I love to explain math. You have been warned.

My prior Deviation, Recursive Snowflake, was made from a tiling of diamonds, rhombi with their small angles measuring 60 degrees. It originated from me playing with Penrose tiles. One set of Penrose tiles are rombi in two shapes: 72 degree rhombi and 36 degree rhombi that fit together to tile the plane in an aperiodic fashion. The proof that the tiling covers the entire plane but never becomes periodic relies on inflation. In the Penrose inflation, every fat 72-degree rhombus is replaced with two smaller fat rhombi and one smaller thin rhombus and every thin 36-degree rhombus is replaced with one smaller fat rhombus and one smaller thin rhombus.

I generalized the inflation to 60-degree rhombi. Since the diamond tiles the plane all by itself, its inflation is pretty simple. A diamond is replaced by three smaller diamonds, as show in the illustration Inflation of the Diamond. The first diamond is uninflated. The second diamond has one inflation. In that inflation, each edge is covered by a 57.735% scale diamond with its long axis along the edge. This leaves a central hole that is the size and shape of another 57.735% scale diamond, so we fill the hole. Everything is limited by the footprint of the original diamond, so the four diamonds along the edges are reduced to half diamonds. Four half diamonds and one intact diamond add up to three diamonds.

In the next step of the inflation, we see several half diamonds that merged with their neighboring half diamonds to form full diamonds. In the step after that, the pattern of the planar tiliing is evident. If I had drawn another step of the inflation, it would blend into the background pattern of small white diamonds.

In studying the inflation of the Penrose tiles, I discovered a handy visualization technique of drawing the tiles in layers with some of the inflation tiles superimposed over the original tiles. I used that technique in creating Recursive Snowflake. In the snowflake I added only two tiles in each new layer for each tile in the previous layer. Adding three would have mostly covered the layer below. I mimicked the reversal of direction that occurs with the Penrose rhombi. I could have done the Recursive Snowflake as a Recursive Star using the fat Penrose rhimbi, and it would have been true to the Penrose inflation. But I prefer snowflakes.