Monday, February 11, 2013

rhombic patch

Sometimes reasoning about balloon twisting patterns can have interesting non-balloon results.  In this case, I was thinking about a rhombic triacontahedron--a ball that has a surface of 30 identical rhombi.  Doodling 3D things on paper can be tricky so I normally flatten them to 2D by distorting the shapes but keeping the connections (unlike a net this is like a stereographic projection putting usually one face as the perimeter of the rest: here the outer curvy lines represent one rhombus).   From a doodle in Notablity (I like this iPad app's grid), I  created some interesting incidental patterns that may suggest cubes etc.  Since I was in Kathmandu, where there are a lot of T-shirt embroidery folk, I got them to stitch this up for me for less than $10.  I really like the colouring of the regions.  I may give it some more thought but I'm happy for now.

The pattern that you can look for is the alternating of vertices with degree 5 and vertices with degree 3.  Precisely, every degree 3 vertex (where three black lines meet) is surrounded by degree 5 vertices and vice versa.

Perhaps, I'll post the original pattern as well.

BTW, here is an example of a rhombic triacontahedron:

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