Saturday, February 16, 2013

heptagons bugging out

Lots of twists with this pattern of heptagons and squares.  There are two squares that ended up having to be removed since they were overlapping but you may still be able to spot the 90 degree angles.

The bug pattern is new.  I had a bug pattern a while ago that used 6 heptagons but this one is very different.  One easy way to see the difference is that the bug is surrounded by 8 heptagons.  The other one also used triangles.

Maybe it would better viewed from the side...

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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