Cubes are kind of boring. A rhombic dodecahedron, on the other hand, has a certain je ne sais quoi that I like better. What’s your favorite, and why?
If we’re talking 3D I always like the Tetrahedron for its lack of parallel faces. In more than 3D then hypercubes are the only things I can get my head arround.
hexadecahedral geodesic, even if it’s not geometrically possible.
“Hexadecahedral Geodesic” isn’t terribly descriptive, is it?
It’s similar to a truncated icosahedron, the standard “buckyball”, in that it’s made up entirely of hexagons and pentagons, but it’s the other way around. Buckyballs surround pentagons (10) with hexagons (20). The Hexadecahedron, OTOH, has hexagons (4) surrounded by pentagons (12).
An easy way to visualize it. Visualize a dodecahedron (12 faces, all pentagons). Break it apart into four 3-pentagon pieces, so you end up with pointy pentagonal pyramids. Truncate the points of a tetrahedron with the 3-pentagon fragments, until the pentagon edges away from the points touch, turning the face of the tetrahedron from a triangle into a hexagon.
This doesn’t actually work, though, because the edges of the pentagons don’t all end up in the same plane, so it can’t be made of regular pentagons and hexagons. It’s damn close, though, so it’s possible to make models of them using origami. It’s also fun to connect the hexadecahedra together by thier hexagonal faces and building dodecahedra this way (I.E. with a hexadecahedron at each vertex, the hexagonal faces of each hexadecahedron being centered on the edges)
Do 4D solids count as polyhedrons?
If so, my favorite is a simple hypercube.
Since reading the Phantom Tollbooth, I’ve always been fond of regular dodecahedrons.
Polyhedra are strictly 3D. Polytopes may exist in any number of dimensions.
I rather liked the Zocchihedron, just because before I saw one, I wouldn’t have thought it was possible in three-dimensional space…
Polly Hedron would be a cool user name for someone who is a math dork. 
That’s all.
Oh.
In that case my favorite polyhedron is the dodecahedron. But the hypercube remains my favorite polytope.
amarinth: I was going to say the same thing. A merry band of nerds are we.
A close second would be the isocahedron, giving shape to the humble 20-sided die.
I’ve always loved the pentagonal hexecontahedron (12 groups of 5 irregular pentagons each), one of the Archimedian duals.
I’m quite fond of the shining trapezohedron.
And her best friend would have to be Molly Cule, the chemistry dork.
Ooooh! The Controvert! My (current) favorite polyhedron is a rhombic dodecahedron too!
I’m working on a project at the Science Museum called Inner Space. It teaches about crystal structures, including the simple cubic, the body-centered cubic, and the tetrahedral crystal structures. It’s a rhombic dodecahedron-shaped room, and all the walls are mirrors! ( The floor is not a mirror, and there is a doorway in one face that is not mirrored, but otherwise…mirrors!) In each vertex is a light, and along each intersection of a pair of faces is a string of lights. There is a light at the center of the figure too.
Different combinations of lights produce different crystal structures, and they appear to be virtually infinite! You step into this crystal room, listen to pretty music, sound effects, and narration, and imagine you are inside of an infinite crystal lattice! It is way, way cool.
Get this…the lights are digitally controlled red, green, and blue LEDs! They are made by Color Kinetics. My job is to program the light system and make the light show and audio program play nice together. Oh man, it is sooo fun!
If you like polyhedrons, **Inner Space ** is a must-see! Opening in June 2004.
It’s not very interesting (unless you are me, which you are not, or are into astral projection) but I have always favored the humble pyramid. I am a big fan of threes and triangles really do it for me.
They don’t have to be convex, do they? My favorite is the Great Dodcahedron, the first image here . Cool rotational Java at that link.
This shape is the basis of the puzzle known as Alexander’s Star.