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I am trying to find out if there is at least one polyhedron that tessellates for each valid edge number. I have found one for all edge numbers except 10 and 13. Here is my thought process so far.

Solving this problem with polygons is easy and is necessary for extension to polyhedrons. We add and subtract shapes to either a triangle or square. When these are used as prisms, this allows us to get 3n for n<=3. If we add and subtract a square pyramid this gets us to 3n+8. Doing the same with an irregular triangular pyramid give us 3n+10. This gives us nine unsolved cases: 6,8,10,11,13,14, and 16 and (there are no polyhedrons with 7 edges). For six we have the hill tetrahedron, for eight we have a pyramiddle (square pyramid 1/6th volume of a cube), for 11 we have the hemicube, for 14 we have a Gyrobifastigium., for 16 we have a ten-of-diamonds dodecahedron. There are two types of polyhedrons with 10 edges: tetragonal antiwedge (chiral) and pentagonal pyramids. There are 22 with 13 edges, but I could only find 17. Here are three examples of polyhedrons with 13 edges. I could not show all of them since that seemed to be to large of a file size four polyhedrons made from adding and subrating polyhedrons from prisms

Thanks

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    $\begingroup$ Can you say more about the prisms and prisms with square pyramids that you're imagining? I don't see what your construction is $\endgroup$ Commented Sep 12, 2023 at 1:06
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    $\begingroup$ I updated this post with new pictures showing four polygons. two based on triangular prisms (top), two based on cubes (bottom). two with wedges (left) and two with pyramids (right). $\endgroup$ Commented Sep 12, 2023 at 17:39
  • $\begingroup$ Ah, I see - thanks for clarifying! Where did you find the diagrams for your original image? I'm struggling to locate a good table of small polyhedra by their incidence structures. $\endgroup$ Commented Sep 12, 2023 at 22:16
  • $\begingroup$ en.wikipedia.org/wiki/Heptahedron there are 12 polyhedron here with 13 edges J49 makes 13. aparently i miscounted when I said i found 17. $\endgroup$ Commented Sep 13, 2023 at 15:04
  • $\begingroup$ The wikipedia article for hexahedra mentions an additional 10-edged possibility under the "Concave" section, although it seems pretty tricky to find such a shape that would fill space. en.wikipedia.org/wiki/Hexahedron $\endgroup$ Commented Sep 15, 2023 at 6:41

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Consider the following heptacube, given by gluing cubes to all sides of a central unit cube:

enter image description here

This polycube tiles space: center a copy at all points $(x,y,z)$ for which $x+2y+3x=0\pmod{7}$ and observe that every point with integer coordinates is either such a center or is a single taxicab move away from a unique center.

Cut it into six pieces, each containing an outer cube and a square pyramid from a face of the inner cube into its center. This produces a "house" shape (a cube with a square pyramid on top) that also tiles space.

Now, cut this "house" shape into quarters to produce polyhedra that look like the below:

enter image description here

This is a 13-edged polyhedron that fills space.

I don't yet have any solutions for 10, and it isn't clear to me if there are any (though I suspect so); a restricted question that seems interesting is to ask whether any of a particular combinatorial class of polyhedron can tile space (e.g. can any pentagonal pyramid).

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    $\begingroup$ Beautiful construction! $\endgroup$ Commented Sep 17, 2023 at 18:20

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