Symmetrohedra

What makes a polyhedron interesting or attractive? This is obviously an open-ended question with many possible answers. One way to approach the problem of designing interesting polyhedra is to note our natural affinity for symmetry and regularity in art and design. We may then describe a family of polyhedra by choosing some set of symmetry and regularity properties we think might be interesting and seeing what polyhedra have those properties. The properties we choose determine who gets invited to the party. The trick is to find a set of properties that open the door just wide enough to let in a bunch of cool, interesting polyhedra, without opening it so wide that all the boring, annoying polyhedra show up too.

The problem with working this way is that most the obvious choices of properties are already spoken for. Lots of people have worked on this problem before, and they've figured out some well-known families of polyhedra. Here's a table summarizing some of those families. Click on the names for more information.

  All faces regular High symmetry Convex Congruent faces Transitive vertices
Platonic X X X X X
Kepler-Poinsot X X   X X
Archimedean X X X   X
Uniform X X     X
Johnson X   X    
Deltahedra X   (X) X  

Because all of these families are so well-known, we have to start looking further afield to come up with interesting new polyhedra. We knew that we liked regular faces and convexity, but as you can see above that combination is captured by the Johnson solids. So we decided to loosen up on "all faces regular", and asked only that many faces be regular. Obviously, this property is a bit fuzzy, so there's no right way to enumerate all such polyhedra. Instead, we devise an explicit notation and construction technique that yields polyhedra in this family. Because our construction builds polyhedra by starting with a symmetry group, we call them Symmetrohedra:

  All faces regular High symmetry Convex Congruent faces Transitive vertices
Symmetrohedra   X X    

What follows is a gallery of symmetrohedra, both well-known and new. All images were produced using a modified version of a polyhedron viewer written by Doug Zongker. I added code to produce semi-transparent SVG renderings that were then turned into images using Batik.

Each solid is given with a symbol. The symbols use a notation explained in the Bridges 2001 paper. This notation has since been superseded by a more general one that I developed for a later paper about Islamic star patterns – indeed, Symmetrohedra were inspired by an attempt to express a rich class of template tilings for drawing star patterns.

Platonic and Archimedean Solids

Note that Symmetrohedra can't be used to represent all 18 Platonic and Archimedean solids. The way we defined them, the resulting polyhedra will have all the symmetries of one of the polyhedral groups generated by reflections, and so we can't represent the two snub Archimedeans (though we could with a suitable extension to our notation). The other 16 are all possible.

[full-size PNG] [SVG]
Symbol: T(1,*,*,e)
Name: Tetrahedron
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Symbol: O(1,*,*,e)
Name: Cube
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Symbol: O(*,1,*,e)
Name: Octahedron
Photo not available Symbol: I(1,*,*,e)
Name: Dodecahedron
Photo not available Symbol: I(*,1,*,e)
Name: Icosahedron
[full-size PNG] [SVG]
Symbol: O(1,1,*,e)
Name: Cuboctahedron
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Symbol: I(1,1,*,e)
Name: Icosidodecahedron
[full-size PNG]
Symbol: T(2,1,*,e)
Name: Truncated Tetrahedron
[full-size PNG]
Symbol: O(2,1,*,e)
Name: Truncated Cube
[full-size PNG]
Symbol: O(1,2,*,e)
Name: Truncated Octahedron
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Symbol: I(2,1,*,e)
Name: Truncated Dodecahedron
[full-size PNG] [SVG]
Symbol: I(1,2,*,e)
Name: Truncated Icosahedron
Notes: Also known as the 'Buckyball'
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Symbol: O(1,1,*,1)
Name: Rhombicuboctahedron
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Symbol: I(1,1,*,e)
Name: Rhombicosidodecahedron
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Symbol: O(2,2,*,e)
Name: Truncated Cuboctahedron
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Symbol: I(2,2,*,e)
Name: Truncated Icosidodecahedron

Interesting and New Polyhedra

The following list of polyhedra is not intended to be definitive or complete in some way. It's merely a sampling of interesting polyhedra we encountered that can be expressed as Symmetrohedra. Many of these polyhedra have never been documented before to our knowledge.

I have dozens more examples on file for this section. I'll add them in as time permits.

[full-size PNG] [SVG]
Symbol: T(2,3,*,e)
Name: Bowtie Tetrahedron
[full-size PNG] [SVG]
Symbol: T(1,2,*,1)
Name: Alternate Bowtie Tetrahedron
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Symbol: O(2,3,*,e)
Name: Bowtie Octahedron
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Symbol: O(1,2,*,1)
Name: Alternate Bowtie Octahedron
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Symbol: O(3,2,*,e)
Name: Bowtie Cube
[full-size PNG] [SVG]
Symbol: O(2,1,*,1)
Name: Alternate Bowtie Cube
[full-size PNG] [SVG]
Symbol: I(2,3,*,e)
Name: Bowtie Icosahedron
Notes:
  • It was a vision of this polyhedron that was the original inspiration for Symmetrohedra.
  • The Bowtie Icosahedron has been built as a stainless steel wire sculpture by Leigh Boileau.
  • I have also created a separate PDF folding net for this solid.
[full-size PNG] [SVG]
Symbol: I(1,2,*,1)
Name: Alternate Bowtie Icosahedron Notes: This solid is a "near miss" in the sense that it's almost a Johnson solid, except that some of the faces are not quite regular. It's close enough that you can construct it out of paper using regular faces.

More information about near misses

[full-size PNG] [SVG]
Symbol: I(3,2,*,e)
Name: Bowtie Dodecahedron
[full-size PNG] [SVG]
Symbol: I(2,1,*,1)
Name: Alternate Bowtie Dodecahedron

Papers


Craig S. Kaplan Last updated: