Theme
One point away
Constructions that work perfectly except at a single exceptional place, and what is done about it.
Geometry
One cone, four curves
The circle, the ellipse, the parabola and the hyperbola look like four separate objects with four separate equations. They are one object, cut at four angles.
TopologyA sphere is a plane plus one point
Remove a single point from a sphere and what is left can be flattened out to cover an infinite plane exactly. The construction is one straight line, repeated.
TopologySomething always stays put
Stir a cup of coffee however violently and let it settle. Some molecule is exactly where it started. Crumple a map and drop it on the region it depicts, and one point lies over the place it names.
TopologyNothing on a sphere can be combed flat
Point an arrow along the surface at every place on a sphere, continuously, and somewhere an arrow has to vanish. On a doughnut it can be done. The difference between the two is a number that was already known from counting corners.