Hi all,
Just dropping to mention a curious fact that I discovered today: the Hasse diagram of the face lattice of an n-simplex is isomorphic to the edges and vertices of an (n+1)-hypercube.
For example, given a triangle ABC, we have the set ABC, then the edges, AB, AC, and BC; and then the vertices, A, B, and C, and finally the empty set. If you draw the Hasse diagram, you'll see that there are 8 nodes, corresponding with 8 vertices, and 12 subset relationships, corresponding with 12 edges.
The case of the triangle by itself is unremarkable, and may be attributed to coincidence... but if you draw the Hasse diagram of the face lattice of a tetrahedron, you'll see that it's isomorphic to the edges and vertices of a tetracube.
In general, an n-simplex's face lattice gives rise to a Hasse diagram that is isomorphic to an (n+1)-cube. This is a rather remarkable fact, and quite unexpected to me. What do people think about this strange connection?
(Some explanations for those who are unfamiliar with face lattices: to understand a what a lattice is, wrt to polytopes, consider the set V of the vertices of, say, a cube. We may represent each face as a 4-element subset of V, containing those vertices that define the face, and we may represent each edge as a 2-element subset of V, and each vertex as a singleton set containing the vertex. Now, add the empty set to the mix, and you have a collection of subsets of V, including V itself (which represents the entire cube). Now, to make the Hasse diagram of this structure, represent each of these subsets of V as a node, and connect two nodes A and B if A is a subset of B. But omit the edges between vertices and V, since it's implied (the subset relation is transitive). This gives you a graph, the Hasse diagram with a number of nodes and some edges connecting them, describing the face lattice of the cube.)
Now, I've also wondered what the Hasse diagram of a square may represent, and I get a graph that looks like the dual of a square antiprism. I haven't tried to draw the Hasse diagram of a cube yet, but it might turn out to be an interesting shape.