Higher-dimensional antiprisms

[quickfur]

Most of us here already realize that one of the best ways of generalizing antiprisms to higher dimensions is to analyze the top/bottom polygons of the 3D antiprisms as duals, rather than rotated copies of each other. This allows convenient definition of higher-dimensional antiprisms such as the cube antiprism (x4o3o || o4o3x), and so forth. Since the dual of a polytope always exists (though it may not always be CRF), this construction will always yield a polytope, that moreover preserves the symmetries of the base polytope.

Today I started wondering, though: does this definition really allow unbounded extension of the concept of antiprisms to higher dimensions? Specifically, I considered the line of n-cube antiprisms. One thing that immediately occurred to me is that the circumradius of the n-cube increases without bound as n increases, but the circumradius of the n-cross remains constantly at 1/√2. So there must exist some value of n above which the n-cube antiprism can no longer be CRF-able, because it would not be possible for the lacing edges to be unit length. A quick test computation revealed that this threshold is as low as 5: the tesseract antiprism is the last CRF in the series! The circumradius of the 5-cube is too large for unit lacing edges with the 5-cross, and therefore the 6D 5-cube antiprism cannot be CRF.

This leaves not many options left for constructing antiprisms (of this type). It's already well-known that the n-simplex antiprism always exists, and is CRF -- in fact, it's regular, and is identical to the (n+1)-cross. Among the uniform polytopes, this seems to be the only one that continues indefinitely. In 5D there's also the 24-cell antiprism which should be CRF, probably scaliform? I'm almost certain the 120-cell/600-cell antiprism cannot be CRF, due to the large discrepancy in circumradii. I don't know of many 5D CRFs that have CRF duals, but it would seem there wouldn't be very many of them, so they wouldn't have corresponding higher-dimensional CRF antiprisms.

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.

Less symmetric antiprisms may go a bit further. I haven't studied them as much yet. In 4D there are the n-pyramid antiprisms, which are just antiprism bipyramids. But I don't know of any non-regular self-dual 4D CRFs that could serve as the basis for 5D CRF antiprisms. Generally it seems the higher the dimension, the harder it is to construct CRF antiprisms of this type.

[wendy]

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.

As to antiprisms generally, the sloping faces are the elements of the top and bottom, which are generally unrelated, except to lie in orthogonal spaces (and orthogonal to the sloping edge). So in an orthotope-measure antiprism, the opposites are various x-cubes and (N-x-1)-simplexes in pyramid product. The count of faces (aka "facets") is always the count of total surtopes of a base, for a N-cube or N-cross, this amounts to \(3^n-1\).

[username5243]

The 120-cell antiprism cannot be CRF because it would have to contain 120 dodecahedral pyramids (from a cell of the 120-cell to the opposing 600-cell vertex), which cannot be made CRF.

[wendy]

Pentagonal pyramids are indeed CRF, and the icosahedron-dodecahedron occurs as ring 1,2 of the {3,3,5} vertex-first.

But in 4d, you need a fairly big step to get from {3,3,5} to {5,3,3}, and a single unit-edge is too short for the job. You need edges of length f to go from x3o3o5o to o3x3o5o, the next smallest.

[quickfur]

I'm trying to think of 4D CRFs whose duals are also CRF (since these would generate CRF 5D antiprisms). So far, I haven't come up with any self-dual examples besides the regular cases of the 5-cell and 24-cell. But self-duality is not really a requirement; the cube pyramid, for example, would have as its dual the octahedral pyramid. So these two could form a CRF 5D antiprism.

In general, a CRF pyramid whose base has a CRF dual would qualify. Unfortunately, for the case of n-cube pyramids (resp. n-cross pyramids), the cube pyramid antiprism is the last CRF member of the series, because for n=4, the 4-cube's circumradius is equal to its edge length, which means the 4-cube pyramid is degenerate (zero-height) if made CRF. So it does not generate a full-dimensioned antiprism under this construction; and for n=5, the 5-cube's circumradius exceeds its edge length, so the 5-cube pyramid cannot be CRF.

Interestingly, there's a series of "chained pyramids" (i.e., pyramid of lower dimensional pyramids) that seems promising. Given some self-dual basis pyramid P, such as any of the 3D CRF pyramids, we can iteratively construct their higher-dimensional equivalents. Take the square pyramid, for example. It's dual is just the square pyramid. So if we build the 4D square-pyramid pyramid, it's dual would also be a square-pyramid pyramid (it's a dual square-pyramid pyramid, which is identical). So we can extend this to arbitrary dimensions, and for each dimension we have a self-dual polytope. Therefore, their corresponding antiprisms must also exist, and ought to be CRF-able.

For the simplest case of P=triangular pyramid (tetrahedron), we obtain the simplex family, which generates the n-crosses as antiprisms. For P=square pyramid, we have something new that also exists across dimensions and is CRF, thus generating another infinite family of antiprisms. Ditto for P=pentagonal pyramid.

In 4D, the 24-cell is self-dual, and might seem promising as the basis for another series of chained pyramids and CRF antiprisms; however, the 24-cell's circumradius is equal to its edge length, so the CRF 24-cell pyramid is degenerate and does not produce any interesting new antiprisms or chained pyramids.

[Klitzing]

In 5D there's also the 24-cell antiprism which should be CRF, probably scaliform?

Right you are, cf. icoap.
--- rk

[Klitzing]

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.

And here is tessap.
--- rk

[quickfur]

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.)

[Klitzing]

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.

That one is known as jakaljak.
--- rk

[Klitzing]

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.)

Cf. here for direct links to n-py-aps and n-cu-"ap"s.
--- rk

[Klitzing]

Lately the idea of an antiprism (i.e. using regular bases, which are vice versas duals), extrapolated to higher dimensions, became broadened to what now is called an alterprism. Here base polytopes are to be selected from any (undecorated) group diagram, which shows up an additional outer symmetry (i.e. symmetry of the diagram itself). Then the other base would simply be the very same polytope (as on the first base), but being provided in an "alternate" representation. This could be a reflection of a linear graph (as for the antiprisms), it could be a reflection of the arms of a bifurcated graph, it could be a rotation of a circular graph or a gyration of a tridental graph (with alike legs), etc.

Previously Polyhedrondude provided here acronyms according to their segmentotopal description only, i.e. as X-al-X (X atop alternate X). Now this can be shortened into a mere X-a (X alterprism). Obviously alterprisms necessarily are already scaliform - at least.

For instance the first scaliform polychoron ever, tutcup = tut atop inverted tut = xo3xx3ox&#x now might be seen as a first example, i.e. as tut alterprism too. Also hin (hemipenteract), which is known to be hex atop gyro hex, might be reviewed alternatively as a hex alterprism. But this extends to non-convex polytopes as well. Thus the alterprism of siid = x3x3o5/2*a would be possible as well. That siid alterprism already is known too: siidcup.

But what is even more interesting here, we are now prepared to consider alterprisms of alterprisms to. Those then get called altersquarisms (X-as), altercubisms (X-ac), altertessisms (X-at).

Thus I looked into tutas today. I.e. tutas = xo3xx3ox&#x || ox3xx3xo&#x. As tutcup = tuta itself was 4D, tutas will be 5D. The respective height surely will be that of tutcup (on either side). By means of that mutual displacement and the individual circumradius of tutcup, the circumradius of tutas can be calculated to sqrt(13/8). At least now it becomes evident, that there ought be some relation to siphin = x3o3o *b3x3o, which has the very same circumradius. Looking deeper into this, it is known that siphin = hex || gyro rit || rit || gyro hex. Thus its medial segment is rita = rit || gyro rit, the rit alterprism. And rit = x3o3x *b3o itself is known to be rit = tet || tut || inv tut || dual tet. That is, the medial segment therefrom is nothing but tutcup = tuta. Thus putting that all together we obtain that tutas will be nothing else than the cyclodiminishing of siphin (its central portion wrt. lace city display)!

Having said all this, you might be interested into the elemental structure of tutas, respectively which portions of siphins elements would remain in here. For that reason I just managed to fiddle out its incidence matrix. Here it comes:

Code: Select all

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


--- rk

[quickfur]

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?

[username5243]

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?

That's just the demipenteract itself, formed from alternating the vertices of the penteract (which is just a tesseract prism).

Several of the other tesseractic truncates (the rectified tesseract, truncated hexadecachoron, and bitruncated tesseract) form alterprisms in 5D along similar lines. Most of these are diminishings of some of the demipenteract's truncates, as in fact tutcup itself is a diminishing of the rectified tesseract (with two opposite tetrahedra removed).

Many of the non-double-symmetry truncates of the pentachoral and icositetrachoral families can do this too. (In fact all but the direct xxoo-type truncates, which would require lateral cells which can't be made CRF.) THe pentachoral case is just the triacontiditeron (5D cross polytope), the one of the rectified pentachoron is the dodecateron (birectified hexateron). The others will all be scaliform polytera (some of the pentachoral cases can also be obtained from certain diminishing of uniform polytera).

In 6D and up, the simplex and demicube families will keep producing these - in fact the n-demicube alterprism will just be an (n+1)-demicube. There'll also be 7D ones based on the E6 symmetry group.

[quickfur]

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]

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?

Yes, you get the following three, in each case the polychora have to be viewed with demitesseract symmetry.

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.

[Klitzing]

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.

These are thexa (old: thexag thex), rita (old: ritag rit), taha (old: tahgtah) respectively.

--- rk

[Klitzing]

Found a severe restriction for altersquarisms etc.:

okay, now I've seen that icoas indeed cannot exist. - Why? - Well the height of icoap is smaller than 1/sqrt(2). Thus additional (unit) edges, which cross the lace city display (i.e. inserting here a full featured - or even only a pseudo - icope), cannot be inserted, they would be bound to stab out somewhere like a too long spoke. - Btw. we could well consider ricoas then, as there the height of ricoa is larger than 1/sqrt(2). But there again s.th. like ricoas (singular!) does not exist either. This now is because the 2 opposite ricoas (attention: "-s" indicates plural form only!) could not remain within a flat lace city plane if connected by according diagonal edges, rather they would wobble up into a lace simplex instead (i.e. of one dimension plus) as that square then only is the shadow of an according disphenoid. - That is, altersquarisms can be constructed at most if the lace city side, i.e. the according alterprism height, is 1/sqrt(2) exactly, I fear. - Thus tutas was a good chosen first try after all, hehe.

--- rk