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Roughly, a cubical complex is like a simplicial complex except all the pieces glued together are combinatorial cubes of various dimensions. A cubical sphere is a cubical complex that is homeomorphic to a sphere. I have encountered papers that distinguish between cubical spheres and cubical polytopes, but I do not understand the distinction. Is there a distinction already in $\mathbb{R}^3$? If so, could anyone provide an example? A reference to clear definitions would suffice as well. Thanks!

My understanding is that, say, the rhombic triacontahedron is both a cubical polytope and a cubical sphere in $\mathbb{R}^3$:
           Rhombic triacontahedron
          Image from Wikipedia article

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Nice image. XD +1 –  Patrick Da Silva Jul 21 '11 at 0:57
I think the distinction is that you don't allow "holes" in a cubical sphere. So, for example (I hope I make myself clear): take nine dice and arrange them as a $3 \times 3$-square. Remove the one in the middle. The surface of this certainly gives you a cubical complex that doesn't deserve the name sphere, as it is homeomorphic to a torus. –  t.b. Jul 21 '11 at 1:03
@Theo: So is your torus then classified as a cubical polytope? –  Joseph O'Rourke Jul 21 '11 at 1:04
I would say so. The square faces can be glued to a polytope along their edges. (A polytope doesn't include convexity assumptions, as far as I know). –  t.b. Jul 21 '11 at 1:06
How do you define "combinatorial cubes"? –  Michael Lugo Jul 21 '11 at 1:12

1 Answer 1

(Too long for a comment): Here are some thoughts on this circle of ideas though I am not sure what distinctions have been made with these terms in the literature. The diagram you show is a 3-polytope whose surface is built up of 2-dimensional combinatorial cubes, namely 4--gons. However, it is not clear that this 3-polytope or similar 3-polytopes including their interior points can be always be decomposed as 3-cubes that meet along faces. The note on this page which talks about combinatorial cubes: shows a diagram of as 4-cube but it also can be thought of a 3-cube whose interior has been cut up into other combinatorial 3-cubes. As regards the torus, there are some graphs which will not embed on a sphere but will embed on a torus. Now one can ask if one can embed in 3-space a surface (topologically a torus) with flat faces so that the vertex edge graph of this surface is the given graph. When this is possible it is common to call the resulting surface a toroidal polytope. The adjective toroidal overcomes the usage of polytope to be something convex.

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Thanks, Joe, I think you are absolutely correct in focusing on building the 2-surface from glued 3-cubes, as opposed to just looking at the surface alone. (I think in my example it accidentally works out, because it is a zonotope...) –  Joseph O'Rourke Jul 21 '11 at 17:34
Joe: The dual of any 4-valent 3-polytope will have all of its faces 4-gons but I would be surprised if all of these 3-polytopes can be decomposed into 3-cubes. –  Joseph Malkevitch Jul 22 '11 at 2:53
Perhaps this thesis might be of interest: –  Joseph Malkevitch Jul 31 '11 at 13:19
Thanks, Joe, I will look at it! (Sorry for not revisiting this in so long...) –  Joseph O'Rourke Aug 28 '11 at 19:58

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