# space-filling polyhedra

Which polyhedra fill space such that each vertex has the same number of neighbours? (besides extruded triangle, square, or hexagon)

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Perhaps look at the Mathworld article on Space-Filling Polyhedra, and then sharpen your question to whatever is not answered there? mathworld.wolfram.com/Space-FillingPolyhedron.html –  Joseph O'Rourke Apr 28 '11 at 11:51
What's an extruded triangle? Ah, I guess that would be a triangular prism. Never mind. –  Gerry Myerson Apr 28 '11 at 12:01
@Joseph: I looked there but couldn't find a reference to the number of neighbouring vertices. Or are all vertices of all space-filling polyhedra connected to the same number of vertices? –  Thomas Apr 28 '11 at 12:17
Sorry, what I meant was: In a polyhedron where each face has the same number of edges, are all vertices connected to the same number of vertices when the polyhedra are stacked together to fill the space? –  Thomas Apr 28 '11 at 12:30
It sounds likely that you have some bad assumptions about which polyhedra tile space. Please be a lot more clear about what you know and what you are asking. –  Douglas Zare Apr 28 '11 at 16:37