The pinwheel

Start from a rectangle made of two triangles. At every cut each right triangle with legs 1 and 2 splits into five smaller copies of itself, so there are five times as many. Each is coloured by the direction its long leg points, out from the right angle, brighter if it is only turned, darker if it is flipped. Drag Cuts, or press play: the colours multiply cut by cut (from the fifth cut on, the view zooms in by √5 at every cut, so the triangles stay about the same size on screen; the corner and the compass always count the whole rectangle). Before any cut there are 2 directions; after n cuts there are 4n: 4, 8, 12…, growing without end. The reason is in the five copies: some are turned by an angle α ≈ 26.565° that is no fraction of a full turn, so no number of turns brings them back. Switch to Compass and the directions are marked round a circle, turned-only outside, flipped inside; a direction may have both ticks and is counted once. The ticks crowd closer, and the largest gap drops from 90° after the first cut, sometimes holding for a few cuts, to about 12° after twelve, and to nothing as the cuts go on. A pattern that repeats like wallpaper can point in only finitely many directions, so this one does not repeat that way.