From: Hop David Subject: Re: Filling up space Date: Tue, 23 Jan 2001 08:51:45 -0700 Newsgroups: sci.math Summary: Regular solids which tile R^3 J wrote: > Is there a Platonic solid besides the cube which can fill up infinite space > without leaving any gaps? And what about Archimedean solids? I agree with Colin. The cube is the only Platonic solid and the truncated octahedron the only platonic solid. Octahedra alternating with tetrahedra fill space. Some time ago I had asked about polyhedra that tile space and got this response: ----------------------- Hop David observes that Euclidean 3-space can be monohedrally tiled with cubes, rhombic dodecahedra, truncated octahedra, triangular prisms, and hexagonal prisms and asks whether there are others. Leaving aside the tiling by rhombic dodecahedra, which is the dual of a uniform tiling by regular tetrahedra and octahedra, there are just five uniform monohedral tilings of Euclidean 3-space. These are the regular tiling {4, 3, 4}, with cubic cells; the "bitruncated" {4, 3, 4}, with truncated octahedral cells; the dual tilings {3, 6} x {oo} and {6, 3} x {oo}, with triangular or hexagonal prismatic cells; and one other tiling, by triangular prisms arranged in a different way. To construct the last tiling, start with {3, 6} x {oo} and note that the square faces of the prisms lie in sets of parallel planes. Turn every other layer of prisms between a pair of adjacent planes through a right angle. Every vertex of the tiling is surrounded as before by twelve triangular prisms, but now six are oriented one way and six another way. Since the cell polyhedra are uniform and the vertices are all alike, the tiling is uniform. One final twist: Two triangular prisms can be glued together along a common square face to form a "gyrobifastigium," a solid bounded by four equilateral triangles and four squares, one of the 92 nonuniform convex polyhedra with regular faces. So by fusing pairs of prisms in the last tiling, we can create a monohedral tiling by gyrobifastigia. This solid is the only nonuniform convex regular-faced space-filing polyhedron. Norman -- Hop http://clowder.net/hop/index.html ============================================================================== From: Hop David Subject: Re: Filling up space Date: Tue, 23 Jan 2001 09:05:33 -0700 Newsgroups: sci.math Hop David wrote: > J wrote: > > > Is there a Platonic solid besides the cube which can fill up infinite space > > without leaving any gaps? And what about Archimedean solids? > > I agree with Colin. The cube is the only Platonic solid and the truncated > octahedron the only platonic solid. ^^^^^^ oops! should be Archimedean -- Hop http://clowder.net/hop/index.html ============================================================================== From: mathwft@math.canterbury.ac.nz (Bill Taylor) Subject: Re: Filling up space Date: 29 Jan 2001 06:09:23 GMT Newsgroups: sci.math Great little post here by Hop David who writes: |> One final twist: Two triangular prisms can be glued together along a |> common square face to form a "gyrobifastigium," Great name! Where does it come from? I note that DH made a subtle little pun here: the gyro-thingy can be made from a 60^-diamond-ended prism, slicing along the square diagonal, *twisting* it through 90^ and re-glueing. Great final twist indeed! |> four equilateral triangles and four squares, one of the 92 nonuniform convex |> polyhedra with regular faces. Yes, aren't they sometimes called Jacobson polyhedra? Josephson? Something like that. Just to give yet more pangs of jealousy to that part of the math world that cares about such things, let me recount the following. A late colleague of mine here, Derrek Breach, a noted combinatorialist, was also a keen modeller in cardboard. He constructed, over the years and decades, what might be the finest collection of mathematical models anywhere, certainly to my knowledge. It is definitely more extensive than the collections in the British museum and at Havard, and in beautiful colours like them. He bequeathed the collection to the University of Canterbury, and our math department has (some of) them on permanent display here. Anyone can drop in and see them any time. If you're passing through Christchurch NZ it's worth a visit. If you can arrange for it (through me or someone else) get a guided tour through the back room and see them all! Among other things, there is a complete set of the 92 J-whatever polyhedra! ,--. | .:`. (We also have maybe the only glass Boyes' surface in the world!) | .::`. ^^^^^^T^^^^^^^ \ . .::\ | \ .:`. <------------------------------' \ . .::\ \ .::\ \ . .,-`----..._____ \ ( . . `""--._ \ . `.__....:..:..:.......) \ /:. `>::::::::::::_,-' ,'`-':. / `""""""""""' /:. . / /:. . ,' _ /:. . / _| |___ =============================================== /:. . ,' _| _| _|_ Bill Taylor W.Taylor@math.canterbury.ac.nz /. . ,' |___|_ | _|___ =============================================== ( ,' |_| |_ _| Text tiles make very poor textiles. \_.' |___| | =============================================== W F C T |_|