I drew this fractal triforce during math class because I was really bored. Then I decide to find a way to calculate the number of triangles there was in my drawing. There’s different level of precision, at the first level there’s is 1 triangle, at the second there’s 5, at the third there’s 17…

We quickly understand that the number of triangles of one level of precision is equal to the previous number of triangles more 4 multiplied by a power of three increasingly higher.

- First lvl: 1

- Second lvl: 1+4.1 = 5

- Third lvl: 5+4.3 = 17

- Fourth lvl: 17+4.9 = 53

We can transform the formula for something simpler and which avoids having to calculate each previous levels of triangles for calculate the level we want. After some time I find this:

3^{n-1}+2.(3^{n-2}+3^{n-3}+…+3^{n-n})

I drew a fractal triforce of a level of precision of 7.

729+2.(243+81+27+9+3+1) = 1457

So there’s 1457 triangles is this drawing.

If you simply want to know the number of triangles of one size it’s just 4.3^{n-2}, and n is the level of precision of the kind of triangles you want to know the number. For example the there’s 12 triangles of the third level of precision and 108 triangles of the fifth level of precision. But the weird thing is that if you want to know the number of triangle of the first level of precision the result is 1,33333…

I don’t know where is the mistake but I like the result like that. It explains why just one equilateral triangle looks so beautiful. It’s because there’s not really one triangle but one and a third which is hidden.

Yeah, I know I have a strange way to have fun during math class but equilateral triangles are so full of emotions.