Am I being stupid? Isn't this mind-numbingly obvious? Collapse the shape into a function describing the area of the left side if you were to cut the shape at that X coordinate in that orientation. On one end the function has value zero and on the other its value is the area of the shape. Is there any reason why this function might not be continuous?
>Collapse the shape into a function describing the area of the left side if you were to cut the shape at that X coordinate in that orientation. On one end the function has value zero and on the other its value is the area of the shape. Is there any reason why this function might not be continuous?
Fair point, I used AI to quickly build the interactive widget and English isn't my native language so ofc I used it to help edit the text. I'm looking at the code right now, what did you see that was broken?
Seems fixed now. But when I posted my comment, on mobile, the triangle bisector appeared to be measuring whether you were bisecting the long leg of the triangle, very much not the area.
I wanted to like this, but the article is rife with Claudisms. I don't really care if you use LLMs to generate interesting articles, but maybe re-write it, or prompt it to not produce output like this?
1. The Guarantee: A Smooth Sweep Across Flatland
2. The Surprising Part: Existence Does Not Help You Find It
3. Why the Obvious Answer Fails: The Center-of-Mass Trap
4. It Gets Stranger: The Ham Sandwich Theorem
5. How I Turned the Math into a Game
6. Why This Is Harder Than It Looks
At least there's nothing loadbearing about it.
I still don't know which huge part of the training corpus led to this style. It had to be something massive that I never interact with in real life. Maybe marketing presentations? Pitch meetings? LinkedIn posts?
Where did the LLMs start to learn to talk like this? Are there actual examples from the pre-LLM era that anyone can point to that resulted in this style of output?
How about a circular shape that has an odd number of spiral shaped lobes around the outside.
No straight line cut can cut it in half because all straight line cuts that give 50% area on each side of the line would slice the spirals into disconnected shapes.
Good point, but "cutting in half" here refers to total area, not keeping each half as a single connected piece. Slicing through spirals will definetely cut one side into multiple disconnected fragments, but as the straight line sweeps across the shape, the sum of the areas on the left side still grows continuously from 0% to 100%, so it is guaranteed to cross the exact 50% mark at some point.
Am I being stupid? Isn't this mind-numbingly obvious? Collapse the shape into a function describing the area of the left side if you were to cut the shape at that X coordinate in that orientation. On one end the function has value zero and on the other its value is the area of the shape. Is there any reason why this function might not be continuous?
>Isn't this mind-numbingly obvious?
Yeah, I think so.
>Collapse the shape into a function describing the area of the left side if you were to cut the shape at that X coordinate in that orientation. On one end the function has value zero and on the other its value is the area of the shape. Is there any reason why this function might not be continuous?
Not for this reason, though...
Well, I'm listening.
Along similar lines, if you hang the shape by a string, the line of the string will go through the center of mass, and you can cut along that line.
The math behind the interactive bits is terribly broken. You may want to add testing to the prompt that wrote this article.
Fair point, I used AI to quickly build the interactive widget and English isn't my native language so ofc I used it to help edit the text. I'm looking at the code right now, what did you see that was broken?
Seems fixed now. But when I posted my comment, on mobile, the triangle bisector appeared to be measuring whether you were bisecting the long leg of the triangle, very much not the area.
Yes, it was a floating point rounding issue when slicing holes. Just pushed a fix, thank you for pointing it out
I wanted to like this, but the article is rife with Claudisms. I don't really care if you use LLMs to generate interesting articles, but maybe re-write it, or prompt it to not produce output like this?
1. The Guarantee: A Smooth Sweep Across Flatland 2. The Surprising Part: Existence Does Not Help You Find It 3. Why the Obvious Answer Fails: The Center-of-Mass Trap 4. It Gets Stranger: The Ham Sandwich Theorem 5. How I Turned the Math into a Game 6. Why This Is Harder Than It Looks
At least there's nothing loadbearing about it.
I still don't know which huge part of the training corpus led to this style. It had to be something massive that I never interact with in real life. Maybe marketing presentations? Pitch meetings? LinkedIn posts?
Where did the LLMs start to learn to talk like this? Are there actual examples from the pre-LLM era that anyone can point to that resulted in this style of output?
The game is very nice: https://bisecto.com/
Thank you
AI can't do math. An AI-generated article discussing how to do this in which it's painfully obvious it's wrong is just sad.
Please, do better.
How about a circular shape that has an odd number of spiral shaped lobes around the outside.
No straight line cut can cut it in half because all straight line cuts that give 50% area on each side of the line would slice the spirals into disconnected shapes.
Good point, but "cutting in half" here refers to total area, not keeping each half as a single connected piece. Slicing through spirals will definetely cut one side into multiple disconnected fragments, but as the straight line sweeps across the shape, the sum of the areas on the left side still grows continuously from 0% to 100%, so it is guaranteed to cross the exact 50% mark at some point.
The math behind why you can always find a perfect 50/50 cut, even for weird shapes.