Composites of Antiprisms and Prisms

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Introduction

In January 2026 Jim McNeill sent me an email pointing out the self augmented heptagonal antiprism that is on his website [MCNE1]I noticed the ring that lacked triangles was a bit wider than expected, which I thought was interesting and lead me to further investigate. Eventually it lead to a series of polyhedra that I refer to as tri-composites of antiprisms, or TCA in short.

Since 2006 I have had an interest for polyhedra with heptagons. My fascination had to do with the fact that of the regular n-gons the heptagon is the first gap that appears for polyhedra that don't have an n-fold symmetry axis. There are, of course, some obvious examples where e.g. heptagonal prisms are used to augment polyhedra with with Platonic symmetries, e.g. the cube. This feels a bit like a cheat though. Regular heptagons don't appear in the Johnson solids either.

At the Bridges conference in London in 2006 I met Scott Vorthmann and when I talked to Scott about this he mentioned that in higher dimensions there are polytopes belonging to Coxeter's E8 symmetry group, e.g. the 421 of the Gosset's polytopes, and that the E8 Coxeter group has elements of order seven. He mentioned that perhaps by some smart projection one could find something with heptagons. That motivated me not to give up, though I am not very familiar with polytopes and higher dimensions. It seems that Scott and I share an interest, e.g. check out his visualisation of heptagons maths.

My ultimate goal was to find some polyhedron with a Platonic symmetry that involved regular heptagons. I felt that it was best to start by loosing up some requirements to be able to find at least something. Then I could search that space in the hope to find solutions where the property became exactly how I wanted. I decided to start with equilateral heptagons and see whether some of the solutions would become regular. I didn't find any such solution, though I did find the 'near' miss shown on the left.

Later I tried to let go of the flatness requirement instead: I searched for any polyhedra consisting of regular heptagons that were folded over one more more diagonals. I found out that this search space was enormous and I restricted it by requiring that only regular folded heptagons were used. Obviously none of those consisted of flat regular heptagons. Note that my search was done numerically and one would still need to proof that the solutions are really regular folded heptagons. For some I could proof indirectly that solutions exist for regular heptagons. Some models of these can be found here.

As stated I was concentrating on Platonic symmetries. I thought it would be obvious that heptagons would appear in polyhedra with 7-fold symmetries, but I was completely taken by surprise when I saw the arrangements of heptagons and heptagrams in Mason Green's small and great supersemicupola. I think those are really cool objects and in 2025 I built paper models of the supersemicupolae.

This page describes a way to construct polyhedra using prisms or antiprisms that are based on {m/n}-gons. Depending on the value 'm' the resulting polyhedron belongs to the symmetry group D4C4 or D4xI in case prisms are used and D3C3 or D3xI in case antiprisms are used.

From Jim McNeill's self augmented heptagonal antiprism I noticed that three antiprisms share a total of six triangles. This is actually a general property of any antiprism. For the compound of any three antiprisms based on {m/n}-gons, with a 3-fold symmetry axis through the centre of a triangle 'A' and parallel to the {m/n}-gons, the triangles sharing an edge with triangle 'A' are completely shared with one of the other antiprisms. These triangles, indicated as type 'B', occur in pairs.

Consider for example m=7 and n=1 then you get a .In this compound the yellow triangles, which are of type 'A', are three triangles from different antiprisms and the red triangles, which are of type 'B' are covered by two triangles.

If for triangle 'A' there is an opposite triangle parallel to triangle 'A', like for the given example above. then there are six pairs of these triangles of type 'B', and the symmetry of the compound belongs to the algebraic group D3xI. This means there is a 3-fold symmetry axis, three mirror planes sharing the 3-fold axis, three half-turns for which the axes lie in an equator plane, a central inversion, and two rotary reflections. This can easily be verified in the example above for the heptagons. Since for m = 3 an octahedron is obtained, let's assume that m > 3.

If there is only one triangle of type 'A', i.e. there is no triangle parallel to triangle 'A', then there are three pairs of triangles of type 'B'. See for example the case for m=6 and n=1: . The symmetry of this compound belongs to the algebraic group D3C3, which is a subgroup of D3xI. and it has the 3-fold axis and the three mirror planes sharing the axis.

To understand that the triangles of type 'B' are completely shared between the different antiprisms, position one {m/n} antiprism in such a way that you are looking straight into the gravity point or centre of triangle 'A'. Now say that the z-axis passes that centre and that its origin is at the centre of the antiprism. In that case the following can be observed:

  1. The z-axis is orthogonal to triangle 'A'
  2. There are three neighbouring triangles that share an edge with the base triangle. Name the two of the shared edges 'a' and 'b' in such a way that 'a' is mapped on 'b' by a 120° rotation of the antiprism around the z-axis and name the remaining edge 'c'
  3. Call two of the neighbouring triangles A and B in such a way that 'a' is an edge of 'A' and 'b' is an edge of 'B'.
  4. Note that the dihedral angles between 'A' and the base triangle and between 'B' and the base triangle are the same.
  5. This means that triangle 'A' is mapped on triangle 'B' by the 120° rotation around the z-axis, while triangle 'B' is mapped on a triangle 'C' attached to edge 'c'.
  6. Using the same logic, the opposite rotation will map 'A' on 'C' and 'B' on 'A'.

If you interpret the shared edges of the shared triangles as one, then you can remove triangles until each edge is only shared by two faces. By removing all yellow triangles and keeping one red triangle in the examples above, the following polyhedra is obtained: the and the . This method can be applied to any {m/n}-prism.

If for a tri-composite antiprism with base {m/n} the values of 'm' and 'n' share a common divisor 'd' then the result will be a compound of:

For example the and the , both of which consist of a with additional square antiprisms, three for the former and six for the latter.

As mentioned two different symmetries can be obtained. The prismatic symmetry D3C3 occurs when 'n' is odd and 'm' even or vice versa. The higher symmetry D3xI occurs either when both 'm' and 'n' are odd or when both are even. Note that in the latter case the result is also a compound.

The table below summarises the TCA polyhedra for some {m/n} polygons

Base Polygon Link to Model

Faceting of a small stellated dodecahedron

Tri-composite of square antiprisms with three complete square antiprisms added.

Tri-composite of pentagonal antiprisms with three complete pentagonal antiprisms added.

Tri-composite of pentagrammic antiprisms with three complete pentagrammic antiprisms added.

Tri-composite of pentagrammic crossed antiprisms with three complete pentagrammic crossed antiprisms added.

A similar, though easier, method can be used with prisms. One can make a two-compound of prisms be letting them share a square and their gravity point. Depending on whether the amount of vertices of the base is even or add, one or two squares are shared. The edges of these squares are shared with 4 faces, which means that the pair of squares can be removed. This way the compound is turned into a polyhedron.

If 'm' and 'n' specifying the {m/n}-base share a common divisor, then the polygon is a compound and then the whole polyhedron is a compound as well.

Base Polygon Symmetry Link to Model

Also a {2/1} sesquicupola (blended gyro bicupola)		 D4C4
D4C4
D4C4

This is the minimal torroid according to Alex Doskey		 D4xI

Compound of 2 BCP based on triangle		 D4xI
D4C4
D4C4
D4C4
D4xI
D4xI
D4C4
D4C4

Compound of 3 BCP based on triangle		 D4C4
D4C4
D4xI

compound of 2 BCP based on pentagon		 D4xI
D4xI
D4xI

Anything related to m = 4 isn't included in the table, since if m = 4 a cube is obtained and in that case the two-composite will share all faces. As a consequence any {d×4/d×n} isn't included either. Obviously {2n/n} isn't included either since such prisms are just n squares meeting in an n-fold symmetry axis.

Thanks to Jim McNeill for sharing his self augmented heptagonal antiprism, which was the direct source of inspiration of this discovery. Jim also gave some feed-back and direction on these pages and he found some of these polyhedra, not knowing I had sent sent OFF files of these already. Thanks to Scott Vorthmann for fruitfull discussions. Thanks to Don Romano, Piotr Pawlikowski and Ulrich Mikloweit for feed-back and inspiration.

References

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Last Updated

2026-05-29, 23:08 CET