From Futility Closet:
The ordinary tetrahedron, or triangular pyramid, has no diagonals — every pair of vertices is joined by an edge. How many other polyhedra have this feature? In 1949, Hungarian topologist Ákos Császár found the specimen above, which has 7 vertices, 14 faces, and 21 edges.
But so far these two are the only residents in this particular zoo. “It isn’t known if there are any other polyhedra in which every pair of vertices is joined by an edge,” writes David Darling in The Universal Book of Mathematics. “The next possible figure would have 12 faces, 66 edges, 44 vertices, and 6 holes, but this seems an unlikely configuration — as, indeed, to an even greater extent, does any more complex member of this curious family.”
A few of you might know that I’m very fond of making polyhedrons out of cardstock. It takes a lot of time with the really complicated ones (though not nearly as complex as the one above!!) but it’s a rewarding task and the result is (hopefully) pleasing to the eye. If you are interested in trying your hand at making some yourself this summer, this is a very good source for templates.