In 30 Sekunden
3Blue1Brown introduces a hexagonal tiling puzzle: rotate any three rhombuses that form a smaller hexagon, then ask whether repeated rotations can connect every valid tiling. The short poses the question and points to a longer video; it does not give the solution.
- 1Build the tilingTile a hexagon with equal rhombuses whose angles are 60° and 120°.
- 2Rotate a local hexagonThree rhombuses forming a smaller hexagon can be rotated by 60°.
- 3Ask about reachabilityCan a sequence of these moves transform any tiling into any other?
Das Wichtigste
- The large hexagon is tiled with congruent 60°–120° rhombuses.
- Three tiles that form a small hexagon can be rotated by 60° to create a new tiling.
- The puzzle asks whether those local moves connect every possible tiling; the short leaves the answer open.
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The tiles and the move
A large hexagon is tiled with rhombuses. Three tiles forming a small hexagon can be rotated together.
Here is a tiling puzzle. Start with a hexagon whose side length is a whole-number multiple of the side of a small tile. Fill it with congruent rhombuses, each with angles of 60° and 120°.
Whenever three rhombuses form a smaller hexagon, rotate that little hexagon by 60°. The move creates a different local tiling and can reveal other small hexagons that were not present before, making further rotations possible.
The question
The puzzle asks whether every tiling is reachable from every other. The short directs viewers to a longer video for the underlying insight.
Using only these moves, can you always get from any tiling of the large hexagon to any other? The short leaves the question for the viewer and points to a longer video with several puzzles and unexpected solutions. The presenter suggests that seeing the picture in the right way reveals the key insight.
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