# What is a cubical sphere?

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 $\mathbb{R}^3$:

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