## Planar tilings based on Goursat tetrahedra

Higher-dimensional geometry (previously "Polyshapes").

### Planar tilings based on Goursat tetrahedra

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

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

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

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
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### Re: Planar tilings based on Goursat tetrahedra

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

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

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

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

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

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; ABATriangle B: 54.736, 70.529, 90; ABCTriangle C: 54.736, 70.529, 90; DCBTriangle D: 54.736, 60,     90; DCDVertices:A-54.736 & B-70.529 & C-54.736Pattern: AABCCBTriangle with angles 120, 125.265, 125.265A-60Pattern: AAAAAATriangle with angles 109.472, 109.472, 109.472A-90 & B-90Pattern: AABBQuadrangle with angles 109.472, 125.265, 120, 125.265B-54.736 & C-70.529 & D-54.736Pattern: BBCDDCTriangle with angles 120, 125.265, 125.265C-90 & D-90Pattern: CCDDQuadrangle with angles 109.472, 125.265, 120, 125.265D-60Pattern: DDDDDDEquilateral triangle with angle 109.472Repeating 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; BAATriangle B: 35.264, 70.529, 90; ABCTriangle 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; DDDVertices:A-45Pattern: AAAAAAAASquare with angle 109.472A-54.736 & B-70.529 & C-54.736Pattern: AABCCBTriangle with angles 90, 90, 109.472A-90 & B-90Pattern: AABBQuadrangle with angles 90, 125.265, 109.472, 125.265B-35.264 & C-54.736Pattern: BBCCBBCCSquare with angle 125.265C-90Pattern: CCCCSquare with angle 109.472D-45Pattern: DDDDDDDDSquare with angle 120D-60Pattern: DDDDDDEquilateral triangle with angle 90D-90Pattern: DDDDRhombus with angles 90, 120, 90, 120Repeating 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; BAATriangle B: 20.905, 70.529, 90; ABCTriangle C: 37.377, 54.736, 90; CDBTriangle D: 31.717, 60,     90; DCDVertices:A-36Pattern: AAAAAAAAAARegular pentagon with angle 109.472A-54.736 & B-70.529 & C-54.736Pattern: AABCCBTriangle with angles 58.282, 58.282, 72A-90 & B-90Pattern: AABBQuadrangle with angles 41.810, 125.265, 72, 125.265B-20.905 & C-37.377 & D-31.717Pattern: BBCDDCBBCDDCHexagon with angles 120, 125.265, 125.265, 120, 125.265, 125.265C-90 & D-90Pattern: CCDDQuadrangle with angles 69.094, 109.472, 69.094, 120D-60Pattern: DDDDDDEquilateral triangle with angle 63.434Repeating 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 BPlane 2: tiled by combination of C and DTriangle A: 35.264, 60,     90; ABATriangle B: 45,     54.736, 90; ABB Triangle C: 45,     54.736, 90; DCCTriangle D: 35.264, 60,     90; DCDVertices:A-35.264 & B-54.736Pattern: AABBAABBRhombus with angles 90, 120, 90, 120A-60Pattern: AAAAAAEquilateral triangle with angle 70.528A-90 & B-90Pattern: AABBQuadrangle with angles 90, 90, 90, 120B-45Pattern: BBBBBBBBSquare with angle 109.472C-45Pattern: CCCCCCCCSquare with angle 109.472C-54.736 & D-35.264Pattern: CCDDCCDDRhombus with angles 90, 120, 90, 120C-90 & D-90Pattern: CCDDQuadrangle with angles 90, 90, 90, 120D-60Pattern: DDDDDDEquilateral triangle with angle 70.529Repeating 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    ; AABTriangle B: 70.529, 70.529, 70.529; ABCTriangle C: 54.736, 54.736, 90    ; CCBTriangle D: 54.736, 54.736, 90    ; DDBVertices:A-54.736 & B-70.529 & C-54.736Pattern: AABCCBQuadrangle with angles 125.265, 125.265, 125.265, 125.265 (not square, apparently)A-54.736 & B-70.529 & D-54.736Pattern: AABDDBQuadrangle with angles 125.265, 125.265, 125.265, 125.265 (not square, apparently)A-90Pattern: AAAASquare with angle 109.472B-70.529 & C-54.736 & D-54.736Pattern: BCCBDDQuadrangle with angles 125.265, 125.265, 125.265, 125.265 (not square, apparently)C-90Pattern: CCCCSquare with angle 109.472D-90Pattern: DDDDSquare with angle 109.472Repeating 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 APlane 2: tiled by combination of B and CPlane 3: tiled solely by DTriangle A: 45,     45,     90; AAATriangle B: 35.264, 54.736, 90; BBCTriangle C: 35.264, 54.736, 90; CCBTriangle D: 45,     45,     90; DDDVertices:A-45Pattern: AAAAAAAASquare with angle 90A-45Pattern: AAAAAAAASquare 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-90Pattern: AAAASquare with angle 90B-35.264 & C-54.736Pattern: BBCCRectangle with angle 90B-54.736 & C-35.264Pattern: BBCCRectangle with angle 90B-90Pattern: BBBBRhombus with angles 70.529, 109.472, 70.529, 109.472C-90Pattern: CCCCRhombus with angles 70.529, 109.472, 70.529, 109.472D-45Pattern: DDDDDDDDSquare with angle 90D-45Pattern: DDDDDDDDSquare with angle 90D-90Pattern: DDDDSquare with angle 90Repeating 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 CPlane 2: tiled solely by DTriangle A: 35.264, 54.736, 90    ; AABTriangle B: 54.736, 54.736, 70.529; ACB (double of A or C triangle)Triangle C: 35.264, 54.736, 90    ; CCBTriangle D: 45,     45,     90    ; DDDVertices:A-35.264 & B-54.736Pattern: AABBAABBHexagon with angles 109.472, 125.565, 125.565, 109.472, 125.565, 125.565A-54.736 & B-70.529 & C-54.736Pattern: AABCCBSquare with angle 90A-90Pattern: AAAARhombus with angles 70.529, 109.472, 70.529, 109.472B-54.736 & C-35.264Pattern: BBCCBBCCHexagon with angles 109.472, 125.565, 125.565, 109.472, 125.565, 125.565C-90Pattern: CCCCRhombus with angles 70.529, 109.472, 70.529, 109.472D-45Pattern: DDDDDDDDSquare with angle 90D-45Pattern: DDDDDDDDSquare with angle 90D-90Pattern: DDDDSquare with angle 90Repeating unit 1: Rectangle of angle 90. Composed of 1 A, 1 B, 1 C.Repeating unit 2: Triangle D.Cyclical 3333Triangles 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; BDATriangle B: 54.736, 54.736, 70.529; ACBTriangle C: 54.736, 54.736, 70.529; BDCTriangle D: 54.736, 54.736, 70.529; ACDVertices:A-54.736 & B-70.529 & C-54.736Pattern: AABCCBHexagon with angles 109.472, 125.565, 125.565, 109.472, 125.565, 125.565A-54.736 & C-54.736 & D-70.529Pattern: AADCCDHexagon with angles 109.472, 125.565, 125.565, 109.472, 125.565, 125.565A-70.529 & B-54.736 & D-54.736Pattern: ABBADDHexagon with angles 109.472, 125.565, 125.565, 109.472, 125.565, 125.565B-54.736 & C-70.529 & D-54.736Pattern: BBCDDCHexagon with angles 109.472, 125.565, 125.565, 109.472, 125.565, 125.565Repeating unit: An infinite strip formed by repeating triangles A, B, C, D.{4,3,5}Plane 1: tiled solely by APlane 2: tiled by combination of B, C, and DTriangle A: 36,     45,     90; AAATriangle B: 20.905, 54.736, 90; BBCTriangle C: 35.264, 37.377, 90; DCBTriangle D: 31.717, 45,     90; DCDVertices:A-36Pattern: AAAAAAAAAARegular pentagon with angle 90A-45Pattern: AAAAAAAASquare with angle 72A-90Pattern: AAAARhombus with angles 72, 90, 72, 90B-20.905 & C-37.377 & D-31.717Pattern: BBCDDCBBCDDCRight-angled hexagon, not regularB-54.736 & C-35.264Pattern: BBCCBBCCRectangle with angle 58.282B-90Pattern: BBBBRhombus with angles 41.810, 109.472, 41.810, 109.472C-90 & D-90Pattern: CCDDQuadrangle with angles 69.095, 70.529, 90, 69.095D-45Pattern: DDDDDDDDSquare with angle 63.434Repeating 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; ABATriangle B: 20.905, 37.377, 90; ABCTriangle C: 20.905, 37.377, 90; DCBTriangle D: 31.717, 36,     90; DCCVertices:A-31.717 & B-37.377 & C-20.905Pattern: AABCCBAABCCBHexagon with angles 58.282, 58.282, 72, 58.282, 58.282, 72A-36Pattern: AAAAAAAAAARegular pentagon with angle 63.434A-90 & B-90Pattern: AABBQuadrangle with angles 41.810, 69.095, 72, 69.095B-20.905 & C-37.377 & D-31.717Pattern: BBCDDCBBCDDCHexagon with angles 58.282, 58.282, 72, 58.282, 58.282, 72C-90 & D-90Pattern: CCDDQuadrangle with angles 41.810, 69.095, 72, 69.095D-36Pattern: DDDDDDDDDDRegular pentagon with angle 63.434Repeating 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; ABATriangle B: 31.717, 37.377, 90; ABCTriangle C: 31.717, 37.377, 90; DCBTriangle D: 20.905,     60, 90; DCDVertices:A-20.905 & B-37.377 & C-31.717Pattern: AABCCBAABCCBHexagon with angles 69.095, 69.095, 120, 69.095, 69.095, 120A-60Pattern: AAAAAAEquilateral triangle with angle 41.810A-90 & B-90Pattern: AABBQuadrangle with angles 58.282, 63.434, 58.282 and 120B-31.717 & C-37.377 & D-20.905Pattern: BBCDDCBBCDDCHexagon with angles 69.095, 69.095, 120, 69.095, 69.095, 120C-90 & D-90Pattern: CCDDQuadrangle with angles 58.282, 63.434, 58.282 and 120D-60Pattern: AAAAAAEquilateral triangle with angle 41.810Repeating 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; AABTriangle B: 37.377, 37.377, 70.529; ACDTriangle C: 20.905, 54.736,     90; CCBTriangle D: 31.717, 31.717,     90; DDBVertices:A-20.905 & B-37.377 & D-31.717Pattern: AABDDBAABDDBOctagon with alternating angles of 69.095 and 125.565A-54.736 & B-70.529 & C-54.736Pattern: AABCCBRectangle with angle 58.282A-90Pattern: AAAARhombus with angles 41.810, 109.472, 41.810, 109.472B-37.377 & C-20.905 & D-31.717Pattern: BCCBDDBCCBDDOctagon with alternating angles of 69.095 and 125.565C-90Pattern: CCCCRhombus with angles 41.810, 109.472, 41.810, 109.472D-90Pattern: DDDDSquare with angle 63.434Repeating unit: Right-angled pentagon; not regular. Composed of 1 A, 1 B, 1 C, 1 D.Cyclical 3334A 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; BAATriangle B: 35.264, 54.736, 70.529; CABTriangle C: 35.264, 54.736, 70.529; BDCTriangle D: 45,     54.736, 54.736; CDDVertices:A-45Pattern: AAAAAAAARegular octagon with angle 109.472A-54.736 & B-35.264Pattern: AABBAABBOctagon with angles 90, 125.565, 109.472, 125.565, 90, 125.565, 109.472, 125.565A-54.736 & B-70.529 & C-54.736Pattern: AABCCBHexagon with angles 70.529, 125.565, 90, 90, 90, 125.565B-54.736 & C-70.529 & D-54.736Pattern: BBCDDCHexagon with angles 70.529, 125.565, 90, 90, 90, 125.565C-35.264 & D-54.736Pattern: CCDDCCDDOctagon with angles 90, 125.565, 109.472, 125.565, 90, 125.565, 109.472, 125.565D-45Pattern: DDDDDDDDRegular octagon with angle 109.472Repeating unit: Quadrangle with angles 45, 90, 45, 90. Composed of 1 A, 1 B, 1 C, 1 D.Cyclical 3335A 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; BADTriangle B: 20.905, 54.736, 70.529; CABTriangle C: 20.905, 54.736, 70.529; BDCTriangle D: 31.717, 37.377, 54.736; CDAVertices:A-31.717 & C-20.905 & D-37.377Pattern: AADCCDAADCCDDodecagon with angles 69.095, 109.472, 69.095, 125.565, 109.472, 125.565, 69.095, 109.472, 69.095, 125.565, 109.472, 125.565A-37.377 & B-20.905 & D-31.717Pattern: ABBADDABBADDDodecagon with angles 69.095, 109.472, 69.095, 125.565, 109.472, 125.565, 69.095, 109.472, 69.095, 125.565, 109.472, 125.565A-54.736 & B-70.529 & C-54.736Pattern: AABCCDHexagon with angles 41.810, 125.565, 58.282, 63.434, 58.282, 125.565B-54.736 & C-70.529 & D-54.736Pattern: BBCDDCHexagon with angles 41.810, 125.565, 58.282, 63.434, 58.282, 125.565Repeating unit: An infinite strip formed by repeating triangles A, B, C, D. Its two edges form right-angled pseudogons.Cyclical 3434Triangles A, B, C, and D all identical.Plane 1: tiled by combination of A and BPlane 2: tiled by combination of C and DTriangle A: 35.264, 45, 54.736; ABATriangle B: 35.264, 45, 54.736; BABTriangle C: 35.264, 45, 54.736; CDCTriangle D: 35.264, 45, 54.736; DCDVertices:A-35.264 & B-54.736Pattern: AABBAABBRight-angled octagon, not regularA-45Pattern: AAAAAAAAOctagon with alternating angles 70.529 and 125.565A-54.736 & B-35.264Pattern: AABBAABBRight-angled octagon, not regularB-45Pattern: BBBBBBBBOctagon with alternating angles 70.529 and 125.565C-35.264 & D-54.736Pattern: CCDDCCDDRight-angled octagon, not regularC-45Pattern: CCCCCCCCOctagon with alternating angles 70.529 and 125.565C-54.736 & D-35.264Pattern: CCDDCCDDRight-angled octagon, not regularD-45Pattern: DDDDDDDDOctagon with alternating angles 70.529 and 125.565Repeating 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 3435A 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; BDATriangle B: 20.905, 45,     54.736; BABTriangle C: 20.905, 45,     54.736; CDCTriangle D: 31.717, 35.264, 37.377; CADVertices:A-31.717 & C-20.905 & D-37.377Pattern: AADCCDAADCCDDodecagon with angles 69.095, 70.529, 69.095, 90, 90, 90, 69.095, 70.529, 69.095, 90, 90, 90A-35.264 & B-54.736Pattern: AABBAABBOctagon with angles 58.282, 63.434, 58.282, 90, 58.282, 63.434, 58.282, 90A-37.377 & B-20,905 & D-31.717Pattern: ABBADDABBADDDodecagon with angles 69.095, 70.529, 69.095, 90, 90, 90, 69.095, 70.529, 69.095, 90, 90, 90B-45Pattern: BBBBBBBBOctagon with alternating angles 41.810 and 109.472C-45Pattern: CCCCCCCCOctagon with alternating angles 41.810 and 109.472C-54.736 & D-35.264Pattern: CCDDCCDDOctagon with angles 58.282, 63.434, 58.282, 90, 58.282, 63.434, 58.282, 90Repeating unit: Hexagon with angles 45, 90, 90, 45, 90, 90. Composed of 1 A, 1 B, 1 C, 1 D.Cyclical 3535Triangles 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; DBATriangle B: 20.905, 31.717, 37.377; CABTriangle C: 20.905, 31.717, 37.377; BDCTriangle D: 20.905, 31.717, 37.377; ACDVertices:A-20.905 & C-31.717 & D-37.377Pattern: AADCCDAADCCDDodecagon with angles 41.810, 58.282, 69.095, 63.434, 69.095, 58.282, 41.810, 58.282, 69.095, 63.434, 69.095, 58.282A-31.717 & B-37.377 & C-20.905Pattern: AABCCBAABCCBDodecagon with angles 41.810, 58.282, 69.095, 63.434, 69.095, 58.282, 41.810, 58.282, 69.095, 63.434, 69.095, 58.282A-37.377 & B-20.905 & D-31.717Pattern: ABBADDABBADDDodecagon with angles 41.810, 58.282, 69.095, 63.434, 69.095, 58.282, 41.810, 58.282, 69.095, 63.434, 69.095, 58.282B-31.717 & C-37.377 & D-20.905Pattern: BBCDDCBBCDDCDodecagon with angles 41.810, 58.282, 69.095, 63.434, 69.095, 58.282, 41.810, 58.282, 69.095, 63.434, 69.095, 58.282Repeating 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; BAATriangle B: 0,  70.529, 90; ABCTriangle C: 0,  54.736, 90; CCBTriangle D: 0,  60,     90; DDDVertices:A-30Pattern: AAAAAAAAAAAARegular hexagon with angle 109.472A-54.736 & B-70.529 & C-54.736Pattern: AABCCBTriangle with angles 0, 0, 60A-90 & B-90Pattern: AABBQuadrangle with angles 0, 125.565, 60, 125.565B-0 & C-0Pattern: BBCC...Apeirogon with angle 125.565, not regularC-90Pattern: CCCCRhombus with angles 0, 109.472, 0, 109.472D-0Pattern: D...Regular apeirogon with angle 120D-60Pattern: DDDDDDEquilateral triangle with angle 0D-90Pattern: DDDDRhombus with angles 0, 120, 0, 120Repeating 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; AAATriangle B: 0,  54.736, 90; BBCTriangle C: 0,  35.264, 90; CCBTriangle D: 0,  45,     90; DDDVertices:A-30Pattern: AAAAAAAAAAAARight-angled dodecagonA-45Pattern: AAAAAAAARegular octagon with angle 60A-90Pattern: AAAARhombus with angles 60, 90, 60, 90B-0Pattern: B...Regular apeirogon with angle 109.472B-54.736 & C-35.264Pattern: BBCCBBCCRectangle with angle 0B-90 & C-90Pattern: BBCCQuadrangle with angles 0, 90, 0, 90C-0Pattern: C...Regular apeirogon with angle 70.529D-0Pattern: D...Right-angled apeirogonD-45Pattern: DDDDDDDDSquare with angle 0D-90Pattern: DDDDRhombus with angles 0, 90, 0, 90Repeating 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; BAATriangle B: 0,  37.377, 90; ABCTriangle C: 0,  20.905, 90; CCBTriangle D: 0,  36,     90; DDDVertices:A-30Pattern: AAAAAAAARegular hexagon with angle 63.434A-31.717 & B-37.377 & C-20.905Pattern: AABCCBAABCCBHexagon with angles 0, 0, 60, 0, 0, 60A-90 & B-90Pattern: AABBQuadrangle with angles 0, 69.095, 60, 69.095B-0 & C-0Pattern: BBCC...Apeirogon with angle 58.282, not regularC-90Pattern: CCCCRhombus with angles 0, 41.810, 0, 41.810D-0Pattern: D...Regular apeirogon with angle 72D-36Pattern: DDDDDDDDDDRegular pentagon with angle 0D-90Pattern: DDDDRhombus with angles 0, 72, 0, 72Repeating 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 APlane 2: tiled by combination of B and CPlane 3: tiled solely by DTriangle A: 0, 30, 90; AAATriangle B: 0, 0,  90; BBCTriangle C: 0, 0,  90; CCBTriangle D: 0, 30, 90; DDDVertices:A-0Pattern: A...Regular apeirogon with angle 60A-30Pattern: AAAAAAAAAAAARegular hexagon with angle 0A-90Pattern: AAAARhombus with angles 0, 60, 0, 60B-0 & C-0Pattern: BBCC...Regular apeirogon with angle 0B-0 & C-0Pattern: BBCC...Regular apeirogon with angle 0B-90Pattern: BBBBSquare with angle 0C-90Pattern: CCCCSquare with angle 0D-0Pattern: D...Regular apeirogon with angle 60D-30Pattern: DDDDDDDDDDDDRegular hexagon with angle 0D-90Pattern: DDDDRhombus with angles 0, 60, 0, 60Repeating 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; ABATriangle B: 0,      54.736, 90; ABBTriangle C: 0,      45,     90; CCCTriangle D: 0,      60,     90; CCCVertices:A-35.264 & B-54.736Rhombus with angles 0, 90, 0, 90A-45Pattern: AAAAAAAASquare with angle 70.529A-90 & B-90Pattern: AABBQuadrangle with angles 0, 90, 90, 90B-0Pattern: B...Regular apeirogon with angle 109.472C-0Pattern: C...Right-angled apeirogonC-45Pattern: CCCCCCCCSquare with angle 0C-90Pattern: CCCCRhombus with angles 0, 90, 0, 90D-0Pattern: D...Regular apeirogon with angle 120D-60Pattern: DDDDDDEquilateral triangle with angle 0D-90Pattern: DDDDRhombus with angles 0, 120, 0, 120Repeating 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; AAATriangle B: 0, 0,  90; BBBTriangle C: 0, 0,  90; CCCTriangle D: 0, 45, 90; DDDVertices:A-0Pattern: A...Right-angled apeirogonA-45Pattern: AAAAAAAASquare with angle 0A-90Pattern: AAAARhombus with angles 0, 90, 0, 90B-0Pattern: B...Regular apeirogon with angle 0B-0Pattern: B...Regular apeirogon with angle 0B-90Pattern: BBBBSquare with angle 0C-0Pattern: C...Regular apeirogon with angle 0C-0Pattern: C...Regular apeirogon with angle 0C-90Pattern: CCCCSquare with angle 0D-0Pattern: D...Right-angled apeirogonD-45Pattern: DDDDDDDDSquare with angle 0D-90Pattern: DDDDRhombus with angles 0, 90, 0, 90Repeating 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 BPlane 2: tiled by combination of C and DTriangle A: 0, 60, 90; ABATriangle B: 0, 0,  90; ABBTriangle C: 0, 0,  90; CDCTriangle D: 0, 60, 90; DCDVertices:A-0 & B-0Pattern: AABB...Apeirogon with alternating angles 0 and 120A-60Pattern: AAAAAAEquilateral triangle with angle 0A-90 & B-90Pattern: AABBQuadrangle with angles 0, 0, 0, 120B-0Pattern: B...Regular apeirogon with angle 0C-0Pattern: C...Regular apeirogon with angle 0C-0 & D-0Pattern: CCDD...Apeirogon with alternating angles 0 and 120C-90 & D-90Pattern: CCDDQuadrangle with angles 0, 0, 0, 120D-60Pattern: DDDDDDEquilateral triangle with angle 0Repeating 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 336A and C triangles identical.Plane 1: tiled by combination of A, B, and CPlane 2: tiled solely by DTriangle A: 0, 54.736, 90    ; AABTriangle B: 0, 0,      70.529; ACBTriangle C: 0, 54.736, 90    ; CCBTriangle D: 0, 0,      90    ; DDDVertices:A-0 & B-0Pattern: AABB...Apeirogon with alternating angles 0 and 109.472A-54.736 & B-70.529 & C-54.736Pattern: AABCCBHexagon with all angles 0, not regularA-90Pattern: AAAARhombus with angles 0, 109.472, 0, 109.472B-0 & C-0Pattern: CCDD...Apeirogon with alternating angles 0 and 109.472C-90Pattern: CCCCRhombus with angles 0, 109.472, 0, 109.472D-0Pattern: D...Regular apeirogon with angle 0D-0Pattern: D...Regular apeirogon with angle 0D-90Pattern: DDDDSquare with angle 0Repeating unit 1: Quadrangle with angles 0, 0, 90, 90. Composed of 1 A, 1 B, 1 C.Repeating unit 2: Triangle D.Branched 344C and D triangles identical.Plane 1: tiled by combination of A and BPlane 2: tiled solely by CPlane 3: tiled solely by DTriangle 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-54.736Pattern: AABBHexagon with angles 0, 90, 90, 0, 90, 90A-35.264 & B-54.736Pattern: AABBHexagon with angles 0, 90, 90, 0, 90, 90A-90Pattern: AAAASquare with angle 70.529B-0Pattern: B...Regular apeirogon with angle 109.472C-0Pattern: C...Right-angled apeirogonC-45Pattern: CCCCCCCCSquare with angle 0C-90Pattern: CCCCRhombus with angles 0, 90, 0, 90D-0Pattern: D...Right-angled apeirogonD-45Pattern: DDDDDDDDSquare with angle 0D-90Pattern: DDDDRhombus with angles 0, 90, 0, 90Repeating 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 444A, C and D triangles identical.Plane 1: tiled solely by APlane 2: tiled solely by BPlane 3: tiled solely by CPlane 4: tiled solely by DTriangle A: 0, 0, 90; AAATriangle B: 0, 0, 0 ; BBBTriangle C: 0, 0, 90; CCCTriangle D: 0, 0, 90; DDDVertices:A-0Pattern: A...Regular apeirogon with angle 0A-0Pattern: A...Regular apeirogon with angle 0A-90Pattern: AAAASquare with angle 0B-0Pattern: B...Regular apeirogon with angle 0B-0Pattern: B...Regular apeirogon with angle 0B-0Pattern: B...Regular apeirogon with angle 0C-0Pattern: C...Regular apeirogon with angle 0C-0Pattern: C...Regular apeirogon with angle 0C-90Pattern: CCCCSquare with angle 0D-0Pattern: D...Regular apeirogon with angle 0D-0Pattern: D...Regular apeirogon with angle 0D-90Pattern: CCCCSquare with angle 0Repeating unit 1: Triangle A.Repeating unit 2: Triangle B.Repeating unit 3: Triangle C.Repeating unit 4: Triangle D.Cyclical 3336A 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; BAATriangle B: 0, 54.736, 70.529; CABTriangle C: 0, 54.736, 70.529; BDCTriangle D: 0, 0,      54.736; DCDVertices:A-0Pattern: A...Regular apeirogon with angle 0A-0 & B-0Pattern: AABB...Apeirogon with angles 0, 125.565, 109.472, 125.565, 0...A-54.736 & B-70.529 & C-54.736Pattern: AABCCBHexagon with angles 0, 0, 0, 125.565, 0, 125.565B-54.736 & C-70.529 & D-54.736Pattern: AABCCBHexagon with angles 0, 0, 0, 125.565, 0, 125.565C-0 & D-0Pattern: CCDD...Apeirogon with angles 0, 125.565, 109.472, 125.565, 0...D-0Pattern: D...Regular apeirogon with angle 0Repeating unit: Rectangle with angle 0. Composed of 1 A, 1 B, 1 C, 1 D.Cyclical 3436A and D triangles identical.B and C triangles identical.Plane 1: tiled by combination of A and BPlane 2: tiled by combination of C and DTriangle A: 0, 0,  35.264; BAATriangle B: 0, 45, 54.736; ABATriangle C: 0, 45, 54.736; CDCTriangle D: 0, 0,  35.264; DCDVertices:A-0Pattern: A...Apeirogon with alternating angles 0 and 70.529A-0 & B-0Pattern: AABB...Apeirogon with angles 0, 90, 90, 90...A-35.264 & B-54.736Pattern: AABBAABBOctagon with angles 0, 0, 0, 90, 0, 0, 0, 90B-45Pattern: BBBBBBBBOctagon with alternating angles 0 and 109.472C-0 & D-0Pattern: CCDD...Apeirogon with angles 0, 90, 90, 90...C-45Pattern: CCCCCCCCOctagon with alternating angles 0 and 109.472C-54.736 & D-35.264Pattern: CCDDCCDDOctagon with angles 0, 0, 0, 90, 0, 0, 0, 90D-0Pattern: D...Apeirogon with alternating angles 0 and 70.529Repeating 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 3536A 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; BAATriangle B: 0, 31.717, 37.377; CABTriangle C: 0, 31.717, 37.377; BDCTriangle D: 0, 0,      20.905; DCDVertices:A-0Pattern: A...Apeirogon with alternating angles 0 and 41.810A-0 & B-0Pattern: AABB...Apeirogon with angles 0, 58.282, 63.434, 58.282, 0...A-20.905 & B-37.377 & C-31.717Pattern: AABCCBAABCCBDodecagon with angles 0, 0, 0, 69.095, 0, 69.095, 0, 0, 0, 69.095, 0, 69.095B-31.717 & C-37.377 & D-20.905Pattern: BBCDDCBBCDDCDodecagon with angles 0, 0, 0, 69.095, 0, 69.095, 0, 0, 0, 69.095, 0, 69.095C-0 & D-0Pattern: CCDD...Apeirogon with angles 0, 58.282, 63.434, 58.282, 0...D-0Pattern: D...Apeirogon with alternating angles 0 and 41.810Repeating unit: Hexagon with angles 0, 0, 90, 0, 0, 90. Composed of 1 A, 1 B, 1 C, 1 D.Cyclical 3636Triangles A, B, C, and D all identical.Plane 1: tiled by combination of A and BPlane 2: tiled by combination of C and DTriangle A: 0, 0, 0; BAATriangle B: 0, 0, 0; ABBTriangle C: 0, 0, 0; CCDTriangle D: 0, 0, 0; DDCVertices:A-0Pattern: A...Regular apeirogon with angle 0A-0 & B-0Pattern: AABB...Regular apeirogon with angle 0A-0 & B-0Pattern: AABB...Regular apeirogon with angle 0B-0Pattern: B...Regular apeirogon with angle 0C-0Pattern: C...Regular apeirogon with angle 0C-0 & D-0Pattern: CCDD...Regular apeirogon with angle 0C-0 & D-0Pattern: CCDD...Regular apeirogon with angle 0D-0Pattern: D...Regular apeirogon with angle 0Repeating 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 3344A and C triangles identicalPlane 1: tiled by combination of A, B, and C.Plane 2: tiled solely by D.Triangle A: 0,      54.736, 54.736; BAATriangle B: 35.264, 35.264, 70.529; ACBTriangle C: 0,      54.736, 54.736; BCCTriangle D: 0,      45,     45    ; DDDVertices:A-0Pattern: A...Regular apeirogon with angle 109.472A-54.736 & B-35.264Pattern: AABBAABBOctagon with angles 0, 125.565, 70.529, 125.565, 0, 125.565, 70.529, 125.565A-54.736 & B-70.529 & C-54.736Pattern: AABCCBHexagon with angles 0, 90, 90, 0, 90, 90B-35.264 & C-54.736Pattern: BBCCBBCCOctagon with angles 0, 125.565, 70.529, 125.565, 0, 125.565, 70.529, 125.565C-0Pattern: C...Regular apeirogon with angle 109.472D-0Pattern: D...Right-angled apeirogonD-45Pattern: DDDDDDDDOctagon with alternating angles 0 and 90D-45Pattern: DDDDDDDDOctagon with alternating angles 0 and 90Repeating unit 1: Quadrangle with angles 0, 0, 90, 90. Composed of 1 A, 1 B, 1 C.Repeating unit 2: Triangle D.Cyclical 3444A and B triangles identicalC and D triangles identicalPlane 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; BAATriangle B: 0, 35.264, 54.736; ABBTriangle C: 0, 0,      45    ; CCCTriangle D: 0, 0,      45    ; DDDVertices:A-0Pattern: A...Apeirogon with alternating angles 70.529 and 109.472A-35.264 & B-54.736Pattern: AABBAABBOctagon with angles 0, 90, 0, 90, 0, 90, 0, 90A-54.736 & B-35.264Pattern: AABBAABBOctagon with angles 0, 90, 0, 90, 0, 90, 0, 90B-0Pattern: B...Apeirogon with alternating angles 70.529 and 109.472C-0Pattern: C...Apeirogon with alternating angles 0 and 90C-0Pattern: C...Apeirogon with alternating angles 0 and 90C-45Patttern: CCCCCCCCRegular octagon with angle 0D-0Pattern: D...Apeirogon with alternating angles 0 and 90D-0Pattern: D...Apeirogon with alternating angles 0 and 90D-45Patttern: DDDDDDDDRegular octagon with angle 0Repeating 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 4444Triangles 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; AAATriangle B: 0, 0, 0; BBBTriangle C: 0, 0, 0; CCCTriangle D: 0, 0, 0; DDDVertices:A-0Pattern: A...Regular apeirogon with angle 0A-0Pattern: A...Regular apeirogon with angle 0A-0Pattern: A...Regular apeirogon with angle 0B-0Pattern: B...Regular apeirogon with angle 0B-0Pattern: B...Regular apeirogon with angle 0B-0Pattern: B...Regular apeirogon with angle 0C-0Pattern: C...Regular apeirogon with angle 0C-0Pattern: C...Regular apeirogon with angle 0C-0Pattern: C...Regular apeirogon with angle 0D-0Pattern: D...Regular apeirogon with angle 0D-0Pattern: D...Regular apeirogon with angle 0D-0Pattern: D...Regular apeirogon with angle 0Repeating unit 1: Triangle A.Repeating unit 2: Triangle B.Repeating unit 3: Triangle C.Repeating unit 4: Triangle D.Triangle with added 3-branchA and B triangles identical.One plane tiled by combination of A, B, C, and D.Triangle A: 0,      54.736,     90; ABCTriangle B: 0,      54.736,     90; BACTriangle C: 0,      70.529, 70.529; DABTriangle D: 54.736, 54.736,     60; DDCVertices:A-0 & B-0 & C-0Pattern: ABC...Apeirogon with angle 125.565; not regularA-54.736 & C-70.529 & D-54.736Pattern: AACDDCPentagon with angles 0, 0, 125.565, 120, 125.565A-90 & B-90Pattern: AABBRhombus with angles 0, 109.472, 0, 109.472B-54.736 & C-70.529 & D-54.736Pattern: BBCDDCPentagon with angles 0, 0, 125.565, 120, 125.565D-60Pattern: DDDDDDRegular hexagon with angle 109.472Repeating 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-branchA 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    ; ABCTriangle B: 0,  35.264, 90    ; BACTriangle C: 0,  54.736, 54.736; CABTriangle D: 45, 45,     60    ; DDDVertices:A-0 & B-0 & C-0Pattern: ABC...Non-regular right-angled apeirogonA-35.264 & C-54.736Pattern: AACCAACCHexagon with angles 0, 0, 109.472, 0, 0, 109.472A-90 & B-90Pattern: AABBRhombus with angles 0, 70.529, 0, 70.529B-35.264 & C-54.736Pattern: BBCCBBCCHexagon with angles 0, 0, 109.472, 0, 0, 109.472D-45Pattern: DDDDDDDDOctagon with alternating angles 90 and 120D-45Pattern: DDDDDDDDOctagon with alternating angles 90 and 120D-60Pattern: DDDDDDRight-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-branchA and B triangles identical.One plane tiled by combination of A, B, C, and D.Triangle A: 0,      20.905, 90    ; ABCTriangle B: 0,      20.905, 90    ; BACTriangle C: 0,      37.377, 37.377; DABTriangle D: 31.717, 31.717, 60    ; DDCVertices:A-0 & B-0 & C-0Pattern: ABC...Apeirogon with angle 58.282, not regularA-20.905 & C-37,377 & D-31.317Pattern: AACDDCAACDDCDecagon with angles 0, 0, 69.094, 120, 69.094, 0, 0, 69.094, 120, 69.094A-90 & B-90Pattern: AABBRhombus with angles 0, 41.810, 0, 41.810B-20.905 & C-37.377 & 31.717Pattern: BBCDDCBBCDDCDecagon with angles 0, 0, 69.094, 120, 69.094, 0, 0, 69.094, 120, 69.094D-60Pattern: DDDDDDRegular hexagon with angle 63.434Repeating 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-branchA 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; BACTriangle B: 0a, 0d, 90c; ABCTriangle C: 0a, 0b, 0d ; ABCTriangle D: 0a, 0b, 60c; DDDVertices:A-0 & B-0 & C-0Pattern: ABC...Apeirogon with angle 0A-0 & C-0Pattern: AACC...Apeirogon with angle 0A-90 & B-90Pattern: AABBSquare with angle 0B-0 & C-0Pattern: BBCC...Apeirogon with angle 0D-0Pattern: D...Apeirogon with alternating angles 0 and 120D-0Pattern: D...Apeirogon with alternating angles 0 and 120D-60Pattern: DDDDDDRegular hexagon with angle 0Repeating 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 trianglesA 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; ABCTriangle B: 0, 0,      70.529; ADCTriangle C: 0, 0,      70.529; ADBTriangle D: 0, 54.736, 54.736; DCBVertices:A-0 & B-0 & C-0Pattern: ABC...Apeirogon with angles 0, 125.565, 125.565...A-54.736 & B-70.529 & D-54.736Pattern: AABDDBHexagon with angles 0, 0, 109.472, 0, 0, 109.472A-54.736 & C-70.529 & D-54.736Pattern: AACDDCHexagon with angles 0, 0, 109.472, 0, 0, 109.472B-0 & C-0 & D-0Pattern: 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-edgesTriangles A, B, C, and D all identical.One plane tiled by combination of A, B, C, and D.Triangle A: 0, 0, 0; BCDTriangle B: 0, 0, 0; ACDTriangle C: 0, 0, 0; ABDTriangle D: 0, 0, 0; ABCVertices:A-0 & B-0 & C-0Pattern: ABC...Regular apeirogon with angle 0A-0 & B-0 & D-0Pattern: ABC...Regular apeirogon with angle 0A-0 & C-0 & D-0Pattern: ABC...Regular apeirogon with angle 0B-0 & C-0 & D-0Pattern: ABC...Regular apeirogon with angle 0Repeating 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 ABTriangle AC: 54.736, 65.905, 90; AC AC ADTriangle AD: 48.190, 70.529, 90; AE BD ACTriangle AE: 54.736, 54.736, 90; AD BE AETriangle BC: 52.239, 75.522, 90; AB BC CDTriangle BD: 65.905, 65.905, 90; AD BE BDTriangle BE: 48.190, 70.529, 90; BD AE CETriangle CD: 52.239, 75.522, 90; DE CD BCTriangle CE: 54.736, 65.905, 90; CE CE BETriangle DE: 52.239, 60,     90; DE CD DERepeating 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 ABTriangle AC: 35.264, 65.905, 90; AC AC ADTriangle AD: 48.190, 54.736, 90; AD BD ACTriangle AE: 45,     54.736, 90; BE AE AETriangle BC: 30,     75.522, 90; AB BC CDTriangle BD: 45,     65.905, 90; AD BD BDTriangle BE: 35.264, 70.529, 90; AE BE CETriangle CD: 52.239, 60,     90; CD CD BCTriangle CE: 54.736, 54.736, 90; CE CE BETriangle DE: 45,     60,     90; DE DE DERepeating 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.DemipenteracticA, 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 BCTriangle AC: 65.905, 65.905, 70.529; AD AE ACTriangle AD: 48.190, 54.736, 90    ; AD BD ACTriangle AE: 48.190, 54.736, 90    ; AE BE ACTriangle BC: 75.522, 75.522, 90    ; CD CE ABTriangle BD: 45,     65.905, 90    ; AD BD BDTriangle BE: 45,     65.905, 90    ; AE BE BETriangle CD: 52.239, 60,     90    ; CE CD BCTriangle CE: 52.239, 60,     90    ; CD CE BCTriangle DE: 45,     60,     90    ; DE DE DERepeating 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 ABTriangle AC: 35.264, 54.736, 90; AC AC ADTriangle AD: 35.264, 54.736, 90; AD AD ACTriangle AE: 45,     45,     90; AE AE AETriangle BC: 30,     60,     90; BC BC CDTriangle BD: 45,     45,     90; BD BD BDTriangle BE: 35.264, 54.736, 90; BE BE CETriangle CD: 30,     60,     90; CD CD BCTriangle CE: 35.264, 54.736, 90; CE CE BETriangle DE: 45,     45,     90; DE DE DERepeating unit 1: Triangle ABRepeating unit 2: Rectangle made from AC and AD.Repeating unit 3: Triangle AERepeating unit 4: Rectangle made from BC and CD.Repeating unit 5: Triangle BDRepeating unit 6: Rectangle made from BE and CE.Repeating unit 7: Triangle DEF~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 ABTriangle AC: 45,     45,     90; AC AC ACTriangle AD: 35.264, 54.736, 90; AE BD ADTriangle AE: 35.264, 54.736, 90; BE AD AETriangle BC: 30,     60,     90; BC AB BCTriangle BD: 35.264, 54.736, 90; AD BE BDTriangle BE: 19.471, 70.529, 90; AE BD CETriangle CD: 45,     45,     90; CD CD CDTriangle CE: 35.264, 54.736, 90; CE CE BETriangle DE: 30,     60,     90; DE DE DERepeating 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 ABTriangle AC: 54.736, 54.736, 70.529; AD AE ACTriangle AD: 35.264, 54.736, 90    ; AD AD ACTriangle AE: 35.264, 54.736, 90    ; AE AE ACTriangle BC: 60,     60,     60    ; BC CD CETriangle BD: 45,     45,     90    ; BD BD BDTriangle BE: 45,     45,     90    ; BE BE BETriangle CD: 30,     60,     90    ; CD CD BCTriangle CE: 30,     60,     90    ; CE CE BCTriangle DE: 45,     45,     90    ; DE DE DERepeating 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 BETriangle AC: 45, 45, 90; AC AC ACTriangle AD: 45, 45, 90; AD AD ADTriangle AE: 45, 45, 90; AE AE AETriangle BC: 60, 60, 60; AB BD BETriangle BD: 60, 60, 60; AB BC BETriangle BE: 60, 60, 60; AB BC BDTriangle CD: 45, 45, 90; CD CD CDTriangle CE: 45, 45, 90; CE CE CETriangle DE: 45, 45, 90; DE DE DERepeating 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 33333AB, 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 ABTriangle AC: 48.190, 65.905, 65.905; AC AD CETriangle AD: 48.190, 65.905, 65.905; AD AC BDTriangle AE: 52.239, 52.239, 75.522; AB DE AETriangle BC: 52.239, 52.239, 75.522; AB CD BCTriangle BD: 48.190, 65.905, 65.905; BD AD BETriangle BE: 48.190, 65.905, 65.905; BE BD CETriangle CD: 52.239, 52.239, 75.522; BC DE CDTriangle CE: 48.190, 65.905, 65.905; CE AC BETriangle DE: 52.239, 52.239, 75.522; AE CD DERepeating 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 ABTriangle AC: 20.905, 65.905, 90; AC AC ADTriangle AD: 37.377, 48.190, 90; BD AE ACTriangle AE: 31.717, 54.736, 90; BE AD AETriangle BC: 7.761,  75.522, 90; AB BC CDTriangle BD: 13.283, 65.905, 90; AD BE BDTriangle BE: 10.812, 70.529, 90; AE BD CETriangle CD: 22.239, 52.239, 90; CD CD BCTriangle CE: 20.905, 54.736, 90; CE CE BETriangle DE: 18,     60,     90; DE DE DERepeating 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 ABTriangle AC: 20.905, 54.736, 90; AC AC ADTriangle AD: 35.264, 37.377, 90; AE AD ACTriangle AE: 31.717, 45,     90; AE AD AETriangle BC: 7.761,  60,     90; BC BC CDTriangle BD: 13.283, 45,     90; BD BE BDTriangle BE: 10.812, 54.736, 90; BE BD CETriangle CD: 22.239, 30,     90; CD CD BC Triangle CE: 20.905, 35.264, 90; CE CE BETriangle DE: 18,     45,     90; DE DE DERepeating 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 ABTriangle AC: 20.905, 20.905, 90; AC AC ADTriangle AD: 10.812, 37.377, 90; AE BD ACTriangle AE: 31.717, 31.717, 90; AD BE AETriangle BC: 7.761,  22.239, 90; BC CD BCTriangle BD: 13.283, 13.283, 90; AD BE BDTriangle BE: 10.812, 37.377, 90; AE BD CETriangle CD: 7.761,  22.239, 90; CD BC CDTriangle CE: 20.905, 20.905, 90; CE CE BETriangle DE: 18,     36,     90; DE DE DERepeating 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 3AD 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 ABTriangle AC: 20.905, 20.905, 70.529; AD AE ACTriangle AD: 10.812, 54.736, 90    ; AD BD ACTriangle AE: 10.812, 54.736, 90    ; AE BE ACTriangle BC: 22.239, 22.239, 60    ; CD CE BCTriangle BD: 13.283, 45,     90    ; BD AD BDTriangle BE: 13.283, 45,     90    ; BE AE BETriangle CD: 7.761,  60,     90    ; CE CD BCTriangle CE: 7.761,  60,     90    ; CD CE BCTriangle DE: 36,     45,     90    ; DE DE DERepeating 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 33334AE is 4, AB, BC, CD, and DE are 3AB 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 ABTriangle AC: 35.264, 45,     65.905; AC AD ACTriangle AD: 35.264, 48.190, 54.736; AC BD ADTriangle AE: 45,     45,     45    ; AE AE AETriangle BC: 30,     52.239, 70.529; CD AB BCTriangle BD: 19.471, 65.905, 65.905; BD AD BETriangle BE: 35.264, 48.190, 54.736; CE BD BETriangle CD: 30,     52.239, 70.529; BC DE CDTriangle CE: 35.264, 45,     65.905; CE BE CETriangle DE: 30,     52.239, 60    ; CD DE DERepeating 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.
Marek14
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### Re: Planar tilings based on Goursat tetrahedra

{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

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.
Marek14
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### Re: Planar tilings based on Goursat tetrahedra

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.
The dream you dream alone is only a dream
the dream we dream together is reality.

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### Re: Planar tilings based on Goursat tetrahedra

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 4-6-6-4-6-6opposites: 4/4, 6a/6btruncated cuboctahedron:4-6: 144.735610 - 35.264390 - arccos(sqrt(2/3))4-8: 135 - 456-8: 125.264390 - 54.735610 - arccos(1/sqrt(3))cycle 1: 4-6-8-6-4-6-8-6cycle 2: 4-8-4-8-4-8-4-8opposites: 4/4, 6/6, 8/8truncated 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: 4-6-10-4-10-6-4-6-10-4-10-6opposites: 4/4, 6/6, 10/10o{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 = 180cycle 2: |hp1-6-to1-6-to2-6-hp2| - 52.239 + 75.522 + 52.239 = 180opposites: to1/to2, hp1/hp2o{3,3,4}truncated octahedron/hexagonal prism 150 - 30truncated octahedron/octagonal prism 135 - 45truncated octahedron/truncated cuboctahedron 120 - 60hexagonal 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 - 45cycle 1: |to-4-op| - 45cycle 2: |op-4-hp-4-tco| - 35.264 + 54.736 = 90cycle 3: |hp-6-to-6-tco| - 30 + 60 = 90cycle 4: |op-8-tco| - 45opposites: to/to, hp/hp/ op/op, tco/tcoo{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 - 18cycle 1: |to-4-dp-4-hp-4-tid| - 13.283 + 10.812 + 20.905 = 45cycle 2: |hp-6-to-6-tid| - 7.761 + 22.239 = 30cycle 3: |dp-10-tid| - 18opposites: to/to, hp/hp, dp/dp, tid/tido{3,4,3}truncated cuboctahedron 1/hexagonal prism 1 150 - 30truncated cuboctahedron 1/hexagonal prism 2 144.735610 - 35.264390 - arccos(sqrt[2/3])truncated cuboctahedron 1/truncated cuboctahedron 2 135 - 45hexagonal 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 - 30cycle 1: |tco1-4-hp2-4-hp1-4-tco2| - 35.264 + 19.471 + 35.264 = 90cycle 2: |tco1-6-hp1| - 30cycle 3: |hp2-6-tco2| - 30cycle 4: |tco1-8-tco2| - 45opposites: tco1/tco1, tco2/tco2, hp1/hp1, hp2/hp2o(b333)truncated octahedron 1/truncated octahedron 2 120 - 60truncated octahedron 1/cube 135 - 45truncated octahedron 1/truncated octahedron 3 120 - 60truncated octahedron 2/cube 135 - 45truncated octahedron 2/truncated octahedron 3 120 - 60cube/truncated octahedron 3 135 - 45cycle 1: |to1-4-c| - 45cycle 2: |to2-4-c| - 45cycle 3: |c-4-to3| - 45cycle 4: (to1-6-to2-6-to3-6-) - 60 + 60 + 60 = 180opposites: 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; ABATriangle B: 54.736, 70.529, 90; ABCTriangle C: 54.736, 70.529, 90; DCBTriangle D: 54.736, 60,     90; DCDVertices:A-54.736 & B-70.529 & C-54.736Pattern: AABCCBTriangle with angles 120, 125.265, 125.265A-60Pattern: AAAAAATriangle with angles 109.472, 109.472, 109.472A-90 & B-90Pattern: AABBQuadrangle with angles 109.472, 125.265, 120, 125.265B-54.736 & C-70.529 & D-54.736Pattern: BBCDDCTriangle with angles 120, 125.265, 125.265C-90 & D-90Pattern: CCDDQuadrangle with angles 109.472, 125.265, 120, 125.265D-60Pattern: DDDDDDEquilateral triangle with angle 109.472Repeating 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; BAATriangle B: 35.264, 70.529, 90; ABCTriangle 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; DDDVertices:A-45Pattern: AAAAAAAASquare with angle 109.472A-54.736 & B-70.529 & C-54.736Pattern: AABCCBTriangle with angles 90, 90, 109.472A-90 & B-90Pattern: AABBQuadrangle with angles 90, 125.265, 109.472, 125.265B-35.264 & C-54.736Pattern: BBCCBBCCSquare with angle 125.265C-90Pattern: CCCCSquare with angle 109.472D-45Pattern: DDDDDDDDSquare with angle 120D-60Pattern: DDDDDDEquilateral triangle with angle 90D-90Pattern: DDDDRhombus with angles 90, 120, 90, 120Repeating 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; BAATriangle B: 20.905, 70.529, 90; ABCTriangle C: 37.377, 54.736, 90; CDBTriangle D: 31.717, 60,     90; DCDVertices:A-36Pattern: AAAAAAAAAARegular pentagon with angle 109.472A-54.736 & B-70.529 & C-54.736Pattern: AABCCBTriangle with angles 58.282, 58.282, 72A-90 & B-90Pattern: AABBQuadrangle with angles 41.810, 125.265, 72, 125.265B-20.905 & C-37.377 & D-31.717Pattern: BBCDDCBBCDDCHexagon with angles 120, 125.265, 125.265, 120, 125.265, 125.265C-90 & D-90Pattern: CCDDQuadrangle with angles 69.094, 109.472, 69.094, 120D-60Pattern: DDDDDDEquilateral triangle with angle 63.434Repeating 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 BPlane 2: tiled by combination of C and DTriangle A: 35.264, 60,     90; ABATriangle B: 45,     54.736, 90; ABB Triangle C: 45,     54.736, 90; DCCTriangle D: 35.264, 60,     90; DCDVertices:A-35.264 & B-54.736Pattern: AABBAABBRhombus with angles 90, 120, 90, 120A-60Pattern: AAAAAAEquilateral triangle with angle 70.528A-90 & B-90Pattern: AABBQuadrangle with angles 90, 90, 90, 120B-45Pattern: BBBBBBBBSquare with angle 109.472C-45Pattern: CCCCCCCCSquare with angle 109.472C-54.736 & D-35.264Pattern: CCDDCCDDRhombus with angles 90, 120, 90, 120C-90 & D-90Pattern: CCDDQuadrangle with angles 90, 90, 90, 120D-60Pattern: DDDDDDEquilateral triangle with angle 70.529Repeating 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    ; AABTriangle B: 70.529, 70.529, 70.529; ABCTriangle C: 54.736, 54.736, 90    ; CCBTriangle D: 54.736, 54.736, 90    ; DDBVertices:A-54.736 & B-70.529 & C-54.736Pattern: AABCCBQuadrangle with angles 125.265, 125.265, 125.265, 125.265 (not square, apparently)A-54.736 & B-70.529 & D-54.736Pattern: AABDDBQuadrangle with angles 125.265, 125.265, 125.265, 125.265 (not square, apparently)A-90Pattern: AAAASquare with angle 109.472B-70.529 & C-54.736 & D-54.736Pattern: BCCBDDQuadrangle with angles 125.265, 125.265, 125.265, 125.265 (not square, apparently)C-90Pattern: CCCCSquare with angle 109.472D-90Pattern: DDDDSquare with angle 109.472Repeating 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 APlane 2: tiled by combination of B and CPlane 3: tiled solely by DTriangle A: 45,     45,     90; AAATriangle B: 35.264, 54.736, 90; BBCTriangle C: 35.264, 54.736, 90; CCBTriangle D: 45,     45,     90; DDDVertices:A-45Pattern: AAAAAAAASquare with angle 90A-45Pattern: AAAAAAAASquare 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-90Pattern: AAAASquare with angle 90B-35.264 & C-54.736Pattern: BBCCRectangle with angle 90B-54.736 & C-35.264Pattern: BBCCRectangle with angle 90B-90Pattern: BBBBRhombus with angles 70.529, 109.472, 70.529, 109.472C-90Pattern: CCCCRhombus with angles 70.529, 109.472, 70.529, 109.472D-45Pattern: DDDDDDDDSquare with angle 90D-45Pattern: DDDDDDDDSquare with angle 90D-90Pattern: DDDDSquare with angle 90Repeating 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 CPlane 2: tiled solely by DTriangle A: 35.264, 54.736, 90    ; AABTriangle B: 54.736, 54.736, 70.529; ACB (double of A or C triangle)Triangle C: 35.264, 54.736, 90    ; CCBTriangle D: 45,     45,     90    ; DDDVertices:A-35.264 & B-54.736Pattern: AABBAABBHexagon with angles 109.472, 125.565, 125.565, 109.472, 125.565, 125.565A-54.736 & B-70.529 & C-54.736Pattern: AABCCBSquare with angle 90A-90Pattern: AAAARhombus with angles 70.529, 109.472, 70.529, 109.472B-54.736 & C-35.264Pattern: BBCCBBCCHexagon with angles 109.472, 125.565, 125.565, 109.472, 125.565, 125.565C-90Pattern: CCCCRhombus with angles 70.529, 109.472, 70.529, 109.472D-45Pattern: DDDDDDDDSquare with angle 90D-45Pattern: DDDDDDDDSquare with angle 90D-90Pattern: DDDDSquare with angle 90Repeating unit 1: Rectangle of angle 90. Composed of 1 A, 1 B, 1 C.Repeating unit 2: Triangle D.Cyclical 3333Triangles 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; BDATriangle B: 54.736, 54.736, 70.529; ACBTriangle C: 54.736, 54.736, 70.529; BDCTriangle D: 54.736, 54.736, 70.529; ACDVertices:A-54.736 & B-70.529 & C-54.736Pattern: AABCCBHexagon with angles 109.472, 125.565, 125.565, 109.472, 125.565, 125.565A-54.736 & C-54.736 & D-70.529Pattern: AADCCDHexagon with angles 109.472, 125.565, 125.565, 109.472, 125.565, 125.565A-70.529 & B-54.736 & D-54.736Pattern: ABBADDHexagon with angles 109.472, 125.565, 125.565, 109.472, 125.565, 125.565B-54.736 & C-70.529 & D-54.736Pattern: BBCDDCHexagon with angles 109.472, 125.565, 125.565, 109.472, 125.565, 125.565Repeating unit: An infinite strip formed by repeating triangles A, B, C, D.{4,3,5}Plane 1: tiled solely by APlane 2: tiled by combination of B, C, and DTriangle A: 36,     45,     90; AAATriangle B: 20.905, 54.736, 90; BBCTriangle C: 35.264, 37.377, 90; DCBTriangle D: 31.717, 45,     90; DCDVertices:A-36Pattern: AAAAAAAAAARegular pentagon with angle 90A-45Pattern: AAAAAAAASquare with angle 72A-90Pattern: AAAARhombus with angles 72, 90, 72, 90B-20.905 & C-37.377 & D-31.717Pattern: BBCDDCBBCDDCRight-angled hexagon, not regularB-54.736 & C-35.264Pattern: BBCCBBCCRectangle with angle 58.282B-90Pattern: BBBBRhombus with angles 41.810, 109.472, 41.810, 109.472C-90 & D-90Pattern: CCDDQuadrangle with angles 69.095, 70.529, 90, 69.095D-45Pattern: DDDDDDDDSquare with angle 63.434Repeating 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; ABATriangle B: 20.905, 37.377, 90; ABCTriangle C: 20.905, 37.377, 90; DCBTriangle D: 31.717, 36,     90; DCCVertices:A-31.717 & B-37.377 & C-20.905Pattern: AABCCBAABCCBHexagon with angles 58.282, 58.282, 72, 58.282, 58.282, 72A-36Pattern: AAAAAAAAAARegular pentagon with angle 63.434A-90 & B-90Pattern: AABBQuadrangle with angles 41.810, 69.095, 72, 69.095B-20.905 & C-37.377 & D-31.717Pattern: BBCDDCBBCDDCHexagon with angles 58.282, 58.282, 72, 58.282, 58.282, 72C-90 & D-90Pattern: CCDDQuadrangle with angles 41.810, 69.095, 72, 69.095D-36Pattern: DDDDDDDDDDRegular pentagon with angle 63.434Repeating 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; ABATriangle B: 31.717, 37.377, 90; ABCTriangle C: 31.717, 37.377, 90; DCBTriangle D: 20.905,     60, 90; DCDVertices:A-20.905 & B-37.377 & C-31.717Pattern: AABCCBAABCCBHexagon with angles 69.095, 69.095, 120, 69.095, 69.095, 120A-60Pattern: AAAAAAEquilateral triangle with angle 41.810A-90 & B-90Pattern: AABBQuadrangle with angles 58.282, 63.434, 58.282 and 120B-31.717 & C-37.377 & D-20.905Pattern: BBCDDCBBCDDCHexagon with angles 69.095, 69.095, 120, 69.095, 69.095, 120C-90 & D-90Pattern: CCDDQuadrangle with angles 58.282, 63.434, 58.282 and 120D-60Pattern: AAAAAAEquilateral triangle with angle 41.810Repeating 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; AABTriangle B: 37.377, 37.377, 70.529; ACDTriangle C: 20.905, 54.736,     90; CCBTriangle D: 31.717, 31.717,     90; DDBVertices:A-20.905 & B-37.377 & D-31.717Pattern: AABDDBAABDDBOctagon with alternating angles of 69.095 and 125.565A-54.736 & B-70.529 & C-54.736Pattern: AABCCBRectangle with angle 58.282A-90Pattern: AAAARhombus with angles 41.810, 109.472, 41.810, 109.472B-37.377 & C-20.905 & D-31.717Pattern: BCCBDDBCCBDDOctagon with alternating angles of 69.095 and 125.565C-90Pattern: CCCCRhombus with angles 41.810, 109.472, 41.810, 109.472D-90Pattern: DDDDSquare with angle 63.434Repeating unit: Right-angled pentagon; not regular. Composed of 1 A, 1 B, 1 C, 1 D.Cyclical 3334A 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; BAATriangle B: 35.264, 54.736, 70.529; CABTriangle C: 35.264, 54.736, 70.529; BDCTriangle D: 45,     54.736, 54.736; CDDVertices:A-45Pattern: AAAAAAAARegular octagon with angle 109.472A-54.736 & B-35.264Pattern: AABBAABBOctagon with angles 90, 125.565, 109.472, 125.565, 90, 125.565, 109.472, 125.565A-54.736 & B-70.529 & C-54.736Pattern: AABCCBHexagon with angles 70.529, 125.565, 90, 90, 90, 125.565B-54.736 & C-70.529 & D-54.736Pattern: BBCDDCHexagon with angles 70.529, 125.565, 90, 90, 90, 125.565C-35.264 & D-54.736Pattern: CCDDCCDDOctagon with angles 90, 125.565, 109.472, 125.565, 90, 125.565, 109.472, 125.565D-45Pattern: DDDDDDDDRegular octagon with angle 109.472Repeating unit: Quadrangle with angles 45, 90, 45, 90. Composed of 1 A, 1 B, 1 C, 1 D.Cyclical 3335A 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; BADTriangle B: 20.905, 54.736, 70.529; CABTriangle C: 20.905, 54.736, 70.529; BDCTriangle D: 31.717, 37.377, 54.736; CDAVertices:A-31.717 & C-20.905 & D-37.377Pattern: AADCCDAADCCDDodecagon with angles 69.095, 109.472, 69.095, 125.565, 109.472, 125.565, 69.095, 109.472, 69.095, 125.565, 109.472, 125.565A-37.377 & B-20.905 & D-31.717Pattern: ABBADDABBADDDodecagon with angles 69.095, 109.472, 69.095, 125.565, 109.472, 125.565, 69.095, 109.472, 69.095, 125.565, 109.472, 125.565A-54.736 & B-70.529 & C-54.736Pattern: AABCCDHexagon with angles 41.810, 125.565, 58.282, 63.434, 58.282, 125.565B-54.736 & C-70.529 & D-54.736Pattern: BBCDDCHexagon with angles 41.810, 125.565, 58.282, 63.434, 58.282, 125.565Repeating unit: An infinite strip formed by repeating triangles A, B, C, D. Its two edges form right-angled pseudogons.Cyclical 3434Triangles A, B, C, and D all identical.Plane 1: tiled by combination of A and BPlane 2: tiled by combination of C and DTriangle A: 35.264, 45, 54.736; ABATriangle B: 35.264, 45, 54.736; BABTriangle C: 35.264, 45, 54.736; CDCTriangle D: 35.264, 45, 54.736; DCDVertices:A-35.264 & B-54.736Pattern: AABBAABBRight-angled octagon, not regularA-45Pattern: AAAAAAAAOctagon with alternating angles 70.529 and 125.565A-54.736 & B-35.264Pattern: AABBAABBRight-angled octagon, not regularB-45Pattern: BBBBBBBBOctagon with alternating angles 70.529 and 125.565C-35.264 & D-54.736Pattern: CCDDCCDDRight-angled octagon, not regularC-45Pattern: CCCCCCCCOctagon with alternating angles 70.529 and 125.565C-54.736 & D-35.264Pattern: CCDDCCDDRight-angled octagon, not regularD-45Pattern: DDDDDDDDOctagon with alternating angles 70.529 and 125.565Repeating 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 3435A 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; BDATriangle B: 20.905, 45,     54.736; BABTriangle C: 20.905, 45,     54.736; CDCTriangle D: 31.717, 35.264, 37.377; CADVertices:A-31.717 & C-20.905 & D-37.377Pattern: AADCCDAADCCDDodecagon with angles 69.095, 70.529, 69.095, 90, 90, 90, 69.095, 70.529, 69.095, 90, 90, 90A-35.264 & B-54.736Pattern: AABBAABBOctagon with angles 58.282, 63.434, 58.282, 90, 58.282, 63.434, 58.282, 90A-37.377 & B-20,905 & D-31.717Pattern: ABBADDABBADDDodecagon with angles 69.095, 70.529, 69.095, 90, 90, 90, 69.095, 70.529, 69.095, 90, 90, 90B-45Pattern: BBBBBBBBOctagon with alternating angles 41.810 and 109.472C-45Pattern: CCCCCCCCOctagon with alternating angles 41.810 and 109.472C-54.736 & D-35.264Pattern: CCDDCCDDOctagon with angles 58.282, 63.434, 58.282, 90, 58.282, 63.434, 58.282, 90Repeating unit: Hexagon with angles 45, 90, 90, 45, 90, 90. Composed of 1 A, 1 B, 1 C, 1 D.Cyclical 3535Triangles 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; DBATriangle B: 20.905, 31.717, 37.377; CABTriangle C: 20.905, 31.717, 37.377; BDCTriangle D: 20.905, 31.717, 37.377; ACDVertices:A-20.905 & C-31.717 & D-37.377Pattern: AADCCDAADCCDDodecagon with angles 41.810, 58.282, 69.095, 63.434, 69.095, 58.282, 41.810, 58.282, 69.095, 63.434, 69.095, 58.282A-31.717 & B-37.377 & C-20.905Pattern: AABCCBAABCCBDodecagon with angles 41.810, 58.282, 69.095, 63.434, 69.095, 58.282, 41.810, 58.282, 69.095, 63.434, 69.095, 58.282A-37.377 & B-20.905 & D-31.717Pattern: ABBADDABBADDDodecagon with angles 41.810, 58.282, 69.095, 63.434, 69.095, 58.282, 41.810, 58.282, 69.095, 63.434, 69.095, 58.282B-31.717 & C-37.377 & D-20.905Pattern: BBCDDCBBCDDCDodecagon with angles 41.810, 58.282, 69.095, 63.434, 69.095, 58.282, 41.810, 58.282, 69.095, 63.434, 69.095, 58.282Repeating 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; BAATriangle B: 0,  70.529, 90; ABCTriangle C: 0,  54.736, 90; CCBTriangle D: 0,  60,     90; DDDVertices:A-30Pattern: AAAAAAAAAAAARegular hexagon with angle 109.472A-54.736 & B-70.529 & C-54.736Pattern: AABCCBTriangle with angles 0, 0, 60A-90 & B-90Pattern: AABBQuadrangle with angles 0, 125.565, 60, 125.565B-0 & C-0Pattern: BBCC...Apeirogon with angle 125.565, not regularC-90Pattern: CCCCRhombus with angles 0, 109.472, 0, 109.472D-0Pattern: D...Regular apeirogon with angle 120D-60Pattern: DDDDDDEquilateral triangle with angle 0D-90Pattern: DDDDRhombus with angles 0, 120, 0, 120Repeating 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; AAATriangle B: 0,  54.736, 90; BBCTriangle C: 0,  35.264, 90; CCBTriangle D: 0,  45,     90; DDDVertices:A-30Pattern: AAAAAAAAAAAARight-angled dodecagonA-45Pattern: AAAAAAAARegular octagon with angle 60A-90Pattern: AAAARhombus with angles 60, 90, 60, 90B-0Pattern: B...Regular apeirogon with angle 109.472B-54.736 & C-35.264Pattern: BBCCBBCCRectangle with angle 0B-90 & C-90Pattern: BBCCQuadrangle with angles 0, 90, 0, 90C-0Pattern: C...Regular apeirogon with angle 70.529D-0Pattern: D...Right-angled apeirogonD-45Pattern: DDDDDDDDSquare with angle 0D-90Pattern: DDDDRhombus with angles 0, 90, 0, 90Repeating 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; BAATriangle B: 0,  37.377, 90; ABCTriangle C: 0,  20.905, 90; CCBTriangle D: 0,  36,     90; DDDVertices:A-30Pattern: AAAAAAAARegular hexagon with angle 63.434A-31.717 & B-37.377 & C-20.905Pattern: AABCCBAABCCBHexagon with angles 0, 0, 60, 0, 0, 60A-90 & B-90Pattern: AABBQuadrangle with angles 0, 69.095, 60, 69.095B-0 & C-0Pattern: BBCC...Apeirogon with angle 58.282, not regularC-90Pattern: CCCCRhombus with angles 0, 41.810, 0, 41.810D-0Pattern: D...Regular apeirogon with angle 72D-36Pattern: DDDDDDDDDDRegular pentagon with angle 0D-90Pattern: DDDDRhombus with angles 0, 72, 0, 72Repeating 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 APlane 2: tiled by combination of B and CPlane 3: tiled solely by DTriangle A: 0, 30, 90; AAATriangle B: 0, 0,  90; BBCTriangle C: 0, 0,  90; CCBTriangle D: 0, 30, 90; DDDVertices:A-0Pattern: A...Regular apeirogon with angle 60A-30Pattern: AAAAAAAAAAAARegular hexagon with angle 0A-90Pattern: AAAARhombus with angles 0, 60, 0, 60B-0 & C-0Pattern: BBCC...Regular apeirogon with angle 0B-0 & C-0Pattern: BBCC...Regular apeirogon with angle 0B-90Pattern: BBBBSquare with angle 0C-90Pattern: CCCCSquare with angle 0D-0Pattern: D...Regular apeirogon with angle 60D-30Pattern: DDDDDDDDDDDDRegular hexagon with angle 0D-90Pattern: DDDDRhombus with angles 0, 60, 0, 60Repeating 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; ABATriangle B: 0,      54.736, 90; ABBTriangle C: 0,      45,     90; CCCTriangle D: 0,      60,     90; CCCVertices:A-35.264 & B-54.736Rhombus with angles 0, 90, 0, 90A-45Pattern: AAAAAAAASquare with angle 70.529A-90 & B-90Pattern: AABBQuadrangle with angles 0, 90, 90, 90B-0Pattern: B...Regular apeirogon with angle 109.472C-0Pattern: C...Right-angled apeirogonC-45Pattern: CCCCCCCCSquare with angle 0C-90Pattern: CCCCRhombus with angles 0, 90, 0, 90D-0Pattern: D...Regular apeirogon with angle 120D-60Pattern: DDDDDDEquilateral triangle with angle 0D-90Pattern: DDDDRhombus with angles 0, 120, 0, 120Repeating 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; AAATriangle B: 0, 0,  90; BBBTriangle C: 0, 0,  90; CCCTriangle D: 0, 45, 90; DDDVertices:A-0Pattern: A...Right-angled apeirogonA-45Pattern: AAAAAAAASquare with angle 0A-90Pattern: AAAARhombus with angles 0, 90, 0, 90B-0Pattern: B...Regular apeirogon with angle 0B-0Pattern: B...Regular apeirogon with angle 0B-90Pattern: BBBBSquare with angle 0C-0Pattern: C...Regular apeirogon with angle 0C-0Pattern: C...Regular apeirogon with angle 0C-90Pattern: CCCCSquare with angle 0D-0Pattern: D...Right-angled apeirogonD-45Pattern: DDDDDDDDSquare with angle 0D-90Pattern: DDDDRhombus with angles 0, 90, 0, 90Repeating 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 BPlane 2: tiled by combination of C and DTriangle A: 0, 60, 90; ABATriangle B: 0, 0,  90; ABBTriangle C: 0, 0,  90; CDCTriangle D: 0, 60, 90; DCDVertices:A-0 & B-0Pattern: AABB...Apeirogon with alternating angles 0 and 120A-60Pattern: AAAAAAEquilateral triangle with angle 0A-90 & B-90Pattern: AABBQuadrangle with angles 0, 0, 0, 120B-0Pattern: B...Regular apeirogon with angle 0C-0Pattern: C...Regular apeirogon with angle 0C-0 & D-0Pattern: CCDD...Apeirogon with alternating angles 0 and 120C-90 & D-90Pattern: CCDDQuadrangle with angles 0, 0, 0, 120D-60Pattern: DDDDDDEquilateral triangle with angle 0Repeating 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 336A and C triangles identical.Plane 1: tiled by combination of A, B, and CPlane 2: tiled solely by DTriangle A: 0, 54.736, 90    ; AABTriangle B: 0, 0,      70.529; ACBTriangle C: 0, 54.736, 90    ; CCBTriangle D: 0, 0,      90    ; DDDVertices:A-0 & B-0Pattern: AABB...Apeirogon with alternating angles 0 and 109.472A-54.736 & B-70.529 & C-54.736Pattern: AABCCBHexagon with all angles 0, not regularA-90Pattern: AAAARhombus with angles 0, 109.472, 0, 109.472B-0 & C-0Pattern: CCDD...Apeirogon with alternating angles 0 and 109.472C-90Pattern: CCCCRhombus with angles 0, 109.472, 0, 109.472D-0Pattern: D...Regular apeirogon with angle 0D-0Pattern: D...Regular apeirogon with angle 0D-90Pattern: DDDDSquare with angle 0Repeating unit 1: Quadrangle with angles 0, 0, 90, 90. Composed of 1 A, 1 B, 1 C.Repeating unit 2: Triangle D.Branched 344C and D triangles identical.Plane 1: tiled by combination of A and BPlane 2: tiled solely by CPlane 3: tiled solely by DTriangle 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-54.736Pattern: AABBHexagon with angles 0, 90, 90, 0, 90, 90A-35.264 & B-54.736Pattern: AABBHexagon with angles 0, 90, 90, 0, 90, 90A-90Pattern: AAAASquare with angle 70.529B-0Pattern: B...Regular apeirogon with angle 109.472C-0Pattern: C...Right-angled apeirogonC-45Pattern: CCCCCCCCSquare with angle 0C-90Pattern: CCCCRhombus with angles 0, 90, 0, 90D-0Pattern: D...Right-angled apeirogonD-45Pattern: DDDDDDDDSquare with angle 0D-90Pattern: DDDDRhombus with angles 0, 90, 0, 90Repeating 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 444A, C and D triangles identical.Plane 1: tiled solely by APlane 2: tiled solely by BPlane 3: tiled solely by CPlane 4: tiled solely by DTriangle A: 0, 0, 90; AAATriangle B: 0, 0, 0 ; BBBTriangle C: 0, 0, 90; CCCTriangle D: 0, 0, 90; DDDVertices:A-0Pattern: A...Regular apeirogon with angle 0A-0Pattern: A...Regular apeirogon with angle 0A-90Pattern: AAAASquare with angle 0B-0Pattern: B...Regular apeirogon with angle 0B-0Pattern: B...Regular apeirogon with angle 0B-0Pattern: B...Regular apeirogon with angle 0C-0Pattern: C...Regular apeirogon with angle 0C-0Pattern: C...Regular apeirogon with angle 0C-90Pattern: CCCCSquare with angle 0D-0Pattern: D...Regular apeirogon with angle 0D-0Pattern: D...Regular apeirogon with angle 0D-90Pattern: CCCCSquare with angle 0Repeating unit 1: Triangle A.Repeating unit 2: Triangle B.Repeating unit 3: Triangle C.Repeating unit 4: Triangle D.Cyclical 3336A 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; BAATriangle B: 0, 54.736, 70.529; CABTriangle C: 0, 54.736, 70.529; BDCTriangle D: 0, 0,      54.736; DCDVertices:A-0Pattern: A...Regular apeirogon with angle 0A-0 & B-0Pattern: AABB...Apeirogon with angles 0, 125.565, 109.472, 125.565, 0...A-54.736 & B-70.529 & C-54.736Pattern: AABCCBHexagon with angles 0, 0, 0, 125.565, 0, 125.565B-54.736 & C-70.529 & D-54.736Pattern: AABCCBHexagon with angles 0, 0, 0, 125.565, 0, 125.565C-0 & D-0Pattern: CCDD...Apeirogon with angles 0, 125.565, 109.472, 125.565, 0...D-0Pattern: D...Regular apeirogon with angle 0Repeating unit: Rectangle with angle 0. Composed of 1 A, 1 B, 1 C, 1 D.Cyclical 3436A and D triangles identical.B and C triangles identical.Plane 1: tiled by combination of A and BPlane 2: tiled by combination of C and DTriangle A: 0, 0,  35.264; BAATriangle B: 0, 45, 54.736; ABATriangle C: 0, 45, 54.736; CDCTriangle D: 0, 0,  35.264; DCDVertices:A-0Pattern: A...Apeirogon with alternating angles 0 and 70.529A-0 & B-0Pattern: AABB...Apeirogon with angles 0, 90, 90, 90...A-35.264 & B-54.736Pattern: AABBAABBOctagon with angles 0, 0, 0, 90, 0, 0, 0, 90B-45Pattern: BBBBBBBBOctagon with alternating angles 0 and 109.472C-0 & D-0Pattern: CCDD...Apeirogon with angles 0, 90, 90, 90...C-45Pattern: CCCCCCCCOctagon with alternating angles 0 and 109.472C-54.736 & D-35.264Pattern: CCDDCCDDOctagon with angles 0, 0, 0, 90, 0, 0, 0, 90D-0Pattern: D...Apeirogon with alternating angles 0 and 70.529Repeating 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 3536A 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; BAATriangle B: 0, 31.717, 37.377; CABTriangle C: 0, 31.717, 37.377; BDCTriangle D: 0, 0,      20.905; DCDVertices:A-0Pattern: A...Apeirogon with alternating angles 0 and 41.810A-0 & B-0Pattern: AABB...Apeirogon with angles 0, 58.282, 63.434, 58.282, 0...A-20.905 & B-37.377 & C-31.717Pattern: AABCCBAABCCBDodecagon with angles 0, 0, 0, 69.095, 0, 69.095, 0, 0, 0, 69.095, 0, 69.095B-31.717 & C-37.377 & D-20.905Pattern: BBCDDCBBCDDCDodecagon with angles 0, 0, 0, 69.095, 0, 69.095, 0, 0, 0, 69.095, 0, 69.095C-0 & D-0Pattern: CCDD...Apeirogon with angles 0, 58.282, 63.434, 58.282, 0...D-0Pattern: D...Apeirogon with alternating angles 0 and 41.810Repeating unit: Hexagon with angles 0, 0, 90, 0, 0, 90. Composed of 1 A, 1 B, 1 C, 1 D.Cyclical 3636Triangles A, B, C, and D all identical.Plane 1: tiled by combination of A and BPlane 2: tiled by combination of C and DTriangle A: 0, 0, 0; BAATriangle B: 0, 0, 0; ABBTriangle C: 0, 0, 0; CCDTriangle D: 0, 0, 0; DDCVertices:A-0Pattern: A...Regular apeirogon with angle 0A-0 & B-0Pattern: AABB...Regular apeirogon with angle 0A-0 & B-0Pattern: AABB...Regular apeirogon with angle 0B-0Pattern: B...Regular apeirogon with angle 0C-0Pattern: C...Regular apeirogon with angle 0C-0 & D-0Pattern: CCDD...Regular apeirogon with angle 0C-0 & D-0Pattern: CCDD...Regular apeirogon with angle 0D-0Pattern: D...Regular apeirogon with angle 0Repeating 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 3344A and C triangles identicalPlane 1: tiled by combination of A, B, and C.Plane 2: tiled solely by D.Triangle A: 0,      54.736, 54.736; BAATriangle B: 35.264, 35.264, 70.529; ACBTriangle C: 0,      54.736, 54.736; BCCTriangle D: 0,      45,     45    ; DDDVertices:A-0Pattern: A...Regular apeirogon with angle 109.472A-54.736 & B-35.264Pattern: AABBAABBOctagon with angles 0, 125.565, 70.529, 125.565, 0, 125.565, 70.529, 125.565A-54.736 & B-70.529 & C-54.736Pattern: AABCCBHexagon with angles 0, 90, 90, 0, 90, 90B-35.264 & C-54.736Pattern: BBCCBBCCOctagon with angles 0, 125.565, 70.529, 125.565, 0, 125.565, 70.529, 125.565C-0Pattern: C...Regular apeirogon with angle 109.472D-0Pattern: D...Right-angled apeirogonD-45Pattern: DDDDDDDDOctagon with alternating angles 0 and 90D-45Pattern: DDDDDDDDOctagon with alternating angles 0 and 90Repeating unit 1: Quadrangle with angles 0, 0, 90, 90. Composed of 1 A, 1 B, 1 C.Repeating unit 2: Triangle D.Cyclical 3444A and B triangles identicalC and D triangles identicalPlane 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; BAATriangle B: 0, 35.264, 54.736; ABBTriangle C: 0, 0,      45    ; CCCTriangle D: 0, 0,      45    ; DDDVertices:A-0Pattern: A...Apeirogon with alternating angles 70.529 and 109.472A-35.264 & B-54.736Pattern: AABBAABBOctagon with angles 0, 90, 0, 90, 0, 90, 0, 90A-54.736 & B-35.264Pattern: AABBAABBOctagon with angles 0, 90, 0, 90, 0, 90, 0, 90B-0Pattern: B...Apeirogon with alternating angles 70.529 and 109.472C-0Pattern: C...Apeirogon with alternating angles 0 and 90C-0Pattern: C...Apeirogon with alternating angles 0 and 90C-45Patttern: CCCCCCCCRegular octagon with angle 0D-0Pattern: D...Apeirogon with alternating angles 0 and 90D-0Pattern: D...Apeirogon with alternating angles 0 and 90D-45Patttern: DDDDDDDDRegular octagon with angle 0Repeating 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 4444Triangles 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; AAATriangle B: 0, 0, 0; BBBTriangle C: 0, 0, 0; CCCTriangle D: 0, 0, 0; DDDVertices:A-0Pattern: A...Regular apeirogon with angle 0A-0Pattern: A...Regular apeirogon with angle 0A-0Pattern: A...Regular apeirogon with angle 0B-0Pattern: B...Regular apeirogon with angle 0B-0Pattern: B...Regular apeirogon with angle 0B-0Pattern: B...Regular apeirogon with angle 0C-0Pattern: C...Regular apeirogon with angle 0C-0Pattern: C...Regular apeirogon with angle 0C-0Pattern: C...Regular apeirogon with angle 0D-0Pattern: D...Regular apeirogon with angle 0D-0Pattern: D...Regular apeirogon with angle 0D-0Pattern: D...Regular apeirogon with angle 0Repeating unit 1: Triangle A.Repeating unit 2: Triangle B.Repeating unit 3: Triangle C.Repeating unit 4: Triangle D.Triangle with added 3-branchA and B triangles identical.One plane tiled by combination of A, B, C, and D.Triangle A: 0,      54.736,     90; ABCTriangle B: 0,      54.736,     90; BACTriangle C: 0,      70.529, 70.529; DABTriangle D: 54.736, 54.736,     60; DDCVertices:A-0 & B-0 & C-0Pattern: ABC...Apeirogon with angle 125.565; not regularA-54.736 & C-70.529 & D-54.736Pattern: AACDDCPentagon with angles 0, 0, 125.565, 120, 125.565A-90 & B-90Pattern: AABBRhombus with angles 0, 109.472, 0, 109.472B-54.736 & C-70.529 & D-54.736Pattern: BBCDDCPentagon with angles 0, 0, 125.565, 120, 125.565D-60Pattern: DDDDDDRegular hexagon with angle 109.472Repeating 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-branchA 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    ; ABCTriangle B: 0,  35.264, 90    ; BACTriangle C: 0,  54.736, 54.736; CABTriangle D: 45, 45,     60    ; DDDVertices:A-0 & B-0 & C-0Pattern: ABC...Non-regular right-angled apeirogonA-35.264 & C-54.736Pattern: AACCAACCHexagon with angles 0, 0, 109.472, 0, 0, 109.472A-90 & B-90Pattern: AABBRhombus with angles 0, 70.529, 0, 70.529B-35.264 & C-54.736Pattern: BBCCBBCCHexagon with angles 0, 0, 109.472, 0, 0, 109.472D-45Pattern: DDDDDDDDOctagon with alternating angles 90 and 120D-45Pattern: DDDDDDDDOctagon with alternating angles 90 and 120D-60Pattern: DDDDDDRight-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-branchA and B triangles identical.One plane tiled by combination of A, B, C, and D.Triangle A: 0,      20.905, 90    ; ABCTriangle B: 0,      20.905, 90    ; BACTriangle C: 0,      37.377, 37.377; DABTriangle D: 31.717, 31.717, 60    ; DDCVertices:A-0 & B-0 & C-0Pattern: ABC...Apeirogon with angle 58.282, not regularA-20.905 & C-37,377 & D-31.317Pattern: AACDDCAACDDCDecagon with angles 0, 0, 69.094, 120, 69.094, 0, 0, 69.094, 120, 69.094A-90 & B-90Pattern: AABBRhombus with angles 0, 41.810, 0, 41.810B-20.905 & C-37.377 & 31.717Pattern: BBCDDCBBCDDCDecagon with angles 0, 0, 69.094, 120, 69.094, 0, 0, 69.094, 120, 69.094D-60Pattern: DDDDDDRegular hexagon with angle 63.434Repeating 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-branchA 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; BACTriangle B: 0a, 0d, 90c; ABCTriangle C: 0a, 0b, 0d ; ABCTriangle D: 0a, 0b, 60c; DDDVertices:A-0 & B-0 & C-0Pattern: ABC...Apeirogon with angle 0A-0 & C-0Pattern: AACC...Apeirogon with angle 0A-90 & B-90Pattern: AABBSquare with angle 0B-0 & C-0Pattern: BBCC...Apeirogon with angle 0D-0Pattern: D...Apeirogon with alternating angles 0 and 120D-0Pattern: D...Apeirogon with alternating angles 0 and 120D-60Pattern: DDDDDDRegular hexagon with angle 0Repeating 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 trianglesA 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; ABCTriangle B: 0, 0,      70.529; ADCTriangle C: 0, 0,      70.529; ADBTriangle D: 0, 54.736, 54.736; DCBVertices:A-0 & B-0 & C-0Pattern: ABC...Apeirogon with angles 0, 125.565, 125.565...A-54.736 & B-70.529 & D-54.736Pattern: AABDDBHexagon with angles 0, 0, 109.472, 0, 0, 109.472A-54.736 & C-70.529 & D-54.736Pattern: AACDDCHexagon with angles 0, 0, 109.472, 0, 0, 109.472B-0 & C-0 & D-0Pattern: 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-edgesTriangles A, B, C, and D all identical.One plane tiled by combination of A, B, C, and D.Triangle A: 0, 0, 0; BCDTriangle B: 0, 0, 0; ACDTriangle C: 0, 0, 0; ABDTriangle D: 0, 0, 0; ABCVertices:A-0 & B-0 & C-0Pattern: ABC...Regular apeirogon with angle 0A-0 & B-0 & D-0Pattern: ABC...Regular apeirogon with angle 0A-0 & C-0 & D-0Pattern: ABC...Regular apeirogon with angle 0B-0 & C-0 & D-0Pattern: ABC...Regular apeirogon with angle 0Repeating 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 ABTriangle AC: 54.736, 65.905, 90; AC AC ADTriangle AD: 48.190, 70.529, 90; AE BD ACTriangle AE: 54.736, 54.736, 90; AD BE AETriangle BC: 52.239, 75.522, 90; AB BC CDTriangle BD: 65.905, 65.905, 90; AD BE BDTriangle BE: 48.190, 70.529, 90; BD AE CETriangle CD: 52.239, 75.522, 90; DE CD BCTriangle CE: 54.736, 65.905, 90; CE CE BETriangle DE: 52.239, 60,     90; DE CD DERepeating 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 ABTriangle AC: 35.264, 65.905, 90; AC AC ADTriangle AD: 48.190, 54.736, 90; AD BD ACTriangle AE: 45,     54.736, 90; BE AE AETriangle BC: 30,     75.522, 90; AB BC CDTriangle BD: 45,     65.905, 90; AD BD BDTriangle BE: 35.264, 70.529, 90; AE BE CETriangle CD: 52.239, 60,     90; CD CD BCTriangle CE: 54.736, 54.736, 90; CE CE BETriangle DE: 45,     60,     90; DE DE DERepeating 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.DemipenteracticA, 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 BCTriangle AC: 65.905, 65.905, 70.529; AD AE ACTriangle AD: 48.190, 54.736, 90    ; AD BD ACTriangle AE: 48.190, 54.736, 90    ; AE BE ACTriangle BC: 75.522, 75.522, 90    ; CD CE ABTriangle BD: 45,     65.905, 90    ; AD BD BDTriangle BE: 45,     65.905, 90    ; AE BE BETriangle CD: 52.239, 60,     90    ; CE CD BCTriangle CE: 52.239, 60,     90    ; CD CE BCTriangle DE: 45,     60,     90    ; DE DE DERepeating 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 ABTriangle AC: 35.264, 54.736, 90; AC AC ADTriangle AD: 35.264, 54.736, 90; AD AD ACTriangle AE: 45,     45,     90; AE AE AETriangle BC: 30,     60,     90; BC BC CDTriangle BD: 45,     45,     90; BD BD BDTriangle BE: 35.264, 54.736, 90; BE BE CETriangle CD: 30,     60,     90; CD CD BCTriangle CE: 35.264, 54.736, 90; CE CE BETriangle DE: 45,     45,     90; DE DE DERepeating unit 1: Triangle ABRepeating unit 2: Rectangle made from AC and AD.Repeating unit 3: Triangle AERepeating unit 4: Rectangle made from BC and CD.Repeating unit 5: Triangle BDRepeating unit 6: Rectangle made from BE and CE.Repeating unit 7: Triangle DEF~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 ABTriangle AC: 45,     45,     90; AC AC ACTriangle AD: 35.264, 54.736, 90; AE BD ADTriangle AE: 35.264, 54.736, 90; BE AD AETriangle BC: 30,     60,     90; BC AB BCTriangle BD: 35.264, 54.736, 90; AD BE BDTriangle BE: 19.471, 70.529, 90; AE BD CETriangle CD: 45,     45,     90; CD CD CDTriangle CE: 35.264, 54.736, 90; CE CE BETriangle DE: 30,     60,     90; DE DE DERepeating 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 ABTriangle AC: 54.736, 54.736, 70.529; AD AE ACTriangle AD: 35.264, 54.736, 90    ; AD AD ACTriangle AE: 35.264, 54.736, 90    ; AE AE ACTriangle BC: 60,     60,     60    ; BC CD CETriangle BD: 45,     45,     90    ; BD BD BDTriangle BE: 45,     45,     90    ; BE BE BETriangle CD: 30,     60,     90    ; CD CD BCTriangle CE: 30,     60,     90    ; CE CE BCTriangle DE: 45,     45,     90    ; DE DE DERepeating 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 BETriangle AC: 45, 45, 90; AC AC ACTriangle AD: 45, 45, 90; AD AD ADTriangle AE: 45, 45, 90; AE AE AETriangle BC: 60, 60, 60; AB BD BETriangle BD: 60, 60, 60; AB BC BETriangle BE: 60, 60, 60; AB BC BDTriangle CD: 45, 45, 90; CD CD CDTriangle CE: 45, 45, 90; CE CE CETriangle DE: 45, 45, 90; DE DE DERepeating 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 33333AB, 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 ABTriangle AC: 48.190, 65.905, 65.905; AC AD CETriangle AD: 48.190, 65.905, 65.905; AD AC BDTriangle AE: 52.239, 52.239, 75.522; AB DE AETriangle BC: 52.239, 52.239, 75.522; AB CD BCTriangle BD: 48.190, 65.905, 65.905; BD AD BETriangle BE: 48.190, 65.905, 65.905; BE BD CETriangle CD: 52.239, 52.239, 75.522; BC DE CDTriangle CE: 48.190, 65.905, 65.905; CE AC BETriangle DE: 52.239, 52.239, 75.522; AE CD DERepeating 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 ABTriangle AC: 20.905, 65.905, 90; AC AC ADTriangle AD: 37.377, 48.190, 90; BD AE ACTriangle AE: 31.717, 54.736, 90; BE AD AETriangle BC: 7.761,  75.522, 90; AB BC CDTriangle BD: 13.283, 65.905, 90; AD BE BDTriangle BE: 10.812, 70.529, 90; AE BD CETriangle CD: 22.239, 52.239, 90; CD CD BCTriangle CE: 20.905, 54.736, 90; CE CE BETriangle DE: 18,     60,     90; DE DE DERepeating 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 ABTriangle AC: 20.905, 54.736, 90; AC AC ADTriangle AD: 35.264, 37.377, 90; AE AD ACTriangle AE: 31.717, 45,     90; AE AD AETriangle BC: 7.761,  60,     90; BC BC CDTriangle BD: 13.283, 45,     90; BD BE BDTriangle BE: 10.812, 54.736, 90; BE BD CETriangle CD: 22.239, 30,     90; CD CD BC Triangle CE: 20.905, 35.264, 90; CE CE BETriangle DE: 18,     45,     90; DE DE DERepeating 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 ABTriangle AC: 20.905, 20.905, 90; AC AC ADTriangle AD: 10.812, 37.377, 90; AE BD ACTriangle AE: 31.717, 31.717, 90; AD BE AETriangle BC: 7.761,  22.239, 90; BC CD BCTriangle BD: 13.283, 13.283, 90; AD BE BDTriangle BE: 10.812, 37.377, 90; AE BD CETriangle CD: 7.761,  22.239, 90; CD BC CDTriangle CE: 20.905, 20.905, 90; CE CE BETriangle DE: 18,     36,     90; DE DE DERepeating 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 3AD 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 ABTriangle AC: 20.905, 20.905, 70.529; AD AE ACTriangle AD: 10.812, 54.736, 90    ; AD BD ACTriangle AE: 10.812, 54.736, 90    ; AE BE ACTriangle BC: 22.239, 22.239, 60    ; CD CE BCTriangle BD: 13.283, 45,     90    ; BD AD BDTriangle BE: 13.283, 45,     90    ; BE AE BETriangle CD: 7.761,  60,     90    ; CE CD BCTriangle CE: 7.761,  60,     90    ; CD CE BCTriangle DE: 36,     45,     90    ; DE DE DERepeating 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 33334AE is 4, AB, BC, CD, and DE are 3AB 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 ABTriangle AC: 35.264, 45,     65.905; AC AD ACTriangle AD: 35.264, 48.190, 54.736; AC BD ADTriangle AE: 45,     45,     45    ; AE AE AETriangle BC: 30,     52.239, 70.529; CD AB BCTriangle BD: 19.471, 65.905, 65.905; BD AD BETriangle BE: 35.264, 48.190, 54.736; CE BD BETriangle CD: 30,     52.239, 70.529; BC DE CDTriangle CE: 35.264, 45,     65.905; CE BE CETriangle DE: 30,     52.239, 60    ; CD DE DERepeating 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 ABTriangle AC: 35.264, 35.264, 90; AC AC ADTriangle AD: 19.471, 54.736, 90; AD BD ACTriangle AE: 35.264, 45,     90; AE BE AETriangle BC: 0,      45,     90; BC BC BCTriangle BD: 0,      35.264, 90; AD BD BDTriangle BE: 0,      54.736, 90; AE BE BETriangle CD: 0,      30,     90; CD CD CDTriangle CE: 0,      45,     90; CE CE CETriangle DE: 0,      60,     90; DE DE DERepeating 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 3Plane 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 BCTriangle AC: 45,     54.736, 65.905; AD AC ACTriangle AD: 35.264, 48.190, 90    ; BD AD ACTriangle AE: 35.264, 45,     90    ; AE BE AETriangle BC: 0,      60,     75.522; AB CD BCTriangle BD: 0,      65.905, 90    ; AD BD BDTriangle BE: 0,      54.736, 90    ; AE BE BETriangle CD: 0,      52.239, 90    ; CD CD BCTriangle CE: 0,      45,     90    ; CE CE CETriangle DE: 0,      60,     90    ; DE DE DERepeating 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 3AD 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 ABTriangle AC: 35.264, 35.264, 70.529; AD AE ACTriangle AD: 19.471, 54.736, 90    ; AD BD ACTriangle AE: 19.471, 54.736, 90    ; AE BD ACTriangle BC: 0,      45,     45    ; BC BC BCTriangle BD: 0,      35.264, 90    ; AD BD BDTriangle BE: 0,      35.264, 90    ; AE BE BETriangle CD: 0,      30,     90    ; CD CD CDTriangle CE: 0,      30,     90    ; CE CE CETriangle DE: 0,      60,     90    ; DE DE DERepeating 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 3Plane 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 ABTriangle AC: 0,      54.736, 54.736; AD AC ACTriangle AD: 35.264, 35.264, 90    ; AD AD ACTriangle AE: 0,      45,     90    ; AE AE AETriangle BC: 0,      0,      60    ; BC CD BCTriangle BD: 0,      45,     90    ; BD BD BDTriangle BE: 0,      0,      90    ; BE BE BETriangle CD: 0,      30,     90    ; CD CD BCTriangle CE: 0,      0,      90    ; CE CE CETriangle DE: 0,      45,     90    ; DE DE DERepeating 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 3334BE is 4, AB, BC, BD are 3AB, 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 ABTriangle AC: 0, 45, 90; AC AC ACTriangle AD: 0, 45, 90; AC AC ACTriangle AE: 0, 0,  90; AE AE AETriangle BC: 0, 0,  60; AB BD BCTriangle BD: 0, 0,  60; AB BC BDTriangle BE: 0, 0,  0 ; BE BE BETriangle CD: 0, 45, 90; CD CD CDTriangle CE: 0, 0,  90; CE CE CETriangle DE: 0, 0,  90; DE DE DERepeating 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 33434AE, CD are 4, AB, BC, DE are 3AB 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 ABTriangle AC: 0,      45,     45    ; AC AC ACTriangle AD: 0,      35.264, 35.264; BD AD ADTriangle AE: 0,      45,     45    ; AE AE AETriangle BC: 30,     30,     60    ; AB BC BCTriangle BD: 19.471, 35.264, 54.736; BD AE BETriangle BE: 19.471, 35.264, 54.736; BE CE BDTriangle CD: 0,      45,     45    ; CD CD CDTriangle CE: 0,      35.264, 35.264; BE CE CETriangle DE: 0,      30,     30    ; DE DE DERepeating 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-branchAB, AC, BD, CD, DE are 3AB 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 BDTriangle AC: 0,      52.239, 90    ; AC AB CDTriangle AD: 0,      65.905, 65.905; AE AD ADTriangle AE: 48.190, 48.190, 70.529; BE CE ADTriangle BC: 0,      45,     90    ; BC BC BCTriangle BD: 0,      60,     75.522; DE AB CDTriangle BE: 45,     54.736, 65.905; AE BE BETriangle CD: 0,      60,     75.522; DE AC BDTriangle CE: 45,     54.736, 65.905; AE CE CETriangle DE: 52.239, 52.239, 60    ; BD CD DERepeating 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-branchDE is 4, AB, AC, BD, CD are 3AB and AC triangles identical.BD and CD triangles identical.BE and CE triangles identical.Plane 1: tiled by combination of AB, AC, BD, and CDPlane 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 BDTriangle AC: 0,      30,     90    ; AC AB CDTriangle AD: 0,      45,     45    ; AD AD ADTriangle AE: 35.264, 35.264, 70.529; BE CE AETriangle BC: 0,      0,      90    ; BC BC BCTriangle BD: 0,      0,      60    ; BD BD CDTriangle BE: 0,      54.736, 54.736; AE BE BETriangle CD: 0,      0,      60    ; CD CD BDTriangle CE: 0,      54.736, 54.736; AE CE CETriangle DE: 0,      45,     45    ; DE DE DERepeating 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 bridgeAB, AD, BC, BE, CD, DE are 3AB, 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 ADTriangle AC: 0, 45, 45; AC AC ACTriangle AD: 0, 0,  60; CD DE ABTriangle AE: 0, 45, 45; AE AE AETriangle BC: 0, 0,  60; AB BE CDTriangle BD: 0, 0,  0 ; BD BD BDTriangle BE: 0, 0,  60; AB BC DETriangle CD: 0, 0,  60; AD DE BCTriangle CE: 0, 45, 45; CE CE CETriangle DE: 0, 0,  60; AD CD BERepeating 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 - 45truncated octahedral prism 1/truncated octahedral prism 2 120 - 60truncated octahedral prism 1/great prismated decachoron 2 108.434949 - 71.565051 - arccos(sqrt[1/10])hexagonal duoprism/truncated octahedral prism 2 135 - 45hexagonal 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 = 180cycle 2: |top1 -c- top2| - 60cycle 3: (gpd1 -hp- top2 -hp- hd -hp- gpd2 -hp- top1 -hp- hd -hp-) - 71.565 + 45 + 63.435 + 71.565 + 45 + 63.435 = 360opposites: 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 - 30truncated octahedral prism/truncated cuboctahedral prism 135 - 45truncated octahedral prism/great prismated tesseract 120 - 60hexagonal-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 - 45cycle 1: |top -to- gpd -to- gpt| - 26.565 + 63.435 = 90cycle 2: |top -c- tcop| - 45cycle 3: |hod -hp- gpd -hp- tcop| - 39.231 + 50.768 = 90cycle 4: |hod -hp- top -hp- gpt| - 30 + 60 = 90cycle 5: |tcop -tco- gpt| - 45cycle 6: |tcop -op- hod -op- gpt| - 35.264 + 54.735 = 90opposites: 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 demipenteractgreat 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 - 45cycle 1: |shd -hp- gpd1 -hp- top -hp- gpd2 -hp- shd| - 39.232 + 50.768 + 50.768 + 39.232 = 180cycle 2: |top -to- ti| - 45cycle 3: |top -c- shd -c- ti| - 35.264 + 54.736 = 90cycle 4: |ti -to- gpd1 -to- gpd2 -to- ti| - 63.435 + 53.130 + 63.435 = 180opposites: gpd1/gpd2, shd/shd, top/top, ti/ti`
Marek14
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### Re: Planar tilings based on Goursat tetrahedra

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. --- rk
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### Re: Planar tilings based on Goursat tetrahedra

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

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

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

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)).
The dream you dream alone is only a dream
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### Re: Planar tilings based on Goursat tetrahedra

I see and how is this matrix obtained?
Marek14
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### Re: Planar tilings based on Goursat tetrahedra

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

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.
Marek14
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### Re: Planar tilings based on Goursat tetrahedra

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}, {0, 0, 0, 0, -Sqrt, 2}}

Inversion got me this matrix:

{{1, 1, 1, 1, 1, 1/Sqrt}, {1, 2, 2, 2, 2, Sqrt}, {1, 2, 3, 3, 3, 3/Sqrt}, {1, 2, 3, 4, 4, 2 Sqrt}, {1, 2, 3, 4, 5, 5/Sqrt}, {1/Sqrt, Sqrt, 3/Sqrt, 2 Sqrt, 5/Sqrt, 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

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

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

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 Marek14
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### Re: Planar tilings based on Goursat tetrahedra

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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