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A Duality-related Question

 
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Nordehylop



Joined: 27 Feb 2008
Posts: 21
Location: Illinois, USA

PostPosted: Thu Feb 28, 2008 9:15 pm    Post subject: A Duality-related Question Reply with quote

Can you take the faces of a polyhedron, along with the faces of its dual, and "rearrange" them into a new polyhedron? I noticed that you can with the Platonic Solids (you get the icosidodecahedron and cuboctahedron this way) but I was wondering if this applies to any (or all) other polyhedra. Possibly only convex ones?
If it works, it would be interesting to make a set of Archimedian/Catalan "quasisemiregular" polyhedra.
It appears that a truncated Truncated Dodecahedron+Triakisicosahedron compound would be this kind of polyhedron.
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robertw
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Joined: 10 Jan 2008
Posts: 394
Location: Melbourne, Australia

PostPosted: Fri Feb 29, 2008 3:26 am    Post subject: Re: A Duality-related Question Reply with quote

In general, no. It's one of many things that only works for Platonics.

The technique can be generalised, but the faces may change (generally to their duals, which are the same for regular polygons).

Have a look at Stella's "Morph Duals by Truncation" view. With the cube, at 50% you'll see the cuboctahedron. Hold down Ctrl+Left-mouse-button and drag the mouse left and right to see other percentages.

See the Dual Morphing section at http://www.software3d.com/PolyNav/PolyNavigator.php#morph

Rob.
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guy



Joined: 11 Feb 2008
Posts: 75
Location: England

PostPosted: Sat Mar 01, 2008 8:59 pm    Post subject: Reply with quote

It depends on whether you want the dual to be exactly like the one Stella provides (there are various kinds of dual).

If you cut the corners off a Platonic polyhedron (truncate it), the cut surfaces are just like the faces of the dual. This is why the cuboctahedron and icosidodecahedron work.

If you cut the corners off most polyhedra, the cut faces have the same number of sides as the dual's faces have, but are not usually exactly the same shape (for example you might get long thin triangles instead of short fat ones). You can do what you ask in this way, but the dual you have used is a distorted version of the one Stella provides.
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Nordehylop



Joined: 27 Feb 2008
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Location: Illinois, USA

PostPosted: Mon Mar 03, 2008 10:37 pm    Post subject: Reply with quote

Thanks for the replies, guys!

I just noticed that the base+dual thing also works for the Kepler-Poinsot solids (dodecadodecahedron and small ditrigonal icosidodecahedron/great icosidodecahedron, possibly others), though this may just be a coincidence, as there are a lot of things that you can do with pentagons/grams and triangles.
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polymorphic



Joined: 17 Feb 2008
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PostPosted: Sun Mar 16, 2008 2:05 pm    Post subject: Reply with quote

Hi,

This gave me an idea. I took a platonic or achimedean and added the base and its dual. Then I did a convex hull. If we do this a number of times we come up with more and more intricate polyhedra.

Any other polyhedra could be used but it must be such that the dual doesn't have infinite faces. If the dual or base completely hide one or the other it will break the chain.

By looks, they remind me of the Waterman polyhedra but of course are something completely different.

Roger
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