( 2 -1 0 0)
(-1 2 -1 0)
( 0 -1 2 -1)
( 0 0 -1 2)
a ( 8 6 4 2)
b ( 6 12 8 4)
c ( 4 8 12 6)
d ( 2 4 6 8)
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
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.
{3,3,3}
A and D triangles identical.
B and C triangles identical.
One plane tiled by combination of A, B, C, and D.
Triangle A: 54.736, 60, 90; ABA
Triangle B: 54.736, 70.529, 90; ABC
Triangle C: 54.736, 70.529, 90; DCB
Triangle D: 54.736, 60, 90; DCD
Vertices:
A-54.736 & B-70.529 & C-54.736
Pattern: AABCCB
Triangle with angles 120, 125.265, 125.265
A-60
Pattern: AAAAAA
Triangle with angles 109.472, 109.472, 109.472
A-90 & B-90
Pattern: AABB
Quadrangle with angles 109.472, 125.265, 120, 125.265
B-54.736 & C-70.529 & D-54.736
Pattern: BBCDDC
Triangle with angles 120, 125.265, 125.265
C-90 & D-90
Pattern: CCDD
Quadrangle with angles 109.472, 125.265, 120, 125.265
D-60
Pattern: DDDDDD
Equilateral triangle with angle 109.472
Repeating unit: Digonal strip of angle 60. Composed of 1 A, 1 B, 1 C and 1 D.
{3,3,4}
Plane 1: tiled by combination of A, B, and C.
Plane 2: tiled solely by D.
Triangle A: 45, 54.736, 90; BAA
Triangle B: 35.264, 70.529, 90; ABC
Triangle C: 54.736, 54.736, 90; CCB - double of triangle A. Note that each of its 54.736 angles belongs to a different type of vertex.
Triangle D: 45, 60, 90; DDD
Vertices:
A-45
Pattern: AAAAAAAA
Square with angle 109.472
A-54.736 & B-70.529 & C-54.736
Pattern: AABCCB
Triangle with angles 90, 90, 109.472
A-90 & B-90
Pattern: AABB
Quadrangle with angles 90, 125.265, 109.472, 125.265
B-35.264 & C-54.736
Pattern: BBCCBBCC
Square with angle 125.265
C-90
Pattern: CCCC
Square with angle 109.472
D-45
Pattern: DDDDDDDD
Square with angle 120
D-60
Pattern: DDDDDD
Equilateral triangle with angle 90
D-90
Pattern: DDDD
Rhombus with angles 90, 120, 90, 120
Repeating unit 1: Triangle with angles 45, 90, 90. Composed of 1 A, 1 B, 1 C.
Repeating unit 2: Triangle D.
{3,3,5}
One plane tiled by combination of A, B, C, and D.
Triangle A: 36, 54.736, 90; BAA
Triangle B: 20.905, 70.529, 90; ABC
Triangle C: 37.377, 54.736, 90; CDB
Triangle D: 31.717, 60, 90; DCD
Vertices:
A-36
Pattern: AAAAAAAAAA
Regular pentagon with angle 109.472
A-54.736 & B-70.529 & C-54.736
Pattern: AABCCB
Triangle with angles 58.282, 58.282, 72
A-90 & B-90
Pattern: AABB
Quadrangle with angles 41.810, 125.265, 72, 125.265
B-20.905 & C-37.377 & D-31.717
Pattern: BBCDDCBBCDDC
Hexagon with angles 120, 125.265, 125.265, 120, 125.265, 125.265
C-90 & D-90
Pattern: CCDD
Quadrangle with angles 69.094, 109.472, 69.094, 120
D-60
Pattern: DDDDDD
Equilateral triangle with angle 63.434
Repeating unit: Triangle with angles 36, 60, 90. Composed of 1 A, 1 B, 1 C, 1 D.
{3,4,3}
A and D triangles identical.
B and C triangles identical.
Plane 1: tiled by combination of A and B
Plane 2: tiled by combination of C and D
Triangle A: 35.264, 60, 90; ABA
Triangle B: 45, 54.736, 90; ABB
Triangle C: 45, 54.736, 90; DCC
Triangle D: 35.264, 60, 90; DCD
Vertices:
A-35.264 & B-54.736
Pattern: AABBAABB
Rhombus with angles 90, 120, 90, 120
A-60
Pattern: AAAAAA
Equilateral triangle with angle 70.528
A-90 & B-90
Pattern: AABB
Quadrangle with angles 90, 90, 90, 120
B-45
Pattern: BBBBBBBB
Square with angle 109.472
C-45
Pattern: CCCCCCCC
Square with angle 109.472
C-54.736 & D-35.264
Pattern: CCDDCCDD
Rhombus with angles 90, 120, 90, 120
C-90 & D-90
Pattern: CCDD
Quadrangle with angles 90, 90, 90, 120
D-60
Pattern: DDDDDD
Equilateral triangle with angle 70.529
Repeating unit 1: Triangle with angles 45, 60, 90. Composed of 1 A, 1 B.
Repeating unit 2: Triangle with angles 45, 60, 90. Composed of 1 C, 1 D.
Branched 333 (demitesseractic)
A, C, and D triangles identical.
One plane tiled by combination of A, B, C, and D.
Triangle A: 54.736, 54.736, 90 ; AAB
Triangle B: 70.529, 70.529, 70.529; ABC
Triangle C: 54.736, 54.736, 90 ; CCB
Triangle D: 54.736, 54.736, 90 ; DDB
Vertices:
A-54.736 & B-70.529 & C-54.736
Pattern: AABCCB
Quadrangle with angles 125.265, 125.265, 125.265, 125.265 (not square, apparently)
A-54.736 & B-70.529 & D-54.736
Pattern: AABDDB
Quadrangle with angles 125.265, 125.265, 125.265, 125.265 (not square, apparently)
A-90
Pattern: AAAA
Square with angle 109.472
B-70.529 & C-54.736 & D-54.736
Pattern: BCCBDD
Quadrangle with angles 125.265, 125.265, 125.265, 125.265 (not square, apparently)
C-90
Pattern: CCCC
Square with angle 109.472
D-90
Pattern: DDDD
Square with angle 109.472
Repeating unit: Equilateral triangle of angle 90. Composed of 1 A, 1 B, 1 C and 1 D.
{4,3,4}
A and D triangles identical.
B and C triangles identical.
Plane 1: tiled solely by A
Plane 2: tiled by combination of B and C
Plane 3: tiled solely by D
Triangle A: 45, 45, 90; AAA
Triangle B: 35.264, 54.736, 90; BBC
Triangle C: 35.264, 54.736, 90; CCB
Triangle D: 45, 45, 90; DDD
Vertices:
A-45
Pattern: AAAAAAAA
Square with angle 90
A-45
Pattern: AAAAAAAA
Square with angle 90
(The two 45 angles at A look identical within the plane, but differ in how other tetrahedron faces are connected to them.)
A-90
Pattern: AAAA
Square with angle 90
B-35.264 & C-54.736
Pattern: BBCC
Rectangle with angle 90
B-54.736 & C-35.264
Pattern: BBCC
Rectangle with angle 90
B-90
Pattern: BBBB
Rhombus with angles 70.529, 109.472, 70.529, 109.472
C-90
Pattern: CCCC
Rhombus with angles 70.529, 109.472, 70.529, 109.472
D-45
Pattern: DDDDDDDD
Square with angle 90
D-45
Pattern: DDDDDDDD
Square with angle 90
D-90
Pattern: DDDD
Square with angle 90
Repeating unit 1: Triangle A.
Repeating unit 2: Rectangle of angle 90. Composed of 1 B, 1 C.
Repeating unit 3: Triangle D.
Branched 334 (tetrahedral/octahedral honeycomb)
A and C triangles identical.
Plane 1: tiled by combination of A, B, and C
Plane 2: tiled solely by D
Triangle A: 35.264, 54.736, 90 ; AAB
Triangle B: 54.736, 54.736, 70.529; ACB (double of A or C triangle)
Triangle C: 35.264, 54.736, 90 ; CCB
Triangle D: 45, 45, 90 ; DDD
Vertices:
A-35.264 & B-54.736
Pattern: AABBAABB
Hexagon with angles 109.472, 125.565, 125.565, 109.472, 125.565, 125.565
A-54.736 & B-70.529 & C-54.736
Pattern: AABCCB
Square with angle 90
A-90
Pattern: AAAA
Rhombus with angles 70.529, 109.472, 70.529, 109.472
B-54.736 & C-35.264
Pattern: BBCCBBCC
Hexagon with angles 109.472, 125.565, 125.565, 109.472, 125.565, 125.565
C-90
Pattern: CCCC
Rhombus with angles 70.529, 109.472, 70.529, 109.472
D-45
Pattern: DDDDDDDD
Square with angle 90
D-45
Pattern: DDDDDDDD
Square with angle 90
D-90
Pattern: DDDD
Square with angle 90
Repeating unit 1: Rectangle of angle 90. Composed of 1 A, 1 B, 1 C.
Repeating unit 2: Triangle D.
Cyclical 3333
Triangles A, B, C, and D all identical.
One plane tiled by combination of A, B, C, and D.
Triangle A: 54.736, 54.736, 70.529; BDA
Triangle B: 54.736, 54.736, 70.529; ACB
Triangle C: 54.736, 54.736, 70.529; BDC
Triangle D: 54.736, 54.736, 70.529; ACD
Vertices:
A-54.736 & B-70.529 & C-54.736
Pattern: AABCCB
Hexagon with angles 109.472, 125.565, 125.565, 109.472, 125.565, 125.565
A-54.736 & C-54.736 & D-70.529
Pattern: AADCCD
Hexagon with angles 109.472, 125.565, 125.565, 109.472, 125.565, 125.565
A-70.529 & B-54.736 & D-54.736
Pattern: ABBADD
Hexagon with angles 109.472, 125.565, 125.565, 109.472, 125.565, 125.565
B-54.736 & C-70.529 & D-54.736
Pattern: BBCDDC
Hexagon with angles 109.472, 125.565, 125.565, 109.472, 125.565, 125.565
Repeating unit: An infinite strip formed by repeating triangles A, B, C, D.
{4,3,5}
Plane 1: tiled solely by A
Plane 2: tiled by combination of B, C, and D
Triangle A: 36, 45, 90; AAA
Triangle B: 20.905, 54.736, 90; BBC
Triangle C: 35.264, 37.377, 90; DCB
Triangle D: 31.717, 45, 90; DCD
Vertices:
A-36
Pattern: AAAAAAAAAA
Regular pentagon with angle 90
A-45
Pattern: AAAAAAAA
Square with angle 72
A-90
Pattern: AAAA
Rhombus with angles 72, 90, 72, 90
B-20.905 & C-37.377 & D-31.717
Pattern: BBCDDCBBCDDC
Right-angled hexagon, not regular
B-54.736 & C-35.264
Pattern: BBCCBBCC
Rectangle with angle 58.282
B-90
Pattern: BBBB
Rhombus with angles 41.810, 109.472, 41.810, 109.472
C-90 & D-90
Pattern: CCDD
Quadrangle with angles 69.095, 70.529, 90, 69.095
D-45
Pattern: DDDDDDDD
Square with angle 63.434
Repeating unit: Quadrangle with angles 45, 90, 90, 90. Composed of 1 B, 1 C, 1 D.
{5,3,5}
A and D triangles identical.
B and C triangles identical.
One plane tiled by combination of A, B, C, and D.
Triangle A: 31.717, 36, 90; ABA
Triangle B: 20.905, 37.377, 90; ABC
Triangle C: 20.905, 37.377, 90; DCB
Triangle D: 31.717, 36, 90; DCC
Vertices:
A-31.717 & B-37.377 & C-20.905
Pattern: AABCCBAABCCB
Hexagon with angles 58.282, 58.282, 72, 58.282, 58.282, 72
A-36
Pattern: AAAAAAAAAA
Regular pentagon with angle 63.434
A-90 & B-90
Pattern: AABB
Quadrangle with angles 41.810, 69.095, 72, 69.095
B-20.905 & C-37.377 & D-31.717
Pattern: BBCDDCBBCDDC
Hexagon with angles 58.282, 58.282, 72, 58.282, 58.282, 72
C-90 & D-90
Pattern: CCDD
Quadrangle with angles 41.810, 69.095, 72, 69.095
D-36
Pattern: DDDDDDDDDD
Regular pentagon with angle 63.434
Repeating unit: Quadrangle with angles 36, 90, 36, 90. Composed of 1 A, 1 B, 1 C, 1 D.
{3,5,3}
A and D triangles identical.
B and C triangles identical.
One plane tiled by combination of A, B, C, and D.
Triangle A: 20.905, 60, 90; ABA
Triangle B: 31.717, 37.377, 90; ABC
Triangle C: 31.717, 37.377, 90; DCB
Triangle D: 20.905, 60, 90; DCD
Vertices:
A-20.905 & B-37.377 & C-31.717
Pattern: AABCCBAABCCB
Hexagon with angles 69.095, 69.095, 120, 69.095, 69.095, 120
A-60
Pattern: AAAAAA
Equilateral triangle with angle 41.810
A-90 & B-90
Pattern: AABB
Quadrangle with angles 58.282, 63.434, 58.282 and 120
B-31.717 & C-37.377 & D-20.905
Pattern: BBCDDCBBCDDC
Hexagon with angles 69.095, 69.095, 120, 69.095, 69.095, 120
C-90 & D-90
Pattern: CCDD
Quadrangle with angles 58.282, 63.434, 58.282 and 120
D-60
Pattern: AAAAAA
Equilateral triangle with angle 41.810
Repeating unit: Quadrangle with angles 60, 90, 60, 90. Composed of 1 A, 1 B, 1 C, 1 D.
Branched 335 (tetrahedral/icosahedral honeycomb)
A and C triangles identical.
One plane tiled by combination of A, B, C, and D.
Triangle A: 20.905, 54.736, 90; AAB
Triangle B: 37.377, 37.377, 70.529; ACD
Triangle C: 20.905, 54.736, 90; CCB
Triangle D: 31.717, 31.717, 90; DDB
Vertices:
A-20.905 & B-37.377 & D-31.717
Pattern: AABDDBAABDDB
Octagon with alternating angles of 69.095 and 125.565
A-54.736 & B-70.529 & C-54.736
Pattern: AABCCB
Rectangle with angle 58.282
A-90
Pattern: AAAA
Rhombus with angles 41.810, 109.472, 41.810, 109.472
B-37.377 & C-20.905 & D-31.717
Pattern: BCCBDDBCCBDD
Octagon with alternating angles of 69.095 and 125.565
C-90
Pattern: CCCC
Rhombus with angles 41.810, 109.472, 41.810, 109.472
D-90
Pattern: DDDD
Square with angle 63.434
Repeating unit: Right-angled pentagon; not regular. Composed of 1 A, 1 B, 1 C, 1 D.
Cyclical 3334
A and D triangles identical.
B and C triangles identical.
One plane tiled by combination of A, B, C, and D.
Triangle A: 45, 54.736, 54.736; BAA
Triangle B: 35.264, 54.736, 70.529; CAB
Triangle C: 35.264, 54.736, 70.529; BDC
Triangle D: 45, 54.736, 54.736; CDD
Vertices:
A-45
Pattern: AAAAAAAA
Regular octagon with angle 109.472
A-54.736 & B-35.264
Pattern: AABBAABB
Octagon with angles 90, 125.565, 109.472, 125.565, 90, 125.565, 109.472, 125.565
A-54.736 & B-70.529 & C-54.736
Pattern: AABCCB
Hexagon with angles 70.529, 125.565, 90, 90, 90, 125.565
B-54.736 & C-70.529 & D-54.736
Pattern: BBCDDC
Hexagon with angles 70.529, 125.565, 90, 90, 90, 125.565
C-35.264 & D-54.736
Pattern: CCDDCCDD
Octagon with angles 90, 125.565, 109.472, 125.565, 90, 125.565, 109.472, 125.565
D-45
Pattern: DDDDDDDD
Regular octagon with angle 109.472
Repeating unit: Quadrangle with angles 45, 90, 45, 90. Composed of 1 A, 1 B, 1 C, 1 D.
Cyclical 3335
A and D triangles identical.
B and C triangles identical.
One plane tiled by combination of A, B, C, and D.
Triangle A: 31.717, 37.377, 54.736; BAD
Triangle B: 20.905, 54.736, 70.529; CAB
Triangle C: 20.905, 54.736, 70.529; BDC
Triangle D: 31.717, 37.377, 54.736; CDA
Vertices:
A-31.717 & C-20.905 & D-37.377
Pattern: AADCCDAADCCD
Dodecagon with angles 69.095, 109.472, 69.095, 125.565, 109.472, 125.565, 69.095, 109.472, 69.095, 125.565, 109.472, 125.565
A-37.377 & B-20.905 & D-31.717
Pattern: ABBADDABBADD
Dodecagon with angles 69.095, 109.472, 69.095, 125.565, 109.472, 125.565, 69.095, 109.472, 69.095, 125.565, 109.472, 125.565
A-54.736 & B-70.529 & C-54.736
Pattern: AABCCD
Hexagon with angles 41.810, 125.565, 58.282, 63.434, 58.282, 125.565
B-54.736 & C-70.529 & D-54.736
Pattern: BBCDDC
Hexagon with angles 41.810, 125.565, 58.282, 63.434, 58.282, 125.565
Repeating unit: An infinite strip formed by repeating triangles A, B, C, D. Its two edges form right-angled pseudogons.
Cyclical 3434
Triangles A, B, C, and D all identical.
Plane 1: tiled by combination of A and B
Plane 2: tiled by combination of C and D
Triangle A: 35.264, 45, 54.736; ABA
Triangle B: 35.264, 45, 54.736; BAB
Triangle C: 35.264, 45, 54.736; CDC
Triangle D: 35.264, 45, 54.736; DCD
Vertices:
A-35.264 & B-54.736
Pattern: AABBAABB
Right-angled octagon, not regular
A-45
Pattern: AAAAAAAA
Octagon with alternating angles 70.529 and 125.565
A-54.736 & B-35.264
Pattern: AABBAABB
Right-angled octagon, not regular
B-45
Pattern: BBBBBBBB
Octagon with alternating angles 70.529 and 125.565
C-35.264 & D-54.736
Pattern: CCDDCCDD
Right-angled octagon, not regular
C-45
Pattern: CCCCCCCC
Octagon with alternating angles 70.529 and 125.565
C-54.736 & D-35.264
Pattern: CCDDCCDD
Right-angled octagon, not regular
D-45
Pattern: DDDDDDDD
Octagon with alternating angles 70.529 and 125.565
Repeating unit 1: Quadrangle with angles 45, 90, 45, 90. Composed of 1 A, 1 B.
Repeating unit 2: Quadrangle with angles 45, 90, 45, 90. Composed of 1 C, 1 D.
Cyclical 3435
A and D triangles identical.
B and C triangles identical.
One plane tiled by combination of A, B, C, and D.
Triangle A: 31.717, 35.264, 37.377; BDA
Triangle B: 20.905, 45, 54.736; BAB
Triangle C: 20.905, 45, 54.736; CDC
Triangle D: 31.717, 35.264, 37.377; CAD
Vertices:
A-31.717 & C-20.905 & D-37.377
Pattern: AADCCDAADCCD
Dodecagon with angles 69.095, 70.529, 69.095, 90, 90, 90, 69.095, 70.529, 69.095, 90, 90, 90
A-35.264 & B-54.736
Pattern: AABBAABB
Octagon with angles 58.282, 63.434, 58.282, 90, 58.282, 63.434, 58.282, 90
A-37.377 & B-20,905 & D-31.717
Pattern: ABBADDABBADD
Dodecagon with angles 69.095, 70.529, 69.095, 90, 90, 90, 69.095, 70.529, 69.095, 90, 90, 90
B-45
Pattern: BBBBBBBB
Octagon with alternating angles 41.810 and 109.472
C-45
Pattern: CCCCCCCC
Octagon with alternating angles 41.810 and 109.472
C-54.736 & D-35.264
Pattern: CCDDCCDD
Octagon with angles 58.282, 63.434, 58.282, 90, 58.282, 63.434, 58.282, 90
Repeating unit: Hexagon with angles 45, 90, 90, 45, 90, 90. Composed of 1 A, 1 B, 1 C, 1 D.
Cyclical 3535
Triangles A, B, C, and D all identical.
One plane tiled by combination of A, B, C, and D.
Triangle A: 20.905, 31.717, 37.377; DBA
Triangle B: 20.905, 31.717, 37.377; CAB
Triangle C: 20.905, 31.717, 37.377; BDC
Triangle D: 20.905, 31.717, 37.377; ACD
Vertices:
A-20.905 & C-31.717 & D-37.377
Pattern: AADCCDAADCCD
Dodecagon with angles 41.810, 58.282, 69.095, 63.434, 69.095, 58.282, 41.810, 58.282, 69.095, 63.434, 69.095, 58.282
A-31.717 & B-37.377 & C-20.905
Pattern: AABCCBAABCCB
Dodecagon with angles 41.810, 58.282, 69.095, 63.434, 69.095, 58.282, 41.810, 58.282, 69.095, 63.434, 69.095, 58.282
A-37.377 & B-20.905 & D-31.717
Pattern: ABBADDABBADD
Dodecagon with angles 41.810, 58.282, 69.095, 63.434, 69.095, 58.282, 41.810, 58.282, 69.095, 63.434, 69.095, 58.282
B-31.717 & C-37.377 & D-20.905
Pattern: BBCDDCBBCDDC
Dodecagon with angles 41.810, 58.282, 69.095, 63.434, 69.095, 58.282, 41.810, 58.282, 69.095, 63.434, 69.095, 58.282
Repeating unit: An infinite strip formed by repeating triangles A, B, C, D. Its two edges form right-angled pseudogons.
{3,3,6}
Plane 1: tiled by combination of A, B, and C.
Plane 2: tiled solely by D.
Triangle A: 30, 54.736, 90; BAA
Triangle B: 0, 70.529, 90; ABC
Triangle C: 0, 54.736, 90; CCB
Triangle D: 0, 60, 90; DDD
Vertices:
A-30
Pattern: AAAAAAAAAAAA
Regular hexagon with angle 109.472
A-54.736 & B-70.529 & C-54.736
Pattern: AABCCB
Triangle with angles 0, 0, 60
A-90 & B-90
Pattern: AABB
Quadrangle with angles 0, 125.565, 60, 125.565
B-0 & C-0
Pattern: BBCC...
Apeirogon with angle 125.565, not regular
C-90
Pattern: CCCC
Rhombus with angles 0, 109.472, 0, 109.472
D-0
Pattern: D...
Regular apeirogon with angle 120
D-60
Pattern: DDDDDD
Equilateral triangle with angle 0
D-90
Pattern: DDDD
Rhombus with angles 0, 120, 0, 120
Repeating unit 1: Triangle with angles 0, 30, 90. Composed of 1 A, 1 B, 1 C.
Repeating unit 2: Triangle D.
{4,3,6}
Plane 1: tiled solely by A.
Plane 2: tiled by combination of B and C.
Plane 3: tiled solely by D.
Triangle A: 30, 45, 90; AAA
Triangle B: 0, 54.736, 90; BBC
Triangle C: 0, 35.264, 90; CCB
Triangle D: 0, 45, 90; DDD
Vertices:
A-30
Pattern: AAAAAAAAAAAA
Right-angled dodecagon
A-45
Pattern: AAAAAAAA
Regular octagon with angle 60
A-90
Pattern: AAAA
Rhombus with angles 60, 90, 60, 90
B-0
Pattern: B...
Regular apeirogon with angle 109.472
B-54.736 & C-35.264
Pattern: BBCCBBCC
Rectangle with angle 0
B-90 & C-90
Pattern: BBCC
Quadrangle with angles 0, 90, 0, 90
C-0
Pattern: C...
Regular apeirogon with angle 70.529
D-0
Pattern: D...
Right-angled apeirogon
D-45
Pattern: DDDDDDDD
Square with angle 0
D-90
Pattern: DDDD
Rhombus with angles 0, 90, 0, 90
Repeating unit 1: Triangle A.
Repeating unit 2: Triangle with angles 0, 0, 90. Composed of 1 B and 1 C.
Repeating unit 3: Triangle D.
{5,3,6}
Plane 1: tiled by combination of A, B, and C.
Plane 2: tiled solely by D.
Triangle A: 30, 31.717, 90; BAA
Triangle B: 0, 37.377, 90; ABC
Triangle C: 0, 20.905, 90; CCB
Triangle D: 0, 36, 90; DDD
Vertices:
A-30
Pattern: AAAAAAAA
Regular hexagon with angle 63.434
A-31.717 & B-37.377 & C-20.905
Pattern: AABCCBAABCCB
Hexagon with angles 0, 0, 60, 0, 0, 60
A-90 & B-90
Pattern: AABB
Quadrangle with angles 0, 69.095, 60, 69.095
B-0 & C-0
Pattern: BBCC...
Apeirogon with angle 58.282, not regular
C-90
Pattern: CCCC
Rhombus with angles 0, 41.810, 0, 41.810
D-0
Pattern: D...
Regular apeirogon with angle 72
D-36
Pattern: DDDDDDDDDD
Regular pentagon with angle 0
D-90
Pattern: DDDD
Rhombus with angles 0, 72, 0, 72
Repeating unit 1: Quadrangle with angles 0, 30, 90, 90. Composed of 1 A, 1 B, 1 C.
Repeating unit 2: Triangle D.
{6,3,6}
A and D triangles identical.
B and C triangles identical.
Plane 1: tiled solely by A
Plane 2: tiled by combination of B and C
Plane 3: tiled solely by D
Triangle A: 0, 30, 90; AAA
Triangle B: 0, 0, 90; BBC
Triangle C: 0, 0, 90; CCB
Triangle D: 0, 30, 90; DDD
Vertices:
A-0
Pattern: A...
Regular apeirogon with angle 60
A-30
Pattern: AAAAAAAAAAAA
Regular hexagon with angle 0
A-90
Pattern: AAAA
Rhombus with angles 0, 60, 0, 60
B-0 & C-0
Pattern: BBCC...
Regular apeirogon with angle 0
B-0 & C-0
Pattern: BBCC...
Regular apeirogon with angle 0
B-90
Pattern: BBBB
Square with angle 0
C-90
Pattern: CCCC
Square with angle 0
D-0
Pattern: D...
Regular apeirogon with angle 60
D-30
Pattern: DDDDDDDDDDDD
Regular hexagon with angle 0
D-90
Pattern: DDDD
Rhombus with angles 0, 60, 0, 60
Repeating unit 1: Triangle A.
Repeating unit 2: Rhombus of angles 0, 90, 0, 90. Composed of 1 B, 1 C.
Repeating unit 3: Triangle D.
{3,4,4}
Plane 1: tiled by combination of A and B.
Plane 2: tiled solely by C.
Plane 3: tiled solely by D.
Triangle A: 35.264, 45, 90; ABA
Triangle B: 0, 54.736, 90; ABB
Triangle C: 0, 45, 90; CCC
Triangle D: 0, 60, 90; CCC
Vertices:
A-35.264 & B-54.736
Rhombus with angles 0, 90, 0, 90
A-45
Pattern: AAAAAAAA
Square with angle 70.529
A-90 & B-90
Pattern: AABB
Quadrangle with angles 0, 90, 90, 90
B-0
Pattern: B...
Regular apeirogon with angle 109.472
C-0
Pattern: C...
Right-angled apeirogon
C-45
Pattern: CCCCCCCC
Square with angle 0
C-90
Pattern: CCCC
Rhombus with angles 0, 90, 0, 90
D-0
Pattern: D...
Regular apeirogon with angle 120
D-60
Pattern: DDDDDD
Equilateral triangle with angle 0
D-90
Pattern: DDDD
Rhombus with angles 0, 120, 0, 120
Repeating unit 1: Triangle with angles 0, 45, 90. Composed of 1 A and 1 B.
Repeating unit 2: Triangle C.
Repeating unit 3: Triangle D.
{4,4,4}
A and D triangles identical.
B and C triangles identical.
Plane 1: tiled solely by A.
Plane 2: tiled solely by B.
Plane 3: tiled solely by C.
Plane 4: tiled solely by D.
Triangle A: 0, 45, 90; AAA
Triangle B: 0, 0, 90; BBB
Triangle C: 0, 0, 90; CCC
Triangle D: 0, 45, 90; DDD
Vertices:
A-0
Pattern: A...
Right-angled apeirogon
A-45
Pattern: AAAAAAAA
Square with angle 0
A-90
Pattern: AAAA
Rhombus with angles 0, 90, 0, 90
B-0
Pattern: B...
Regular apeirogon with angle 0
B-0
Pattern: B...
Regular apeirogon with angle 0
B-90
Pattern: BBBB
Square with angle 0
C-0
Pattern: C...
Regular apeirogon with angle 0
C-0
Pattern: C...
Regular apeirogon with angle 0
C-90
Pattern: CCCC
Square with angle 0
D-0
Pattern: D...
Right-angled apeirogon
D-45
Pattern: DDDDDDDD
Square with angle 0
D-90
Pattern: DDDD
Rhombus with angles 0, 90, 0, 90
Repeating unit 1: Triangle A.
Repeating unit 2: Triangle B.
Repeating unit 3: Triangle C.
Repeating unit 4: Triangle D.
{3,6,3}
A and D triangles identical.
B and C triangles identical.
Plane 1: tiled by combination of A and B
Plane 2: tiled by combination of C and D
Triangle A: 0, 60, 90; ABA
Triangle B: 0, 0, 90; ABB
Triangle C: 0, 0, 90; CDC
Triangle D: 0, 60, 90; DCD
Vertices:
A-0 & B-0
Pattern: AABB...
Apeirogon with alternating angles 0 and 120
A-60
Pattern: AAAAAA
Equilateral triangle with angle 0
A-90 & B-90
Pattern: AABB
Quadrangle with angles 0, 0, 0, 120
B-0
Pattern: B...
Regular apeirogon with angle 0
C-0
Pattern: C...
Regular apeirogon with angle 0
C-0 & D-0
Pattern: CCDD...
Apeirogon with alternating angles 0 and 120
C-90 & D-90
Pattern: CCDD
Quadrangle with angles 0, 0, 0, 120
D-60
Pattern: DDDDDD
Equilateral triangle with angle 0
Repeating unit 1: Triangle with angles 0, 0, 60. Composed of 1 A, 1 B.
Repeating unit 2: Triangle with angles 0, 0, 60. Composed of 1 C, 1 D.
Branched 336
A and C triangles identical.
Plane 1: tiled by combination of A, B, and C
Plane 2: tiled solely by D
Triangle A: 0, 54.736, 90 ; AAB
Triangle B: 0, 0, 70.529; ACB
Triangle C: 0, 54.736, 90 ; CCB
Triangle D: 0, 0, 90 ; DDD
Vertices:
A-0 & B-0
Pattern: AABB...
Apeirogon with alternating angles 0 and 109.472
A-54.736 & B-70.529 & C-54.736
Pattern: AABCCB
Hexagon with all angles 0, not regular
A-90
Pattern: AAAA
Rhombus with angles 0, 109.472, 0, 109.472
B-0 & C-0
Pattern: CCDD...
Apeirogon with alternating angles 0 and 109.472
C-90
Pattern: CCCC
Rhombus with angles 0, 109.472, 0, 109.472
D-0
Pattern: D...
Regular apeirogon with angle 0
D-0
Pattern: D...
Regular apeirogon with angle 0
D-90
Pattern: DDDD
Square with angle 0
Repeating unit 1: Quadrangle with angles 0, 0, 90, 90. Composed of 1 A, 1 B, 1 C.
Repeating unit 2: Triangle D.
Branched 344
C and D triangles identical.
Plane 1: tiled by combination of A and B
Plane 2: tiled solely by C
Plane 3: tiled solely by D
Triangle A: 35.264, 35.264, 90 ;
Triangle B: 0, 54.736, 54.736;
Triangle C: 0, 45, 90 ;
Triangle D: 0, 45, 90 ;
Vertices:
A-35.264 & B-54.736
Pattern: AABB
Hexagon with angles 0, 90, 90, 0, 90, 90
A-35.264 & B-54.736
Pattern: AABB
Hexagon with angles 0, 90, 90, 0, 90, 90
A-90
Pattern: AAAA
Square with angle 70.529
B-0
Pattern: B...
Regular apeirogon with angle 109.472
C-0
Pattern: C...
Right-angled apeirogon
C-45
Pattern: CCCCCCCC
Square with angle 0
C-90
Pattern: CCCC
Rhombus with angles 0, 90, 0, 90
D-0
Pattern: D...
Right-angled apeirogon
D-45
Pattern: DDDDDDDD
Square with angle 0
D-90
Pattern: DDDD
Rhombus with angles 0, 90, 0, 90
Repeating unit 1: Quadrangle with angles 0, 90, 90, 90. Composed of 1 A and 1 B.
Repeating unit 2: Triangle C.
Repeating unit 3: Triangle D.
Branched 444
A, C and D triangles identical.
Plane 1: tiled solely by A
Plane 2: tiled solely by B
Plane 3: tiled solely by C
Plane 4: tiled solely by D
Triangle A: 0, 0, 90; AAA
Triangle B: 0, 0, 0 ; BBB
Triangle C: 0, 0, 90; CCC
Triangle D: 0, 0, 90; DDD
Vertices:
A-0
Pattern: A...
Regular apeirogon with angle 0
A-0
Pattern: A...
Regular apeirogon with angle 0
A-90
Pattern: AAAA
Square with angle 0
B-0
Pattern: B...
Regular apeirogon with angle 0
B-0
Pattern: B...
Regular apeirogon with angle 0
B-0
Pattern: B...
Regular apeirogon with angle 0
C-0
Pattern: C...
Regular apeirogon with angle 0
C-0
Pattern: C...
Regular apeirogon with angle 0
C-90
Pattern: CCCC
Square with angle 0
D-0
Pattern: D...
Regular apeirogon with angle 0
D-0
Pattern: D...
Regular apeirogon with angle 0
D-90
Pattern: CCCC
Square with angle 0
Repeating unit 1: Triangle A.
Repeating unit 2: Triangle B.
Repeating unit 3: Triangle C.
Repeating unit 4: Triangle D.
Cyclical 3336
A and D triangles identical.
B and C triangles identical.
One plane tiled by combination of A, B, C, and D.
Triangle A: 0, 0, 54.736; BAA
Triangle B: 0, 54.736, 70.529; CAB
Triangle C: 0, 54.736, 70.529; BDC
Triangle D: 0, 0, 54.736; DCD
Vertices:
A-0
Pattern: A...
Regular apeirogon with angle 0
A-0 & B-0
Pattern: AABB...
Apeirogon with angles 0, 125.565, 109.472, 125.565, 0...
A-54.736 & B-70.529 & C-54.736
Pattern: AABCCB
Hexagon with angles 0, 0, 0, 125.565, 0, 125.565
B-54.736 & C-70.529 & D-54.736
Pattern: AABCCB
Hexagon with angles 0, 0, 0, 125.565, 0, 125.565
C-0 & D-0
Pattern: CCDD...
Apeirogon with angles 0, 125.565, 109.472, 125.565, 0...
D-0
Pattern: D...
Regular apeirogon with angle 0
Repeating unit: Rectangle with angle 0. Composed of 1 A, 1 B, 1 C, 1 D.
Cyclical 3436
A and D triangles identical.
B and C triangles identical.
Plane 1: tiled by combination of A and B
Plane 2: tiled by combination of C and D
Triangle A: 0, 0, 35.264; BAA
Triangle B: 0, 45, 54.736; ABA
Triangle C: 0, 45, 54.736; CDC
Triangle D: 0, 0, 35.264; DCD
Vertices:
A-0
Pattern: A...
Apeirogon with alternating angles 0 and 70.529
A-0 & B-0
Pattern: AABB...
Apeirogon with angles 0, 90, 90, 90...
A-35.264 & B-54.736
Pattern: AABBAABB
Octagon with angles 0, 0, 0, 90, 0, 0, 0, 90
B-45
Pattern: BBBBBBBB
Octagon with alternating angles 0 and 109.472
C-0 & D-0
Pattern: CCDD...
Apeirogon with angles 0, 90, 90, 90...
C-45
Pattern: CCCCCCCC
Octagon with alternating angles 0 and 109.472
C-54.736 & D-35.264
Pattern: CCDDCCDD
Octagon with angles 0, 0, 0, 90, 0, 0, 0, 90
D-0
Pattern: D...
Apeirogon with alternating angles 0 and 70.529
Repeating unit 1: Quadrangle with angles 0, 0, 45, 90. Composed of 1 A and 1 B.
Repeating unit 2: Quadrangle with angles 0, 0, 45, 90. Composed of 1 A and 1 B.
Cyclical 3536
A and D triangles identical.
B and C triangles identical.
One plane tiled by combination of A, B, C, and D.
Triangle A: 0, 0, 20.905; BAA
Triangle B: 0, 31.717, 37.377; CAB
Triangle C: 0, 31.717, 37.377; BDC
Triangle D: 0, 0, 20.905; DCD
Vertices:
A-0
Pattern: A...
Apeirogon with alternating angles 0 and 41.810
A-0 & B-0
Pattern: AABB...
Apeirogon with angles 0, 58.282, 63.434, 58.282, 0...
A-20.905 & B-37.377 & C-31.717
Pattern: AABCCBAABCCB
Dodecagon with angles 0, 0, 0, 69.095, 0, 69.095, 0, 0, 0, 69.095, 0, 69.095
B-31.717 & C-37.377 & D-20.905
Pattern: BBCDDCBBCDDC
Dodecagon with angles 0, 0, 0, 69.095, 0, 69.095, 0, 0, 0, 69.095, 0, 69.095
C-0 & D-0
Pattern: CCDD...
Apeirogon with angles 0, 58.282, 63.434, 58.282, 0...
D-0
Pattern: D...
Apeirogon with alternating angles 0 and 41.810
Repeating unit: Hexagon with angles 0, 0, 90, 0, 0, 90. Composed of 1 A, 1 B, 1 C, 1 D.
Cyclical 3636
Triangles A, B, C, and D all identical.
Plane 1: tiled by combination of A and B
Plane 2: tiled by combination of C and D
Triangle A: 0, 0, 0; BAA
Triangle B: 0, 0, 0; ABB
Triangle C: 0, 0, 0; CCD
Triangle D: 0, 0, 0; DDC
Vertices:
A-0
Pattern: A...
Regular apeirogon with angle 0
A-0 & B-0
Pattern: AABB...
Regular apeirogon with angle 0
A-0 & B-0
Pattern: AABB...
Regular apeirogon with angle 0
B-0
Pattern: B...
Regular apeirogon with angle 0
C-0
Pattern: C...
Regular apeirogon with angle 0
C-0 & D-0
Pattern: CCDD...
Regular apeirogon with angle 0
C-0 & D-0
Pattern: CCDD...
Regular apeirogon with angle 0
D-0
Pattern: D...
Regular apeirogon with angle 0
Repeating unit 1: Square with angle 0. Composed of 1 A, 1 B.
Repeating unit 2: Square with angle 0. Composed of 1 C, 1 D.
Cyclical 3344
A and C triangles identical
Plane 1: tiled by combination of A, B, and C.
Plane 2: tiled solely by D.
Triangle A: 0, 54.736, 54.736; BAA
Triangle B: 35.264, 35.264, 70.529; ACB
Triangle C: 0, 54.736, 54.736; BCC
Triangle D: 0, 45, 45 ; DDD
Vertices:
A-0
Pattern: A...
Regular apeirogon with angle 109.472
A-54.736 & B-35.264
Pattern: AABBAABB
Octagon with angles 0, 125.565, 70.529, 125.565, 0, 125.565, 70.529, 125.565
A-54.736 & B-70.529 & C-54.736
Pattern: AABCCB
Hexagon with angles 0, 90, 90, 0, 90, 90
B-35.264 & C-54.736
Pattern: BBCCBBCC
Octagon with angles 0, 125.565, 70.529, 125.565, 0, 125.565, 70.529, 125.565
C-0
Pattern: C...
Regular apeirogon with angle 109.472
D-0
Pattern: D...
Right-angled apeirogon
D-45
Pattern: DDDDDDDD
Octagon with alternating angles 0 and 90
D-45
Pattern: DDDDDDDD
Octagon with alternating angles 0 and 90
Repeating unit 1: Quadrangle with angles 0, 0, 90, 90. Composed of 1 A, 1 B, 1 C.
Repeating unit 2: Triangle D.
Cyclical 3444
A and B triangles identical
C and D triangles identical
Plane 1: tiled by combination of A and B.
Plane 2: tiled solely by C.
Plane 3: tiled solely by D.
Triangle A: 0, 35.264, 54.736; BAA
Triangle B: 0, 35.264, 54.736; ABB
Triangle C: 0, 0, 45 ; CCC
Triangle D: 0, 0, 45 ; DDD
Vertices:
A-0
Pattern: A...
Apeirogon with alternating angles 70.529 and 109.472
A-35.264 & B-54.736
Pattern: AABBAABB
Octagon with angles 0, 90, 0, 90, 0, 90, 0, 90
A-54.736 & B-35.264
Pattern: AABBAABB
Octagon with angles 0, 90, 0, 90, 0, 90, 0, 90
B-0
Pattern: B...
Apeirogon with alternating angles 70.529 and 109.472
C-0
Pattern: C...
Apeirogon with alternating angles 0 and 90
C-0
Pattern: C...
Apeirogon with alternating angles 0 and 90
C-45
Patttern: CCCCCCCC
Regular octagon with angle 0
D-0
Pattern: D...
Apeirogon with alternating angles 0 and 90
D-0
Pattern: D...
Apeirogon with alternating angles 0 and 90
D-45
Patttern: DDDDDDDD
Regular octagon with angle 0
Repeating unit 1: Quadrangle with angles 0, 90, 0, 90. Composed of 1 A and 1 B.
Repeating unit 2: Triangle C.
Repeating unit 3: Triangle D.
Cyclical 4444
Triangles A, B, C, and D all identical.
Plane 1: tiled solely by A.
Plane 2: tiled solely by B.
Plane 3: tiled solely by C.
Plane 4: tiled solely by D.
Triangle A: 0, 0, 0; AAA
Triangle B: 0, 0, 0; BBB
Triangle C: 0, 0, 0; CCC
Triangle D: 0, 0, 0; DDD
Vertices:
A-0
Pattern: A...
Regular apeirogon with angle 0
A-0
Pattern: A...
Regular apeirogon with angle 0
A-0
Pattern: A...
Regular apeirogon with angle 0
B-0
Pattern: B...
Regular apeirogon with angle 0
B-0
Pattern: B...
Regular apeirogon with angle 0
B-0
Pattern: B...
Regular apeirogon with angle 0
C-0
Pattern: C...
Regular apeirogon with angle 0
C-0
Pattern: C...
Regular apeirogon with angle 0
C-0
Pattern: C...
Regular apeirogon with angle 0
D-0
Pattern: D...
Regular apeirogon with angle 0
D-0
Pattern: D...
Regular apeirogon with angle 0
D-0
Pattern: D...
Regular apeirogon with angle 0
Repeating unit 1: Triangle A.
Repeating unit 2: Triangle B.
Repeating unit 3: Triangle C.
Repeating unit 4: Triangle D.
Triangle with added 3-branch
A and B triangles identical.
One plane tiled by combination of A, B, C, and D.
Triangle A: 0, 54.736, 90; ABC
Triangle B: 0, 54.736, 90; BAC
Triangle C: 0, 70.529, 70.529; DAB
Triangle D: 54.736, 54.736, 60; DDC
Vertices:
A-0 & B-0 & C-0
Pattern: ABC...
Apeirogon with angle 125.565; not regular
A-54.736 & C-70.529 & D-54.736
Pattern: AACDDC
Pentagon with angles 0, 0, 125.565, 120, 125.565
A-90 & B-90
Pattern: AABB
Rhombus with angles 0, 109.472, 0, 109.472
B-54.736 & C-70.529 & D-54.736
Pattern: BBCDDC
Pentagon with angles 0, 0, 125.565, 120, 125.565
D-60
Pattern: DDDDDD
Regular hexagon with angle 109.472
Repeating unit: Regular apeirogon with angle 60. Basically the full A-0 B-0 C-0 vertex with triangle D capping the finite sides of triangles C. A quadrangle with angles 0, 90, 60, 90 can also be considered a repeating unit, but when reflecting through a 0-90 side, triangles A and B will switch.
Triangle with added 4-branch
A and B triangles identical.
Plane 1: tiled by combination of A, B, and C.
Plane 2: tiled solely by D.
Triangle A: 0, 35.264, 90 ; ABC
Triangle B: 0, 35.264, 90 ; BAC
Triangle C: 0, 54.736, 54.736; CAB
Triangle D: 45, 45, 60 ; DDD
Vertices:
A-0 & B-0 & C-0
Pattern: ABC...
Non-regular right-angled apeirogon
A-35.264 & C-54.736
Pattern: AACCAACC
Hexagon with angles 0, 0, 109.472, 0, 0, 109.472
A-90 & B-90
Pattern: AABB
Rhombus with angles 0, 70.529, 0, 70.529
B-35.264 & C-54.736
Pattern: BBCCBBCC
Hexagon with angles 0, 0, 109.472, 0, 0, 109.472
D-45
Pattern: DDDDDDDD
Octagon with alternating angles 90 and 120
D-45
Pattern: DDDDDDDD
Octagon with alternating angles 90 and 120
D-60
Pattern: DDDDDD
Right-angled hexagon.
Repeating unit 1: The basic one is a pentagon with angles 0, 90, 90, 90, 90, but reflecting through a 0-90 side will switch A and B. Full repeating unit is a non-regular right-angled apeirogon.
Repeating unit 2: Triangle D.
Triangle with added 5-branch
A and B triangles identical.
One plane tiled by combination of A, B, C, and D.
Triangle A: 0, 20.905, 90 ; ABC
Triangle B: 0, 20.905, 90 ; BAC
Triangle C: 0, 37.377, 37.377; DAB
Triangle D: 31.717, 31.717, 60 ; DDC
Vertices:
A-0 & B-0 & C-0
Pattern: ABC...
Apeirogon with angle 58.282, not regular
A-20.905 & C-37,377 & D-31.317
Pattern: AACDDCAACDDC
Decagon with angles 0, 0, 69.094, 120, 69.094, 0, 0, 69.094, 120, 69.094
A-90 & B-90
Pattern: AABB
Rhombus with angles 0, 41.810, 0, 41.810
B-20.905 & C-37.377 & 31.717
Pattern: BBCDDCBBCDDC
Decagon with angles 0, 0, 69.094, 120, 69.094, 0, 0, 69.094, 120, 69.094
D-60
Pattern: DDDDDD
Regular hexagon with angle 63.434
Repeating unit: The basic one is a hexagon with angles 0, 90, 90, 60, 90, 90, but reflecting through a 0-90 side will switch A and B. Full repeating unit is an apeirogon with repeating angle sequence 60, 90, 90.
Triangle with added 6-branch
A and B triangles identical.
Plane 1: tiled by combination of A, B, and C.
Plane 2: tiled solely by D.
Triangle A: 0b, 0d, 90c; BAC
Triangle B: 0a, 0d, 90c; ABC
Triangle C: 0a, 0b, 0d ; ABC
Triangle D: 0a, 0b, 60c; DDD
Vertices:
A-0 & B-0 & C-0
Pattern: ABC...
Apeirogon with angle 0
A-0 & C-0
Pattern: AACC...
Apeirogon with angle 0
A-90 & B-90
Pattern: AABB
Square with angle 0
B-0 & C-0
Pattern: BBCC...
Apeirogon with angle 0
D-0
Pattern: D...
Apeirogon with alternating angles 0 and 120
D-0
Pattern: D...
Apeirogon with alternating angles 0 and 120
D-60
Pattern: DDDDDD
Regular hexagon with angle 0
Repeating unit 1: The basic one is a pentagon with angles 0, 0, 90, 0, 90, but reflecting through a 0-90 side will switch A and B. Full repeating unit is an apeirogon with angle 0.
Repeating unit 2: Triangle D.
Two fused triangles
A and D triangles identical.
B and C triangles identical.
One plane tiled by combination of A, B, C, and D.
Triangle A: 0, 54.736, 54.736; ABC
Triangle B: 0, 0, 70.529; ADC
Triangle C: 0, 0, 70.529; ADB
Triangle D: 0, 54.736, 54.736; DCB
Vertices:
A-0 & B-0 & C-0
Pattern: ABC...
Apeirogon with angles 0, 125.565, 125.565...
A-54.736 & B-70.529 & D-54.736
Pattern: AABDDB
Hexagon with angles 0, 0, 109.472, 0, 0, 109.472
A-54.736 & C-70.529 & D-54.736
Pattern: AACDDC
Hexagon with angles 0, 0, 109.472, 0, 0, 109.472
B-0 & C-0 & D-0
Pattern: BCD...
Apeirogon with angles 0, 125.565, 125.565...
Repeating unit: The notion of repeating unit starts breaking down a bit here. The ABD or ACD vertices work, with some swaps caused by reflection.
Tetrahedron of 3-edges
Triangles A, B, C, and D all identical.
One plane tiled by combination of A, B, C, and D.
Triangle A: 0, 0, 0; BCD
Triangle B: 0, 0, 0; ACD
Triangle C: 0, 0, 0; ABD
Triangle D: 0, 0, 0; ABC
Vertices:
A-0 & B-0 & C-0
Pattern: ABC...
Regular apeirogon with angle 0
A-0 & B-0 & D-0
Pattern: ABC...
Regular apeirogon with angle 0
A-0 & C-0 & D-0
Pattern: ABC...
Regular apeirogon with angle 0
B-0 & C-0 & D-0
Pattern: ABC...
Regular apeirogon with angle 0
Repeating unit: Each of the ideal triangles can be considered a repeating unit; they are just differently labeled.
{3,3,3,3}
AB and DE triangles identical.
AC and CE triangles identical.
AD and BE triangles identical.
BC and CD triangles identical.
Plane 1: tiled by combination of AB, BC, CD, and DE.
Plane 2: tiled by combination of AC, AD, AE, BD, BE, and CE.
Triangle AB: 52.239, 60, 90; AB BC AB
Triangle AC: 54.736, 65.905, 90; AC AC AD
Triangle AD: 48.190, 70.529, 90; AE BD AC
Triangle AE: 54.736, 54.736, 90; AD BE AE
Triangle BC: 52.239, 75.522, 90; AB BC CD
Triangle BD: 65.905, 65.905, 90; AD BE BD
Triangle BE: 48.190, 70.529, 90; BD AE CE
Triangle CD: 52.239, 75.522, 90; DE CD BC
Triangle CE: 54.736, 65.905, 90; CE CE BE
Triangle DE: 52.239, 60, 90; DE CD DE
Repeating unit 1: Digonal strip of angle 60, made from AB, BC, CD, DE.
Repeating unit 2: Digonal strip of angle 90, made from AC, AD, AE, BD, BE, CE.
{3,3,3,4}
Plane 1: tiled by combination of AB, BC, and CD.
Plane 2: tiled by combination of AC, AD, and BD.
Plane 3: tiled by combination of AE, BE, and CE.
Plane 4: tiled solely by DE.
Triangle AB: 45, 52.239, 90; BC AB AB
Triangle AC: 35.264, 65.905, 90; AC AC AD
Triangle AD: 48.190, 54.736, 90; AD BD AC
Triangle AE: 45, 54.736, 90; BE AE AE
Triangle BC: 30, 75.522, 90; AB BC CD
Triangle BD: 45, 65.905, 90; AD BD BD
Triangle BE: 35.264, 70.529, 90; AE BE CE
Triangle CD: 52.239, 60, 90; CD CD BC
Triangle CE: 54.736, 54.736, 90; CE CE BE
Triangle DE: 45, 60, 90; DE DE DE
Repeating unit 1: Equilateral triangle with angle 90, made from AB, BC, and CD.
Repeating unit 2: Triangle with angles 45, 90, 90, made from AC, AD, and BD.
Repeating unit 3: Triangle with angles 45, 90, 90, made from AE, BE, and CE.
Repeating unit 4: Triangle DE.
Demipenteractic
A, B are one branch, C is the center, D and E are ends of short branches.
AD and AE triangles identical.
BD and BE triangles identical.
CD and CE triangles identical.
Plane 1: tiled by combination of AB, BC, CD, and CE.
Plane 2: tiled by combination of AC, AD, AE, BD, and BE.
Plane 3: tiled solely by DE.
Triangle AB: 52.239, 52.239, 90 ; AB AB BC
Triangle AC: 65.905, 65.905, 70.529; AD AE AC
Triangle AD: 48.190, 54.736, 90 ; AD BD AC
Triangle AE: 48.190, 54.736, 90 ; AE BE AC
Triangle BC: 75.522, 75.522, 90 ; CD CE AB
Triangle BD: 45, 65.905, 90 ; AD BD BD
Triangle BE: 45, 65.905, 90 ; AE BE BE
Triangle CD: 52.239, 60, 90 ; CE CD BC
Triangle CE: 52.239, 60, 90 ; CD CE BC
Triangle DE: 45, 60, 90 ; DE DE DE
Repeating unit 1: Digonal strip of angle 90, made from AB, BC, CD, and CE (two of each). Half of it, equilateral triangle of angle 90, works, but swaps some labels when reflecting.
Repeating unit 1: Digonal strip of angle 45, made from AC, AD, AE, BD, and BE.
Repeating unit 3: Triangle DE.
C~4 {4,3,3,4}
AB and DE triangles identical.
AC and CE triangles identical.
AD and BE triangles identical.
BC and CD triangles identical.
Plane 1: tiled solely by AB.
Plane 2: tiled by combination of AC and AD.
Plane 3: tiled solely by AE.
Plane 4: tiled by combination of BC and CD.
Plane 5: tiled solely by BD.
Plane 6: tiled by combination of BE and CE.
Plane 7: tiled solely by DE.
Triangle AB: 45, 45, 90; AB AB AB
Triangle AC: 35.264, 54.736, 90; AC AC AD
Triangle AD: 35.264, 54.736, 90; AD AD AC
Triangle AE: 45, 45, 90; AE AE AE
Triangle BC: 30, 60, 90; BC BC CD
Triangle BD: 45, 45, 90; BD BD BD
Triangle BE: 35.264, 54.736, 90; BE BE CE
Triangle CD: 30, 60, 90; CD CD BC
Triangle CE: 35.264, 54.736, 90; CE CE BE
Triangle DE: 45, 45, 90; DE DE DE
Repeating unit 1: Triangle AB
Repeating unit 2: Rectangle made from AC and AD.
Repeating unit 3: Triangle AE
Repeating unit 4: Rectangle made from BC and CD.
Repeating unit 5: Triangle BD
Repeating unit 6: Rectangle made from BE and CE.
Repeating unit 7: Triangle DE
F~4 {3,3,4,3}
Plane 1: tiled by combination of AB and BC.
Plane 2: tiled solely by AC.
Plane 3: tiled by combination of AD, AE, BD, BE, and CE.
Plane 4: tiled solely by CD.
Plane 5: tiled solely by DE.
Triangle AB: 30, 60, 90; AB BC AB
Triangle AC: 45, 45, 90; AC AC AC
Triangle AD: 35.264, 54.736, 90; AE BD AD
Triangle AE: 35.264, 54.736, 90; BE AD AE
Triangle BC: 30, 60, 90; BC AB BC
Triangle BD: 35.264, 54.736, 90; AD BE BD
Triangle BE: 19.471, 70.529, 90; AE BD CE
Triangle CD: 45, 45, 90; CD CD CD
Triangle CE: 35.264, 54.736, 90; CE CE BE
Triangle DE: 30, 60, 90; DE DE DE
Repeating unit 1: Equilateral triangle made from AB and BC.
Repeating unit 2: Triangle AC.
Repeating unit 3: This is an interesting one. The five triangles AD, AE, BD, BE, and CE form a rectangle. Four of these five triangles are similar; only the "central" one, BE, has a different shape.
Repeating unit 4: Triangle CD.
Repeating unit 5: Triangle DE.
B~4 Branched Euclidean group (half of tesseractic honeycomb)
Same marking as demipenteractic, AB branch is 4, BC, CD and DE are 3.
AD and AE triangles identical.
BD and BE triangles identical.
CD and CE triangles identical.
Plane 1: tiled solely by AB.
Plane 2: tiled by combination of AC, AD, and AE.
Plane 3: tiled by combination of BC, CD, and CE.
Plane 4: tiled solely by BD.
Plane 5: tiled solely by BE.
Plane 6: tiled solely by DE.
Triangle AB: 45, 45, 90 ; AB AB AB
Triangle AC: 54.736, 54.736, 70.529; AD AE AC
Triangle AD: 35.264, 54.736, 90 ; AD AD AC
Triangle AE: 35.264, 54.736, 90 ; AE AE AC
Triangle BC: 60, 60, 60 ; BC CD CE
Triangle BD: 45, 45, 90 ; BD BD BD
Triangle BE: 45, 45, 90 ; BE BE BE
Triangle CD: 30, 60, 90 ; CD CD BC
Triangle CE: 30, 60, 90 ; CE CE BC
Triangle DE: 45, 45, 90 ; DE DE DE
Repeating unit 1: Triangle AB.
Repeating unit 2: Rectangle made from triangles AC, AD, and AE.
Repeating unit 3: Rectangle made from triangles BC, CD, and CE, two of each. Half of it works as well, but swaps some labels when reflecting.
Repeating unit 4: Triangle BE.
Repeating unit 5: Triangle BE.
Repeating unit 6: Triangle DE.
D~4 Cross group (quarter of tesseractic honeycomb)
AB, BC, BD, BE branches are 3.
AB, BC, BC, BE triangles identical.
AC, AD, AE, CD, CE, DE triangles identical.
Plane 1: tiled by combination of AB, BC, BD, and BE.
Plane 2: tiled solely by AC.
Plane 3: tiled solely by AD.
Plane 4: tiled solely by AE.
Plane 5: tiled solely by CD.
Plane 6: tiled solely by CE.
Plane 7: tiled solely by DE.
Triangle AB: 60, 60, 60; BC BD BE
Triangle AC: 45, 45, 90; AC AC AC
Triangle AD: 45, 45, 90; AD AD AD
Triangle AE: 45, 45, 90; AE AE AE
Triangle BC: 60, 60, 60; AB BD BE
Triangle BD: 60, 60, 60; AB BC BE
Triangle BE: 60, 60, 60; AB BC BD
Triangle CD: 45, 45, 90; CD CD CD
Triangle CE: 45, 45, 90; CE CE CE
Triangle DE: 45, 45, 90; DE DE DE
Repeating unit 1: Plane 1 is tiled by equilateral triangles with a particular 4-coloring such as that each vertex has triangles of three colors around it, with opposite pairs colored alike, and each triangle has three vertices with different color combinations.
Repeating unit 2: Triangle AC.
Repeating unit 3: Triangle AD.
Repeating unit 4: Triangle AE.
Repeating unit 5: Triangle CD.
Repeating unit 6: Triangle CE.
Repeating unit 7: Triangle DE.
A~4 Cyclical 33333
AB, AE, BC, CD, DE triangles identical.
AC, AD, BD, BE, CE triangles identical.
Plane 1: tiled by combination of AB, AE, BC, CD, and DE.
Plane 2: tiled by combination of AC, AD, BD, BE, and CE.
Triangle AB: 52.239, 52.239, 75.522; AE BC AB
Triangle AC: 48.190, 65.905, 65.905; AC AD CE
Triangle AD: 48.190, 65.905, 65.905; AD AC BD
Triangle AE: 52.239, 52.239, 75.522; AB DE AE
Triangle BC: 52.239, 52.239, 75.522; AB CD BC
Triangle BD: 48.190, 65.905, 65.905; BD AD BE
Triangle BE: 48.190, 65.905, 65.905; BE BD CE
Triangle CD: 52.239, 52.239, 75.522; BC DE CD
Triangle CE: 48.190, 65.905, 65.905; CE AC BE
Triangle DE: 52.239, 52.239, 75.522; AE CD DE
Repeating unit 1: Triangles form strips where five "colors" repeat endlessly.
Repeating unit 2: Triangles form strips where five "colors" repeat endlessly.
H4 {3,3,3,5}
Plane 1: tiled by combination of AB, BC, and CD.
Plane 2: tiled by combination of AC, AD, AE, BD, BE, and CE.
Plane 3: tiled solely by DE.
Triangle AB: 36, 52.239, 90; BC AB AB
Triangle AC: 20.905, 65.905, 90; AC AC AD
Triangle AD: 37.377, 48.190, 90; BD AE AC
Triangle AE: 31.717, 54.736, 90; BE AD AE
Triangle BC: 7.761, 75.522, 90; AB BC CD
Triangle BD: 13.283, 65.905, 90; AD BE BD
Triangle BE: 10.812, 70.529, 90; AE BD CE
Triangle CD: 22.239, 52.239, 90; CD CD BC
Triangle CE: 20.905, 54.736, 90; CE CE BE
Triangle DE: 18, 60, 90; DE DE DE
Repeating unit 1: Triangle with angles 30, 36, 90, made from AC, BC, and CD.
Repeating unit 2: Quadrangle with angles 45, 90, 90, 90, made from AC, AD, AE, BD, BE, and CE.
Repeating unit 3: Triangle DE.
BH4 {4,3,3,5}
Plane 1: tiled solely by AB.
Plane 2: tiled by combination of AC, AD, and AE.
Plane 3: tiled by combination of BC and CD.
Plane 4: tiled by combination of BD, BE, and CE.
Plane 5: tiled solely by DE.
Triangle AB: 36, 45, 90; AB AB AB
Triangle AC: 20.905, 54.736, 90; AC AC AD
Triangle AD: 35.264, 37.377, 90; AE AD AC
Triangle AE: 31.717, 45, 90; AE AD AE
Triangle BC: 7.761, 60, 90; BC BC CD
Triangle BD: 13.283, 45, 90; BD BE BD
Triangle BE: 10.812, 54.736, 90; BE BD CE
Triangle CD: 22.239, 30, 90; CD CD BC
Triangle CE: 20.905, 35.264, 90; CE CE BE
Triangle DE: 18, 45, 90; DE DE DE
Repeating unit 1: Triangle AB.
Repeating unit 2: Quadrangle with angles 45, 90, 90, 90, made from AC, AD, and AE.
Repeating unit 3: Quadrangle with angles 30, 90, 90, 90, made from BC and CD.
Repeating unit 4: Quadrangle with angles 45, 45, 90, 90, made from BD, BE, and CE.
Repeating unit 5: Triangle DE.
K4 {5,3,3,5}
AB and DE triangles identical.
AC and CE triangles identical.
AD and BE triangles identical.
BC and CD triangles identical.
Plane 1: tiled solely by AB.
Plane 2: tiled by combination of AC, AD, AE, BD, BE, and CE.
Plane 3: tiled by combination of BC and CD.
Plane 4: tiled solely by DE.
Triangle AB: 18, 36, 90; AB AB AB
Triangle AC: 20.905, 20.905, 90; AC AC AD
Triangle AD: 10.812, 37.377, 90; AE BD AC
Triangle AE: 31.717, 31.717, 90; AD BE AE
Triangle BC: 7.761, 22.239, 90; BC CD BC
Triangle BD: 13.283, 13.283, 90; AD BE BD
Triangle BE: 10.812, 37.377, 90; AE BD CE
Triangle CD: 7.761, 22.239, 90; CD BC CD
Triangle CE: 20.905, 20.905, 90; CE CE BE
Triangle DE: 18, 36, 90; DE DE DE
Repeating unit 1: Triangle AB.
Repeating unit 3: Hexagon with angles 45, 45, 90, 90, 90, 90, made from AC, AD, AE, BD, BE, and CE.
Repeating unit 3: Quadrangle with angles 30, 90, 30, 90, made from BC and CD.
Repeating unit 4: Triangle DE.
DH4 Branched half of {4,3,3,5}
AB is 5, BC, CD and CE are 3
AD and AE triangles identical.
BD and BE triangles identical.
CD and CE triangles identical.
Plane 1: tiled solely by AB.
Plane 2: tiled by combination of AC, AD, AE, BD, and BE.
Plane 3: tiled combination of BC, CD, and CE.
Plane 4: tiled solely by DE.
Triangle AB: 18, 18, 90 ; AB AB AB
Triangle AC: 20.905, 20.905, 70.529; AD AE AC
Triangle AD: 10.812, 54.736, 90 ; AD BD AC
Triangle AE: 10.812, 54.736, 90 ; AE BE AC
Triangle BC: 22.239, 22.239, 60 ; CD CE BC
Triangle BD: 13.283, 45, 90 ; BD AD BD
Triangle BE: 13.283, 45, 90 ; BE AE BE
Triangle CD: 7.761, 60, 90 ; CE CD BC
Triangle CE: 7.761, 60, 90 ; CD CE BC
Triangle DE: 36, 45, 90 ; DE DE DE
Repeating unit 1: Triangle AB.
Repeating unit 2: Quadrangle with all angles 45, made from AC, AD, AE, BD, and BE.
Repeating unit 3: Quadrangle with all angles 30, made from BC, CD, and CE, two of each. Half of it works as well, but swaps some labels when reflecting.
Repeating unit 4: Triangle DE.
AF4 Cyclical 33334
AE is 4, AB, BC, CD, and DE are 3
AB and DE triangles identical.
AC and CE triangles identical.
AD and BE triangles identical.
BC and CD triangles identical.
Plane 1: tiled by combination of AB, BC, CD, and DE.
Plane 2: tiled by combination of AC, AD, BD, BE, and CE.
Plane 3: tiled solely by AE.
Triangle AB: 30, 52.239, 60 ; BC AB AB
Triangle AC: 35.264, 45, 65.905; AC AD AC
Triangle AD: 35.264, 48.190, 54.736; AC BD AD
Triangle AE: 45, 45, 45 ; AE AE AE
Triangle BC: 30, 52.239, 70.529; CD AB BC
Triangle BD: 19.471, 65.905, 65.905; BD AD BE
Triangle BE: 35.264, 48.190, 54.736; CE BD BE
Triangle CD: 30, 52.239, 70.529; BC DE CD
Triangle CE: 35.264, 45, 65.905; CE BE CE
Triangle DE: 30, 52.239, 60 ; CD DE DE
Repeating unit 1: Quadrangle with angles 30, 90, 30, 90, made from AB, BC, CD, and DE.
Repeating unit 1: Quadrangle with angles 45, 45, 90, 90, made from AC, AD, BD, BE, and CE.
Repeating unit 3: Triangle AE.
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.
truncated octahedron:
4-6: 125.264390 - 54.735610 - arccos(1/sqrt(3))
6-6: 109.471221 - 70.528779 - arccos(1/3)
cycle 4-6-6-4-6-6
opposites: 4/4, 6a/6b
truncated cuboctahedron:
4-6: 144.735610 - 35.264390 - arccos(sqrt(2/3))
4-8: 135 - 45
6-8: 125.264390 - 54.735610 - arccos(1/sqrt(3))
cycle 1: 4-6-8-6-4-6-8-6
cycle 2: 4-8-4-8-4-8-4-8
opposites: 4/4, 6/6, 8/8
truncated icosidodecahedron:
4-6: 159.094843 - 20.905157 - arccos(sqrt[(3+sqrt(5))/6])
4-10: 148.282526 - 31.717474 - arccos(sqrt[(5+sqrt(5))/10])
6-10: 142.622632 - 37.377368 - arccos(sqrt[(5+2 sqrt(5))/15])
cycle: 4-6-10-4-10-6-4-6-10-4-10-6
opposites: 4/4, 6/6, 10/10
o{3,3,3}
truncated octahedron 1/hexagonal prism 1
127.761244 - 52.238756 - arccos(sqrt[3/8])
truncated octahedron 1/hexagonal prism 2
114.094843 - 65.905157 - arccos(sqrt[1/6])
truncated octahedron 1/truncated octahedron 2
104.477512 - 75.522488 - arccos(1/4)
hexagonal prism 1/hexagonal prism 2
131.810315 - 48.189685 - arccos(2/3)
hexagonal prism 1/truncated octahedron 2
114.094843 - 65.905157 - arccos(sqrt[1/6])
hexagonal prism 2/truncated octahedron 2
127.761244 - 52.238756 - arccos(sqrt[3/8])
cycle 1: |to1-4-hp2-4-hp1-4-to2| - 65.905 + 48.190 + 65.905 = 180
cycle 2: |hp1-6-to1-6-to2-6-hp2| - 52.239 + 75.522 + 52.239 = 180
opposites: to1/to2, hp1/hp2
o{3,3,4}
truncated octahedron/hexagonal prism
150 - 30
truncated octahedron/octagonal prism
135 - 45
truncated octahedron/truncated cuboctahedron
120 - 60
hexagonal prism/octagonal prism
144.735610 - 35.264390 - arccos(sqrt[2/3])
hexagonal prism/truncated cuboctahedron
125.264390 - 54.735610 - arccos(1/sqrt(3))
octagonal prism/truncated cuboctahedron
135 - 45
cycle 1: |to-4-op| - 45
cycle 2: |op-4-hp-4-tco| - 35.264 + 54.736 = 90
cycle 3: |hp-6-to-6-tco| - 30 + 60 = 90
cycle 4: |op-8-tco| - 45
opposites: to/to, hp/hp/ op/op, tco/tco
o{3,3,5}
truncated octahedron/hexagonal prism
172.238756 - 7.761244 - arccos(sqrt[9+3*sqrt(5)]/4)
truncated octahedron/decagonal prism
166.717474 - 13.282526 - arccos(sqrt[(5+2 sqrt(5))/10])
truncated octahedron/truncated icosidodecahedron
157.761244 - 22.238756 - arccos(sqrt[7+3sqrt(5)]/4)
hexagonal prism/decagonal prism
169.187683 - 10.812317 - arccos(sqrt[(10+2*sqrt(5))/15])
hexagonal prism/truncated icosidodecahedron
159.094843 - 20.905157 - arccos(sqrt[(3+sqrt(5))/6])
decagonal prism/truncated icosidodecahedron
162 - 18
cycle 1: |to-4-dp-4-hp-4-tid| - 13.283 + 10.812 + 20.905 = 45
cycle 2: |hp-6-to-6-tid| - 7.761 + 22.239 = 30
cycle 3: |dp-10-tid| - 18
opposites: to/to, hp/hp, dp/dp, tid/tid
o{3,4,3}
truncated cuboctahedron 1/hexagonal prism 1
150 - 30
truncated cuboctahedron 1/hexagonal prism 2
144.735610 - 35.264390 - arccos(sqrt[2/3])
truncated cuboctahedron 1/truncated cuboctahedron 2
135 - 45
hexagonal prism 1/hexagonal prism 2
160.528779 - 19.471221 - arccos(sqrt[8/9])
hexagonal prism 1/truncated cuboctahedron 2
144.735610 - 35.264390 - arccos(sqrt[2/3])
hexagonal prism 2/truncated cuboctahedron 2
150 - 30
cycle 1: |tco1-4-hp2-4-hp1-4-tco2| - 35.264 + 19.471 + 35.264 = 90
cycle 2: |tco1-6-hp1| - 30
cycle 3: |hp2-6-tco2| - 30
cycle 4: |tco1-8-tco2| - 45
opposites: tco1/tco1, tco2/tco2, hp1/hp1, hp2/hp2
o(b333)
truncated octahedron 1/truncated octahedron 2
120 - 60
truncated octahedron 1/cube
135 - 45
truncated octahedron 1/truncated octahedron 3
120 - 60
truncated octahedron 2/cube
135 - 45
truncated octahedron 2/truncated octahedron 3
120 - 60
cube/truncated octahedron 3
135 - 45
cycle 1: |to1-4-c| - 45
cycle 2: |to2-4-c| - 45
cycle 3: |c-4-to3| - 45
cycle 4: (to1-6-to2-6-to3-6-) - 60 + 60 + 60 = 180
opposites: to1/to1, to2/to2, to3/to3, c/c
{3,3,3}
A and D triangles identical.
B and C triangles identical.
One plane tiled by combination of A, B, C, and D.
Triangle A: 54.736, 60, 90; ABA
Triangle B: 54.736, 70.529, 90; ABC
Triangle C: 54.736, 70.529, 90; DCB
Triangle D: 54.736, 60, 90; DCD
Vertices:
A-54.736 & B-70.529 & C-54.736
Pattern: AABCCB
Triangle with angles 120, 125.265, 125.265
A-60
Pattern: AAAAAA
Triangle with angles 109.472, 109.472, 109.472
A-90 & B-90
Pattern: AABB
Quadrangle with angles 109.472, 125.265, 120, 125.265
B-54.736 & C-70.529 & D-54.736
Pattern: BBCDDC
Triangle with angles 120, 125.265, 125.265
C-90 & D-90
Pattern: CCDD
Quadrangle with angles 109.472, 125.265, 120, 125.265
D-60
Pattern: DDDDDD
Equilateral triangle with angle 109.472
Repeating unit: Digonal strip of angle 60. Composed of 1 A, 1 B, 1 C and 1 D.
{3,3,4}
Plane 1: tiled by combination of A, B, and C.
Plane 2: tiled solely by D.
Triangle A: 45, 54.736, 90; BAA
Triangle B: 35.264, 70.529, 90; ABC
Triangle C: 54.736, 54.736, 90; CCB - double of triangle A. Note that each of its 54.736 angles belongs to a different type of vertex.
Triangle D: 45, 60, 90; DDD
Vertices:
A-45
Pattern: AAAAAAAA
Square with angle 109.472
A-54.736 & B-70.529 & C-54.736
Pattern: AABCCB
Triangle with angles 90, 90, 109.472
A-90 & B-90
Pattern: AABB
Quadrangle with angles 90, 125.265, 109.472, 125.265
B-35.264 & C-54.736
Pattern: BBCCBBCC
Square with angle 125.265
C-90
Pattern: CCCC
Square with angle 109.472
D-45
Pattern: DDDDDDDD
Square with angle 120
D-60
Pattern: DDDDDD
Equilateral triangle with angle 90
D-90
Pattern: DDDD
Rhombus with angles 90, 120, 90, 120
Repeating unit 1: Triangle with angles 45, 90, 90. Composed of 1 A, 1 B, 1 C.
Repeating unit 2: Triangle D.
{3,3,5}
One plane tiled by combination of A, B, C, and D.
Triangle A: 36, 54.736, 90; BAA
Triangle B: 20.905, 70.529, 90; ABC
Triangle C: 37.377, 54.736, 90; CDB
Triangle D: 31.717, 60, 90; DCD
Vertices:
A-36
Pattern: AAAAAAAAAA
Regular pentagon with angle 109.472
A-54.736 & B-70.529 & C-54.736
Pattern: AABCCB
Triangle with angles 58.282, 58.282, 72
A-90 & B-90
Pattern: AABB
Quadrangle with angles 41.810, 125.265, 72, 125.265
B-20.905 & C-37.377 & D-31.717
Pattern: BBCDDCBBCDDC
Hexagon with angles 120, 125.265, 125.265, 120, 125.265, 125.265
C-90 & D-90
Pattern: CCDD
Quadrangle with angles 69.094, 109.472, 69.094, 120
D-60
Pattern: DDDDDD
Equilateral triangle with angle 63.434
Repeating unit: Triangle with angles 36, 60, 90. Composed of 1 A, 1 B, 1 C, 1 D.
{3,4,3}
A and D triangles identical.
B and C triangles identical.
Plane 1: tiled by combination of A and B
Plane 2: tiled by combination of C and D
Triangle A: 35.264, 60, 90; ABA
Triangle B: 45, 54.736, 90; ABB
Triangle C: 45, 54.736, 90; DCC
Triangle D: 35.264, 60, 90; DCD
Vertices:
A-35.264 & B-54.736
Pattern: AABBAABB
Rhombus with angles 90, 120, 90, 120
A-60
Pattern: AAAAAA
Equilateral triangle with angle 70.528
A-90 & B-90
Pattern: AABB
Quadrangle with angles 90, 90, 90, 120
B-45
Pattern: BBBBBBBB
Square with angle 109.472
C-45
Pattern: CCCCCCCC
Square with angle 109.472
C-54.736 & D-35.264
Pattern: CCDDCCDD
Rhombus with angles 90, 120, 90, 120
C-90 & D-90
Pattern: CCDD
Quadrangle with angles 90, 90, 90, 120
D-60
Pattern: DDDDDD
Equilateral triangle with angle 70.529
Repeating unit 1: Triangle with angles 45, 60, 90. Composed of 1 A, 1 B.
Repeating unit 2: Triangle with angles 45, 60, 90. Composed of 1 C, 1 D.
Branched 333 (demitesseractic)
A, C, and D triangles identical.
One plane tiled by combination of A, B, C, and D.
Triangle A: 54.736, 54.736, 90 ; AAB
Triangle B: 70.529, 70.529, 70.529; ABC
Triangle C: 54.736, 54.736, 90 ; CCB
Triangle D: 54.736, 54.736, 90 ; DDB
Vertices:
A-54.736 & B-70.529 & C-54.736
Pattern: AABCCB
Quadrangle with angles 125.265, 125.265, 125.265, 125.265 (not square, apparently)
A-54.736 & B-70.529 & D-54.736
Pattern: AABDDB
Quadrangle with angles 125.265, 125.265, 125.265, 125.265 (not square, apparently)
A-90
Pattern: AAAA
Square with angle 109.472
B-70.529 & C-54.736 & D-54.736
Pattern: BCCBDD
Quadrangle with angles 125.265, 125.265, 125.265, 125.265 (not square, apparently)
C-90
Pattern: CCCC
Square with angle 109.472
D-90
Pattern: DDDD
Square with angle 109.472
Repeating unit: Equilateral triangle of angle 90. Composed of 1 A, 1 B, 1 C and 1 D.
{4,3,4}
A and D triangles identical.
B and C triangles identical.
Plane 1: tiled solely by A
Plane 2: tiled by combination of B and C
Plane 3: tiled solely by D
Triangle A: 45, 45, 90; AAA
Triangle B: 35.264, 54.736, 90; BBC
Triangle C: 35.264, 54.736, 90; CCB
Triangle D: 45, 45, 90; DDD
Vertices:
A-45
Pattern: AAAAAAAA
Square with angle 90
A-45
Pattern: AAAAAAAA
Square with angle 90
(The two 45 angles at A look identical within the plane, but differ in how other tetrahedron faces are connected to them.)
A-90
Pattern: AAAA
Square with angle 90
B-35.264 & C-54.736
Pattern: BBCC
Rectangle with angle 90
B-54.736 & C-35.264
Pattern: BBCC
Rectangle with angle 90
B-90
Pattern: BBBB
Rhombus with angles 70.529, 109.472, 70.529, 109.472
C-90
Pattern: CCCC
Rhombus with angles 70.529, 109.472, 70.529, 109.472
D-45
Pattern: DDDDDDDD
Square with angle 90
D-45
Pattern: DDDDDDDD
Square with angle 90
D-90
Pattern: DDDD
Square with angle 90
Repeating unit 1: Triangle A.
Repeating unit 2: Rectangle of angle 90. Composed of 1 B, 1 C.
Repeating unit 3: Triangle D.
Branched 334 (tetrahedral/octahedral honeycomb)
A and C triangles identical.
Plane 1: tiled by combination of A, B, and C
Plane 2: tiled solely by D
Triangle A: 35.264, 54.736, 90 ; AAB
Triangle B: 54.736, 54.736, 70.529; ACB (double of A or C triangle)
Triangle C: 35.264, 54.736, 90 ; CCB
Triangle D: 45, 45, 90 ; DDD
Vertices:
A-35.264 & B-54.736
Pattern: AABBAABB
Hexagon with angles 109.472, 125.565, 125.565, 109.472, 125.565, 125.565
A-54.736 & B-70.529 & C-54.736
Pattern: AABCCB
Square with angle 90
A-90
Pattern: AAAA
Rhombus with angles 70.529, 109.472, 70.529, 109.472
B-54.736 & C-35.264
Pattern: BBCCBBCC
Hexagon with angles 109.472, 125.565, 125.565, 109.472, 125.565, 125.565
C-90
Pattern: CCCC
Rhombus with angles 70.529, 109.472, 70.529, 109.472
D-45
Pattern: DDDDDDDD
Square with angle 90
D-45
Pattern: DDDDDDDD
Square with angle 90
D-90
Pattern: DDDD
Square with angle 90
Repeating unit 1: Rectangle of angle 90. Composed of 1 A, 1 B, 1 C.
Repeating unit 2: Triangle D.
Cyclical 3333
Triangles A, B, C, and D all identical.
One plane tiled by combination of A, B, C, and D.
Triangle A: 54.736, 54.736, 70.529; BDA
Triangle B: 54.736, 54.736, 70.529; ACB
Triangle C: 54.736, 54.736, 70.529; BDC
Triangle D: 54.736, 54.736, 70.529; ACD
Vertices:
A-54.736 & B-70.529 & C-54.736
Pattern: AABCCB
Hexagon with angles 109.472, 125.565, 125.565, 109.472, 125.565, 125.565
A-54.736 & C-54.736 & D-70.529
Pattern: AADCCD
Hexagon with angles 109.472, 125.565, 125.565, 109.472, 125.565, 125.565
A-70.529 & B-54.736 & D-54.736
Pattern: ABBADD
Hexagon with angles 109.472, 125.565, 125.565, 109.472, 125.565, 125.565
B-54.736 & C-70.529 & D-54.736
Pattern: BBCDDC
Hexagon with angles 109.472, 125.565, 125.565, 109.472, 125.565, 125.565
Repeating unit: An infinite strip formed by repeating triangles A, B, C, D.
{4,3,5}
Plane 1: tiled solely by A
Plane 2: tiled by combination of B, C, and D
Triangle A: 36, 45, 90; AAA
Triangle B: 20.905, 54.736, 90; BBC
Triangle C: 35.264, 37.377, 90; DCB
Triangle D: 31.717, 45, 90; DCD
Vertices:
A-36
Pattern: AAAAAAAAAA
Regular pentagon with angle 90
A-45
Pattern: AAAAAAAA
Square with angle 72
A-90
Pattern: AAAA
Rhombus with angles 72, 90, 72, 90
B-20.905 & C-37.377 & D-31.717
Pattern: BBCDDCBBCDDC
Right-angled hexagon, not regular
B-54.736 & C-35.264
Pattern: BBCCBBCC
Rectangle with angle 58.282
B-90
Pattern: BBBB
Rhombus with angles 41.810, 109.472, 41.810, 109.472
C-90 & D-90
Pattern: CCDD
Quadrangle with angles 69.095, 70.529, 90, 69.095
D-45
Pattern: DDDDDDDD
Square with angle 63.434
Repeating unit: Quadrangle with angles 45, 90, 90, 90. Composed of 1 B, 1 C, 1 D.
{5,3,5}
A and D triangles identical.
B and C triangles identical.
One plane tiled by combination of A, B, C, and D.
Triangle A: 31.717, 36, 90; ABA
Triangle B: 20.905, 37.377, 90; ABC
Triangle C: 20.905, 37.377, 90; DCB
Triangle D: 31.717, 36, 90; DCC
Vertices:
A-31.717 & B-37.377 & C-20.905
Pattern: AABCCBAABCCB
Hexagon with angles 58.282, 58.282, 72, 58.282, 58.282, 72
A-36
Pattern: AAAAAAAAAA
Regular pentagon with angle 63.434
A-90 & B-90
Pattern: AABB
Quadrangle with angles 41.810, 69.095, 72, 69.095
B-20.905 & C-37.377 & D-31.717
Pattern: BBCDDCBBCDDC
Hexagon with angles 58.282, 58.282, 72, 58.282, 58.282, 72
C-90 & D-90
Pattern: CCDD
Quadrangle with angles 41.810, 69.095, 72, 69.095
D-36
Pattern: DDDDDDDDDD
Regular pentagon with angle 63.434
Repeating unit: Quadrangle with angles 36, 90, 36, 90. Composed of 1 A, 1 B, 1 C, 1 D.
{3,5,3}
A and D triangles identical.
B and C triangles identical.
One plane tiled by combination of A, B, C, and D.
Triangle A: 20.905, 60, 90; ABA
Triangle B: 31.717, 37.377, 90; ABC
Triangle C: 31.717, 37.377, 90; DCB
Triangle D: 20.905, 60, 90; DCD
Vertices:
A-20.905 & B-37.377 & C-31.717
Pattern: AABCCBAABCCB
Hexagon with angles 69.095, 69.095, 120, 69.095, 69.095, 120
A-60
Pattern: AAAAAA
Equilateral triangle with angle 41.810
A-90 & B-90
Pattern: AABB
Quadrangle with angles 58.282, 63.434, 58.282 and 120
B-31.717 & C-37.377 & D-20.905
Pattern: BBCDDCBBCDDC
Hexagon with angles 69.095, 69.095, 120, 69.095, 69.095, 120
C-90 & D-90
Pattern: CCDD
Quadrangle with angles 58.282, 63.434, 58.282 and 120
D-60
Pattern: AAAAAA
Equilateral triangle with angle 41.810
Repeating unit: Quadrangle with angles 60, 90, 60, 90. Composed of 1 A, 1 B, 1 C, 1 D.
Branched 335 (tetrahedral/icosahedral honeycomb)
A and C triangles identical.
One plane tiled by combination of A, B, C, and D.
Triangle A: 20.905, 54.736, 90; AAB
Triangle B: 37.377, 37.377, 70.529; ACD
Triangle C: 20.905, 54.736, 90; CCB
Triangle D: 31.717, 31.717, 90; DDB
Vertices:
A-20.905 & B-37.377 & D-31.717
Pattern: AABDDBAABDDB
Octagon with alternating angles of 69.095 and 125.565
A-54.736 & B-70.529 & C-54.736
Pattern: AABCCB
Rectangle with angle 58.282
A-90
Pattern: AAAA
Rhombus with angles 41.810, 109.472, 41.810, 109.472
B-37.377 & C-20.905 & D-31.717
Pattern: BCCBDDBCCBDD
Octagon with alternating angles of 69.095 and 125.565
C-90
Pattern: CCCC
Rhombus with angles 41.810, 109.472, 41.810, 109.472
D-90
Pattern: DDDD
Square with angle 63.434
Repeating unit: Right-angled pentagon; not regular. Composed of 1 A, 1 B, 1 C, 1 D.
Cyclical 3334
A and D triangles identical.
B and C triangles identical.
One plane tiled by combination of A, B, C, and D.
Triangle A: 45, 54.736, 54.736; BAA
Triangle B: 35.264, 54.736, 70.529; CAB
Triangle C: 35.264, 54.736, 70.529; BDC
Triangle D: 45, 54.736, 54.736; CDD
Vertices:
A-45
Pattern: AAAAAAAA
Regular octagon with angle 109.472
A-54.736 & B-35.264
Pattern: AABBAABB
Octagon with angles 90, 125.565, 109.472, 125.565, 90, 125.565, 109.472, 125.565
A-54.736 & B-70.529 & C-54.736
Pattern: AABCCB
Hexagon with angles 70.529, 125.565, 90, 90, 90, 125.565
B-54.736 & C-70.529 & D-54.736
Pattern: BBCDDC
Hexagon with angles 70.529, 125.565, 90, 90, 90, 125.565
C-35.264 & D-54.736
Pattern: CCDDCCDD
Octagon with angles 90, 125.565, 109.472, 125.565, 90, 125.565, 109.472, 125.565
D-45
Pattern: DDDDDDDD
Regular octagon with angle 109.472
Repeating unit: Quadrangle with angles 45, 90, 45, 90. Composed of 1 A, 1 B, 1 C, 1 D.
Cyclical 3335
A and D triangles identical.
B and C triangles identical.
One plane tiled by combination of A, B, C, and D.
Triangle A: 31.717, 37.377, 54.736; BAD
Triangle B: 20.905, 54.736, 70.529; CAB
Triangle C: 20.905, 54.736, 70.529; BDC
Triangle D: 31.717, 37.377, 54.736; CDA
Vertices:
A-31.717 & C-20.905 & D-37.377
Pattern: AADCCDAADCCD
Dodecagon with angles 69.095, 109.472, 69.095, 125.565, 109.472, 125.565, 69.095, 109.472, 69.095, 125.565, 109.472, 125.565
A-37.377 & B-20.905 & D-31.717
Pattern: ABBADDABBADD
Dodecagon with angles 69.095, 109.472, 69.095, 125.565, 109.472, 125.565, 69.095, 109.472, 69.095, 125.565, 109.472, 125.565
A-54.736 & B-70.529 & C-54.736
Pattern: AABCCD
Hexagon with angles 41.810, 125.565, 58.282, 63.434, 58.282, 125.565
B-54.736 & C-70.529 & D-54.736
Pattern: BBCDDC
Hexagon with angles 41.810, 125.565, 58.282, 63.434, 58.282, 125.565
Repeating unit: An infinite strip formed by repeating triangles A, B, C, D. Its two edges form right-angled pseudogons.
Cyclical 3434
Triangles A, B, C, and D all identical.
Plane 1: tiled by combination of A and B
Plane 2: tiled by combination of C and D
Triangle A: 35.264, 45, 54.736; ABA
Triangle B: 35.264, 45, 54.736; BAB
Triangle C: 35.264, 45, 54.736; CDC
Triangle D: 35.264, 45, 54.736; DCD
Vertices:
A-35.264 & B-54.736
Pattern: AABBAABB
Right-angled octagon, not regular
A-45
Pattern: AAAAAAAA
Octagon with alternating angles 70.529 and 125.565
A-54.736 & B-35.264
Pattern: AABBAABB
Right-angled octagon, not regular
B-45
Pattern: BBBBBBBB
Octagon with alternating angles 70.529 and 125.565
C-35.264 & D-54.736
Pattern: CCDDCCDD
Right-angled octagon, not regular
C-45
Pattern: CCCCCCCC
Octagon with alternating angles 70.529 and 125.565
C-54.736 & D-35.264
Pattern: CCDDCCDD
Right-angled octagon, not regular
D-45
Pattern: DDDDDDDD
Octagon with alternating angles 70.529 and 125.565
Repeating unit 1: Quadrangle with angles 45, 90, 45, 90. Composed of 1 A, 1 B.
Repeating unit 2: Quadrangle with angles 45, 90, 45, 90. Composed of 1 C, 1 D.
Cyclical 3435
A and D triangles identical.
B and C triangles identical.
One plane tiled by combination of A, B, C, and D.
Triangle A: 31.717, 35.264, 37.377; BDA
Triangle B: 20.905, 45, 54.736; BAB
Triangle C: 20.905, 45, 54.736; CDC
Triangle D: 31.717, 35.264, 37.377; CAD
Vertices:
A-31.717 & C-20.905 & D-37.377
Pattern: AADCCDAADCCD
Dodecagon with angles 69.095, 70.529, 69.095, 90, 90, 90, 69.095, 70.529, 69.095, 90, 90, 90
A-35.264 & B-54.736
Pattern: AABBAABB
Octagon with angles 58.282, 63.434, 58.282, 90, 58.282, 63.434, 58.282, 90
A-37.377 & B-20,905 & D-31.717
Pattern: ABBADDABBADD
Dodecagon with angles 69.095, 70.529, 69.095, 90, 90, 90, 69.095, 70.529, 69.095, 90, 90, 90
B-45
Pattern: BBBBBBBB
Octagon with alternating angles 41.810 and 109.472
C-45
Pattern: CCCCCCCC
Octagon with alternating angles 41.810 and 109.472
C-54.736 & D-35.264
Pattern: CCDDCCDD
Octagon with angles 58.282, 63.434, 58.282, 90, 58.282, 63.434, 58.282, 90
Repeating unit: Hexagon with angles 45, 90, 90, 45, 90, 90. Composed of 1 A, 1 B, 1 C, 1 D.
Cyclical 3535
Triangles A, B, C, and D all identical.
One plane tiled by combination of A, B, C, and D.
Triangle A: 20.905, 31.717, 37.377; DBA
Triangle B: 20.905, 31.717, 37.377; CAB
Triangle C: 20.905, 31.717, 37.377; BDC
Triangle D: 20.905, 31.717, 37.377; ACD
Vertices:
A-20.905 & C-31.717 & D-37.377
Pattern: AADCCDAADCCD
Dodecagon with angles 41.810, 58.282, 69.095, 63.434, 69.095, 58.282, 41.810, 58.282, 69.095, 63.434, 69.095, 58.282
A-31.717 & B-37.377 & C-20.905
Pattern: AABCCBAABCCB
Dodecagon with angles 41.810, 58.282, 69.095, 63.434, 69.095, 58.282, 41.810, 58.282, 69.095, 63.434, 69.095, 58.282
A-37.377 & B-20.905 & D-31.717
Pattern: ABBADDABBADD
Dodecagon with angles 41.810, 58.282, 69.095, 63.434, 69.095, 58.282, 41.810, 58.282, 69.095, 63.434, 69.095, 58.282
B-31.717 & C-37.377 & D-20.905
Pattern: BBCDDCBBCDDC
Dodecagon with angles 41.810, 58.282, 69.095, 63.434, 69.095, 58.282, 41.810, 58.282, 69.095, 63.434, 69.095, 58.282
Repeating unit: An infinite strip formed by repeating triangles A, B, C, D. Its two edges form right-angled pseudogons.
{3,3,6}
Plane 1: tiled by combination of A, B, and C.
Plane 2: tiled solely by D.
Triangle A: 30, 54.736, 90; BAA
Triangle B: 0, 70.529, 90; ABC
Triangle C: 0, 54.736, 90; CCB
Triangle D: 0, 60, 90; DDD
Vertices:
A-30
Pattern: AAAAAAAAAAAA
Regular hexagon with angle 109.472
A-54.736 & B-70.529 & C-54.736
Pattern: AABCCB
Triangle with angles 0, 0, 60
A-90 & B-90
Pattern: AABB
Quadrangle with angles 0, 125.565, 60, 125.565
B-0 & C-0
Pattern: BBCC...
Apeirogon with angle 125.565, not regular
C-90
Pattern: CCCC
Rhombus with angles 0, 109.472, 0, 109.472
D-0
Pattern: D...
Regular apeirogon with angle 120
D-60
Pattern: DDDDDD
Equilateral triangle with angle 0
D-90
Pattern: DDDD
Rhombus with angles 0, 120, 0, 120
Repeating unit 1: Triangle with angles 0, 30, 90. Composed of 1 A, 1 B, 1 C.
Repeating unit 2: Triangle D.
{4,3,6}
Plane 1: tiled solely by A.
Plane 2: tiled by combination of B and C.
Plane 3: tiled solely by D.
Triangle A: 30, 45, 90; AAA
Triangle B: 0, 54.736, 90; BBC
Triangle C: 0, 35.264, 90; CCB
Triangle D: 0, 45, 90; DDD
Vertices:
A-30
Pattern: AAAAAAAAAAAA
Right-angled dodecagon
A-45
Pattern: AAAAAAAA
Regular octagon with angle 60
A-90
Pattern: AAAA
Rhombus with angles 60, 90, 60, 90
B-0
Pattern: B...
Regular apeirogon with angle 109.472
B-54.736 & C-35.264
Pattern: BBCCBBCC
Rectangle with angle 0
B-90 & C-90
Pattern: BBCC
Quadrangle with angles 0, 90, 0, 90
C-0
Pattern: C...
Regular apeirogon with angle 70.529
D-0
Pattern: D...
Right-angled apeirogon
D-45
Pattern: DDDDDDDD
Square with angle 0
D-90
Pattern: DDDD
Rhombus with angles 0, 90, 0, 90
Repeating unit 1: Triangle A.
Repeating unit 2: Triangle with angles 0, 0, 90. Composed of 1 B and 1 C.
Repeating unit 3: Triangle D.
{5,3,6}
Plane 1: tiled by combination of A, B, and C.
Plane 2: tiled solely by D.
Triangle A: 30, 31.717, 90; BAA
Triangle B: 0, 37.377, 90; ABC
Triangle C: 0, 20.905, 90; CCB
Triangle D: 0, 36, 90; DDD
Vertices:
A-30
Pattern: AAAAAAAA
Regular hexagon with angle 63.434
A-31.717 & B-37.377 & C-20.905
Pattern: AABCCBAABCCB
Hexagon with angles 0, 0, 60, 0, 0, 60
A-90 & B-90
Pattern: AABB
Quadrangle with angles 0, 69.095, 60, 69.095
B-0 & C-0
Pattern: BBCC...
Apeirogon with angle 58.282, not regular
C-90
Pattern: CCCC
Rhombus with angles 0, 41.810, 0, 41.810
D-0
Pattern: D...
Regular apeirogon with angle 72
D-36
Pattern: DDDDDDDDDD
Regular pentagon with angle 0
D-90
Pattern: DDDD
Rhombus with angles 0, 72, 0, 72
Repeating unit 1: Quadrangle with angles 0, 30, 90, 90. Composed of 1 A, 1 B, 1 C.
Repeating unit 2: Triangle D.
{6,3,6}
A and D triangles identical.
B and C triangles identical.
Plane 1: tiled solely by A
Plane 2: tiled by combination of B and C
Plane 3: tiled solely by D
Triangle A: 0, 30, 90; AAA
Triangle B: 0, 0, 90; BBC
Triangle C: 0, 0, 90; CCB
Triangle D: 0, 30, 90; DDD
Vertices:
A-0
Pattern: A...
Regular apeirogon with angle 60
A-30
Pattern: AAAAAAAAAAAA
Regular hexagon with angle 0
A-90
Pattern: AAAA
Rhombus with angles 0, 60, 0, 60
B-0 & C-0
Pattern: BBCC...
Regular apeirogon with angle 0
B-0 & C-0
Pattern: BBCC...
Regular apeirogon with angle 0
B-90
Pattern: BBBB
Square with angle 0
C-90
Pattern: CCCC
Square with angle 0
D-0
Pattern: D...
Regular apeirogon with angle 60
D-30
Pattern: DDDDDDDDDDDD
Regular hexagon with angle 0
D-90
Pattern: DDDD
Rhombus with angles 0, 60, 0, 60
Repeating unit 1: Triangle A.
Repeating unit 2: Rhombus of angles 0, 90, 0, 90. Composed of 1 B, 1 C.
Repeating unit 3: Triangle D.
{3,4,4}
Plane 1: tiled by combination of A and B.
Plane 2: tiled solely by C.
Plane 3: tiled solely by D.
Triangle A: 35.264, 45, 90; ABA
Triangle B: 0, 54.736, 90; ABB
Triangle C: 0, 45, 90; CCC
Triangle D: 0, 60, 90; CCC
Vertices:
A-35.264 & B-54.736
Rhombus with angles 0, 90, 0, 90
A-45
Pattern: AAAAAAAA
Square with angle 70.529
A-90 & B-90
Pattern: AABB
Quadrangle with angles 0, 90, 90, 90
B-0
Pattern: B...
Regular apeirogon with angle 109.472
C-0
Pattern: C...
Right-angled apeirogon
C-45
Pattern: CCCCCCCC
Square with angle 0
C-90
Pattern: CCCC
Rhombus with angles 0, 90, 0, 90
D-0
Pattern: D...
Regular apeirogon with angle 120
D-60
Pattern: DDDDDD
Equilateral triangle with angle 0
D-90
Pattern: DDDD
Rhombus with angles 0, 120, 0, 120
Repeating unit 1: Triangle with angles 0, 45, 90. Composed of 1 A and 1 B.
Repeating unit 2: Triangle C.
Repeating unit 3: Triangle D.
{4,4,4}
A and D triangles identical.
B and C triangles identical.
Plane 1: tiled solely by A.
Plane 2: tiled solely by B.
Plane 3: tiled solely by C.
Plane 4: tiled solely by D.
Triangle A: 0, 45, 90; AAA
Triangle B: 0, 0, 90; BBB
Triangle C: 0, 0, 90; CCC
Triangle D: 0, 45, 90; DDD
Vertices:
A-0
Pattern: A...
Right-angled apeirogon
A-45
Pattern: AAAAAAAA
Square with angle 0
A-90
Pattern: AAAA
Rhombus with angles 0, 90, 0, 90
B-0
Pattern: B...
Regular apeirogon with angle 0
B-0
Pattern: B...
Regular apeirogon with angle 0
B-90
Pattern: BBBB
Square with angle 0
C-0
Pattern: C...
Regular apeirogon with angle 0
C-0
Pattern: C...
Regular apeirogon with angle 0
C-90
Pattern: CCCC
Square with angle 0
D-0
Pattern: D...
Right-angled apeirogon
D-45
Pattern: DDDDDDDD
Square with angle 0
D-90
Pattern: DDDD
Rhombus with angles 0, 90, 0, 90
Repeating unit 1: Triangle A.
Repeating unit 2: Triangle B.
Repeating unit 3: Triangle C.
Repeating unit 4: Triangle D.
{3,6,3}
A and D triangles identical.
B and C triangles identical.
Plane 1: tiled by combination of A and B
Plane 2: tiled by combination of C and D
Triangle A: 0, 60, 90; ABA
Triangle B: 0, 0, 90; ABB
Triangle C: 0, 0, 90; CDC
Triangle D: 0, 60, 90; DCD
Vertices:
A-0 & B-0
Pattern: AABB...
Apeirogon with alternating angles 0 and 120
A-60
Pattern: AAAAAA
Equilateral triangle with angle 0
A-90 & B-90
Pattern: AABB
Quadrangle with angles 0, 0, 0, 120
B-0
Pattern: B...
Regular apeirogon with angle 0
C-0
Pattern: C...
Regular apeirogon with angle 0
C-0 & D-0
Pattern: CCDD...
Apeirogon with alternating angles 0 and 120
C-90 & D-90
Pattern: CCDD
Quadrangle with angles 0, 0, 0, 120
D-60
Pattern: DDDDDD
Equilateral triangle with angle 0
Repeating unit 1: Triangle with angles 0, 0, 60. Composed of 1 A, 1 B.
Repeating unit 2: Triangle with angles 0, 0, 60. Composed of 1 C, 1 D.
Branched 336
A and C triangles identical.
Plane 1: tiled by combination of A, B, and C
Plane 2: tiled solely by D
Triangle A: 0, 54.736, 90 ; AAB
Triangle B: 0, 0, 70.529; ACB
Triangle C: 0, 54.736, 90 ; CCB
Triangle D: 0, 0, 90 ; DDD
Vertices:
A-0 & B-0
Pattern: AABB...
Apeirogon with alternating angles 0 and 109.472
A-54.736 & B-70.529 & C-54.736
Pattern: AABCCB
Hexagon with all angles 0, not regular
A-90
Pattern: AAAA
Rhombus with angles 0, 109.472, 0, 109.472
B-0 & C-0
Pattern: CCDD...
Apeirogon with alternating angles 0 and 109.472
C-90
Pattern: CCCC
Rhombus with angles 0, 109.472, 0, 109.472
D-0
Pattern: D...
Regular apeirogon with angle 0
D-0
Pattern: D...
Regular apeirogon with angle 0
D-90
Pattern: DDDD
Square with angle 0
Repeating unit 1: Quadrangle with angles 0, 0, 90, 90. Composed of 1 A, 1 B, 1 C.
Repeating unit 2: Triangle D.
Branched 344
C and D triangles identical.
Plane 1: tiled by combination of A and B
Plane 2: tiled solely by C
Plane 3: tiled solely by D
Triangle A: 35.264, 35.264, 90 ;
Triangle B: 0, 54.736, 54.736;
Triangle C: 0, 45, 90 ;
Triangle D: 0, 45, 90 ;
Vertices:
A-35.264 & B-54.736
Pattern: AABB
Hexagon with angles 0, 90, 90, 0, 90, 90
A-35.264 & B-54.736
Pattern: AABB
Hexagon with angles 0, 90, 90, 0, 90, 90
A-90
Pattern: AAAA
Square with angle 70.529
B-0
Pattern: B...
Regular apeirogon with angle 109.472
C-0
Pattern: C...
Right-angled apeirogon
C-45
Pattern: CCCCCCCC
Square with angle 0
C-90
Pattern: CCCC
Rhombus with angles 0, 90, 0, 90
D-0
Pattern: D...
Right-angled apeirogon
D-45
Pattern: DDDDDDDD
Square with angle 0
D-90
Pattern: DDDD
Rhombus with angles 0, 90, 0, 90
Repeating unit 1: Quadrangle with angles 0, 90, 90, 90. Composed of 1 A and 1 B.
Repeating unit 2: Triangle C.
Repeating unit 3: Triangle D.
Branched 444
A, C and D triangles identical.
Plane 1: tiled solely by A
Plane 2: tiled solely by B
Plane 3: tiled solely by C
Plane 4: tiled solely by D
Triangle A: 0, 0, 90; AAA
Triangle B: 0, 0, 0 ; BBB
Triangle C: 0, 0, 90; CCC
Triangle D: 0, 0, 90; DDD
Vertices:
A-0
Pattern: A...
Regular apeirogon with angle 0
A-0
Pattern: A...
Regular apeirogon with angle 0
A-90
Pattern: AAAA
Square with angle 0
B-0
Pattern: B...
Regular apeirogon with angle 0
B-0
Pattern: B...
Regular apeirogon with angle 0
B-0
Pattern: B...
Regular apeirogon with angle 0
C-0
Pattern: C...
Regular apeirogon with angle 0
C-0
Pattern: C...
Regular apeirogon with angle 0
C-90
Pattern: CCCC
Square with angle 0
D-0
Pattern: D...
Regular apeirogon with angle 0
D-0
Pattern: D...
Regular apeirogon with angle 0
D-90
Pattern: CCCC
Square with angle 0
Repeating unit 1: Triangle A.
Repeating unit 2: Triangle B.
Repeating unit 3: Triangle C.
Repeating unit 4: Triangle D.
Cyclical 3336
A and D triangles identical.
B and C triangles identical.
One plane tiled by combination of A, B, C, and D.
Triangle A: 0, 0, 54.736; BAA
Triangle B: 0, 54.736, 70.529; CAB
Triangle C: 0, 54.736, 70.529; BDC
Triangle D: 0, 0, 54.736; DCD
Vertices:
A-0
Pattern: A...
Regular apeirogon with angle 0
A-0 & B-0
Pattern: AABB...
Apeirogon with angles 0, 125.565, 109.472, 125.565, 0...
A-54.736 & B-70.529 & C-54.736
Pattern: AABCCB
Hexagon with angles 0, 0, 0, 125.565, 0, 125.565
B-54.736 & C-70.529 & D-54.736
Pattern: AABCCB
Hexagon with angles 0, 0, 0, 125.565, 0, 125.565
C-0 & D-0
Pattern: CCDD...
Apeirogon with angles 0, 125.565, 109.472, 125.565, 0...
D-0
Pattern: D...
Regular apeirogon with angle 0
Repeating unit: Rectangle with angle 0. Composed of 1 A, 1 B, 1 C, 1 D.
Cyclical 3436
A and D triangles identical.
B and C triangles identical.
Plane 1: tiled by combination of A and B
Plane 2: tiled by combination of C and D
Triangle A: 0, 0, 35.264; BAA
Triangle B: 0, 45, 54.736; ABA
Triangle C: 0, 45, 54.736; CDC
Triangle D: 0, 0, 35.264; DCD
Vertices:
A-0
Pattern: A...
Apeirogon with alternating angles 0 and 70.529
A-0 & B-0
Pattern: AABB...
Apeirogon with angles 0, 90, 90, 90...
A-35.264 & B-54.736
Pattern: AABBAABB
Octagon with angles 0, 0, 0, 90, 0, 0, 0, 90
B-45
Pattern: BBBBBBBB
Octagon with alternating angles 0 and 109.472
C-0 & D-0
Pattern: CCDD...
Apeirogon with angles 0, 90, 90, 90...
C-45
Pattern: CCCCCCCC
Octagon with alternating angles 0 and 109.472
C-54.736 & D-35.264
Pattern: CCDDCCDD
Octagon with angles 0, 0, 0, 90, 0, 0, 0, 90
D-0
Pattern: D...
Apeirogon with alternating angles 0 and 70.529
Repeating unit 1: Quadrangle with angles 0, 0, 45, 90. Composed of 1 A and 1 B.
Repeating unit 2: Quadrangle with angles 0, 0, 45, 90. Composed of 1 A and 1 B.
Cyclical 3536
A and D triangles identical.
B and C triangles identical.
One plane tiled by combination of A, B, C, and D.
Triangle A: 0, 0, 20.905; BAA
Triangle B: 0, 31.717, 37.377; CAB
Triangle C: 0, 31.717, 37.377; BDC
Triangle D: 0, 0, 20.905; DCD
Vertices:
A-0
Pattern: A...
Apeirogon with alternating angles 0 and 41.810
A-0 & B-0
Pattern: AABB...
Apeirogon with angles 0, 58.282, 63.434, 58.282, 0...
A-20.905 & B-37.377 & C-31.717
Pattern: AABCCBAABCCB
Dodecagon with angles 0, 0, 0, 69.095, 0, 69.095, 0, 0, 0, 69.095, 0, 69.095
B-31.717 & C-37.377 & D-20.905
Pattern: BBCDDCBBCDDC
Dodecagon with angles 0, 0, 0, 69.095, 0, 69.095, 0, 0, 0, 69.095, 0, 69.095
C-0 & D-0
Pattern: CCDD...
Apeirogon with angles 0, 58.282, 63.434, 58.282, 0...
D-0
Pattern: D...
Apeirogon with alternating angles 0 and 41.810
Repeating unit: Hexagon with angles 0, 0, 90, 0, 0, 90. Composed of 1 A, 1 B, 1 C, 1 D.
Cyclical 3636
Triangles A, B, C, and D all identical.
Plane 1: tiled by combination of A and B
Plane 2: tiled by combination of C and D
Triangle A: 0, 0, 0; BAA
Triangle B: 0, 0, 0; ABB
Triangle C: 0, 0, 0; CCD
Triangle D: 0, 0, 0; DDC
Vertices:
A-0
Pattern: A...
Regular apeirogon with angle 0
A-0 & B-0
Pattern: AABB...
Regular apeirogon with angle 0
A-0 & B-0
Pattern: AABB...
Regular apeirogon with angle 0
B-0
Pattern: B...
Regular apeirogon with angle 0
C-0
Pattern: C...
Regular apeirogon with angle 0
C-0 & D-0
Pattern: CCDD...
Regular apeirogon with angle 0
C-0 & D-0
Pattern: CCDD...
Regular apeirogon with angle 0
D-0
Pattern: D...
Regular apeirogon with angle 0
Repeating unit 1: Square with angle 0. Composed of 1 A, 1 B.
Repeating unit 2: Square with angle 0. Composed of 1 C, 1 D.
Cyclical 3344
A and C triangles identical
Plane 1: tiled by combination of A, B, and C.
Plane 2: tiled solely by D.
Triangle A: 0, 54.736, 54.736; BAA
Triangle B: 35.264, 35.264, 70.529; ACB
Triangle C: 0, 54.736, 54.736; BCC
Triangle D: 0, 45, 45 ; DDD
Vertices:
A-0
Pattern: A...
Regular apeirogon with angle 109.472
A-54.736 & B-35.264
Pattern: AABBAABB
Octagon with angles 0, 125.565, 70.529, 125.565, 0, 125.565, 70.529, 125.565
A-54.736 & B-70.529 & C-54.736
Pattern: AABCCB
Hexagon with angles 0, 90, 90, 0, 90, 90
B-35.264 & C-54.736
Pattern: BBCCBBCC
Octagon with angles 0, 125.565, 70.529, 125.565, 0, 125.565, 70.529, 125.565
C-0
Pattern: C...
Regular apeirogon with angle 109.472
D-0
Pattern: D...
Right-angled apeirogon
D-45
Pattern: DDDDDDDD
Octagon with alternating angles 0 and 90
D-45
Pattern: DDDDDDDD
Octagon with alternating angles 0 and 90
Repeating unit 1: Quadrangle with angles 0, 0, 90, 90. Composed of 1 A, 1 B, 1 C.
Repeating unit 2: Triangle D.
Cyclical 3444
A and B triangles identical
C and D triangles identical
Plane 1: tiled by combination of A and B.
Plane 2: tiled solely by C.
Plane 3: tiled solely by D.
Triangle A: 0, 35.264, 54.736; BAA
Triangle B: 0, 35.264, 54.736; ABB
Triangle C: 0, 0, 45 ; CCC
Triangle D: 0, 0, 45 ; DDD
Vertices:
A-0
Pattern: A...
Apeirogon with alternating angles 70.529 and 109.472
A-35.264 & B-54.736
Pattern: AABBAABB
Octagon with angles 0, 90, 0, 90, 0, 90, 0, 90
A-54.736 & B-35.264
Pattern: AABBAABB
Octagon with angles 0, 90, 0, 90, 0, 90, 0, 90
B-0
Pattern: B...
Apeirogon with alternating angles 70.529 and 109.472
C-0
Pattern: C...
Apeirogon with alternating angles 0 and 90
C-0
Pattern: C...
Apeirogon with alternating angles 0 and 90
C-45
Patttern: CCCCCCCC
Regular octagon with angle 0
D-0
Pattern: D...
Apeirogon with alternating angles 0 and 90
D-0
Pattern: D...
Apeirogon with alternating angles 0 and 90
D-45
Patttern: DDDDDDDD
Regular octagon with angle 0
Repeating unit 1: Quadrangle with angles 0, 90, 0, 90. Composed of 1 A and 1 B.
Repeating unit 2: Triangle C.
Repeating unit 3: Triangle D.
Cyclical 4444
Triangles A, B, C, and D all identical.
Plane 1: tiled solely by A.
Plane 2: tiled solely by B.
Plane 3: tiled solely by C.
Plane 4: tiled solely by D.
Triangle A: 0, 0, 0; AAA
Triangle B: 0, 0, 0; BBB
Triangle C: 0, 0, 0; CCC
Triangle D: 0, 0, 0; DDD
Vertices:
A-0
Pattern: A...
Regular apeirogon with angle 0
A-0
Pattern: A...
Regular apeirogon with angle 0
A-0
Pattern: A...
Regular apeirogon with angle 0
B-0
Pattern: B...
Regular apeirogon with angle 0
B-0
Pattern: B...
Regular apeirogon with angle 0
B-0
Pattern: B...
Regular apeirogon with angle 0
C-0
Pattern: C...
Regular apeirogon with angle 0
C-0
Pattern: C...
Regular apeirogon with angle 0
C-0
Pattern: C...
Regular apeirogon with angle 0
D-0
Pattern: D...
Regular apeirogon with angle 0
D-0
Pattern: D...
Regular apeirogon with angle 0
D-0
Pattern: D...
Regular apeirogon with angle 0
Repeating unit 1: Triangle A.
Repeating unit 2: Triangle B.
Repeating unit 3: Triangle C.
Repeating unit 4: Triangle D.
Triangle with added 3-branch
A and B triangles identical.
One plane tiled by combination of A, B, C, and D.
Triangle A: 0, 54.736, 90; ABC
Triangle B: 0, 54.736, 90; BAC
Triangle C: 0, 70.529, 70.529; DAB
Triangle D: 54.736, 54.736, 60; DDC
Vertices:
A-0 & B-0 & C-0
Pattern: ABC...
Apeirogon with angle 125.565; not regular
A-54.736 & C-70.529 & D-54.736
Pattern: AACDDC
Pentagon with angles 0, 0, 125.565, 120, 125.565
A-90 & B-90
Pattern: AABB
Rhombus with angles 0, 109.472, 0, 109.472
B-54.736 & C-70.529 & D-54.736
Pattern: BBCDDC
Pentagon with angles 0, 0, 125.565, 120, 125.565
D-60
Pattern: DDDDDD
Regular hexagon with angle 109.472
Repeating unit: Regular apeirogon with angle 60. Basically the full A-0 B-0 C-0 vertex with triangle D capping the finite sides of triangles C. A quadrangle with angles 0, 90, 60, 90 can also be considered a repeating unit, but when reflecting through a 0-90 side, triangles A and B will switch.
Triangle with added 4-branch
A and B triangles identical.
Plane 1: tiled by combination of A, B, and C.
Plane 2: tiled solely by D.
Triangle A: 0, 35.264, 90 ; ABC
Triangle B: 0, 35.264, 90 ; BAC
Triangle C: 0, 54.736, 54.736; CAB
Triangle D: 45, 45, 60 ; DDD
Vertices:
A-0 & B-0 & C-0
Pattern: ABC...
Non-regular right-angled apeirogon
A-35.264 & C-54.736
Pattern: AACCAACC
Hexagon with angles 0, 0, 109.472, 0, 0, 109.472
A-90 & B-90
Pattern: AABB
Rhombus with angles 0, 70.529, 0, 70.529
B-35.264 & C-54.736
Pattern: BBCCBBCC
Hexagon with angles 0, 0, 109.472, 0, 0, 109.472
D-45
Pattern: DDDDDDDD
Octagon with alternating angles 90 and 120
D-45
Pattern: DDDDDDDD
Octagon with alternating angles 90 and 120
D-60
Pattern: DDDDDD
Right-angled hexagon.
Repeating unit 1: The basic one is a pentagon with angles 0, 90, 90, 90, 90, but reflecting through a 0-90 side will switch A and B. Full repeating unit is a non-regular right-angled apeirogon.
Repeating unit 2: Triangle D.
Triangle with added 5-branch
A and B triangles identical.
One plane tiled by combination of A, B, C, and D.
Triangle A: 0, 20.905, 90 ; ABC
Triangle B: 0, 20.905, 90 ; BAC
Triangle C: 0, 37.377, 37.377; DAB
Triangle D: 31.717, 31.717, 60 ; DDC
Vertices:
A-0 & B-0 & C-0
Pattern: ABC...
Apeirogon with angle 58.282, not regular
A-20.905 & C-37,377 & D-31.317
Pattern: AACDDCAACDDC
Decagon with angles 0, 0, 69.094, 120, 69.094, 0, 0, 69.094, 120, 69.094
A-90 & B-90
Pattern: AABB
Rhombus with angles 0, 41.810, 0, 41.810
B-20.905 & C-37.377 & 31.717
Pattern: BBCDDCBBCDDC
Decagon with angles 0, 0, 69.094, 120, 69.094, 0, 0, 69.094, 120, 69.094
D-60
Pattern: DDDDDD
Regular hexagon with angle 63.434
Repeating unit: The basic one is a hexagon with angles 0, 90, 90, 60, 90, 90, but reflecting through a 0-90 side will switch A and B. Full repeating unit is an apeirogon with repeating angle sequence 60, 90, 90.
Triangle with added 6-branch
A and B triangles identical.
Plane 1: tiled by combination of A, B, and C.
Plane 2: tiled solely by D.
Triangle A: 0b, 0d, 90c; BAC
Triangle B: 0a, 0d, 90c; ABC
Triangle C: 0a, 0b, 0d ; ABC
Triangle D: 0a, 0b, 60c; DDD
Vertices:
A-0 & B-0 & C-0
Pattern: ABC...
Apeirogon with angle 0
A-0 & C-0
Pattern: AACC...
Apeirogon with angle 0
A-90 & B-90
Pattern: AABB
Square with angle 0
B-0 & C-0
Pattern: BBCC...
Apeirogon with angle 0
D-0
Pattern: D...
Apeirogon with alternating angles 0 and 120
D-0
Pattern: D...
Apeirogon with alternating angles 0 and 120
D-60
Pattern: DDDDDD
Regular hexagon with angle 0
Repeating unit 1: The basic one is a pentagon with angles 0, 0, 90, 0, 90, but reflecting through a 0-90 side will switch A and B. Full repeating unit is an apeirogon with angle 0.
Repeating unit 2: Triangle D.
Two fused triangles
A and D triangles identical.
B and C triangles identical.
One plane tiled by combination of A, B, C, and D.
Triangle A: 0, 54.736, 54.736; ABC
Triangle B: 0, 0, 70.529; ADC
Triangle C: 0, 0, 70.529; ADB
Triangle D: 0, 54.736, 54.736; DCB
Vertices:
A-0 & B-0 & C-0
Pattern: ABC...
Apeirogon with angles 0, 125.565, 125.565...
A-54.736 & B-70.529 & D-54.736
Pattern: AABDDB
Hexagon with angles 0, 0, 109.472, 0, 0, 109.472
A-54.736 & C-70.529 & D-54.736
Pattern: AACDDC
Hexagon with angles 0, 0, 109.472, 0, 0, 109.472
B-0 & C-0 & D-0
Pattern: BCD...
Apeirogon with angles 0, 125.565, 125.565...
Repeating unit: The notion of repeating unit starts breaking down a bit here. The ABD or ACD vertices work, with some swaps caused by reflection.
Tetrahedron of 3-edges
Triangles A, B, C, and D all identical.
One plane tiled by combination of A, B, C, and D.
Triangle A: 0, 0, 0; BCD
Triangle B: 0, 0, 0; ACD
Triangle C: 0, 0, 0; ABD
Triangle D: 0, 0, 0; ABC
Vertices:
A-0 & B-0 & C-0
Pattern: ABC...
Regular apeirogon with angle 0
A-0 & B-0 & D-0
Pattern: ABC...
Regular apeirogon with angle 0
A-0 & C-0 & D-0
Pattern: ABC...
Regular apeirogon with angle 0
B-0 & C-0 & D-0
Pattern: ABC...
Regular apeirogon with angle 0
Repeating unit: Each of the ideal triangles can be considered a repeating unit; they are just differently labeled.
{3,3,3,3}
AB and DE triangles identical.
AC and CE triangles identical.
AD and BE triangles identical.
BC and CD triangles identical.
Plane 1: tiled by combination of AB, BC, CD, and DE.
Plane 2: tiled by combination of AC, AD, AE, BD, BE, and CE.
Triangle AB: 52.239, 60, 90; AB BC AB
Triangle AC: 54.736, 65.905, 90; AC AC AD
Triangle AD: 48.190, 70.529, 90; AE BD AC
Triangle AE: 54.736, 54.736, 90; AD BE AE
Triangle BC: 52.239, 75.522, 90; AB BC CD
Triangle BD: 65.905, 65.905, 90; AD BE BD
Triangle BE: 48.190, 70.529, 90; BD AE CE
Triangle CD: 52.239, 75.522, 90; DE CD BC
Triangle CE: 54.736, 65.905, 90; CE CE BE
Triangle DE: 52.239, 60, 90; DE CD DE
Repeating unit 1: Digonal strip of angle 60, made from AB, BC, CD, DE.
Repeating unit 2: Digonal strip of angle 90, made from AC, AD, AE, BD, BE, CE.
{3,3,3,4}
Plane 1: tiled by combination of AB, BC, and CD.
Plane 2: tiled by combination of AC, AD, and BD.
Plane 3: tiled by combination of AE, BE, and CE.
Plane 4: tiled solely by DE.
Triangle AB: 45, 52.239, 90; BC AB AB
Triangle AC: 35.264, 65.905, 90; AC AC AD
Triangle AD: 48.190, 54.736, 90; AD BD AC
Triangle AE: 45, 54.736, 90; BE AE AE
Triangle BC: 30, 75.522, 90; AB BC CD
Triangle BD: 45, 65.905, 90; AD BD BD
Triangle BE: 35.264, 70.529, 90; AE BE CE
Triangle CD: 52.239, 60, 90; CD CD BC
Triangle CE: 54.736, 54.736, 90; CE CE BE
Triangle DE: 45, 60, 90; DE DE DE
Repeating unit 1: Equilateral triangle with angle 90, made from AB, BC, and CD.
Repeating unit 2: Triangle with angles 45, 90, 90, made from AC, AD, and BD.
Repeating unit 3: Triangle with angles 45, 90, 90, made from AE, BE, and CE.
Repeating unit 4: Triangle DE.
Demipenteractic
A, B are one branch, C is the center, D and E are ends of short branches.
AD and AE triangles identical.
BD and BE triangles identical.
CD and CE triangles identical.
Plane 1: tiled by combination of AB, BC, CD, and CE.
Plane 2: tiled by combination of AC, AD, AE, BD, and BE.
Plane 3: tiled solely by DE.
Triangle AB: 52.239, 52.239, 90 ; AB AB BC
Triangle AC: 65.905, 65.905, 70.529; AD AE AC
Triangle AD: 48.190, 54.736, 90 ; AD BD AC
Triangle AE: 48.190, 54.736, 90 ; AE BE AC
Triangle BC: 75.522, 75.522, 90 ; CD CE AB
Triangle BD: 45, 65.905, 90 ; AD BD BD
Triangle BE: 45, 65.905, 90 ; AE BE BE
Triangle CD: 52.239, 60, 90 ; CE CD BC
Triangle CE: 52.239, 60, 90 ; CD CE BC
Triangle DE: 45, 60, 90 ; DE DE DE
Repeating unit 1: Digonal strip of angle 90, made from AB, BC, CD, and CE (two of each). Half of it, equilateral triangle of angle 90, works, but swaps some labels when reflecting.
Repeating unit 1: Digonal strip of angle 45, made from AC, AD, AE, BD, and BE.
Repeating unit 3: Triangle DE.
C~4 {4,3,3,4}
AB and DE triangles identical.
AC and CE triangles identical.
AD and BE triangles identical.
BC and CD triangles identical.
Plane 1: tiled solely by AB.
Plane 2: tiled by combination of AC and AD.
Plane 3: tiled solely by AE.
Plane 4: tiled by combination of BC and CD.
Plane 5: tiled solely by BD.
Plane 6: tiled by combination of BE and CE.
Plane 7: tiled solely by DE.
Triangle AB: 45, 45, 90; AB AB AB
Triangle AC: 35.264, 54.736, 90; AC AC AD
Triangle AD: 35.264, 54.736, 90; AD AD AC
Triangle AE: 45, 45, 90; AE AE AE
Triangle BC: 30, 60, 90; BC BC CD
Triangle BD: 45, 45, 90; BD BD BD
Triangle BE: 35.264, 54.736, 90; BE BE CE
Triangle CD: 30, 60, 90; CD CD BC
Triangle CE: 35.264, 54.736, 90; CE CE BE
Triangle DE: 45, 45, 90; DE DE DE
Repeating unit 1: Triangle AB
Repeating unit 2: Rectangle made from AC and AD.
Repeating unit 3: Triangle AE
Repeating unit 4: Rectangle made from BC and CD.
Repeating unit 5: Triangle BD
Repeating unit 6: Rectangle made from BE and CE.
Repeating unit 7: Triangle DE
F~4 {3,3,4,3}
Plane 1: tiled by combination of AB and BC.
Plane 2: tiled solely by AC.
Plane 3: tiled by combination of AD, AE, BD, BE, and CE.
Plane 4: tiled solely by CD.
Plane 5: tiled solely by DE.
Triangle AB: 30, 60, 90; AB BC AB
Triangle AC: 45, 45, 90; AC AC AC
Triangle AD: 35.264, 54.736, 90; AE BD AD
Triangle AE: 35.264, 54.736, 90; BE AD AE
Triangle BC: 30, 60, 90; BC AB BC
Triangle BD: 35.264, 54.736, 90; AD BE BD
Triangle BE: 19.471, 70.529, 90; AE BD CE
Triangle CD: 45, 45, 90; CD CD CD
Triangle CE: 35.264, 54.736, 90; CE CE BE
Triangle DE: 30, 60, 90; DE DE DE
Repeating unit 1: Equilateral triangle made from AB and BC.
Repeating unit 2: Triangle AC.
Repeating unit 3: This is an interesting one. The five triangles AD, AE, BD, BE, and CE form a rectangle. Four of these five triangles are similar; only the "central" one, BE, has a different shape.
Repeating unit 4: Triangle CD.
Repeating unit 5: Triangle DE.
B~4 Branched Euclidean group (half of tesseractic honeycomb)
Same marking as demipenteractic, AB branch is 4, BC, CD and DE are 3.
AD and AE triangles identical.
BD and BE triangles identical.
CD and CE triangles identical.
Plane 1: tiled solely by AB.
Plane 2: tiled by combination of AC, AD, and AE.
Plane 3: tiled by combination of BC, CD, and CE.
Plane 4: tiled solely by BD.
Plane 5: tiled solely by BE.
Plane 6: tiled solely by DE.
Triangle AB: 45, 45, 90 ; AB AB AB
Triangle AC: 54.736, 54.736, 70.529; AD AE AC
Triangle AD: 35.264, 54.736, 90 ; AD AD AC
Triangle AE: 35.264, 54.736, 90 ; AE AE AC
Triangle BC: 60, 60, 60 ; BC CD CE
Triangle BD: 45, 45, 90 ; BD BD BD
Triangle BE: 45, 45, 90 ; BE BE BE
Triangle CD: 30, 60, 90 ; CD CD BC
Triangle CE: 30, 60, 90 ; CE CE BC
Triangle DE: 45, 45, 90 ; DE DE DE
Repeating unit 1: Triangle AB.
Repeating unit 2: Rectangle made from triangles AC, AD, and AE.
Repeating unit 3: Rectangle made from triangles BC, CD, and CE, two of each. Half of it works as well, but swaps some labels when reflecting.
Repeating unit 4: Triangle BE.
Repeating unit 5: Triangle BE.
Repeating unit 6: Triangle DE.
D~4 Cross group (quarter of tesseractic honeycomb)
AB, BC, BD, BE branches are 3.
AB, BC, BC, BE triangles identical.
AC, AD, AE, CD, CE, DE triangles identical.
Plane 1: tiled by combination of AB, BC, BD, and BE.
Plane 2: tiled solely by AC.
Plane 3: tiled solely by AD.
Plane 4: tiled solely by AE.
Plane 5: tiled solely by CD.
Plane 6: tiled solely by CE.
Plane 7: tiled solely by DE.
Triangle AB: 60, 60, 60; BC BD BE
Triangle AC: 45, 45, 90; AC AC AC
Triangle AD: 45, 45, 90; AD AD AD
Triangle AE: 45, 45, 90; AE AE AE
Triangle BC: 60, 60, 60; AB BD BE
Triangle BD: 60, 60, 60; AB BC BE
Triangle BE: 60, 60, 60; AB BC BD
Triangle CD: 45, 45, 90; CD CD CD
Triangle CE: 45, 45, 90; CE CE CE
Triangle DE: 45, 45, 90; DE DE DE
Repeating unit 1: Plane 1 is tiled by equilateral triangles with a particular 4-coloring such as that each vertex has triangles of three colors around it, with opposite pairs colored alike, and each triangle has three vertices with different color combinations.
Repeating unit 2: Triangle AC.
Repeating unit 3: Triangle AD.
Repeating unit 4: Triangle AE.
Repeating unit 5: Triangle CD.
Repeating unit 6: Triangle CE.
Repeating unit 7: Triangle DE.
A~4 Cyclical 33333
AB, AE, BC, CD, DE triangles identical.
AC, AD, BD, BE, CE triangles identical.
Plane 1: tiled by combination of AB, AE, BC, CD, and DE.
Plane 2: tiled by combination of AC, AD, BD, BE, and CE.
Triangle AB: 52.239, 52.239, 75.522; AE BC AB
Triangle AC: 48.190, 65.905, 65.905; AC AD CE
Triangle AD: 48.190, 65.905, 65.905; AD AC BD
Triangle AE: 52.239, 52.239, 75.522; AB DE AE
Triangle BC: 52.239, 52.239, 75.522; AB CD BC
Triangle BD: 48.190, 65.905, 65.905; BD AD BE
Triangle BE: 48.190, 65.905, 65.905; BE BD CE
Triangle CD: 52.239, 52.239, 75.522; BC DE CD
Triangle CE: 48.190, 65.905, 65.905; CE AC BE
Triangle DE: 52.239, 52.239, 75.522; AE CD DE
Repeating unit 1: Triangles form strips where five "colors" repeat endlessly.
Repeating unit 2: Triangles form strips where five "colors" repeat endlessly.
H4 {3,3,3,5}
Plane 1: tiled by combination of AB, BC, and CD.
Plane 2: tiled by combination of AC, AD, AE, BD, BE, and CE.
Plane 3: tiled solely by DE.
Triangle AB: 36, 52.239, 90; BC AB AB
Triangle AC: 20.905, 65.905, 90; AC AC AD
Triangle AD: 37.377, 48.190, 90; BD AE AC
Triangle AE: 31.717, 54.736, 90; BE AD AE
Triangle BC: 7.761, 75.522, 90; AB BC CD
Triangle BD: 13.283, 65.905, 90; AD BE BD
Triangle BE: 10.812, 70.529, 90; AE BD CE
Triangle CD: 22.239, 52.239, 90; CD CD BC
Triangle CE: 20.905, 54.736, 90; CE CE BE
Triangle DE: 18, 60, 90; DE DE DE
Repeating unit 1: Triangle with angles 30, 36, 90, made from AC, BC, and CD.
Repeating unit 2: Quadrangle with angles 45, 90, 90, 90, made from AC, AD, AE, BD, BE, and CE.
Repeating unit 3: Triangle DE.
BH4 {4,3,3,5}
Plane 1: tiled solely by AB.
Plane 2: tiled by combination of AC, AD, and AE.
Plane 3: tiled by combination of BC and CD.
Plane 4: tiled by combination of BD, BE, and CE.
Plane 5: tiled solely by DE.
Triangle AB: 36, 45, 90; AB AB AB
Triangle AC: 20.905, 54.736, 90; AC AC AD
Triangle AD: 35.264, 37.377, 90; AE AD AC
Triangle AE: 31.717, 45, 90; AE AD AE
Triangle BC: 7.761, 60, 90; BC BC CD
Triangle BD: 13.283, 45, 90; BD BE BD
Triangle BE: 10.812, 54.736, 90; BE BD CE
Triangle CD: 22.239, 30, 90; CD CD BC
Triangle CE: 20.905, 35.264, 90; CE CE BE
Triangle DE: 18, 45, 90; DE DE DE
Repeating unit 1: Triangle AB.
Repeating unit 2: Quadrangle with angles 45, 90, 90, 90, made from AC, AD, and AE.
Repeating unit 3: Quadrangle with angles 30, 90, 90, 90, made from BC and CD.
Repeating unit 4: Quadrangle with angles 45, 45, 90, 90, made from BD, BE, and CE.
Repeating unit 5: Triangle DE.
K4 {5,3,3,5}
AB and DE triangles identical.
AC and CE triangles identical.
AD and BE triangles identical.
BC and CD triangles identical.
Plane 1: tiled solely by AB.
Plane 2: tiled by combination of AC, AD, AE, BD, BE, and CE.
Plane 3: tiled by combination of BC and CD.
Plane 4: tiled solely by DE.
Triangle AB: 18, 36, 90; AB AB AB
Triangle AC: 20.905, 20.905, 90; AC AC AD
Triangle AD: 10.812, 37.377, 90; AE BD AC
Triangle AE: 31.717, 31.717, 90; AD BE AE
Triangle BC: 7.761, 22.239, 90; BC CD BC
Triangle BD: 13.283, 13.283, 90; AD BE BD
Triangle BE: 10.812, 37.377, 90; AE BD CE
Triangle CD: 7.761, 22.239, 90; CD BC CD
Triangle CE: 20.905, 20.905, 90; CE CE BE
Triangle DE: 18, 36, 90; DE DE DE
Repeating unit 1: Triangle AB.
Repeating unit 3: Hexagon with angles 45, 45, 90, 90, 90, 90, made from AC, AD, AE, BD, BE, and CE.
Repeating unit 3: Quadrangle with angles 30, 90, 30, 90, made from BC and CD.
Repeating unit 4: Triangle DE.
DH4 Branched half of {4,3,3,5}
AB is 5, BC, CD and CE are 3
AD and AE triangles identical.
BD and BE triangles identical.
CD and CE triangles identical.
Plane 1: tiled solely by AB.
Plane 2: tiled by combination of AC, AD, AE, BD, and BE.
Plane 3: tiled combination of BC, CD, and CE.
Plane 4: tiled solely by DE.
Triangle AB: 18, 18, 90 ; AB AB AB
Triangle AC: 20.905, 20.905, 70.529; AD AE AC
Triangle AD: 10.812, 54.736, 90 ; AD BD AC
Triangle AE: 10.812, 54.736, 90 ; AE BE AC
Triangle BC: 22.239, 22.239, 60 ; CD CE BC
Triangle BD: 13.283, 45, 90 ; BD AD BD
Triangle BE: 13.283, 45, 90 ; BE AE BE
Triangle CD: 7.761, 60, 90 ; CE CD BC
Triangle CE: 7.761, 60, 90 ; CD CE BC
Triangle DE: 36, 45, 90 ; DE DE DE
Repeating unit 1: Triangle AB.
Repeating unit 2: Quadrangle with all angles 45, made from AC, AD, AE, BD, and BE.
Repeating unit 3: Quadrangle with all angles 30, made from BC, CD, and CE, two of each. Half of it works as well, but swaps some labels when reflecting.
Repeating unit 4: Triangle DE.
AF4 Cyclical 33334
AE is 4, AB, BC, CD, and DE are 3
AB and DE triangles identical.
AC and CE triangles identical.
AD and BE triangles identical.
BC and CD triangles identical.
Plane 1: tiled by combination of AB, BC, CD, and DE.
Plane 2: tiled by combination of AC, AD, BD, BE, and CE.
Plane 3: tiled solely by AE.
Triangle AB: 30, 52.239, 60 ; BC AB AB
Triangle AC: 35.264, 45, 65.905; AC AD AC
Triangle AD: 35.264, 48.190, 54.736; AC BD AD
Triangle AE: 45, 45, 45 ; AE AE AE
Triangle BC: 30, 52.239, 70.529; CD AB BC
Triangle BD: 19.471, 65.905, 65.905; BD AD BE
Triangle BE: 35.264, 48.190, 54.736; CE BD BE
Triangle CD: 30, 52.239, 70.529; BC DE CD
Triangle CE: 35.264, 45, 65.905; CE BE CE
Triangle DE: 30, 52.239, 60 ; CD DE DE
Repeating unit 1: Quadrangle with angles 30, 90, 30, 90, made from AB, BC, CD, and DE.
Repeating unit 1: Quadrangle with angles 45, 45, 90, 90, made from AC, AD, BD, BE, and CE.
Repeating unit 3: Triangle AE.
R4 {3,4,3,4}
Plane 1: tiled solely by AB.
Plane 2: tiled by combination of AC, AD, and BD.
Plane 3: tiled by combination of AE and BE.
Plane 4: tiled solely by BC.
Plane 5: tiled solely by CD.
Plane 6: tiled solely by CE.
Plane 7: tiled solely by DE.
Triangle AB: 30, 45, 90; AB AB AB
Triangle AC: 35.264, 35.264, 90; AC AC AD
Triangle AD: 19.471, 54.736, 90; AD BD AC
Triangle AE: 35.264, 45, 90; AE BE AE
Triangle BC: 0, 45, 90; BC BC BC
Triangle BD: 0, 35.264, 90; AD BD BD
Triangle BE: 0, 54.736, 90; AE BE BE
Triangle CD: 0, 30, 90; CD CD CD
Triangle CE: 0, 45, 90; CE CE CE
Triangle DE: 0, 60, 90; DE DE DE
Repeating unit 1: Triangle AB.
Repeating unit 2: Quadrangle with angles 0, 90, 90, 90, made from AC, AD, and BD.
Repeating unit 3: Triangle with angles 0, 45, 90, made from AE and BE.
Repeating unit 4: Triangle BC.
Repeating unit 5: Triangle CD.
Repeating unit 6: Triangle CE.
Repeating unit 7: Triangle DE.
S4 Branched 33(34)
CE is 4, AB, BC, CD is 3
Plane 1: tiled by combination of AB, BC, and CD.
Plane 2: tiled by combination of AC, AD, and BD.
Plane 3: tiled by combination of AE and BE.
Plane 4: tiled solely by CE.
Plane 5: tiled solely by DE.
Triangle AB: 30, 52.239, 90 ; AB AB BC
Triangle AC: 45, 54.736, 65.905; AD AC AC
Triangle AD: 35.264, 48.190, 90 ; BD AD AC
Triangle AE: 35.264, 45, 90 ; AE BE AE
Triangle BC: 0, 60, 75.522; AB CD BC
Triangle BD: 0, 65.905, 90 ; AD BD BD
Triangle BE: 0, 54.736, 90 ; AE BE BE
Triangle CD: 0, 52.239, 90 ; CD CD BC
Triangle CE: 0, 45, 90 ; CE CE CE
Triangle DE: 0, 60, 90 ; DE DE DE
Repeating unit 1: Quadrangle with angles 0, 90, 90, 90, made from AB, BC, and CD.
Repeating unit 2: Triangle with angles 0, 45, 90, made from AC, AD, and BD.
Repeating unit 3: Triangle with angles 0, 45, 90, made from AE and BE.
Repeating unit 4: Triangle CE.
Repeating unit 5: Triangle DE.
O4 Branched 34(33)
BC is 4, AB, CD, CE is 3
AD and AE triangles identical.
BD and BE triangles identical.
CD and CE triangles identical.
Plane 1: tiled solely by AB.
Plane 2: tiled by combination of AC, AD, AE, BD, and BE.
Plane 3: tiled solely by BC.
Plane 4: tiled solely by CD.
Plane 5: tiled solely by CE.
Plane 6: tiled solely by DE.
Triangle AB: 30, 30, 90 ; AB AB AB
Triangle AC: 35.264, 35.264, 70.529; AD AE AC
Triangle AD: 19.471, 54.736, 90 ; AD BD AC
Triangle AE: 19.471, 54.736, 90 ; AE BD AC
Triangle BC: 0, 45, 45 ; BC BC BC
Triangle BD: 0, 35.264, 90 ; AD BD BD
Triangle BE: 0, 35.264, 90 ; AE BE BE
Triangle CD: 0, 30, 90 ; CD CD CD
Triangle CE: 0, 30, 90 ; CE CE CE
Triangle DE: 0, 60, 90 ; DE DE DE
Repeating unit 1: Triangle AB.
Repeating unit 2: Quadrangle with angles 0, 0, 90, 90, made from AC, AD, AE, BD, and BE.
Repeating unit 3: Triangle BC.
Repeating unit 4: Triangle CD.
Repeating unit 5: Triangle CE.
Repeating unit 6: Triangle DE.
N4 Branched 43(34)
AB, CE is 4, BC, CD is 3
Plane 1: tiled solely by AB.
Plane 2: tiled by combination of AC and AD.
Plane 3: tiled solely by AE.
Plane 4: tiled combination of BC and CD.
Plane 5: tiled solely by BD.
Plane 6: tiled solely by BE.
Plane 7: tiled solely by CE.
Plane 8: tiled solely by DE.
Triangle AB: 0, 45, 90 ; AB AB AB
Triangle AC: 0, 54.736, 54.736; AD AC AC
Triangle AD: 35.264, 35.264, 90 ; AD AD AC
Triangle AE: 0, 45, 90 ; AE AE AE
Triangle BC: 0, 0, 60 ; BC CD BC
Triangle BD: 0, 45, 90 ; BD BD BD
Triangle BE: 0, 0, 90 ; BE BE BE
Triangle CD: 0, 30, 90 ; CD CD BC
Triangle CE: 0, 0, 90 ; CE CE CE
Triangle DE: 0, 45, 90 ; DE DE DE
Repeating unit 1: Triangle AB.
Repeating unit 2: Quadrangle with angles 0, 90, 90, 90, made from AC and AD.
Repeating unit 3: Triangle AE.
Repeating unit 4: Quadrangle with angles 0, 0, 90, 90, made from BC and CD.
Repeating unit 5: Triangle BD.
Repeating unit 6: Triangle BE.
Repeating unit 7: Triangle CE.
Repeating unit 8: Triangle DE.
M4 Branched 3334
BE is 4, AB, BC, BD are 3
AB, BC, and BD triangles identical.
AC, AD, and CD triangles identical.
AE, CE, and DE triangles identical.
Plane 1: tiled by combination of AB, BC, and BD.
Plane 2: tiled solely by AC.
Plane 3: tiled solely by AD.
Plane 4: tiled solely by AE.
Plane 5: tiled solely by BE.
Plane 6: tiled solely by CD.
Plane 7: tiled solely by CE.
Plane 8: tiled solely by DE.
Triangle AB: 0, 0, 60; BC BD AB
Triangle AC: 0, 45, 90; AC AC AC
Triangle AD: 0, 45, 90; AC AC AC
Triangle AE: 0, 0, 90; AE AE AE
Triangle BC: 0, 0, 60; AB BD BC
Triangle BD: 0, 0, 60; AB BC BD
Triangle BE: 0, 0, 0 ; BE BE BE
Triangle CD: 0, 45, 90; CD CD CD
Triangle CE: 0, 0, 90; CE CE CE
Triangle DE: 0, 0, 90; DE DE DE
Repeating unit 1: Regular hexagon with angle 0, made of 2 copies of AB, BC, and BD. Half works if we allow for reflection swaps.
Repeating unit 2: Triangle AC.
Repeating unit 3: Triangle AD.
Repeating unit 4: Triangle AE.
Repeating unit 5: Triangle BE.
Repeating unit 6: Triangle CD.
Repeating unit 7: Triangle CE.
Repeating unit 8: Triangle DE.
FR4 Cyclical 33434
AE, CD are 4, AB, BC, DE are 3
AB and BC triangles identical.
AD and CE triangles identical.
AE and CD triangles identical.
BD and BE triangles identical.
Plane 1: tiled with combination of AB and BC.
Plane 2: tiled solely by AC.
Plane 3: tiled by combination of AD, BD, BE, and CE.
Plane 4: tiled solely by AE.
Plane 5: tiled solely by CD.
Plane 6: tiled solely by DE.
Triangle AB: 30, 30, 60 ; AB BC AB
Triangle AC: 0, 45, 45 ; AC AC AC
Triangle AD: 0, 35.264, 35.264; BD AD AD
Triangle AE: 0, 45, 45 ; AE AE AE
Triangle BC: 30, 30, 60 ; AB BC BC
Triangle BD: 19.471, 35.264, 54.736; BD AE BE
Triangle BE: 19.471, 35.264, 54.736; BE CE BD
Triangle CD: 0, 45, 45 ; CD CD CD
Triangle CE: 0, 35.264, 35.264; BE CE CE
Triangle DE: 0, 30, 30 ; DE DE DE
Repeating unit 1: Quadrangle with angles 60, 90, 60, 90, made from AB and BC.
Repeating unit 2: Triangle AC.
Repeating unit 3: Quadrangle with angles 0, 90, 0, 90, made from AD, BD, BE, and CE.
Repeating unit 4: Triangle AE.
Repeating unit 5: Triangle CD.
Repeating unit 6: Triangle DE.
P4 Square with added 3-branch
AB, AC, BD, CD, DE are 3
AB and AC triangles identical.
BD and CD triangles identical.
BE and CE triangles identical.
Plane 1: tiled by combination of AB, AC, BD, CD, and DE.
Plane 2: tiled by combination of AD, AE, BE, and CE.
Plane 3: tiled solely by BC.
Triangle AB: 0, 52.239, 90 ; AB AC BD
Triangle AC: 0, 52.239, 90 ; AC AB CD
Triangle AD: 0, 65.905, 65.905; AE AD AD
Triangle AE: 48.190, 48.190, 70.529; BE CE AD
Triangle BC: 0, 45, 90 ; BC BC BC
Triangle BD: 0, 60, 75.522; DE AB CD
Triangle BE: 45, 54.736, 65.905; AE BE BE
Triangle CD: 0, 60, 75.522; DE AC BD
Triangle CE: 45, 54.736, 65.905; AE CE CE
Triangle DE: 52.239, 52.239, 60 ; BD CD DE
Repeating unit 1: If we count AB/AC and BD/CD as reflections, we get a quadrangle with angles 0, 0, 90, 90, made from AB, AC, BD, CD, and DE. With only true reflection, we get an infinite branching structure.
Repeating unit 2: Triangle with angles 0, 45, 45, made from AD, AE, BE, and CE.
Repeating unit 3: Triangle BC.
BP4 Square with added 4-branch
DE is 4, AB, AC, BD, CD are 3
AB and AC triangles identical.
BD and CD triangles identical.
BE and CE triangles identical.
Plane 1: tiled by combination of AB, AC, BD, and CD
Plane 2: tiled solely by AD.
Plane 3: tiled by combination of AE, BE, and CE.
Plane 4: tiled solely by BC.
Plane 5: tiled solely by DE.
Triangle AB: 0, 30, 90 ; AB AC BD
Triangle AC: 0, 30, 90 ; AC AB CD
Triangle AD: 0, 45, 45 ; AD AD AD
Triangle AE: 35.264, 35.264, 70.529; BE CE AE
Triangle BC: 0, 0, 90 ; BC BC BC
Triangle BD: 0, 0, 60 ; BD BD CD
Triangle BE: 0, 54.736, 54.736; AE BE BE
Triangle CD: 0, 0, 60 ; CD CD BD
Triangle CE: 0, 54.736, 54.736; AE CE CE
Triangle DE: 0, 45, 45 ; DE DE DE
Repeating unit 1: AB, AC, BD, and CD form a pentagon with angles 0, 0, 0, 90, 90. Reflection through 0-0 sides swaps BD and CD.
Repeating unit 2: Triangle AD.
Repeating unit 3: Quadrangle with angles 0, 0, 90, 90, made of AE, BE, and CE.
DP4 Square with bridge
AB, AD, BC, BE, CD, DE are 3
AB, AD, BC, BE, CD, and CE triangles identical.
AC, AE, and CE triangles identical.
Plane 1: tiled by combination of AB, AD, BC, BE, CD, and CE.
Plane 2: tiled solely by AC.
Plane 3: tiled solely by AE.
Plane 4: tiled solely by BD.
Plane 5: tiled solely by CE.
Triangle AB: 0, 0, 60; BC BE AD
Triangle AC: 0, 45, 45; AC AC AC
Triangle AD: 0, 0, 60; CD DE AB
Triangle AE: 0, 45, 45; AE AE AE
Triangle BC: 0, 0, 60; AB BE CD
Triangle BD: 0, 0, 0 ; BD BD BD
Triangle BE: 0, 0, 60; AB BC DE
Triangle CD: 0, 0, 60; AD DE BC
Triangle CE: 0, 45, 45; CE CE CE
Triangle DE: 0, 0, 60; AD CD BE
Repeating unit 1: All triangles are the same (0, 0, 60), but there are six types. The 60-degree vertices are of two kinds, one with B-triangles around (AB, BC, BE, AB, BC, BE), second with D-triangles (AD, CD, DE, AD, CD, DE). Reflection through 0-0 side always switches between a B-triangle and a D-triangle.
Repeating unit 2: Triangle AC.
Repeating unit 3: Triangle AE.
Repeating unit 4: Triangle BD.
Repeating unit 5: Triangle CE.
o{3,3,3,3}
great prismated decachoron 1/truncated octahedral prism 1
129.231520 - 50.768480 - arccos(sqrt[2/5])
great prismated decachoron 1/hexagonal duoprism
116.565051 - 63.434949 - arccos(sqrt[1/5])
great prismated decachoron 1/truncated octahedral prism 2
108.434949 - 71.565051 - arccos(sqrt[1/10])
great prismated decachoron 1/great prismated decachoron 2
101.536959 - 78.463041 - arccos(1/5)
truncated octahedral prism 1/hexagonal duoprism
135 - 45
truncated octahedral prism 1/truncated octahedral prism 2
120 - 60
truncated octahedral prism 1/great prismated decachoron 2
108.434949 - 71.565051 - arccos(sqrt[1/10])
hexagonal duoprism/truncated octahedral prism 2
135 - 45
hexagonal duoprism/great prismated decachoron 2
116.565051 - 63.434949 - arccos(sqrt[1/5])
truncated octahedral prism 2/great prismated decachoron 2
129.231520 - 50.768480 - arccos(sqrt[2/5])
cycle 1: |top1 -to- gpd1 -to- gpd2 -to- top2| - 50.768 + 78.463 + 50.768 = 180
cycle 2: |top1 -c- top2| - 60
cycle 3: (gpd1 -hp- top2 -hp- hd -hp- gpd2 -hp- top1 -hp- hd -hp-) - 71.565 + 45 + 63.435 + 71.565 + 45 + 63.435 = 360
opposites: gpd1/gpd2, top1/top2, hd/hd
o{3,3,3,4}
great prismated decachoron/truncated octahedral prism
153.434949 - 26.565051 - arccos(sqrt[4/5])
great prismated decachoron/hexagonal-octagonal duoprism
140.768480 - 39.231521 - arccos(sqrt[3/5])
great prismated decachoron/truncated cuboctahedral prism
129.231520 - 50.768480 - arccos(sqrt[2/5])
great prismated decachoron/great prismated tesseract
116.565051 - 63.434949 - arccos(sqrt[1/5])
truncated octahedral prism/hexagonal-octagonal duoprism
150 - 30
truncated octahedral prism/truncated cuboctahedral prism
135 - 45
truncated octahedral prism/great prismated tesseract
120 - 60
hexagonal-octagonal duoprism/truncated cuboctahedral prism
144.735610 - 35.264390 - arccos(sqrt[2/3])
hexagonal-octagonal duoprism/great prismated tesseract
125.264390 - 54.735610 - arccos(1/sqrt(3))
truncated cuboctahedral prism/great prismated tesseract
135 - 45
cycle 1: |top -to- gpd -to- gpt| - 26.565 + 63.435 = 90
cycle 2: |top -c- tcop| - 45
cycle 3: |hod -hp- gpd -hp- tcop| - 39.231 + 50.768 = 90
cycle 4: |hod -hp- top -hp- gpt| - 30 + 60 = 90
cycle 5: |tcop -tco- gpt| - 45
cycle 6: |tcop -op- hod -op- gpt| - 35.264 + 54.735 = 90
opposites: gpd/gpd, top/top, hod/hod, tcop/tcop, gpt/gpt
omnitruncated demipenteract
great prismated decachoron 1/great prismated decachoron 2
126.869898 - 53.130102 - arccos(3/5)
great prismated decachoron 1/square-hexagonal duoprism
140.768480 - 39.231521 - arccos(sqrt[3/5])
great prismated decachoron 1/truncated octahedral prism
129.231520 - 50.768480 - arccos(sqrt[2/5])
great prismated decachoron 1/truncated icositetrachoron
116.565051 - 63.434949 - arccos(sqrt[1/5])
great prismated decachoron 2/square-hexagonal duoprism
140.768480 - 39.231521 - arccos(sqrt[3/5])
great prismated decachoron 2/truncated octahedral prism/
129.231520 - 50.768480 - arccos(sqrt[2/5])
great prismated decachoron 2/truncated icositetrachoron
116.565051 - 63.434949 - arccos(sqrt[1/5])
square-hexagonal duoprism/truncated octahedral prism
144.735610 - 35.264390 - arccos(sqrt[2/3])
square-hexagonal duoprism/truncated icositetrachoron
125.264390 - 54.735610 - arccos(sqrt[1/3])
truncated octahedral prism/truncated icositetrachoron
135 - 45
cycle 1: |shd -hp- gpd1 -hp- top -hp- gpd2 -hp- shd| - 39.232 + 50.768 + 50.768 + 39.232 = 180
cycle 2: |top -to- ti| - 45
cycle 3: |top -c- shd -c- ti| - 35.264 + 54.736 = 90
cycle 4: |ti -to- gpd1 -to- gpd2 -to- ti| - 63.435 + 53.130 + 63.435 = 180
opposites: gpd1/gpd2, shd/shd, top/top, ti/ti
Marek14 wrote:... I don't completely understand Dr. Klitzing dihedral page (where I got the polyhedron/polychoron data), ...
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.
( 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
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.
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
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
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).
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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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