Little squares and big squares

Today I'm playing with size a bit. I have two patterns that seem a lot different but only differ by the size of the squares used. As with my recent trend of elaborating a triangular grid by inserting polygon sequences between the triangles, here we have a triangle-square-triangle being inserted. By changing the size of the square inserted the squares can be made to meet in the middle of the hole produced by the insertion.

The change in the amount of white space and the addition of the pointy features makes the pattern feel a lot different even though they are closely related.

To highlight the path of the insertions and the role of the original triangles, I have included close-ups showing the path of the new loops.

Another Triangular Grid Variation

This pattern uses the same strategy as the last one. In this case, the inserted polygons are a square, a pentagon and a triangle. There seems to a little too much going on to be attractive. The holes in the pattern are reminiscent of the shurikens. If you look back at that pattern it is pretty much the same except the inserted pattern does not include a square. The three different shapes of holes are analogous.

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.

Octagons, Esses, and Zeds

Here's another pattern that comes from a grid of octagons. It would be orange octagons and white square but the focus is on the loops of four octagons. Inserted between every pair of octagons is a triangle(red) and a square(blue). The pattern of insertion is alternating. For instance, if you follow a loop of polygons (clockwise) that surrounds a hole that resembles an S, then you will see a triangle-square that bends to the right where as the next triangle-square bends to the left. In fact, the original white squares (from a octagon&square tiling with vertex 488) are transformed into the distorted S's and Z's seen in the pattern.

TileLand Homework

Here's some tricky homework. Figure out the types of vertices of each of these patterns. Then make a large patch of one of these patterns in TileLand. The patterns are assigned as follows : Sonya, Audriea, Kelly, Bobbi-Sue, Amanda, then Loren.

Mixing Hexagons and Pentagons

Here's a pattern that alternates a loop of six pentagons with the green hexagons. The loops of pentagons have two orientations. The three holes surrounded by the pentagons and hexagons are the most interesting structures to look at. I'll try to make a few variations of this pattern where I replace the hexagons with other polygons.

Unraveling a Pentagon Loop

See if you can construct the process of unraveling a loop of six pentagons. This is a similar process to Jan. 3rd's pattern. For me, I enjoy looking at the holes in this pattern: the s-shapes and a shape similar to the "Cross of Lorraine" (the double cross). Let me know if you can construct this.