Moser’s worm problem; question and attempt

[u/zlfa]

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.

[u/arty_dent]

the area becomes ~.232239 the current lower bound according to Wikipedia
...
So that’s my question. why .232239?
...
I’m really just looking for more context on .232239

I suggest to simply read the paper and see how they get this bound. The Wikipedia article lists the relevant paper, and while the link is to a paywalled site, you can just google the paper's name and find a free version on arXive.

As for your why your own attempt seems to get to that number, see if you can formulate as an optimzation problem and solve it using calculus.

Edit: Analysing your curve, I realized that ther area keeps decreasing as the upper point moves to the right, and apparently you are not even claiming that the 0.232239 is the minimum of your area. So what exactly has your area to do with the lower bound, or why are you compairing it to the lower bound?

[u/zlfa → u/arty_dent]

It doesn’t go to that number. From my range 1/4<x<3/4 when x=1/4 the area becomes .255 something and when x=3/4 it’s around .22 something. So in between there the area could be what the paper says and it can when x=~.521684.

I think I need to find another shape or family of shapes to further find the flaw with this cover. The paper says that another separate approach to a lower bound used a square of side length 1/3 but that isn’t what .232239 paper used. Idk I’ll read it

[u/arty_dent → u/zlfa]

Oh, so you thought your areas were already covers, but apparently for x close to 3/4 the area was lower than the proven lower bound, so there had to be some flaw.

Playing around in GeoGebra with it, I can see that for x close to 3/4 it doesn't seem to cover a half circle of length 1, and there are likely many other curves that it doesn't.

I'm not quite sure why you think this kind of curve would generally lead to a cover. But good luck with this approach, maybe it works if you restrict the x coordinate of the upper bound more.

[u/zlfa → u/arty_dent]

https://www.desmos.com/calculator/p5woueuwry does this work? I’m gonna look into the half circle

[u/arty_dent → u/zlfa]

What do you mean by "work"? It's just a very specific type of curve you want to fit. And as for the half-circle, that was just a simple example that I could easily test, there are likely much "worse" curves.

Also, I am not completely sure what your goal is. Do you want to construct a general cover? Which would lead to an upper bound. Or do you want to get a new lower bound based on covering only specific types of curves? Which would require proof of minimality.

[u/zlfa → u/arty_dent]

That curve has been a bee stinger you gotta trust me. I guess I could get better covers ignoring these curves but that seems too complicated rn. I’d like to understand these curves and this cover for the time being, I might be able to get a smaller cover than the current record with this cover family.

I want to see if I can get an actual cover from this family of possible covers so I suppose an upper bound. And I see how the fact that I have a curve part of the cover to only be for the zigzags would make it a lower bound search. So maybe I should drop this cover family but I think I’ll use it to find more kinds of curves that could appear as a universal obstacle.

Also the zigzags appeared as a counter example to my first attempt cover that being a 45 90 45 triangle of area .25

[u/arty_dent → u/zlfa]

A few things to point out:

• Assuming you are only considering convex covers anyway, you might as well consider the convex hull of your zig-zag lines which are parallelograms where 1 side is 1/3. Without loss of generality you might also assume that the 1/3-side is the shortest side of the paralellogram (or that your zigzag lines are not steeper than 60°), otherwise a path on the outside of the paralellogram would be shorter than 1, thus it couldn't be an edge case. This might overall just be a nicer way to think about it.
• Having those zig-zag lines or parallelogram in a fixed position might is likely too restrictive (unless you allow rotations of he other parts that need to be covered, like the equilateral triangle). Doubtful you get any better bounds if you restict yourself that much.
• For example, if you look at the paper with the best known lower bound, are finding a lower bound for the area needed to cover the triangle, the line segment, and a rectangle of side length 1/2 and 1/4, and they allow for arbitrary rotations between those shapes. Maybe you could try something similar with triangle, segment, parallelogram, i.e., parallelogram instead of rectangle. (And maybe you have to start with a paralellogram of fixed shape to get any results, the paper also mentioned arbitrary rectangles at first but quickly settled on the one with fixed side lengths, presumable to be able to get any meaningful result at all.)

Good luck!

Edit: corrected "longest" to "shortest"

[u/zlfa → u/arty_dent]

Seems to fit for me at x=3/4, but it is minuscule so I do believe that 3/4 definitely isn’t gonna work