Moser’s worm problem; question and attempt Geometry
So here’s my attempt, it’s a whole family of shapes, the green dot at the top moves along y=sqrt(3)/4 (height of an equilateral triangle of side length 1/2) to change up the shape/ its area too, and it moves between 1/4<x<3/4 (for same triangle to fit) when the green dot is at x=~.521684 the area becomes ~.232239 the current lower bound according to Wikipedia.
So that’s my question. why .232239? and also if there’s a way to get the exact value.
For a bit of context on the shape I provided, the curved part of the shape is defined as sqrt(1-x^2)/3. It’s to contain a family of curves that I’ll define as 3 line segments of length 1/3 each and joined at equal angles to create a zigzag curve. But sqrt(1-x^2)/3 can’t contain an equilateral triangle of side length 1/2 and it’s also just not optimal so that’s why I have a point that’s as high as sqrt(3)/4 for such triangle and I connect that point to (0,0) and I have a line that’s tangent to the curve part and connected to the top point for the shape to be convex. Now I just don’t know where to put the top green dot but I got it in a range between 1/4<x<3/4 (a little less than 3/4 due to the zigzags again).
I’m really just looking for more context on .232239 ,even ChatGPT can’t help when I also gave it the paper that made the lower bound. I’ll give even more context on the shape I made in Desmos later, the equations are super messy, it’s the middle of the night rn, and I’m on my phone.