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In Peter Engel's 1991 paper Über Wirkungsbereichsteilungen von kubischer Symmetrie, a 38-sided convex polyhedron is described which apparently can tile space. I'm interested in better understanding this polyhedron and how it fits together with congruent copies of itself, but little information seems to be available on it (e.g. we don't even seem to have a 3D diagram), so I thought I would try and extract the information from the original paper and build a model myself.

However, when I got to what I assume is the important part of the paper, I was rather confused: enter image description here There is a table taking up most of a page, part of which is shown above: it has 38 rows, and Google Translate labels the first and last columns as "face" and "corners in the face", so I would assume that the second column describes the bounding planes making up these faces somehow. But I have no idea how I am supposed to interpret this notation as describing a plane - any German speakers or geometers who might be able to shed light on this notation would be welcome!

If I've misunderstood what this table represents in some way, then I'd like a pointer to whichever part of the paper does describe the polyhedron explicitly.

RavenclawPrefect
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  • It seems to me that, if the right hand column is written cyclically (which seems a reasonable guess), you should be able to at least draw a (combinatorial) net for the shape from that information. Some things are a little confusing though. For instance it looks like the vertices $66$ and $67$ appear in only one face each. Surely that can't be right? – Dan Rust May 14 '21 at 09:47
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    Our 38-sided thing is a Voronoi cell of some initial point $(x,y,z)$ (I suppose the coordinates are somewhere there in the paper) surrounded by 38 other points which are its images under the given symmetry transformations. – Ivan Neretin May 14 '21 at 10:24
  • @DanRust As for 66 and 67, we only see a part of the table. – Ivan Neretin May 14 '21 at 10:28
  • @IvanNeretin that would make a lot of sense given OP said the shape has 38 faces... Thanks. – Dan Rust May 14 '21 at 10:44
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    https://refubium.fu-berlin.de/handle/fub188/10176 has coordinates for a voronoi cell, but I never managed to get the author's program to work. – Ed Pegg May 17 '21 at 01:52
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    A link to this related question, which provides coordinates for the polyhedron and thus mostly obviates this question. – RavenclawPrefect Oct 23 '22 at 08:11

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