MoKai's Ninja Star

I have been thinking about this triangle / star shape for years now, and this semester it is hopefully going to really come together. I made these images with Geogebra, and they start by first portraying just the growth outward.

 Then on the last one I show what it looks like if we inscribe this whole thing scaled down into one of the empty triangles.

Before I started trying to work with GeoGebra I did some drawings by hand, in which I let the design grow inwards as well as outwards. The rule I used was to add all of the triangles of a particular size to the drawing where ever they fit.

 Notice that when I drew these ones I did them all inscribed into an equilateral triangle, as the design would be when placed inside of itself.

 On the last iteration that I drew I realized that the whole design could be inscribed into the irregular hexagon you see below. This will end up being crucial to calculating the areas involved in the design as it iterates towards infinity. The triangular pieces left over when the hexagon is drawn inside a triangle of corresponding size will be the only untouched spaces in a way.