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This question is similar to "Why are triangles, squares and hexagons the only polygons with which it is possible to tile a plane?" published here

But here, instead of a 2D environment, the question is toward a 3D context.

For example i think a possible answer is the cube. Ok, then what else?

I don't know for a 3D hexagon i have yet to be sure of what it is. In my eyes, it's more this than that.

Anyway, now you know what i mean, what are the 3D objects with which it is possible to fullfill a volume?

Regards

user59488
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  • See wiki's entry on Space filling polyedron. You can tessellate 3-d space using cube, hexagonal prism, rhombic dodecahedron, elongated dodecahedron and truncated octahedron with translations only. – achille hui Dec 10 '17 at 21:29
  • Excellent, this is what i were looking for, thanks you for the link. Based on your knowledge, do you think they list all possibilities? – user59488 Dec 10 '17 at 23:21
  • If one allow rotation, there are other polyhedra that can tessellate the space. The wiki entry of Space filling polyhedron has some other examples. Look at wiki entry of Plesiohedron too, it is a special class of Space filling polyhedron which can be viewed as some sort of Voronoi cell. It is known there are finitely many combinatorially distinct types of plesiohedron but the complete list is not known. There are even polyhedron which can tessellate space but all its tilings are aperiodic – achille hui Dec 11 '17 at 01:15

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For regular polyhedra the cube is the only one, using the same sort of argument as in the $2D$ question you linked to. It is the only one that has dihedral angles dividing into $2\pi$. There is a volume filling construction using a mix of regular tetrahedra and regular octahedra.

Your question did not specify regular polyhedra. You can certainly use triangular prisms, parallelapipeds, and hexagonal prisms to turn a $2D$ tiling into filling space.

Parcly Taxel
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Ross Millikan
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  • Thanks you Parcly, that was very helpful. Except cubes, don't these 3D object behind the spoking man also possible combining items to fulfill a volume? https://www.youtube.com/watch?time_continue=51&v=Jr8Iw4o7Gic built with pentagon if i'm not wrong. I feel it works too. – user59488 Dec 10 '17 at 20:05
  • I think Octohedron and dodecahedron could fit in this category. What do you think? – user59488 Dec 10 '17 at 21:10
  • No, because the dihedral angles (the ones between neighboring faces) do not divide into $2\pi$ so you can't put an integer number of them around an edge. – Ross Millikan Dec 10 '17 at 21:12
  • I'm surprised at least for octohedron, i will print it from here https://www.learner.org/interactives/geometry/platonicsolids/ and play with it :), thanks anyway. – user59488 Dec 10 '17 at 21:25
  • The dihedral angle of the regular octahedron is $\arccos (-\frac 13)\approx 109.47^\circ$. You can't pack them into $360^\circ$ – Ross Millikan Dec 11 '17 at 01:45