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Face types and Vertex types

 
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polymorphic



Joined: 17 Feb 2008
Posts: 26

PostPosted: Sun Feb 17, 2008 1:46 pm    Post subject: Face types and Vertex types Reply with quote

Hi Rob,

I have a question pertaining face types and vertex types.

I made a compound which is 15 8/3 dipyramids in icosahedral symmetry. Stella lists 4 face types which are all described identically.

The dual list 4 types of 3 valence vertices all on the same radius.

If there are indeed more than one face type and more than one vertex type (in the dual), how are they different and is there a way to describe it?

http://www.interocitors.com/tmp/Icosahedral_8-3_dipyramids.stel

http://www.interocitors.com/tmp/Icosahedral_8-3_dipyramids.off

http://www.interocitors.com/tmp/Icosahedral_8-3_dipyramids.wrl
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robertw
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Joined: 10 Jan 2008
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Location: Melbourne, Australia

PostPosted: Mon Feb 18, 2008 12:51 pm    Post subject: Re: Face types and Vertex types Reply with quote

By "face type" I refer not just to face shape, but also to how the face fits into the symmetry of the overall model.

For example, the snub cube contains two different triangular face types. Both are equilateral, but one connects to three other triangles at its edges, and the other connects to two triangles and one square.

If two faces are of the same type, then a model can be rotated so that one face moves to the original position of the other while the rest of the model also lines up with its original orientation.

Note that in Stella, mirror images are also counted as two separate face types. This can be useful sometimes, but might be misleading at others. For example, see how the dual of the truncated tetrahedron appears with two differently coloured faces.

Rob.
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polymorphic



Joined: 17 Feb 2008
Posts: 26

PostPosted: Mon Feb 18, 2008 7:16 pm    Post subject: Re: Face types and Vertex types Reply with quote

Hi Rob,

robertw wrote:
Note that in Stella, mirror images are also counted as two separate face types. This can be useful sometimes, but might be misleading at others. For example, see how the dual of the truncated tetrahedron appears with two differently coloured faces.


For me the dual of the truncated tetrahedron is all one color, lists one face type and two vertex types.

I know about mirror images because I think you told me once. This was an issue when enumerating the biform deltahedra. This is also an issue when considering isohedral (one face type) polyhedra.

In the ongoing biform deltahedra collection there is an example, the octaugmented Gike.

http://www.interocitors.com/polyhedra/Deltahedra/Cundy/Coptic/Octaugmented_Gike.off

It lists 3 vertex types, two which are the mirror of each other. They better be or this one is invalid Smile

What might be happening in the 8/3 dipyrmid example in my earlier post above is that it has two face types and the listing also contains two mirrors. But I can't verify this. The dual lists 4 vertex types and it might possibly be there two sets of mirror vertex types. From the listing they are all decribed identically.

You seem to list the mirror faces and vertices together. Maybe the second one could be depicted by "(mirror)". I still an curious what makes them two (or more) different face types when they look so similar, and if there is a way of saying this in a hundred words or less (are they within different symmetries and so on). I certainly couldn't explain it.

By the way, nice forum. There never has been a good place to address Stella questions.

Roger
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robertw
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PostPosted: Tue Feb 19, 2008 3:37 am    Post subject: Reply with quote

Oops, I meant the dual of the truncated cuboctahedron. It's faces come in two types which are reflections of each other.

Standard auto-colouring means that Stella will colour by face-type, unless the model is a compound, then it is coloured as a compound instead. Use "Color->Basic Color Scheme->Color by Face Type" to colour compounds by face type too. Then you can see how they differ.

You can then also use "Color->Basic Color Scheme->Same Color for Chiral Face Pairs" to reduce the number of colours from 4 to 2. Now mirrored faces share the same colour and you can see clearly the two truly different face types. Hopefully with this colouring it becomes clearer that the two face types fit into the model differently.

Looking at the dual it's pretty clear that there are two different vertex types, each coming in two mirrored forms.

Good idea about mentioning the reflection in the face type list. Since the properties shown for the two are the same, maybe I should just show something like "2: Reflection of face type 1", rather than making the existing description even longer. What do you think?

Rob.
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polymorphic



Joined: 17 Feb 2008
Posts: 26

PostPosted: Tue Feb 19, 2008 5:50 pm    Post subject: Reply with quote

Rob,

Thanks for the tip. Once clicking Same Color for Chiral Face Pairs then if the polyhedra is isohedral but has mirror image faces, it will be all one color.

Noting the mirror image by e.g "2: Reflection of face type 1" would spell it out. This same thing applies to mirror vertices as well.

When I was looking at the biform deltahedra I found cases with 2 sets of mirror vertices. I just have to wonder if there can be any more than 2 sets at one time.

Roger
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