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January 28th, 2019, 03:04 PM   #1
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Q: What group has trapezoid-faced hedrons?

Recently, I've been investigating platonic solids, which led to archimedian and catalan solids.

Well, I noticed that the truncated polyhedrons were basically two polyhedra with the same center and keep only the overlap between them. (Subtraction with polyhedra?)

Then looking at the chiral or snub polyhedrons, it seemed like a third polyhedron was introduced to truncate the result of the first two. So what are these polyhedra?

Now, lacking any tools, I'm stuck visualizing these in my head.

It seems to me that these polyhedra are made up of identical trapezoids. The number of trapezoids being three times the number of the base hedron, 3*4 for tetrahedal, 3*8 for octahedral, and 3*20 for icosahedral.

Thus I started looking for the dodecahedron made up of trapezoids, but I can't seem to find anything specific about it.

Given that these polyhedra are composed of all identical faces, I figured there might be a special group they belong to.

Does anyone know anything about them? Is there a special group that contains them? If not, do I get to define and name that group?

Last edited by MystMage; January 28th, 2019 at 03:07 PM.
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January 28th, 2019, 06:47 PM   #2
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https://en.wikipedia.org/wiki/Trapez...c_dodecahedron
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January 28th, 2019, 06:57 PM   #3
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Reminds me of my D&D days...

-Dan
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January 28th, 2019, 08:33 PM   #4
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That one has rhombi, and it doesn't fit anyway.

The shape I'm speaking of is only trapezoids.

Consider the d20 to be the result of a shape that has been truncated by an octohedron. Untruncate it and you get a shape of only trapezoids.

This part is easy enough for me to confirm. I have an actual icosehedron with me so I can easily identify the eight faces that would be the truncating octohedron. So one simply needs to see how untruncating those faces would impact the polyhedron, and for each face you get a triangular expansion on only two neighboring faces making a trapezoid. Every face that expands by removing the truncation does that and turns into a trapezoid leading to a polyhedron of only trapezoids.
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February 1st, 2019, 12:36 AM   #5
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Perhaps I should rephrase the question.
What group is for convex polyhedra of non-regular faces that are all the same?

Last edited by skipjack; February 1st, 2019 at 12:59 AM.
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February 1st, 2019, 01:32 PM   #6
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Quote:
Originally Posted by MystMage View Post
Perhaps I should rephrase the question.
What group is for convex polyhedra of non-regular faces that are all the same?
I'm not sure there are any. Certainly we are working outside the crystallographic point groups. I worked with "quasi-crystals" once upon a time, which have "forbidden" symmetries. (The ones I worked with used two different unit cells.) You might find something there. The problem with your question is that quasi-crystals do not represent groups: they are structured but not periodic.

-Dan

Last edited by topsquark; February 1st, 2019 at 01:34 PM.
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February 2nd, 2019, 04:09 PM   #7
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I'm not familiar with crystallographic point groups, but a quick search says that such a group is a group that can be rotated and/or reflected around a central point that gives the original shape, and such a feature I'm pretty sure applies to the trapezoid polyhedrons I'm tracking down. Of course I can't be certain, but it seems odd to so perfectly cut a platonic solid without similar symmetries.

These polyhedrons I would expect to have the same or similar symmetries to the archimedian/platonic solids that they come from, much like how the symmetries of inverse solids are related (icosahedron and dodecahedron for example).

Further, I'm curious about these solids and whether there are other figures related to them much like how platonic solids are a whole group related to each other.

Do people really only bother with polyhedrons of regular polygons and forgo any other type? I mean, sure at some point things get too chaotic and broad to classify neatly and too broad a group will get infinite results, but I figure certain traits are worth looking at even without all other traits, such as convex polyhedrons of identical irregular faces, which seems limited enough to have a potentially finite group worthy of exploration.

Also, what is a good program for making 3d models of polyhedrons anyway? I'd love to actually show these forms I'm talking about.
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