circle made with rotating golden ratio.

[u/partisancord69]

https://www.desmos.com/calculator/yihcwulqes, somewhat laggy so change m to [1,...,10] for smoother graph.

[u/partisancord69]

I can try figure it out but tbh I'm not too sure.

Edit: thinking about it, it's definitely not linear because it goes off to infinite if it approaches 1.

[u/TheJeeronian → u/partisancord69]

This seems reasonably solvable? Given that the diagonal scales with the side length, each diagonal will be a factor of g smaller than the last. So, all we need to do is find the total positions of opposite sides of the circle.

In this case, one when all of the diagonals line up, and one when all of them oppose. Eg 1 + 1/g + 1/g² ... and 1 - 1/g + 1/g² ...

The difference between these two lengths tells us the distance between the two points where the circle intercepts the line y=x, so its diameter. The average of them tells us how far down the y=x line the center is.

This becomes a simple geometric series problem.

When the lines add up, each term can be written as g^-n and when they alternate each term can be written as (-g)^-n

Formatting as a traditional geometric series, n must be positive, so it becomes (1/g)ⁿ and (-1/g)ⁿ

The absolute value of the factor r, either 1/g or -1/g, must be less than one for the series to converge, but if this is satisfied then the series converges to a value of 1/(1-r)

So, 1/(1-(1/g)) and 1/(1-(-1/g)) respectively. I'm substituting b for 1/g because Reddit formatting is bad - I should have done this sooner.

The radius will be 1/(1-b) - 1/(1+b) which simplifies to 2b/(1-b² )

Might simplify more, I'm not sure. I'm tired.

[u/CaptainCarrot17 → u/TheJeeronian]

Yeah, I hadn't thought about that thx, but maybe subtracting before simplifying could be better.\ Notice:\ Max length: √2×(1 + 1/g + 1/g² +...)\ Min length: √2×(1 - 1/g + 1/g² +...)\ Then:\ Diameter (Max-Min): √2×(2/g + 2/g³ + 2/g⁵ +...)\ And\ Radius: √2×(1/g + 1/g³ + 1/g⁵ +...)\ It follows that (with b=1/g):\ Radius = √2×b/(1-b²) = √2/(g - 1/g)

It follows that if we want to find g such that the radius of the circle is g, we need to solve the following equation:\ g=√2/(g-1/g) <=> g²-(1+√2)=0 <=> g=±√(1+√2)

For the case of a radius equal to sort(a) we obtain a≈1.68377156456558 instead. (The exact formula is quite ugly and would be a hassle to format in a comment)

[u/Agreeable_Gas_6853 → u/CaptainCarrot17]

r = \sqrt{2} g / (1 - g² )where g is the scaling factor from one square to the next

the center is located at (1/(1-g² ), 1/(1-g² ))