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George @ MAKE showcases these really cool examples of mathematical beadwork structures by Kazunori Horibe.

Looking closely at one example, you can see how the surface curvature depends on the structure. Generally, six-sided cycles correspond to an infinite tessellation of hexagons, which makes a flat plane or can be rolled into a cylinder. But in the places where positive curvature (a spherical region) is desired, some pentagons are used instead of hexagons. And in places where negative curvature (a saddle-shaped region) is desired, some heptagons are used instead of hexagons. With this knowledge, the bead designer can control the surface outcome.


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