Monday, December 18, 2006

Two conformations of the upper branch

In the process of weaving either branches of the toroid, it is not hard to notice that the beads on the edge form a large pentagon. It is understandable since there are exactly ten heptagons arranged in a petagonic pattern in the inner-rim. The extension of of the inner-rim outward by weaving more beads into the structure make the pentagon pattern even more vivid.

If we only introduce hexagons, the resulting structure is going to be like a funnel on both side of the inner-rim. But if five pentagons are introduced like the one the following picture, the place where the petagon is located will have a positive intrinsic curvature.
From Craft Projects

The structure now have a bistable potential. By which I mean the five hexagons surrounding each pentagon can have two relatively stable spatial positions. We can easily flip the hexagons as shown in the previous picture into the one like the following picture. From this conformation, we can create the final toroidal structure.
From Craft Projects

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