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Wallpaper Patterns As Slices Through Space

Many interesting patterns in 2 dimensions can be understood as a slice through a lattice in a higher dimension. The simplest example is a slice through a three dimensional checker board. If you slice the corner off a cube at just the right angle you get an equilateral triangle. Slice at the same angle, but deeper into the cube, and you get a hexagon. So the pattern of hexagons and triangles, so common in Islamic art, is just a slice through a 3d checkerboard.

This animation illustrates the idea:

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