# Polyhedra not imagined by Plato.

September 19, 2011 2:19 PM Subscribe

How to name a specific polyhedron?

I'm trying to name a polyhedron with 18 vertices. The overall geometry can be described as a truncated cube inscribed within an octahedron. As such it doesn't conform to any of the platonic or archimedian polyedra, as it as inequivalent vertices. ( The vertices can be classified into two sets, 12 of which make 3 connections and 6 of which make 4. ) The wikipedia page which is pretty good for these solid geometry naming conventions isn't helping me name this one.

While I can post here coordinates and/or images of this particular polyhedron (which is of high symmetry) to get this specific case named (and solve the immediate problem), I'm more interested in resources that can help in general for naming polyhedra that aren't Platonic, Archimedian, or their truncates.

I've asked the local university math department and I'm currently coming up empty on that side, so now I ask.metafilter .

I'm trying to name a polyhedron with 18 vertices. The overall geometry can be described as a truncated cube inscribed within an octahedron. As such it doesn't conform to any of the platonic or archimedian polyedra, as it as inequivalent vertices. ( The vertices can be classified into two sets, 12 of which make 3 connections and 6 of which make 4. ) The wikipedia page which is pretty good for these solid geometry naming conventions isn't helping me name this one.

While I can post here coordinates and/or images of this particular polyhedron (which is of high symmetry) to get this specific case named (and solve the immediate problem), I'm more interested in resources that can help in general for naming polyhedra that aren't Platonic, Archimedian, or their truncates.

I've asked the local university math department and I'm currently coming up empty on that side, so now I ask.metafilter .

Two other major classes of less-regular polyhedra are Catalan solids, which have all identical faces but whose vertices are non-equivalent; and Johnson solids, whose faces are all regular polygons.

posted by Johnny Assay at 4:34 PM on September 19, 2011 [1 favorite]

posted by Johnny Assay at 4:34 PM on September 19, 2011 [1 favorite]

Best answer: This page might help.

I don't understand the description of the solid in your question — after you inscribe a truncated cube in an octahedron, you have one polyhedron inside another. Then what?

posted by stebulus at 5:17 PM on September 19, 2011

I don't understand the description of the solid in your question — after you inscribe a truncated cube in an octahedron, you have one polyhedron inside another. Then what?

posted by stebulus at 5:17 PM on September 19, 2011

Hmmm.... I did crystal shape naming stuff about a million years ago in mineralogy class. Here's the Wikipedia page on the Bravais lattices. Maybe you can go from there and figure out a descriptive name based on one of the crystal systems. (I seem to have repressed most of those memories, or I'd be more help.)

posted by Green Eyed Monster at 7:26 PM on September 19, 2011

posted by Green Eyed Monster at 7:26 PM on September 19, 2011

I could be wrong (probably am), but it sounds like you're describing what Buckminster Fuller called the Vector Equilibrium, also known as a cubeoctohedron.

posted by adamrice at 8:23 PM on September 19, 2011

posted by adamrice at 8:23 PM on September 19, 2011

Response by poster: Thanks, Stebulus. You nailed it with your request for clarification-- after inscribing a truncated cube into an octahedron, you still have an overall octahedron!

What was 'throwing me off' here is that the polyhedron I'm trying to name is formed by the vertices of sulfur atoms in a chemical structure. The 6 vertices that define the octahedron define the overall geometry-- the other 12 vertices that define the inscribed truncated cube don't change the descriptios; All these sulfur atoms are either a vertex or lie on an edge (and therefore are not a vertex) of the overall octahedron. It didn't help my thinking that since this is real data, so things are somewhat distorted from idealized geometry. I'll chalk up my 'denseness' here (and that of the other 2 science/math professionals that looked at this problem) up to 'false vertices' and non-idealized structure of what we are looking at.

It's actually a much simpler problem than I realized.

posted by u2604ab at 8:31 PM on September 19, 2011

What was 'throwing me off' here is that the polyhedron I'm trying to name is formed by the vertices of sulfur atoms in a chemical structure. The 6 vertices that define the octahedron define the overall geometry-- the other 12 vertices that define the inscribed truncated cube don't change the descriptios; All these sulfur atoms are either a vertex or lie on an edge (and therefore are not a vertex) of the overall octahedron. It didn't help my thinking that since this is real data, so things are somewhat distorted from idealized geometry. I'll chalk up my 'denseness' here (and that of the other 2 science/math professionals that looked at this problem) up to 'false vertices' and non-idealized structure of what we are looking at.

It's actually a much simpler problem than I realized.

posted by u2604ab at 8:31 PM on September 19, 2011

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posted by empath at 2:58 PM on September 19, 2011