## Faceted snub cube

- robertw
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**Posts:**544**Joined:**Thu Jan 10, 2008 6:47 am**Location:**Melbourne, Australia-
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### Faceted snub cube

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).

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).

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
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**Posts:**544**Joined:**Thu Jan 10, 2008 6:47 am**Location:**Melbourne, Australia-
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I wouldn't say "only"guy wrote:I bet you only posted this so it would hook me in!

The forum's for anyone and everyone, so don't worry about repeating anything people might already know.I am new to this forum, so I hope I am not repeating stuff the rest of you already know.

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

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.

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.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.

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
**Posts:**21**Joined:**Wed Feb 27, 2008 6:04 pm**Location:**Illinois, USA-
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### I made one

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.

At first I was using 6 colors but had to use 8. 3 faces of each color. Has anyone else made one?

Thanks.

I just love Polyhedra,

Kathy

Kathy

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.

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.

Cheers,

Guy. Guy's polyhedra pages

Guy. Guy's polyhedra pages

### Here it is

I hope you can see this.

http://www.flickr.com/photos/26612180@N ... hotostream

http://www.flickr.com/photos/26612180@N ... hotostream

I just love Polyhedra,

Kathy

Kathy

@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

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.Cheers,

Guy. Guy's polyhedra pages

Guy. Guy's polyhedra pages

### Fun to make

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.

I think a small one would be very difficult to make.

I just love Polyhedra,

Kathy

Kathy

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

Ulrich

### Faceted Snub Cube

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

Kathy

Kathy

I just love Polyhedra,

Kathy

Kathy

### Re: Faceted Snub Cube

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

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