[math-fun] Vertex-regular tessellations of 3-space by regular and Archimedean polyhedra
There are of course 3 regular tilings of the plane by copies of one regular polygon. There are 8 more if several regular polygons are used, with the regularity condition being that the group of tile-preserving isometries of the plane is transitive on the set of all vertices: the tiling is "vertex-regular". (9 more if you distinguish between mirror images, but it's more natural not to.) * * * The natural generalization to 3-space would ask for all vertex-regular tilings of space by: a) any collection of regular polyhedra OR, what is perhaps more interesting, b) any collection of regular and/or Archimedean polyhedra. (There are 13 Archimedean polyhedra if we don't distinguish between mirror images.) Has anyone seen a classification of either case a), or better, case b) ??? --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele
Dan, It should be easy to prove that for a) you only get two tilings (Delone cells of, respectively, Z^3 and A_3 lattices) by enumerating valid combinations of dihedral and solid angles surrounding any vertex. How many cases of b) have you found? Veit On Oct 15, 2010, at 7:31 AM, Dan Asimov wrote:
There are of course 3 regular tilings of the plane by copies of one regular polygon.
There are 8 more if several regular polygons are used, with the regularity condition being that the group of tile-preserving isometries of the plane is transitive on the set of all vertices: the tiling is "vertex-regular". (9 more if you distinguish between mirror images, but it's more natural not to.)
* * *
The natural generalization to 3-space would ask for all vertex-regular tilings of space by:
a) any collection of regular polyhedra
OR, what is perhaps more interesting,
b) any collection of regular and/or Archimedean polyhedra. (There are 13 Archimedean polyhedra if we don't distinguish between mirror images.)
Has anyone seen a classification of either case a), or better, case b) ???
--Dan
_____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Hi Dan, I believe the answer is derivable from looking at ringed subsets of the Euclidean reflection groups. I've derived this in the past, but don't recall the complete details or the counts at the moment. Are you familiar with the construction of the Archimedean polyhedra and tilings from ringed subsets of the reflection groups? It's very pretty. Best, - Scott
-----Original Message----- From: math-fun-bounces@mailman.xmission.com [mailto:math-fun- bounces@mailman.xmission.com] On Behalf Of Dan Asimov Sent: Friday, October 15, 2010 4:32 AM To: math-fun Subject: [math-fun] Vertex-regular tessellations of 3-space by regular and Archimedean polyhedra
There are of course 3 regular tilings of the plane by copies of one regular polygon.
There are 8 more if several regular polygons are used, with the regularity condition being that the group of tile-preserving isometries of the plane is transitive on the set of all vertices: the tiling is "vertex-regular". (9 more if you distinguish between mirror images, but it's more natural not to.)
* * *
The natural generalization to 3-space would ask for all vertex-regular tilings of space by:
a) any collection of regular polyhedra
OR, what is perhaps more interesting,
b) any collection of regular and/or Archimedean polyhedra. (There are 13 Archimedean polyhedra if we don't distinguish between mirror images.)
Has anyone seen a classification of either case a), or better, case b) ???
--Dan
_____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." -- Peter Schickele
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (3)
-
Dan Asimov -
Huddleston, Scott -
Veit Elser