A Duality-related Question

For general discussion of polyhedra, not necessarily Stella-specific.
Post Reply
User avatar
Nordehylop
Posts: 21
Joined: Wed Feb 27, 2008 6:04 pm
Location: Illinois, USA
Contact:

A Duality-related Question

Post by Nordehylop » Thu Feb 28, 2008 9:15 pm

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.
It's always darkest just before it goes pitch black.

User avatar
robertw
Site Admin
Posts: 413
Joined: Thu Jan 10, 2008 6:47 am
Location: Melbourne, Australia
Contact:

Re: A Duality-related Question

Post by robertw » Fri Feb 29, 2008 3:26 am

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/PolyN ... .php#morph

Rob.

User avatar
guy
Posts: 75
Joined: Mon Feb 11, 2008 10:30 am
Location: England
Contact:

Post by guy » Sat Mar 01, 2008 8:59 pm

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.

User avatar
Nordehylop
Posts: 21
Joined: Wed Feb 27, 2008 6:04 pm
Location: Illinois, USA
Contact:

Post by Nordehylop » Mon Mar 03, 2008 10:37 pm

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.
It's always darkest just before it goes pitch black.

polymorphic
Posts: 26
Joined: Sun Feb 17, 2008 12:50 pm
Contact:

Post by polymorphic » Sun Mar 16, 2008 2:05 pm

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

Post Reply