⌈1 0 0 0⌉
|0 1 0 0|
1_M = |0 0 1 0|
⌊0 0 0 1⌋
⌈ 0 0 0 1⌉ ⌈ 0 0 -1 0⌉ ⌈ 0 1 0 0⌉
| 0 0 1 0| | 0 0 0 1| |-1 0 0 0|
I_L = | 0 -1 0 0| , J_L = | 1 0 0 0| , K_L = | 0 0 0 1|
⌊-1 0 0 0⌋ ⌊ 0 -1 0 0⌋ ⌊ 0 0 -1 0⌋
⌈ 0 0 0 -1⌉ ⌈ 0 0 -1 0⌉ ⌈ 0 1 0 0⌉
| 0 0 1 0| | 0 0 0 -1| |-1 0 0 0|
I_R = | 0 -1 0 0| , J_R = | 1 0 0 0| , K_R = | 0 0 0 -1|
⌊ 1 0 0 0⌋ ⌊ 0 1 0 0⌋ ⌊ 0 0 1 0⌋
⌈ cos(nθ) -sin(nθ) 0 0 ⌉
| sin(nθ) cos(nθ) 0 0 |
| 0 0 cos(nθ) -sin(nθ) |
⌊ 0 0 sin(nθ) cos(nθ) ⌋
⌈ cos(nθ) -sin(nθ) 0 0 ⌉ ⌈ 0 0 -cos(nθ) -sin(nθ) ⌉
| sin(nθ) cos(nθ) 0 0 | | 0 0 -sin(nθ) cos(nθ) |
| 0 0 cos(nθ) -sin(nθ) | | cos(nθ) sin(nθ) 0 0 |
⌊ 0 0 sin(nθ) cos(nθ) ⌋ , ⌊ sin(nθ) -cos(nθ) 0 0 ⌋
⌈ 1/2 1/2 -1/2 1/2⌉ ⌈ 1/2 1/2 1/2 1/2⌉ ⌈ 1/√2 1/√2 0 0 ⌉
|-1/2 1/2 1/2 1/2| |-1/2 1/2 1/2 -1/2| |-1/√2 1/√2 0 0 |
| 1/2 -1/2 1/2 1/2| |-1/2 -1/2 1/2 1/2| | 0 0 1/√2 1/√2|
⌊-1/2 -1/2 -1/2 1/2⌋ , ⌊-1/2 1/2 -1/2 1/2⌋ , ⌊ 0 0 -1/√2 1/√2⌋
⌈ 1/2 1/2 -1/2 1/2⌉ ⌈ 1/2 1/2 1/2 1/2⌉ ⌈ φ/2 φ⁻¹/2 0 1/2 ⌉
|-1/2 1/2 1/2 1/2| |-1/2 1/2 1/2 -1/2| |-φ⁻¹/2 φ/2 1/2 0 |
| 1/2 -1/2 1/2 1/2| |-1/2 -1/2 1/2 1/2| | 0 -1/2 φ/2 φ⁻¹/2|
⌊-1/2 -1/2 -1/2 1/2⌋ , ⌊-1/2 1/2 -1/2 1/2⌋ , ⌊-1/2 0 -φ⁻¹/2 φ/2 ⌋
mr_e_man wrote:The result will probably look like a tangled blob of tetrahedra. Perhaps we should take the dual, so that the cells have more distinctive shapes, and only 4 cells meet at each vertex.
x..o....
...x..o.
.o....x.
.x..o...
....x..o
..o....x
..x..o..
o....x..
<cos(0*pi*2/8), sin(0*pi*2/8), cos(0*pi*2/8), sin(0*pi*2/8)>
<cos(1*pi*2/8), sin(1*pi*2/8), cos(3*pi*2/8), sin(3*pi*2/8)>
<cos(2*pi*2/8), sin(2*pi*2/8), cos(6*pi*2/8), sin(6*pi*2/8)>
<cos(3*pi*2/8), sin(3*pi*2/8), cos(9*pi*2/8), sin(9*pi*2/8)>
<cos(4*pi*2/8), sin(4*pi*2/8), cos(12*pi*2/8), sin(12*pi*2/8)>
<cos(5*pi*2/8), sin(5*pi*2/8), cos(15*pi*2/8), sin(15*pi*2/8)>
<cos(6*pi*2/8), sin(6*pi*2/8), cos(18*pi*2/8), sin(18*pi*2/8)>
<cos(7*pi*2/8), sin(7*pi*2/8), cos(21*pi*2/8), sin(21*pi*2/8)>
<cos(0*pi*2/8), sin(0*pi*2/8), cos(3*pi*2/8), sin(3*pi*2/8)>
<cos(1*pi*2/8), sin(1*pi*2/8), cos(6*pi*2/8), sin(6*pi*2/8)>
<cos(2*pi*2/8), sin(2*pi*2/8), cos(9*pi*2/8), sin(9*pi*2/8)>
<cos(3*pi*2/8), sin(3*pi*2/8), cos(12*pi*2/8), sin(12*pi*2/8)>
<cos(4*pi*2/8), sin(4*pi*2/8), cos(15*pi*2/8), sin(15*pi*2/8)>
<cos(5*pi*2/8), sin(5*pi*2/8), cos(18*pi*2/8), sin(18*pi*2/8)>
<cos(6*pi*2/8), sin(6*pi*2/8), cos(21*pi*2/8), sin(21*pi*2/8)>
<cos(7*pi*2/8), sin(7*pi*2/8), cos(24*pi*2/8), sin(24*pi*2/8)>
mr_e_man wrote:If the symmetry group contains a double rotation that's not isoclinic, then applying it repeatedly gives a non-trivial simple rotation. For example, 5*(72°, 90°) = (360°, 450°) = (0°, 90°). So the rotation must be isoclinic: θ = ±ϕ.
quickfur wrote:More ideas along the lines of the first few posts about screw symmetries: IIRC Jonathan Bowers once mentioned the following construction: pick two coprime numbers m and n. Build an m,m-duoprism, and lay out its vertices in an m×m grid. Start with the vertex corresponding to row 0 column 0, and select 1 vertex for each subsequent row, each displaced from the previous by n columns (modulo m). Take the convex hull of the selected vertices. The result will be a swirlprism-like polytope with an m-fold twisting symmetry.
mr_e_man wrote:How about the diminished 8,8-duoprism?
The uniform 8,8-duoprism has 64 vertices of the form (R cos(j*45°), R sin(j*45°), R cos(k*45°), R sin(k*45°)). Delete the 8 vertices where j=k. (Maybe also delete those where j=k+1 and those where j=k+3 (mod , to destroy more symmetry.)
username5243 wrote:A vertex-transitive construction along the lines of "diminished duoprisms" some of us on the Discord found last year is the 5-diminished pentagonal duoprism. It turns out you can inscribe a regular pentachoron in a pentagonal duoprism, as a 5-2 step prism (that is, if the vertices are labeled in a grid from (0,0) to (4,4), the pentachoron has vertices at (0,0), (1,2), (2,4), (3,1), and (4,3)).
The result is vertex transitive with 20 vertices. The pentagonal prism cells of the duoprism each get 2 vertices that are as far apart as possible chopped off (this polyhedron is identical to the vertex figure of the scaliform polychoron "spidrox") and 5 disphenoid cells are inserted under the deleted vertices.
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