## Higher-dimensional antiprisms

Discussion of known convex regular-faced polytopes, including the Johnson solids in 3D, and higher dimensions; and the discovery of new ones.

### Higher-dimensional antiprisms

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.
quickfur
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### Re: Higher-dimensional antiprisms

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 \(3^n-1\).
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wendy
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### Re: Higher-dimensional antiprisms

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.
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### Re: Higher-dimensional antiprisms

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.
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### Re: Higher-dimensional antiprisms

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.
quickfur
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### Re: Higher-dimensional antiprisms

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

Right you are, cf. icoap.
--- rk
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### Re: Higher-dimensional antiprisms

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.

And here is tessap.
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### Re: Higher-dimensional antiprisms

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.)
quickfur
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### Re: Higher-dimensional antiprisms

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.

That one is known as jakaljak.
--- rk
Klitzing
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### Re: Higher-dimensional antiprisms

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.)

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