Wednesday, October 20, 2010

So many choices...

After playing around with the sun burst in the previous entry, I remembered these 30-gons with interiors that are made from nice loops of pentagons and triangles that meet exactly at the centre of the 30-gon.  I'm sure I have already made a bunch of patterns that already have this 30-gon but I haven't been able to find a blog entry with one.

My goal for patterns is to have an interesting and symmetric way of making neighbourhoods for tiling.  These 30-gons turn out to have a fair number of choices on how they can relate to their neighbours.  Essentially, coming up with easy to follow rules for describing the pattern is a way of completely understanding the pattern.  These rules embody what I like about patterns: access to the infinite.

My rule I choose to explore here, has to do with having the internal structures back-to-back.  I'll label the structures: loops and arrows.  The loops refer to the three positive-space structures composed of six pentagons and 5 triangles.  The arrows refer to the three negative space structures (white space) that are pointy and all touch the center of each 30-gon.

To see my rule in affect, imagine the pattern without the pink-centred 30-gons.  All the purle-centred 30-gons are connected with back-to-back arrow structures.  Paths that follow these arrow structures form a hexagonal grid with the purple centres being at the vertices.  A similar story occurs for the loops structures but there is a hitch--there are two types of purple-centred 30-gons: ones to the right of a pink-centred 30-gon and ones to the left.  If we ignore all the purple-centred 30-gons on the right, we end up with similar hexagonal grid formed by paths following the loops.

Seeing the above structures makes my rule seem complete.  However, pairing the pink-centred 30-gons with the purple-centred 30-gons on the right leaves us with only back-to-back pairs of loops with arrows.   Still I am fairly content with this patterns symmetry and it's overall appeal.

I have included this blurred patch that can be used as a background.

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