On 3/15/07, Daniel Asimov <dasimov@earthlink.net> wrote:

Not sure if I'd ever heard of the Szilassi polyhedron before, but (for those in the same boat) it has 7 (nonconvex) polygonal faces each of meets the other 6, is topologically a torus, and it embeds in 3-space.

See the small but accurately interactive graphic at http://mathworld.wolfram.com/SzilassiPolyhedron.html Mentioning the connection with the dual Csaszar polyhedron is http://en.wikipedia.org/wiki/Szilassi_polyhedron There is further discussion of a putative analogous polyhedron with 12 faces, 66 edges, 44 vertices, and 6 holes at http://www.ics.uci.edu/~eppstein/junkyard/szilassi.html --- however, the latest link shown, to http://www.qnet.com/~crux/szilassi.html appears broken.

My favorite torus polyhedron has 7 regular hexagons as faces, any two of which meet each other along an edge.*

This embeds isometrically in R^6 (so that its 84 symmetries are all in the orthogonal group O(6)).

Does anyone know for sure whether 6 is the lowest possible dimension for this?

An element of the Moebius group in 3-space preserves a pencil of Dupin cyclides; so a suitably chosen 3-space embedding of this tiling is invariant under a subgroup of the mixed orthogonal group O(4, 1), where points are represented via pentaspherical coordinates. I suppose it's conceivable that this subgroup can be mapped into O(5), or even O(4). The "Hopf fibration" of Clifford parallels, traced out by isoclinic birotations (where the angles about the two orthogonal axis planes are equal), reminds me greatly of just such a pencil of tori ... We used to have several people on the list who knew all about such things! Here O(p, q) denotes the group of order n = p+q matrices which preserve the quadratic form (x_1)^2 + ... + (x_p)^2 - (x_{p+1})^2 - (x_n)^2. I don't know of any references for these groups, though I presume the Lie group specialists must have encountered them in some guise or other: can anybody else help here? Fred Lunnon

I've always wanted to try: torus(t,u):=[(cos(t)+r)*sin(u),(cos(t)+r)*cos(u),sin(t)]; trefoil(t):=torus(3*t,2*t); Are there values of r s.t. the following is a toroidal heptahedron?: The subset of R^3 "below" six of the seven planes tangent to the torus at trefoil(2*pi*k/7). It would be the union of seven convex hex(at most) ahedra. Anybody have a good halfspace intersector? If it works for trefoil(phi+2*pi*k/7), varying phi would make a cool "smoke ring" animation, if only we had computer algebra that knew better than z-buffering. --rwg STRIPTEASE TAPESTRIES