The space-filling n-hedron sequence

[Ed Pegg]

Q: How many space-filling tetrahedra families are there?  
A: Five.   Hill (1896), Baumgartner (1968), Sommerville (1923) and Goldberg (1972)
https://demonstrations.wolfram.com/SpaceFillingTetrahedra/ 

How about pentahedra to 38-hedra?  The last seems to be 4, according to Peter Engel (1986). 

At https://mathworld.wolfram.com/Space-FillingPolyhedron.html  is a messy paragraph describing some of the enumeration work by Goldberg and others.

I've recently compiled the 3903 space-filling polyhedra from  
Schmitt, M. W. "On Space Groups and Dirichlet-Voronoi Stereohedra."   I'll be presenting them in my Math Games talk next Thursday. I'll be glad to send the mess to anyone early if they want, otherwise it'll be at Wolfram Community the same day. I can also share my compilation of Goldberg papers.

Right off the bat, I know Schmitt's results are also incomplete, since he doesn't have all five tetrahedra. His spacegroup methods only give three tetrahedra.  

I'm going to try to figure out if there's anything in my Schmitt list missing from the Goldberg list for 6-hedra and 7-hedra.

Another thing I know is missing: Pentagonal prisms.  Not all of the 15 tiling pentagon families are in the data.  That leads to another question:  Are there any space-filling polyhedra that are not part of an infinite family?  Are there sporadic polyhedra that cannot be altered?  We know there are two sporadic pentagons.  I think two of the tetrahedra are sporadic.

Anyways, the count of families for n-sided space-filling polyhedra is currently
5, ?, ?, ...., ?, ?, 4  

My semi-meaningless Schmitt counts for 4-hedron to 38-hedron are
{16, 81, 210, 277, 162, 123, 184, 153, 180, 252, 469, 346, 312, 305, 270, 165, 100, 74, 47, 46, 26, 25, 28, 11, 12, 9, 3, 4, 2, 3, 2, 3, 1, 1, 1} 
There are lots of repeats and degeneracies in here.  For example, the 16 tetrahedra boil down to 3.  

This sequence is currently in awful shape, but I believe I know enough now to program out solutions. My code can use a group number and generator point. If a slight variation has the same graph structure, then it's part of a family. Mapping all the families over all the groups and separating polyhedra with identical graphs but different structures is doable, as is linking up polyhedra in different groups. And then bringing in sporadic cases. Seems doable.  

[Ed Pegg]

Most space-filling polyhedra are in families. I believe enumerating the families is within reach. 
But there are also some sporadic cases.  
1.  With tetrahedra, there are 2 sporadic cases that are not within a family.  
2.  For tiling pentagons, there are 2 that are not a part of the 13 other tiling pentagon families. These lead to sporadic pentagonal prisms.

Are there more sporadic cases than these? The 4 cases above were difficult to solve, but the solutions came from well-defined problems.  

--
You received this message because you are subscribed to the Google Groups "SeqFan" group.
To unsubscribe from this group and stop receiving emails from it, send an email to seqfan+un...@googlegroups.com.
To view this discussion visit https://groups.google.com/d/msgid/seqfan/0a6655ab-bd30-4d31-bbf0-fcdc122f0c12n%40googlegroups.com.

[Robin Houston]

Surely we are a long way from having a complete enumeration of space-filling polyhedra?

For one thing, we are still discovering new families of plane-filling polygons, such as the hat and spectre families.

Separately, I was recently surprised to discover that this polyhedron https://skfb.ly/pHGzU fills space, and the smallest repeating unit I could find (that tiles periodically by translation) consists of 64 of them. If this falls into any standard class of space-filling polyhedra, I would be interested to hear about it.

Cheers,

Robin