Six Mix Trix

Here’s an equilateral triangle divided into six smaller triangles:

Equilateral triangle divided into six irregular triangles (Stage #1)


Now keep on dividing:

Stage #2


Stage #3


Stage #4


Stage #5


Equilateral triangle dividing into six irregular triangles (animated)


But what happens if you divide the triangle, then discard some of the sub-triangles, then repeat? You get a self-similar shape called a fractal:

Divide-and-discard stage #1


Stage #2


Stage #3


Stage #4


Stage #5


Stage #6


Triangle fractal (animated)


Here’s another example:

Divide-and-discard stage #1


Stage #2


Stage #3


Stage #4


Stage #5


Stage #6


Stage #7


Triangle fractal (animated)


You can also delay the divide-and-discard to create a more symmetrical fractal, like this:

Delayed divide-and-discard stage #1


Stage #2


Stage #3


Stage #4


Stage #5


Stage #6


Stage #7


Triangle fractal (animated)


What next? You can use trigonometry to turn the cramped triangle into a circle:

Triangular fractal

Circular fractal
(Open in new window for full image)


Triangle-to-circle (animated)


Here’s another example:

Triangular fractal

Circular fractal


Triangle-to-circle (animated)


And below are some more circular fractals converted from triangular fractals. Some of them look like distorted skulls or transdimensional Lovecraftian monsters:

(Open in new window for full image)


















Previous Pre-Posted

Circus Trix — an earlier look at sextally-divided-equilateral-triangle fractals