quickfur wrote:In 5D there's also the 24-cell antiprism which should be CRF, probably scaliform?
quickfur wrote:The 5D tesseract antiprism is an interesting polyteron. It consists of 8 cubical pyramids, 16+32=48 5-cells, and 24 square-pyramid pyramids (square||line), for a total of 80 facets. Initially I thought there'd be some interesting occurrences of lower-dimensional antiprisms in it, but it appears that I may have been mistaken.
wendy wrote:Here is something interesting.
xo3oo3oo3oo3oxBoo, is formed by 2_21 || inverted 2_21, ie \( 2_{21} \mid\mid\ _22_1\) might be supposed to derive from the 5d simplex antiprism. It equates to a 3_21 with opposite vertices removed.
quickfur wrote:On a side note, I wonder if there are interesting CRFs to be obtained by Stott expansion (or similar modifications) of antiprisms. E.g., since the pentagonal pyramid J2 is self-dual, we can form the J2 antiprism. Can we also produce a CRF from analogous stacking of two pentagonal cupolae in parallel hyperplanes (with dual orientations)? If this works, what about two pentagonal rotundae in dual orientations? (I did a quick lookup of your list of segmentochora, and did not find anything involving J6 that would have such a configuration. But in retrospect, such a thing will probably be non-orbiform anyway, and thus wouldn't appear in your list.)
xo3xx3ox&#x || ox3xx3xo&#x → height = 1/sqrt(2) = 0.707107
o.3o.3o. .. .. .. & | 48 | 1 2 4 1 | 2 1 6 4 6 2 | 1 6 2 4 6 2 1 | 2 4 1 2
------------------------------+----+-------------+-------------------+--------------------+----------
x. .. .. .. .. .. & | 2 | 24 * * * | 2 0 4 0 0 0 | 1 4 2 2 0 0 0 | 2 2 1 0
.. x. .. .. .. .. & | 2 | * 48 * * | 1 1 0 2 0 1 | 1 4 0 0 3 0 1 | 2 3 0 1
oo3oo3oo&#x .. .. .. & | 2 | * * 96 * | 0 0 2 1 2 0 | 0 2 1 2 2 1 0 | 1 2 1 1
o.3o.3o. || .o3.o3.o & | 2 | * * * 24 | 0 0 0 0 4 2 | 0 0 0 2 4 2 1 | 0 2 1 2
------------------------------+----+-------------+-------------------+--------------------+----------
x.3x. .. .. .. .. & | 6 | 3 3 0 0 | 16 * * * * * | 1 2 0 0 0 0 0 | 2 1 0 0
.. x.3o. .. .. .. & | 3 | 0 3 0 0 | * 16 * * * * | 1 2 0 0 0 0 1 | 2 2 0 0
xo .. ..&#x .. .. .. & | 3 | 1 0 2 0 | * * 96 * * * | 0 1 1 1 0 0 0 | 1 1 1 0
.. xx ..&#x .. .. .. & | 4 | 0 2 2 0 | * * * 48 * * | 0 2 0 0 2 0 0 | 1 2 0 1
oo3oo3oo&#x || o.3o.3o. & | 3 | 0 0 2 1 | * * * * 96 * | 0 0 0 1 1 1 0 | 0 1 1 1
.. x. .. || .. .x .. & | 4 | 0 2 0 2 | * * * * * 24 | 0 0 0 0 2 0 1 | 0 2 0 1
------------------------------+----+-------------+-------------------+--------------------+----------
x.3x.3o. .. .. .. & | 12 | 6 12 0 0 | 4 4 0 0 0 0 | 4 * * * * * * | 2 0 0 0 tut
xo3xx ..&#x .. .. .. & | 9 | 3 6 6 0 | 1 1 3 3 0 0 | * 32 * * * * * | 1 1 0 0 tricu
xo .. ox&#x .. .. .. & | 4 | 2 0 4 0 | 0 0 4 0 0 0 | * * 24 * * * * | 1 0 1 0 tet
xo .. ..&#x || o.3o.3o. & | 4 | 1 0 4 1 | 0 0 2 0 2 0 | * * * 48 * * * | 0 1 1 0 tet
.. xx ..&#x || .. x. .. & | 6 | 0 3 4 2 | 0 0 0 2 2 1 | * * * * 48 * * | 0 1 0 1 trip
oo3oo3oo&#x || oo3oo3oo&#x | 4 | 0 0 4 2 | 0 0 0 0 4 0 | * * * * * 24 * | 0 0 1 1 tet
.. x.3o. || .. .x3.o & | 6 | 0 6 0 3 | 0 2 0 0 0 3 | * * * * * * 8 | 0 2 0 0 trip
------------------------------+----+-------------+-------------------+--------------------+----------
xo3xx3ox&#x .. .. .. & | 24 | 12 24 24 0 | 8 8 24 12 0 0 | 2 8 6 0 0 0 0 | 4 * * * tutcup = tut || inv tut
xo3xx ..&#x || o.3x. .. & | 12 | 3 9 12 3 | 1 2 6 6 6 3 | 0 2 0 3 3 0 1 | * 16 * * tricuf = {6} || trip
xo .. ox&#x || ox .. xo&#x | 8 | 4 0 16 4 | 0 0 16 0 16 0 | 0 0 4 8 0 4 0 | * * 6 * hex = tet || dual tet
.. xx ..&#x || .. xx ..&#x | 8 | 0 4 8 4 | 0 0 0 4 8 2 | 0 0 0 0 4 2 0 | * * * 12 tepe = tet || tet
quickfur wrote:Very interesting concept! So does that mean in 5D it's possible to construct a 16-cell alterprism? By considering the 16-cell as an alternated tesseract, we can place the two possible alternations in parallel hyperplanes and take their convex hull. Would the result be uniform?
quickfur wrote:Oooh that's a cool way to think of a demipenteract! I guess the alterprisms of various Stott expansions of the 16-cell would produce various "non-trivial" alterprisms, then?
username5243 wrote:Truncated hexadecachoron alterprism has as facets: 2 truncated hexadecachora, 8 tutcups, and 16 octahedron || truncated tetrahedron.
Rectified tesseract alterprism has: 2 rectified tesseracts, 8 hexadecachora, 16 tetrahedron || cuboctahedron, 24 tetrahedral prisms (as square || orthogonal square).
Bitruncated tesseract alterprism has: 2 bitruncated tesseracts, 8 tutcups, 16 truncated tetrahedron || truncated octahedron, 24 tetrahedral prisms.
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