736
Pattern: AACDDC
Hexagon with angles 0, 0, 109.472, 0, 0, 109.472

B-0 & C-0 & D-0
Pattern: BCD...
Apeirogon with angles 0, 125.565, 125.565...

Repeating unit: The notion of repeating unit starts breaking down a bit here. The ABD or ACD vertices work, with some swaps caused by reflection.

Tetrahedron of 3-edges
Triangles A, B, C, and D all identical.
One plane tiled by combination of A, B, C, and D.

Triangle A: 0, 0, 0; BCD
Triangle B: 0, 0, 0; ACD
Triangle C: 0, 0, 0; ABD
Triangle D: 0, 0, 0; ABC

Vertices:
A-0 & B-0 & C-0
Pattern: ABC...
Regular apeirogon with angle 0

A-0 & B-0 & D-0
Pattern: ABC...
Regular apeirogon with angle 0

A-0 & C-0 & D-0
Pattern: ABC...
Regular apeirogon with angle 0

B-0 & C-0 & D-0
Pattern: ABC...
Regular apeirogon with angle 0

Repeating unit: Each of the ideal triangles can be considered a repeating unit; they are just differently labeled.

{3,3,3,3}
AB and DE triangles identical.
AC and CE triangles identical.
AD and BE triangles identical.
BC and CD triangles identical.
Plane 1: tiled by combination of AB, BC, CD, and DE.
Plane 2: tiled by combination of AC, AD, AE, BD, BE, and CE.

Triangle AB: 52.239, 60,     90; AB BC AB
Triangle AC: 54.736, 65.905, 90; AC AC AD
Triangle AD: 48.190, 70.529, 90; AE BD AC
Triangle AE: 54.736, 54.736, 90; AD BE AE
Triangle BC: 52.239, 75.522, 90; AB BC CD
Triangle BD: 65.905, 65.905, 90; AD BE BD
Triangle BE: 48.190, 70.529, 90; BD AE CE
Triangle CD: 52.239, 75.522, 90; DE CD BC
Triangle CE: 54.736, 65.905, 90; CE CE BE
Triangle DE: 52.239, 60,     90; DE CD DE

Repeating unit 1: Digonal strip of angle 60, made from AB, BC, CD, DE.
Repeating unit 2: Digonal strip of angle 90, made from AC, AD, AE, BD, BE, CE.

{3,3,3,4}
Plane 1: tiled by combination of AB, BC, and CD.
Plane 2: tiled by combination of AC, AD, and BD.
Plane 3: tiled by combination of AE, BE, and CE.
Plane 4: tiled solely by DE.

Triangle AB: 45,     52.239, 90; BC AB AB
Triangle AC: 35.264, 65.905, 90; AC AC AD
Triangle AD: 48.190, 54.736, 90; AD BD AC
Triangle AE: 45,     54.736, 90; BE AE AE
Triangle BC: 30,     75.522, 90; AB BC CD
Triangle BD: 45,     65.905, 90; AD BD BD
Triangle BE: 35.264, 70.529, 90; AE BE CE
Triangle CD: 52.239, 60,     90; CD CD BC
Triangle CE: 54.736, 54.736, 90; CE CE BE
Triangle DE: 45,     60,     90; DE DE DE

Repeating unit 1: Equilateral triangle with angle 90, made from AB, BC, and CD.
Repeating unit 2: Triangle with angles 45, 90, 90, made from AC, AD, and BD.
Repeating unit 3: Triangle with angles 45, 90, 90, made from AE, BE, and CE.
Repeating unit 4: Triangle DE.

Demipenteractic
A, B are one branch, C is the center, D and E are ends of short branches.
AD and AE triangles identical.
BD and BE triangles identical.
CD and CE triangles identical.
Plane 1: tiled by combination of AB, BC, CD, and CE.
Plane 2: tiled by combination of AC, AD, AE, BD, and BE.
Plane 3: tiled solely by DE.

Triangle AB: 52.239, 52.239, 90    ; AB AB BC
Triangle AC: 65.905, 65.905, 70.529; AD AE AC
Triangle AD: 48.190, 54.736, 90    ; AD BD AC
Triangle AE: 48.190, 54.736, 90    ; AE BE AC
Triangle BC: 75.522, 75.522, 90    ; CD CE AB
Triangle BD: 45,     65.905, 90    ; AD BD BD
Triangle BE: 45,     65.905, 90    ; AE BE BE
Triangle CD: 52.239, 60,     90    ; CE CD BC
Triangle CE: 52.239, 60,     90    ; CD CE BC
Triangle DE: 45,     60,     90    ; DE DE DE

Repeating unit 1: Digonal strip of angle 90, made from AB, BC, CD, and CE (two of each). Half of it, equilateral triangle of angle 90, works, but swaps some labels when reflecting.
Repeating unit 1: Digonal strip of angle 45, made from AC, AD, AE, BD, and BE.
Repeating unit 3: Triangle DE.

C~4 {4,3,3,4}
AB and DE triangles identical.
AC and CE triangles identical.
AD and BE triangles identical.
BC and CD triangles identical.
Plane 1: tiled solely by AB.
Plane 2: tiled by combination of AC and AD.
Plane 3: tiled solely by AE.
Plane 4: tiled by combination of BC and CD.
Plane 5: tiled solely by BD.
Plane 6: tiled by combination of BE and CE.
Plane 7: tiled solely by DE.

Triangle AB: 45,     45,     90; AB AB AB
Triangle AC: 35.264, 54.736, 90; AC AC AD
Triangle AD: 35.264, 54.736, 90; AD AD AC
Triangle AE: 45,     45,     90; AE AE AE
Triangle BC: 30,     60,     90; BC BC CD
Triangle BD: 45,     45,     90; BD BD BD
Triangle BE: 35.264, 54.736, 90; BE BE CE
Triangle CD: 30,     60,     90; CD CD BC
Triangle CE: 35.264, 54.736, 90; CE CE BE
Triangle DE: 45,     45,     90; DE DE DE

Repeating unit 1: Triangle AB
Repeating unit 2: Rectangle made from AC and AD.
Repeating unit 3: Triangle AE
Repeating unit 4: Rectangle made from BC and CD.
Repeating unit 5: Triangle BD
Repeating unit 6: Rectangle made from BE and CE.
Repeating unit 7: Triangle DE

F~4 {3,3,4,3}
Plane 1: tiled by combination of AB and BC.
Plane 2: tiled solely by AC.
Plane 3: tiled by combination of AD, AE, BD, BE, and CE.
Plane 4: tiled solely by CD.
Plane 5: tiled solely by DE.

Triangle AB: 30,     60,     90; AB BC AB
Triangle AC: 45,     45,     90; AC AC AC
Triangle AD: 35.264, 54.736, 90; AE BD AD
Triangle AE: 35.264, 54.736, 90; BE AD AE
Triangle BC: 30,     60,     90; BC AB BC
Triangle BD: 35.264, 54.736, 90; AD BE BD
Triangle BE: 19.471, 70.529, 90; AE BD CE
Triangle CD: 45,     45,     90; CD CD CD
Triangle CE: 35.264, 54.736, 90; CE CE BE
Triangle DE: 30,     60,     90; DE DE DE

Repeating unit 1: Equilateral triangle made from AB and BC.
Repeating unit 2: Triangle AC.
Repeating unit 3: This is an interesting one. The five triangles AD, AE, BD, BE, and CE form a rectangle. Four of these five triangles are similar; only the "central" one, BE, has a different shape.
Repeating unit 4: Triangle CD.
Repeating unit 5: Triangle DE.

B~4 Branched Euclidean group (half of tesseractic honeycomb)
Same marking as demipenteractic, AB branch is 4, BC, CD and DE are 3.
AD and AE triangles identical.
BD and BE triangles identical.
CD and CE triangles identical.
Plane 1: tiled solely by AB.
Plane 2: tiled by combination of AC, AD, and AE.
Plane 3: tiled by combination of BC, CD, and CE.
Plane 4: tiled solely by BD.
Plane 5: tiled solely by BE.
Plane 6: tiled solely by DE.

Triangle AB: 45,     45,     90    ; AB AB AB
Triangle AC: 54.736, 54.736, 70.529; AD AE AC
Triangle AD: 35.264, 54.736, 90    ; AD AD AC
Triangle AE: 35.264, 54.736, 90    ; AE AE AC
Triangle BC: 60,     60,     60    ; BC CD CE
Triangle BD: 45,     45,     90    ; BD BD BD
Triangle BE: 45,     45,     90    ; BE BE BE
Triangle CD: 30,     60,     90    ; CD CD BC
Triangle CE: 30,     60,     90    ; CE CE BC
Triangle DE: 45,     45,     90    ; DE DE DE

Repeating unit 1: Triangle AB.
Repeating unit 2: Rectangle made from triangles AC, AD, and AE.
Repeating unit 3: Rectangle made from triangles BC, CD, and CE, two of each. Half of it works as well, but swaps some labels when reflecting.
Repeating unit 4: Triangle BE.
Repeating unit 5: Triangle BE.
Repeating unit 6: Triangle DE.

D~4 Cross group (quarter of tesseractic honeycomb)
AB, BC, BD, BE branches are 3.
AB, BC, BC, BE triangles identical.
AC, AD, AE, CD, CE, DE triangles identical.
Plane 1: tiled by combination of AB, BC, BD, and BE.
Plane 2: tiled solely by AC.
Plane 3: tiled solely by AD.
Plane 4: tiled solely by AE.
Plane 5: tiled solely by CD.
Plane 6: tiled solely by CE.
Plane 7: tiled solely by DE.

Triangle AB: 60, 60, 60; BC BD BE
Triangle AC: 45, 45, 90; AC AC AC
Triangle AD: 45, 45, 90; AD AD AD
Triangle AE: 45, 45, 90; AE AE AE
Triangle BC: 60, 60, 60; AB BD BE
Triangle BD: 60, 60, 60; AB BC BE
Triangle BE: 60, 60, 60; AB BC BD
Triangle CD: 45, 45, 90; CD CD CD
Triangle CE: 45, 45, 90; CE CE CE
Triangle DE: 45, 45, 90; DE DE DE

Repeating unit 1: Plane 1 is tiled by equilateral triangles with a particular 4-coloring such as that each vertex has triangles of three colors around it, with opposite pairs colored alike, and each triangle has three vertices with different color combinations.
Repeating unit 2: Triangle AC.
Repeating unit 3: Triangle AD.
Repeating unit 4: Triangle AE.
Repeating unit 5: Triangle CD.
Repeating unit 6: Triangle CE.
Repeating unit 7: Triangle DE.

A~4 Cyclical 33333
AB, AE, BC, CD, DE triangles identical.
AC, AD, BD, BE, CE triangles identical.
Plane 1: tiled by combination of AB, AE, BC, CD, and DE.
Plane 2: tiled by combination of AC, AD, BD, BE, and CE.

Triangle AB: 52.239, 52.239, 75.522; AE BC AB
Triangle AC: 48.190, 65.905, 65.905; AC AD CE
Triangle AD: 48.190, 65.905, 65.905; AD AC BD
Triangle AE: 52.239, 52.239, 75.522; AB DE AE
Triangle BC: 52.239, 52.239, 75.522; AB CD BC
Triangle BD: 48.190, 65.905, 65.905; BD AD BE
Triangle BE: 48.190, 65.905, 65.905; BE BD CE
Triangle CD: 52.239, 52.239, 75.522; BC DE CD
Triangle CE: 48.190, 65.905, 65.905; CE AC BE
Triangle DE: 52.239, 52.239, 75.522; AE CD DE

Repeating unit 1: Triangles form strips where five "colors" repeat endlessly.
Repeating unit 2: Triangles form strips where five "colors" repeat endlessly.

H4 {3,3,3,5}
Plane 1: tiled by combination of AB, BC, and CD.
Plane 2: tiled by combination of AC, AD, AE, BD, BE, and CE.
Plane 3: tiled solely by DE.

Triangle AB: 36,     52.239, 90; BC AB AB
Triangle AC: 20.905, 65.905, 90; AC AC AD
Triangle AD: 37.377, 48.190, 90; BD AE AC
Triangle AE: 31.717, 54.736, 90; BE AD AE
Triangle BC: 7.761,  75.522, 90; AB BC CD
Triangle BD: 13.283, 65.905, 90; AD BE BD
Triangle BE: 10.812, 70.529, 90; AE BD CE
Triangle CD: 22.239, 52.239, 90; CD CD BC
Triangle CE: 20.905, 54.736, 90; CE CE BE
Triangle DE: 18,     60,     90; DE DE DE

Repeating unit 1: Triangle with angles 30, 36, 90, made from AC, BC, and CD.
Repeating unit 2: Quadrangle with angles 45, 90, 90, 90, made from AC, AD, AE, BD, BE, and CE.
Repeating unit 3: Triangle DE.

BH4 {4,3,3,5}
Plane 1: tiled solely by AB.
Plane 2: tiled by combination of AC, AD, and AE.
Plane 3: tiled by combination of BC and CD.
Plane 4: tiled by combination of BD, BE, and CE.
Plane 5: tiled solely by DE.

Triangle AB: 36,     45,     90; AB AB AB
Triangle AC: 20.905, 54.736, 90; AC AC AD
Triangle AD: 35.264, 37.377, 90; AE AD AC
Triangle AE: 31.717, 45,     90; AE AD AE
Triangle BC: 7.761,  60,     90; BC BC CD
Triangle BD: 13.283, 45,     90; BD BE BD
Triangle BE: 10.812, 54.736, 90; BE BD CE
Triangle CD: 22.239, 30,     90; CD CD BC
Triangle CE: 20.905, 35.264, 90; CE CE BE
Triangle DE: 18,     45,     90; DE DE DE

Repeating unit 1: Triangle AB.
Repeating unit 2: Quadrangle with angles 45, 90, 90, 90, made from AC, AD, and AE.
Repeating unit 3: Quadrangle with angles 30, 90, 90, 90, made from BC and CD.
Repeating unit 4: Quadrangle with angles 45, 45, 90, 90, made from BD, BE, and CE.
Repeating unit 5: Triangle DE.

K4 {5,3,3,5}
AB and DE triangles identical.
AC and CE triangles identical.
AD and BE triangles identical.
BC and CD triangles identical.
Plane 1: tiled solely by AB.
Plane 2: tiled by combination of AC, AD, AE, BD, BE, and CE.
Plane 3: tiled by combination of BC and CD.
Plane 4: tiled solely by DE.

Triangle AB: 18,     36,     90; AB AB AB
Triangle AC: 20.905, 20.905, 90; AC AC AD
Triangle AD: 10.812, 37.377, 90; AE BD AC
Triangle AE: 31.717, 31.717, 90; AD BE AE
Triangle BC: 7.761,  22.239, 90; BC CD BC
Triangle BD: 13.283, 13.283, 90; AD BE BD
Triangle BE: 10.812, 37.377, 90; AE BD CE
Triangle CD: 7.761,  22.239, 90; CD BC CD
Triangle CE: 20.905, 20.905, 90; CE CE BE
Triangle DE: 18,     36,     90; DE DE DE

Repeating unit 1: Triangle AB.
Repeating unit 3: Hexagon with angles 45, 45, 90, 90, 90, 90, made from AC, AD, AE, BD, BE, and CE.
Repeating unit 3: Quadrangle with angles 30, 90, 30, 90, made from BC and CD.
Repeating unit 4: Triangle DE.

DH4 Branched half of {4,3,3,5}
AB is 5, BC, CD and CE are 3
AD and AE triangles identical.
BD and BE triangles identical.
CD and CE triangles identical.
Plane 1: tiled solely by AB.
Plane 2: tiled by combination of AC, AD, AE, BD, and BE.
Plane 3: tiled combination of BC, CD, and CE.
Plane 4: tiled solely by DE.

Triangle AB: 18,     18,     90    ; AB AB AB
Triangle AC: 20.905, 20.905, 70.529; AD AE AC
Triangle AD: 10.812, 54.736, 90    ; AD BD AC
Triangle AE: 10.812, 54.736, 90    ; AE BE AC
Triangle BC: 22.239, 22.239, 60    ; CD CE BC
Triangle BD: 13.283, 45,     90    ; BD AD BD
Triangle BE: 13.283, 45,     90    ; BE AE BE
Triangle CD: 7.761,  60,     90    ; CE CD BC
Triangle CE: 7.761,  60,     90    ; CD CE BC
Triangle DE: 36,     45,     90    ; DE DE DE

Repeating unit 1: Triangle AB.
Repeating unit 2: Quadrangle with all angles 45, made from AC, AD, AE, BD, and BE.
Repeating unit 3: Quadrangle with all angles 30, made from BC, CD, and CE, two of each. Half of it works as well, but swaps some labels when reflecting.
Repeating unit 4: Triangle DE.

AF4 Cyclical 33334
AE is 4, AB, BC, CD, and DE are 3
AB and DE triangles identical.
AC and CE triangles identical.
AD and BE triangles identical.
BC and CD triangles identical.
Plane 1: tiled by combination of AB, BC, CD, and DE.
Plane 2: tiled by combination of AC, AD, BD, BE, and CE.
Plane 3: tiled solely by AE.

Triangle AB: 30,     52.239, 60    ; BC AB AB
Triangle AC: 35.264, 45,     65.905; AC AD AC
Triangle AD: 35.264, 48.190, 54.736; AC BD AD
Triangle AE: 45,     45,     45    ; AE AE AE
Triangle BC: 30,     52.239, 70.529; CD AB BC
Triangle BD: 19.471, 65.905, 65.905; BD AD BE
Triangle BE: 35.264, 48.190, 54.736; CE BD BE
Triangle CD: 30,     52.239, 70.529; BC DE CD
Triangle CE: 35.264, 45,     65.905; CE BE CE
Triangle DE: 30,     52.239, 60    ; CD DE DE

Repeating unit 1: Quadrangle with angles 30, 90, 30, 90, made from AB, BC, CD, and DE.
Repeating unit 1: Quadrangle with angles 45, 45, 90, 90, made from AC, AD, BD, BE, and CE.
Repeating unit 3: Triangle AE.

R4 {3,4,3,4}
Plane 1: tiled solely by AB.
Plane 2: tiled by combination of AC, AD, and BD.
Plane 3: tiled by combination of AE and BE.
Plane 4: tiled solely by BC.
Plane 5: tiled solely by CD.
Plane 6: tiled solely by CE.
Plane 7: tiled solely by DE.

Triangle AB: 30,     45,     90; AB AB AB
Triangle AC: 35.264, 35.264, 90; AC AC AD
Triangle AD: 19.471, 54.736, 90; AD BD AC
Triangle AE: 35.264, 45,     90; AE BE AE
Triangle BC: 0,      45,     90; BC BC BC
Triangle BD: 0,      35.264, 90; AD BD BD
Triangle BE: 0,      54.736, 90; AE BE BE
Triangle CD: 0,      30,     90; CD CD CD
Triangle CE: 0,      45,     90; CE CE CE
Triangle DE: 0,      60,     90; DE DE DE

Repeating unit 1: Triangle AB.
Repeating unit 2: Quadrangle with angles 0, 90, 90, 90, made from AC, AD, and BD.
Repeating unit 3: Triangle with angles 0, 45, 90, made from AE and BE.
Repeating unit 4: Triangle BC.
Repeating unit 5: Triangle CD.
Repeating unit 6: Triangle CE.
Repeating unit 7: Triangle DE.

S4 Branched 33(34)
CE is 4, AB, BC, CD is 3
Plane 1: tiled by combination of AB, BC, and CD.
Plane 2: tiled by combination of AC, AD, and BD.
Plane 3: tiled by combination of AE and BE.
Plane 4: tiled solely by CE.
Plane 5: tiled solely by DE.

Triangle AB: 30,     52.239, 90    ; AB AB BC
Triangle AC: 45,     54.736, 65.905; AD AC AC
Triangle AD: 35.264, 48.190, 90    ; BD AD AC
Triangle AE: 35.264, 45,     90    ; AE BE AE
Triangle BC: 0,      60,     75.522; AB CD BC
Triangle BD: 0,      65.905, 90    ; AD BD BD
Triangle BE: 0,      54.736, 90    ; AE BE BE
Triangle CD: 0,      52.239, 90    ; CD CD BC
Triangle CE: 0,      45,     90    ; CE CE CE
Triangle DE: 0,      60,     90    ; DE DE DE

Repeating unit 1: Quadrangle with angles 0, 90, 90, 90, made from AB, BC, and CD.
Repeating unit 2: Triangle with angles 0, 45, 90, made from AC, AD, and BD.
Repeating unit 3: Triangle with angles 0, 45, 90, made from AE and BE.
Repeating unit 4: Triangle CE.
Repeating unit 5: Triangle DE.

O4 Branched 34(33)
BC is 4, AB, CD, CE is 3
AD and AE triangles identical.
BD and BE triangles identical.
CD and CE triangles identical.
Plane 1: tiled solely by AB.
Plane 2: tiled by combination of AC, AD, AE, BD, and BE.
Plane 3: tiled solely by BC.
Plane 4: tiled solely by CD.
Plane 5: tiled solely by CE.
Plane 6: tiled solely by DE.

Triangle AB: 30,     30,     90    ; AB AB AB
Triangle AC: 35.264, 35.264, 70.529; AD AE AC
Triangle AD: 19.471, 54.736, 90    ; AD BD AC
Triangle AE: 19.471, 54.736, 90    ; AE BD AC
Triangle BC: 0,      45,     45    ; BC BC BC
Triangle BD: 0,      35.264, 90    ; AD BD BD
Triangle BE: 0,      35.264, 90    ; AE BE BE
Triangle CD: 0,      30,     90    ; CD CD CD
Triangle CE: 0,      30,     90    ; CE CE CE
Triangle DE: 0,      60,     90    ; DE DE DE

Repeating unit 1: Triangle AB.
Repeating unit 2: Quadrangle with angles 0, 0, 90, 90, made from AC, AD, AE, BD, and BE.
Repeating unit 3: Triangle BC.
Repeating unit 4: Triangle CD.
Repeating unit 5: Triangle CE.
Repeating unit 6: Triangle DE.

N4 Branched 43(34)
AB, CE is 4, BC, CD is 3
Plane 1: tiled solely by AB.
Plane 2: tiled by combination of AC and AD.
Plane 3: tiled solely by AE.
Plane 4: tiled combination of BC and CD.
Plane 5: tiled solely by BD.
Plane 6: tiled solely by BE.
Plane 7: tiled solely by CE.
Plane 8: tiled solely by DE.

Triangle AB: 0,      45,     90    ; AB AB AB
Triangle AC: 0,      54.736, 54.736; AD AC AC
Triangle AD: 35.264, 35.264, 90    ; AD AD AC
Triangle AE: 0,      45,     90    ; AE AE AE
Triangle BC: 0,      0,      60    ; BC CD BC
Triangle BD: 0,      45,     90    ; BD BD BD
Triangle BE: 0,      0,      90    ; BE BE BE
Triangle CD: 0,      30,     90    ; CD CD BC
Triangle CE: 0,      0,      90    ; CE CE CE
Triangle DE: 0,      45,     90    ; DE DE DE

Repeating unit 1: Triangle AB.
Repeating unit 2: Quadrangle with angles 0, 90, 90, 90, made from AC and AD.
Repeating unit 3: Triangle AE.
Repeating unit 4: Quadrangle with angles 0, 0, 90, 90, made from BC and CD.
Repeating unit 5: Triangle BD.
Repeating unit 6: Triangle BE.
Repeating unit 7: Triangle CE.
Repeating unit 8: Triangle DE.

M4 Branched 3334
BE is 4, AB, BC, BD are 3
AB, BC, and BD triangles identical.
AC, AD, and CD triangles identical.
AE, CE, and DE triangles identical.
Plane 1: tiled by combination of AB, BC, and BD.
Plane 2: tiled solely by AC.
Plane 3: tiled solely by AD.
Plane 4: tiled solely by AE.
Plane 5: tiled solely by BE.
Plane 6: tiled solely by CD.
Plane 7: tiled solely by CE.
Plane 8: tiled solely by DE.

Triangle AB: 0, 0,  60; BC BD AB
Triangle AC: 0, 45, 90; AC AC AC
Triangle AD: 0, 45, 90; AC AC AC
Triangle AE: 0, 0,  90; AE AE AE
Triangle BC: 0, 0,  60; AB BD BC
Triangle BD: 0, 0,  60; AB BC BD
Triangle BE: 0, 0,  0 ; BE BE BE
Triangle CD: 0, 45, 90; CD CD CD
Triangle CE: 0, 0,  90; CE CE CE
Triangle DE: 0, 0,  90; DE DE DE

Repeating unit 1: Regular hexagon with angle 0, made of 2 copies of AB, BC, and BD. Half works if we allow for reflection swaps.
Repeating unit 2: Triangle AC.
Repeating unit 3: Triangle AD.
Repeating unit 4: Triangle AE.
Repeating unit 5: Triangle BE.
Repeating unit 6: Triangle CD.
Repeating unit 7: Triangle CE.
Repeating unit 8: Triangle DE.

FR4 Cyclical 33434
AE, CD are 4, AB, BC, DE are 3
AB and BC triangles identical.
AD and CE triangles identical.
AE and CD triangles identical.
BD and BE triangles identical.
Plane 1: tiled with combination of AB and BC.
Plane 2: tiled solely by AC.
Plane 3: tiled by combination of AD, BD, BE, and CE.
Plane 4: tiled solely by AE.
Plane 5: tiled solely by CD.
Plane 6: tiled solely by DE.

Triangle AB: 30,     30,     60    ; AB BC AB
Triangle AC: 0,      45,     45    ; AC AC AC
Triangle AD: 0,      35.264, 35.264; BD AD AD
Triangle AE: 0,      45,     45    ; AE AE AE
Triangle BC: 30,     30,     60    ; AB BC BC
Triangle BD: 19.471, 35.264, 54.736; BD AE BE
Triangle BE: 19.471, 35.264, 54.736; BE CE BD
Triangle CD: 0,      45,     45    ; CD CD CD
Triangle CE: 0,      35.264, 35.264; BE CE CE
Triangle DE: 0,      30,     30    ; DE DE DE

Repeating unit 1: Quadrangle with angles 60, 90, 60, 90, made from AB and BC.
Repeating unit 2: Triangle AC.
Repeating unit 3: Quadrangle with angles 0, 90, 0, 90, made from AD, BD, BE, and CE.
Repeating unit 4: Triangle AE.
Repeating unit 5: Triangle CD.
Repeating unit 6: Triangle DE.

P4 Square with added 3-branch
AB, AC, BD, CD, DE are 3
AB and AC triangles identical.
BD and CD triangles identical.
BE and CE triangles identical.
Plane 1: tiled by combination of AB, AC, BD, CD, and DE.
Plane 2: tiled by combination of AD, AE, BE, and CE.
Plane 3: tiled solely by BC.

Triangle AB: 0,      52.239, 90    ; AB AC BD
Triangle AC: 0,      52.239, 90    ; AC AB CD
Triangle AD: 0,      65.905, 65.905; AE AD AD
Triangle AE: 48.190, 48.190, 70.529; BE CE AD
Triangle BC: 0,      45,     90    ; BC BC BC
Triangle BD: 0,      60,     75.522; DE AB CD
Triangle BE: 45,     54.736, 65.905; AE BE BE
Triangle CD: 0,      60,     75.522; DE AC BD
Triangle CE: 45,     54.736, 65.905; AE CE CE
Triangle DE: 52.239, 52.239, 60    ; BD CD DE

Repeating unit 1: If we count AB/AC and BD/CD as reflections, we get a quadrangle with angles 0, 0, 90, 90, made from AB, AC, BD, CD, and DE. With only true reflection, we get an infinite branching structure.
Repeating unit 2: Triangle with angles 0, 45, 45, made from AD, AE, BE, and CE.
Repeating unit 3: Triangle BC.

BP4 Square with added 4-branch
DE is 4, AB, AC, BD, CD are 3
AB and AC triangles identical.
BD and CD triangles identical.
BE and CE triangles identical.
Plane 1: tiled by combination of AB, AC, BD, and CD
Plane 2: tiled solely by AD.
Plane 3: tiled by combination of AE, BE, and CE.
Plane 4: tiled solely by BC.
Plane 5: tiled solely by DE.

Triangle AB: 0,      30,     90    ; AB AC BD
Triangle AC: 0,      30,     90    ; AC AB CD
Triangle AD: 0,      45,     45    ; AD AD AD
Triangle AE: 35.264, 35.264, 70.529; BE CE AE
Triangle BC: 0,      0,      90    ; BC BC BC
Triangle BD: 0,      0,      60    ; BD BD CD
Triangle BE: 0,      54.736, 54.736; AE BE BE
Triangle CD: 0,      0,      60    ; CD CD BD
Triangle CE: 0,      54.736, 54.736; AE CE CE
Triangle DE: 0,      45,     45    ; DE DE DE

Repeating unit 1: AB, AC, BD, and CD form a pentagon with angles 0, 0, 0, 90, 90. Reflection through 0-0 sides swaps BD and CD.
Repeating unit 2: Triangle AD.
Repeating unit 3: Quadrangle with angles 0, 0, 90, 90, made of AE, BE, and CE.

DP4 Square with bridge
AB, AD, BC, BE, CD, DE are 3
AB, AD, BC, BE, CD, and CE triangles identical.
AC, AE, and CE triangles identical.
Plane 1: tiled by combination of AB, AD, BC, BE, CD, and CE.
Plane 2: tiled solely by AC.
Plane 3: tiled solely by AE.
Plane 4: tiled solely by BD.
Plane 5: tiled solely by CE.

Triangle AB: 0, 0,  60; BC BE AD
Triangle AC: 0, 45, 45; AC AC AC
Triangle AD: 0, 0,  60; CD DE AB
Triangle AE: 0, 45, 45; AE AE AE
Triangle BC: 0, 0,  60; AB BE CD
Triangle BD: 0, 0,  0 ; BD BD BD
Triangle BE: 0, 0,  60; AB BC DE
Triangle CD: 0, 0,  60; AD DE BC
Triangle CE: 0, 45, 45; CE CE CE
Triangle DE: 0, 0,  60; AD CD BE

Repeating unit 1: All triangles are the same (0, 0, 60), but there are six types. The 60-degree vertices are of two kinds, one with B-triangles around (AB, BC, BE, AB, BC, BE), second with D-triangles (AD, CD, DE, AD, CD, DE). Reflection through 0-0 side always switches between a B-triangle and a D-triangle.
Repeating unit 2: Triangle AC.
Repeating unit 3: Triangle AE.
Repeating unit 4: Triangle BD.
Repeating unit 5: Triangle CE.


For 6D, I will need centri angles in hexateron / penteract groups. I assume that demipenteract can be derived from penteract.

I don't completely understand Dr. Klitzing dihedral page (where I got the polyhedron/polychoron data), but fortunately, the most complex case (with 5-fold axis) no longer exists in 5D. That means I can simply construct the polytera explicitly and then find centri angles between their various elements.

The simplest way to model a hexateron, rather than dealing with square roots and such, seems to be to embed it in 6-dimensional space with vertices as 6 combinations of (6,0,0,0,0,0). Edge lengths are not important, since I'm only interested in angles. Its center will then be in (1,1,1,1,1,1). I'll shift the center to the origin, getting coordinates
A: (5,-1,-1,-1,-1,-1)
B: (-1,5,-1,-1,-1,-1)
C: (-1,-1,5,-1,-1,-1)
D: (-1,-1,-1,5,-1,-1)
E: (-1,-1,-1,-1,5,-1)
F: (-1,-1,-1,-1,-1,5)

So, element vectors to use are
Vertex A - (5,-1,-1,-1,-1,-1), length sqrt(30); corresponds to great prismated decachoron in omnitruncate
Center of edge AB - (2,2,-1,-1,-1,-1), length 2 sqrt(3); corresponds to truncated octahedral prism in omnitruncate
Center of face ABC - (1,1,1,-1,-1,-1), length sqrt(6); corresponds to hexagonal duoprism in omnitruncate
Center of cell ABCD - (1/2,1/2,1/2,1/2,-1,-1), length sqrt(3); corresponds to truncated octahedral prism in omnitruncate
Center of teron ABCDE - (1/5,1/5,1/5,1/5,1/5,-1), length sqrt(6/5); corresponds to great prismated decachoron in omnitruncate

Calculation shows that omnitruncated hexateron would have these dihedral and centri angles:
Code: Select all
o{3,3,3,3}
great prismated decachoron 1/truncated octahedral prism 1
129.231520 - 50.768480 - arccos(sqrt[2/5])
great prismated decachoron 1/hexagonal duoprism
116.565051 - 63.434949 - arccos(sqrt[1/5])
great prismated decachoron 1/truncated octahedral prism 2
108.434949 - 71.565051 - arccos(sqrt[1/10])
great prismated decachoron 1/great prismated decachoron 2
101.536959 - 78.463041 - arccos(1/5)
truncated octahedral prism 1/hexagonal duoprism
135 - 45
truncated octahedral prism 1/truncated octahedral prism 2
120 - 60
truncated octahedral prism 1/great prismated decachoron 2
108.434949 - 71.565051 - arccos(sqrt[1/10])
hexagonal duoprism/truncated octahedral prism 2
135 - 45
hexagonal duoprism/great prismated decachoron 2
116.565051 - 63.434949 - arccos(sqrt[1/5])
truncated octahedral prism 2/great prismated decachoron 2
129.231520 - 50.768480 - arccos(sqrt[2/5])
cycle 1: |top1 -to- gpd1 -to- gpd2 -to- top2| - 50.768 + 78.463 + 50.768 = 180
cycle 2: |top1 -c- top2| - 60
cycle 3: (gpd1 -hp- top2 -hp- hd -hp- gpd2 -hp- top1 -hp- hd -hp-) - 71.565 + 45 + 63.435 + 71.565 + 45 + 63.435 = 360
opposites: gpd1/gpd2, top1/top2, hd/hd


Similarly for penteract. This can be easily embedded in 5-space, as 32 combinations of (+-1,+-1,+-1,+-1,+-1).
So we can use these vectors:
A - Vertex: (1,1,1,1,1); corresponds to great prismated decachoron in omnitruncate
B - Edge: (1,1,1,1,0); corresponds to truncated octahedral prism in omnitruncate
C - Face: (1,1,1,0,0); corresponds to hexagonal-octagonal duoprism in omnitruncate
D - Cell: (1,1,0,0,0); corresponds to truncated cuboctahedral prism in omnitruncate
E - Teron: (1,0,0,0,0); corresponds to great prismated tesseract in omnitruncate

Data correspond to this:
Code: Select all
o{3,3,3,4}
great prismated decachoron/truncated octahedral prism
153.434949 - 26.565051 - arccos(sqrt[4/5])
great prismated decachoron/hexagonal-octagonal duoprism
140.768480 - 39.231521 - arccos(sqrt[3/5])
great prismated decachoron/truncated cuboctahedral prism
129.231520 - 50.768480 - arccos(sqrt[2/5])
great prismated decachoron/great prismated tesseract
116.565051 - 63.434949 - arccos(sqrt[1/5])
truncated octahedral prism/hexagonal-octagonal duoprism
150 - 30
truncated octahedral prism/truncated cuboctahedral prism
135 - 45
truncated octahedral prism/great prismated tesseract
120 - 60
hexagonal-octagonal duoprism/truncated cuboctahedral prism
144.735610 - 35.264390 - arccos(sqrt[2/3])
hexagonal-octagonal duoprism/great prismated tesseract
125.264390 - 54.735610 - arccos(1/sqrt(3))
truncated cuboctahedral prism/great prismated tesseract
135 - 45
cycle 1: |top -to- gpd -to- gpt| - 26.565 + 63.435 = 90
cycle 2: |top -c- tcop| - 45
cycle 3: |hod -hp- gpd -hp- tcop| - 39.231 + 50.768 = 90
cycle 4: |hod -hp- top -hp- gpt| - 30 + 60 = 90
cycle 5: |tcop -tco- gpt| - 45
cycle 6: |tcop -op- hod -op- gpt| - 35.264 + 54.735 = 90
opposites: gpd/gpd, top/top, hod/hod, tcop/tcop, gpt/gpt


Demipenteract is obtained from penteract by omitting half the vertices, leaving 16 combinations of 32 combinations of (+-1,+-1,+-1,+-1,+-1) with even number of minus signs.
Vectors are:
A - Vertex - (1,1,1,1,1); corresponds to great prismated decachoron in omnitruncate
B - Pentachoric teron - (3/5,3/5,3/5,3/5,-3/5); corresponds to great prismated decachoron in omnitruncate
C - Edge - (1,1,1,0,0); corresponds to square-hexagonal duoprism in omnitruncate
D - One type of cell - (1,1,0,0,0); corresponds to truncated octahedral prism in omnitruncate
E - Hexadecachoric teron - (1,0,0,0,0); corresponds to truncated 24-cell in omnitruncate

Data are:
Code: Select all
omnitruncated demipenteract
great prismated decachoron 1/great prismated decachoron 2
126.869898 - 53.130102 - arccos(3/5)
great prismated decachoron 1/square-hexagonal duoprism
140.768480 - 39.231521 - arccos(sqrt[3/5])
great prismated decachoron 1/truncated octahedral prism
129.231520 - 50.768480 - arccos(sqrt[2/5])
great prismated decachoron 1/truncated icositetrachoron
116.565051 - 63.434949 - arccos(sqrt[1/5])
great prismated decachoron 2/square-hexagonal duoprism
140.768480 - 39.231521 - arccos(sqrt[3/5])
great prismated decachoron 2/truncated octahedral prism/
129.231520 - 50.768480 - arccos(sqrt[2/5])
great prismated decachoron 2/truncated icositetrachoron
116.565051 - 63.434949 - arccos(sqrt[1/5])
square-hexagonal duoprism/truncated octahedral prism
144.735610 - 35.264390 - arccos(sqrt[2/3])
square-hexagonal duoprism/truncated icositetrachoron
125.264390 - 54.735610 - arccos(sqrt[1/3])
truncated octahedral prism/truncated icositetrachoron
135 - 45
cycle 1: |shd -hp- gpd1 -hp- top -hp- gpd2 -hp- shd| - 39.232 + 50.768 + 50.768 + 39.232 = 180
cycle 2: |top -to- ti| - 45
cycle 3: |top -c- shd -c- ti| - 35.264 + 54.736 = 90
cycle 4: |ti -to- gpd1 -to- gpd2 -to- ti| - 63.435 + 53.130 + 63.435 = 180
opposites: gpd1/gpd2, shd/shd, top/top, ti/ti
Marek14
Pentonian
 
Posts: 1095
Joined: Sat Jul 16, 2005 6:40 pm

Re: Planar tilings based on Goursat tetrahedra

Postby Klitzing » Sun Jun 05, 2016 1:58 pm

Marek14 wrote:... I don't completely understand Dr. Klitzing dihedral page (where I got the polyhedron/polychoron data), ...

In fact, those once where mainly based on the input from that post of Wendy. 8)

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Re: Planar tilings based on Goursat tetrahedra

Postby Marek14 » Sun Jun 05, 2016 3:02 pm

Yes, with this approach, I can find dihedral angles for all three infinite families -- simplexes, cubes and demicubes. How would they look for the 3 Gosset groups in 6D, 7D and 8D? Those are the last remaining cases.
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Re: Planar tilings based on Goursat tetrahedra

Postby wendy » Mon Jun 06, 2016 1:31 am

They work on the radials from the centre of the group, and have nothing to do with whether the group is reglular. In fact, it relies on whether the group is symmetric.
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Re: Planar tilings based on Goursat tetrahedra

Postby Marek14 » Mon Jun 06, 2016 6:50 am

wendy wrote:They work on the radials from the centre of the group, and have nothing to do with whether the group is reglular. In fact, it relies on whether the group is symmetric.


Yes, it's just that I haven't looked at the necessary coordinates yet.
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Re: Planar tilings based on Goursat tetrahedra

Postby wendy » Tue Jun 07, 2016 1:09 am

You really don't have to. The angles involved are the supplements of the face normals, which come in turn from the matrix dot of unit vectors. For the group say 4B (the E_6 = 3,3,3,3,B)

Code: Select all

    (  4  5  6  4  2  3
    (  5 10 12  8  4  6
  2 (  6 12 18 12  6  9
  - (  4  8 12 10  5  8
  3 (  2  4  8  5  4  4
    (  3  6  9  8  4  6


The dot product of two vectors (1,0,0,0,0,0) and (0,1,0,0,0) is 2/3 * 5 We divide this by the square root of the vectors themselv es, ie 2/3 sqrt(4*10) to get 24/40 or sqrt (5/8) This is the cos of the supplement.

So the cosine of the margin angle between faces i and j is -sqrt((ij * ji )/( ii * jj)).
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Re: Planar tilings based on Goursat tetrahedra

Postby Marek14 » Tue Jun 07, 2016 6:28 am

I see and how is this matrix obtained?
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Re: Planar tilings based on Goursat tetrahedra

Postby wendy » Tue Jun 07, 2016 7:39 am

The matrix is obtained by taking the inverse of the dynkin matrix, and multiplying it by the determinate of the dynkin matrix.

Code: Select all
  2  -1  0  0   0   0      4  5  6  4  2  3      3  0  0  0  0  0
-1   2 -1  0   0   0      5 10 12  8  4  6      0  3  0  0  0  0
  0  -1  2 -1   0  -1      6 12 18 12  6  9      0  0  3  0  0  0
  0   0 -1  2  -1   0  *   4  8 12 10  5  6  =   0  0  0  3  0  0
  0   0  0 -1   2   0      2  4  6  5  4  3      0  0  0  0  3  0
  0   0 -1  0   0   2      3  6  9  6  3  6      0  0  0  0  0  3
 
   Dynkin matrix            Stott matrix          3 = schlafli det.


This is the same way that the spreadsheet i wrote for Richard works. I worked all of these matrices by hand, and generally construct these from memory. For the gosset group, from E3 to E9, the process is to write from the bottom left corner to the top left, the numbers from 3 2 4 6 5 4 3 2 1 0, the immediate next is written as a denominator (here, the second 3 is the overflow, the lot is thence multiplied by 2/3. The second process is to write in the bottom right hand corner, the 'animal'. It's a symmetric 3*3 matrix, being (2n-2, n-1, 2n-6 ; n-1, 4, n-3; 2n-6, n-3, n). The rest of the rows are (to and including the diagonal), simply 2x, 3x, 4x, ... of the first column. This will give you the necessary matricies for 7d and 8d as well. For 9d, the resulting matrix is a product of the row and column, it represents an infinite tiling.
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Re: Planar tilings based on Goursat tetrahedra

Postby Marek14 » Tue Jun 07, 2016 8:31 am

I see. I was thrown off by the first matrix -- it seems there is some mistake there since some entries are different from the Stott matrix in your later post.

For my purpose, determining of angles, I guess I don't actually need the correct determinate of the dynkin matrix, since if the matrix is multiplied by any nonzero constant, the expression -sqrt((ij * ji )/( ii * jj)) should stay the same. So I can just invert the Dynkin matrix.
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Re: Planar tilings based on Goursat tetrahedra

Postby Marek14 » Tue Jun 07, 2016 8:16 pm

So far I got good results for 6D simplex.

In case of hyperbolic group, I presume that the cosines will be outside of <-1,1>, so they are cosines of imaginary angle -- is this number, by any chance, equal to the minimum distance of ultrapallel lines in a "triangle" with imaginary angle?

But I'm getting weird results for hexeract.

I used Dynkin matrix:

{{2, -1, 0, 0, 0, 0}, {-1, 2, -1, 0, 0, 0}, {0, -1, 2, -1, 0, 0}, {0, 0, -1, 2, -1, 0}, {0, 0, 0, -1, 2, -Sqrt[2]}, {0, 0, 0, 0, -Sqrt[2], 2}}

Inversion got me this matrix:

{{1, 1, 1, 1, 1, 1/Sqrt[2]}, {1, 2, 2, 2, 2, Sqrt[2]}, {1, 2, 3, 3, 3, 3/Sqrt[2]}, {1, 2, 3, 4, 4, 2 Sqrt[2]}, {1, 2, 3, 4, 5, 5/Sqrt[2]}, {1/Sqrt[2], Sqrt[2], 3/Sqrt[2], 2 Sqrt[2], 5/Sqrt[2], 3}}

I computed angles between all pairs of vectors.

Then I tried to build cycles, i.e. if I go in a straight line through the petons, how will the sequence look? And here's where I got stuck.

I'm doing it with omnitruncated hexeract, so every combination of nodes gives some polytope. On each straight line, I pass from one peton to another through a polychoral wall, and these polychora must be the same for each line. So there is a line that passes through great prismated decachora, or omnitruncated pentachora.

There are two kinds of great prismated decachora in omnitruncated hexeract: one joins omnitruncated hexateron and great prismated decachoric prism (this join is between vectors 1 and 2 and has centri angle 45), and the other joins omnitruncated hexateron and omnitruncated penteract (here, the join is between vectors 1 and 6 and has angle 65.905157).

Both omnitruncated penteract and great prismated decachoric prism are reflective with respect to this polychoron, i.e. when you enter through great prismated decachoric prism and pass through, you will leave through another great prismated decachoric prism.

But omnitruncated hexateron has two kinds and they are opposite each other, which means that the cycle looks like this: |gpdp -gpd- oht -gpd- op|. But the two angles involved do not add to anything reasonable, and their sum should divide 360 evenly...

The angle 65.905157 adds to 90 with 24.094843, which is the 5-6 centri angle. But this join is great prismated tesseract, so it's not compatible.

There must be some mistake in there...
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Re: Planar tilings based on Goursat tetrahedra

Postby wendy » Wed Jun 08, 2016 3:30 am

The matrix i wrote in the earlier post was from memory, and i suspected at the time that the error was there, i did not actually invert the thing.

Regards the second question on cycles, they should all work. Here is the group without the multiplier (2/2).

Code: Select all
  2  2  2  2  2  q
  2  4  4  4  4 2q
  2  4  6  6  6 3q
  2  4  6  8  8 4q
  2  4  6  8 10 5q
  q 2q 3q 4q 5q  6


The cycles go through a and b, the common base is a+b. So we look at the angle a/(a+b) and b/(a+b), and we get eg

4*4 / 10 * 4 and 6*6 / 10*6 these give 4/6 and 6/6. This tells us that the suplements of the dihedral angles adds to right angles, as they are the sin and cos of the same angle. You should find that all of the cycles, inc q*q/2*6 and 5q*5q/10*6 gives 1/6 and 5/6 resp. One should note that these are the sin squares, and the sin²+cos²=1, here tells us that we are dealing with two angles that add to a right angle.
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Re: Planar tilings based on Goursat tetrahedra

Postby wendy » Wed Jun 08, 2016 3:45 am

Regarding the hyperbic case, what happens when you get a value greater than one, is that that the cyclic polygon is centred on a line.

Here is 3,3,3,5. We see the first angle is (4-2f)(4-2f)/(8-4f)(5-3f) or (2-f)/(5-3f) = 2.618033&c. The indicated polygon here is W7.236

From what i can tell, this is the closest approach of the normals between these faces, but the actual faces can intersect at angles less than planar.

Code: Select all
   5-3f  4-2f   3-f    2  f
   4-2f  8-4f   6-2f   4  2f
   3- f  6-2f   9-3f   6  3f
     2    4      6     8  4f
     f    2f    3f    4f  5
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Re: Planar tilings based on Goursat tetrahedra

Postby Marek14 » Wed Jun 08, 2016 4:44 pm

wendy wrote:The matrix i wrote in the earlier post was from memory, and i suspected at the time that the error was there, i did not actually invert the thing.

Regards the second question on cycles, they should all work. Here is the group without the multiplier (2/2).

Code: Select all
  2  2  2  2  2  q
  2  4  4  4  4 2q
  2  4  6  6  6 3q
  2  4  6  8  8 4q
  2  4  6  8 10 5q
  q 2q 3q 4q 5q  6


The cycles go through a and b, the common base is a+b. So we look at the angle a/(a+b) and b/(a+b), and we get eg

4*4 / 10 * 4 and 6*6 / 10*6 these give 4/6 and 6/6. This tells us that the suplements of the dihedral angles adds to right angles, as they are the sin and cos of the same angle. You should find that all of the cycles, inc q*q/2*6 and 5q*5q/10*6 gives 1/6 and 5/6 resp. One should note that these are the sin squares, and the sin²+cos²=1, here tells us that we are dealing with two angles that add to a right angle.


I realized the problem -- the angles were correct, but the labels were wrong. I haven't realized this before because the simplex case was symmetric.

Also, it turns out that in this particular matrix, cosine of angle between i and j, with i<j, always comes out as sqrt(i/j). Weird :)
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Re: Planar tilings based on Goursat tetrahedra

Postby wendy » Thu Jun 09, 2016 8:14 am

It's not all that weird, since what you are looking at is that the tesseract has a cross section in a cube, and a penteract has a tesseract cross-section.
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