EVEN MORE PITHAGOREAN TREE
Hi im David, I was looking a t this when I realized that there were some squares and triangles with the same area in the pithagorean tree.
eventualy the figures get one on top of the other , so I couldn’t actually prove it, I looked on google couldn’t find any thing except for these 2 cases listed in the first image. So I made this (I don’t have the best scanner ok)
then after some time I wrote
Is kind of bizzare but the first one is for the summed area of the saquares on any repetition ,the other one is for the triangles(area) next to any sqare
C2=((A2+B2)n/C2n-2) for sqares
(ba/2) =((A2+B2)n ba/ 2C2n) for triangles
(n is the number of repetitions starting from cero on the first shape and adding one the next time a shape repeats).
The first purple sqare is for when “n” equals cero, the second purple sqare is for when “n” equals one and so on ,
The same with the first triangle for when the formula for triangles has “n=0” then the second one when “n=1”and so on.
Yes now using pascal’s triangle…
We assume that only cases n=0, n=1 on any of the formulas doesn’t have figures with the same area.except when a=b(when there are all figures the same on any repetition) ,when c=0(has no area nor figures ,or when[c=b,a=0]and in the same fashion when [c=a,b=0](beacause both[c=b,a=0]and [c=a,b=0] are lines of sqares).
Notice these formulas are not to get the area of the fractal but to show that the sum of all sqares’ area on any repetition equal the area of the first sqare. and the formula for the triangles shows that the sum of all triangles area on any repetition equal the area of the firs triangle.
I made the hand made images, and the pretty digital ones I got them from google.
If there is something wrong, let me know. is the first time I do something like this.
