Faceted snub cube

[robertw]

I just discovered an interesting little faceting of the snub cube. Has anyone seen it before?

It's interesting because it has just one face-type, an irregular pentagon no less, and it a true polyhedron, not a compound. The only other isohedral facetings (ie having just one face type) appear to be compounds of disphenoids (irregular tetrahedra).

[guy]

Hi Rob, I bet you only posted this so it would hook me in!

I am new to this forum, so I hope I am not repeating stuff the rest of you already know.

Like the snub cube (and all uniform polyhedra), this facetting has vertices all alike within its symmetry - we say that it is isogonal.
As you say, it has faces all alike too - it is isohedral.
A polyhedron which is both isogonal and isohedral is called noble.

Noble polyhedra were first studied in depth by Bruckner, a hundred years ago. Since then, Branko Grünbaum has revived interest in them. I wonder if this one has been described before.

[robertw]

I bet you only posted this so it would hook me in!

I wouldn't say "only"

I am new to this forum, so I hope I am not repeating stuff the rest of you already know.

The forum's for anyone and everyone, so don't worry about repeating anything people might already know.

Like the snub cube (and all uniform polyhedra), this facetting has vertices all alike within its symmetry - we say that it is isogonal.
As you say, it has faces all alike too - it is isohedral.
A polyhedron which is both isogonal and isohedral is called noble.

Oh yeah, somehow I wasn't even thinking about the fact that it was isogonal as well

By the way, the same thing doesn't work with the snub dodecahedron, in case anyone was wondering, because the edges placed diagonally across the pentagons leave a pentagrammic hole to fill, where as those across the square faces of the snub cube meet others to close the shape.

Noble polyhedra were first studied in depth by Bruckner, a hundred years ago. Since then, Branko Grünbaum has revived interest in them. I wonder if this one has been described before.

Yeah, Bruckner's 1906 isogonal isohedra can be found in the Stella Library with Great Stella and Stella4D, but this one was not amongst them.

I just noticed that it is self-dual too, so it is also a stellation of the pentagonal icositetrahedron, dual of the snub cube.

Rob.

[Nordehylop]

Very interesting polyhedron! Since it is a faceting of the snub cube it has a sort of twisted look. It looks almost "compoundish', but it isn't a compound...
This is what I love about Stella!

[Kathy]

I saw this on Wed. and just loved it. Now I am wanting to make more faceted poly.
At first I was using 6 colors but had to use 8. 3 faces of each color. Has anyone else made one?
Thanks.

[guy]

Its self-duality just made me realise that its abstract form appears to be regular, so we could write it as {5 5}

V-E+F = 24-60+24 = -12 so topologically it is a toroid of genus (2-(-12))/2 = 7
(where V = number of vertices, E = number of edges, F = number of faces).

I wonder if this regular abstract toriod is already known. Rob, have you asked around?

Note that with these funny topologies, the symbol {5 5} is not unique to this figure, so if other {5 5} figures have been described, it doesn't necessarily mean that this one has. For example there is a well-known {5 5} tiling of the hyperbolic plane.

[Kathy]

I hope you can see this.
http://www.flickr.com/photos/26612180@N ... hotostream

[guy]

@Kathy, Yes, I can see it fine - and if my clunky box can see it, most anybody can. Lovely model!

As a follow-up to my previous post, I am told that it does not have a regular "abstract" structure. In fact it is possibly the only known noble and self-dual polyhedron that is not regular in structure.

[Kathy]

I had fun making this model because it wasn't obvious what color went where. Also when you look at it you can turn it just a tad and it looks like a totally different model.
I think a small one would be very difficult to make.

[Ulrich]

By the way, the same thing doesn't work with the snub dodecahedron, in case anyone was wondering, because the edges placed diagonally across the pentagons leave a pentagrammic hole to fill, where as those across the square faces of the snub cube meet others to close the shape.

But at least that shape is isogonal so its dual is set up of one kind of hexagons only:



Ulrich

[Ulrich]

This is my model of Robert's facetted snub cube:



I call it "24 scalars".

Ulrich

[robertw]

Ulrich, I like the fish design

Someday I must make a model of this myself!
Rob.

[Kathy]

Ulrich, I see you have internal pieces. I thought this looked like fish also. You do great work.
Kathy

[Ulrich]

Thanks!

This is a picture of my model of the dual of the other one, derived from the snub dodecahedron:



It is made of one kind of crossed hexagons

Ulrich