I was playing around with shapes and have formed a conjecture. Length of the red circular arc $=$ total length of the $n$ green line segments Conjecture: $$\sup{\left(\frac{\text{Area}_1}{\text{Area}_2}\right)}=\sqrt{1-\frac{1}{n^2}}$$ Is my conjecture true, and if so, is it a known thing? I hav...

I'm wondering if anyone has heard or seen a function that looks or behaves like this one. $$f\left(x\right)=\frac{x-b^{\left(\operatorname{floor}\left(\log_{b}x\right)\right)}}{b^{\left(\operatorname{floor}\left(\log_{b}x\right)\ +\ 1\right)}-b^{\left(\operatorname{floor}\left(\log_{b}x\right)\ri...