3D picture of the 38-sided Engel space-filling polyhedron

Question

On page 220 of Peter Engel's Geometric Crystallography, he describes a 38-sided convex polyhedron that can fill space.

I've seen this this accepted as the record in various places, but I've never seen a 3D picture. Has anyone ever managed to make one?

Here is the article where Engel first mentioned his plesiohedron. I'll see if I can obtain a copy... — J. M. ain't a mathematician, Aug 14 '11 at 12:40
I've managed to see Engel's article. No coordinates there, unfortunately (but there might be useful information in there I just haven't seen). My German is no longer as great as it used to be, so if anyone wants to take a stab at constructing the polyhedron, and has good grasp of technical German, I can hook you up with a copy of Engel's article. — J. M. ain't a mathematician, Aug 19 '11 at 03:46

score 7 · Answer 1 · answered Aug 06 '11 at 01:40

7

Here is the figure (I assume it is this one) in Grünbaum's paper to which Joe Malkevitch refers:
Fig. 10: Engel

answered Aug 06 '11 at 01:40

Joseph O'Rourke

4

I'm impressed that thing can fill space. – anon Aug 06 '11 at 02:36
2

I'm familiar with these pictures, but I'll accept them. I'd really like to get a 3D version of the packing. – Silas Pike Oct 03 '11 at 21:27

score 5 · Answer 2 · answered Sep 20 '22 at 13:16

5

Now there is a computed 3D model of it with animation, interactive rotation, and space-filling arrangement of a few polyhedrons:

answered Sep 20 '22 at 13:16

Vitaliy Kaurov

score 5 · Accepted Answer · answered Jul 06 '11 at 20:06

5

There is a "schematic drawing" of the Engel polyhedron in Marjorie Senechal's book Crystalline Symmetries. It appears as Figure 1.14 on page 14.

answered Jul 06 '11 at 20:06

Joseph Malkevitch

1

The schematic drawing is in Engel's book as well, and many other places. Can you turn this schematic into a working 3D image? – Silas Pike Jul 06 '11 at 20:43
Look at Branko Grunbaum's paper here http://www.ams.org/bull/1980-03-03/S0273-0979-1980-14827-2/S0273-0979-1980-14827-2.pdf for something closer to perhaps what you want. – Joseph Malkevitch Jul 06 '11 at 22:29
1

I'm hoping to find something suitable for looking at the 3D graphics of it, including a manipulative 3D object of an Engel38 surrounded by 38 others. I'm also hoping to do 3D printing of it at Shapeways. Grunbaum lists coordinates for the other polyhedra, but not for this one. – Silas Pike Jul 07 '11 at 13:53

3 Answers3