What are the only 3D objects with which it is possible to fullfill a volume?

602 Views Asked by At

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

1

There are 1 best solutions below

5
On

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.