Explore a zonotope tiling

The colored pieces are parallelepipeds that fit together to form a three-dimensional polytope, called a zonotope. Horizontal slices cut these pieces into triangles; the dual graph joins neighboring triangles.

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Try this first

1. Rotate: drag the picture. Scroll or pinch to zoom. On a touchscreen, drag with one finger to rotate and use two fingers to zoom or pan.
2. Assemble the tiles: move tile gap to zero. Then increase it to pull the pieces apart. Slice triangles travel with their tiles.
3. Look inside: choose a slice level k and click Show level k. Lower tile opacity to see it through the tiles.
4. Inspect the graph: select Slice tiles to keep just the tiles meeting your slices, enable dual graph, and lower tile opacity. Return to All tiles to see the full polytope. Focus keeps the other tiles faintly visible.
5. Compare levels: click the numbered slice buttons to turn individual levels on or off. Several can be visible together. Clear slices hides them all.

What the parameters mean

n is the number of generating vectors. Their endpoints form a regular polygon in the plane z = 1. Changing n creates a new tiling and hides all slices. The level k means height z = k before the drawing is centered and scaled; levels 0 and n are single points.

Aspect ratio stretches the vertical direction. Rotation speed starts a slow spin. In Advanced, change the tiling seed to get another tiling: the same n and seed reproduce the same tiling.

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Keyboard shortcuts

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Read the mathematical background.