Friday, June 15, 2007

Triangle Grid Elaboration

This pattern used the same technique of expanding a loop path with alternating squares and triangles. Here, I am using an orange triangular grid of and inserting purple squares and red triangles. The difficulty arises that I cannot expand all the triangular loops in the same way: the neighboring loops affect the possible choices. To help show how the particular loops of six orange triangles are expanded, I have included a zoomed in version below. On the left, I have drawn in the augmented paths of the two loops. On the right, I have filled in the polygons involved with each new loop where the square/triangles pairs are recoloured black and/or blue so as to highlight the choices of the orientations.

This style of elaboration is more complicated when using the triangular grid as a base. This is due to the oddness in the number of sides of the triangle and the asymmetry of square/triangle inserted pieces: there is no simple (uniform) choice of fitting things together like in the octagon case below. But it does produce a more interesting pattern.

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