∞ Möbius Strip
A surface with only one side and one edge. Drag to rotate. Trace a path and watch it return to the start on the "opposite" side — proving there is no opposite side.
Möbius Strip — Topology
Discovered independently by August Möbius and Johann Listing in 1858, the Möbius strip (or band) is the simplest non-orientable surface: it has only one side and one edge. If you paint one surface, you end up coating both sides without lifting the brush.
Half-turns: 1 half-turn → classic Möbius strip (1 side, 1 edge). 2 half-turns → a cylinder (2 sides, 2 edges). 3 half-turns → still 1 side, 1 edge. Odd half-turns always give 1 side; even half-turns give 2 sides.
Cut down the middle? Cutting a 1-twist Möbius strip along its centre gives a single longer loop with 2 twists — not two separate pieces! Cutting ⅓ from the edge gives a Möbius strip plus a separate loop.