The eighteen convex uniform polyhedra.
'Uniform' here means that all the faces are regular polygons and all the vertices are congruent.
The first five are the regular polyhedra, sometimes called the Platonic solids. The other thirteen are sometimes called the Archimedean solids.
More information: Uniform Polyhedron on Wikipedia, Uniform Polyhedron, Platonic Solid, Archimedean Solid on Wolfram MathWorld.
You can tile a sphere exactly with identical copies of a spherical triangle, but only certain triangles work. Those triangles are called Möbius triangles, after August Ferdinand Möbius (the Möbius Strip guy).
A spherical triangle consists of three points on the surface of a sphere joined by straight lines. In spherical geometry, the 'straight lines' are arcs of great circles.
More information: Schwarz triangle on Wikipedia, Triangular Symmetry Group on Wolfram MathWorld.
This one was inspired by a video on the Numberphile YouTube channel.
Construct a random domino tiling of the Aztec Diamond board, using iterated shuffling. Inspired by a video on the Mathologer channel.