27 Lines on a Cubic Surface
This animation by Greg Egan shows 27 lines on a surface defined by cubic equations: the Clebsch surface. It illustrates a remarkable fact: any smooth cubic surface contains 27 lines.
Hoffman–Singleton Graph
This is the Hoffman–Singleton graph, a remarkably symmetrical graph with 50 vertices and 175 edges. There is a beautiful way to construct the Hoffman–Singleton graph by connecting 5 pentagons to 5 pentagrams.