I wrote: << was able to figure out the face structure of the four equal cylinders (which could be said to intersect as do the great diagonals of a cube). Same method should apply to the higherhedra.

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Here's the method (but I've patented it, and you may need to pay me royalties): Suppose n solid infinite cylinders of unit radius are arranged in 3-space so their axes each pass through the origin. Then the solid K where they all intersect has boundary surface N, where N has "faces" that are maximal portions of 2D unit cylinders. Imagine the intersection of N with the unit sphere S^2 about the origin. That will be a number of great circles of S^2. These circles cut each other up into (uncut) great circle arcs. Each of these occurs in just one face of N. So # faces of N = # largest uncut great circle arcs on S^2. The faces will resemble Voronoi regions of the great circle arcs. In a symmetrical situation, e.g., where the axes are the diagonals of a higherhedron, this will be exact (if the faces of N are radially projected onto S^2). E.g., for 4 main diagonals of a cube, we can pretend the cube is a unit sphere, and the great circles will be equators w.r.t. opposite pairs of cube vertices -- giving us 4 regular hexagons inscribed on the cube. Each hexagon is cut in sixths by the other hexagons, giving us 24 "great circle" arcs, hence 24 faces of the intersection solid K. The Voronoi regions on the sphere are each a quadrilateral with one axis of symmetry. --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele