From: David Jones Subject: Definition of Archimedean solids Date: Sat, 21 Apr 2001 13:39:12 +0100 Newsgroups: sci.math Summary: Special families of symmetric polyhedra Hi. Just wanted to check whether I have got a correct definition of Archimedean solids. This is that they are the uniform polyhedra of which every face is a regular polygon, and not all faces are the same shape. 'Uniform' means that there exist symmetry operations that take one vertex through all the other vertices and no other points. The definition is sometimes given that they are the only polyhedra with all sides regular polygons, apart from the Platonic solids, to have the same arrangement of regular polygons at each vertex. This is incorrect because this is also true of the elongated square gyrobicupola (a much less complicated shape than its sound might suggest - see http://polyhedra.wolfram.com/johnson/J37.html). The latter shape also has all edges the same length, so that too is not a property unique to the Archimedean and Platonic solids. I would be interested to know whether there is a term for those polyhedra which have all edges the same length. Also, would this include any other solids other than the Platonics, Archimedeans, and the elongated square gyrobicupola? Many thanks in advance for any help with this! Regards, David -- David Jones ============================================================================== From: Ken.Pledger@vuw.ac.nz (Ken Pledger) Subject: Re: Definition of Archimedean solids Date: Mon, 30 Apr 2001 15:13:27 +1200 Newsgroups: sci.math In article , David Jones wrote: > .... > I would be interested to know whether there is a term for those > polyhedra which have all edges the same length. Also, would this include > any other solids other than the Platonics, Archimedeans, and the > elongated square gyrobicupola?.... Paste together two regular tetrahedra, face to face. You get a triangular dipyramid, whose faces are six equilateral triangles. There are lots more of these things, even with regular polygonal faces. Ken Pledger. ============================================================================== From: David Eppstein Subject: Re: Definition of Archimedean solids Date: Sun, 29 Apr 2001 21:12:49 -0700 Newsgroups: sci.math David Jones wrote: > > I would be interested to know whether there is a term for those > > polyhedra which have all edges the same length. Also, would this include > > any other solids other than the Platonics, Archimedeans, and the > > elongated square gyrobicupola?.... As Ken Pledger wrote, a triangular dipyramid will work; so will a pentagonal dipyramid. Another infinite family of solutions is given by zonohedra with unit-length generators. The rhombic dodecahedron and rhombic triacontahedron are well known symmetric examples, but you can make unit-edge-length zonohedra with much less symmetry. See for pointers. -- David Eppstein UC Irvine Dept. of Information & Computer Science eppstein@ics.uci.edu http://www.ics.uci.edu/~eppstein/