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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 +12011-07-21
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    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.2011-07-21
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    @Theo: So is your torus then classified as a cubical polytope?2011-07-21
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    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).2011-07-21
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    How do you define "combinatorial cubes"?2011-07-21
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    @Theo: Thanks, yes, that is what confuses me. Normally the word _polytope_ would not include a torus, in any dimension. But perhaps that is because "polytope" usually abbreviates "convex polytope." To confuse matters further, there are the "cubical pseudomanifolds"...2011-07-21
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    On the other hand some googling reveals that e.g. [Ziegler](http://page.mi.fu-berlin.de/gmziegler/) uses cubical polytopes only for *convex* polytopes, and he certainly has more authority than I do... (I didn't see your last comment before posting).2011-07-21
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    @Michael: Good question! I think the meaning used for these cubical thingies might be both combinatorial (in terms of poset inclusion) and geometrical. Concerning the latter, a face of a cubical polytope in $\mathbb{R}^3$ should both be a quadrilateral and planar. Otherwise I think authors would precede the terms with the adjective "abstract."2011-07-21
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    So a combinatorial cube is something that has the same inclusions among its faces as the usual $n$-cube?2011-07-21

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