Every shape can be cut exactly in half with one straight line
bisecto.comAm 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.
Sorry, I wasn't challenging your logic, I was implying that it's intuitively obvious - it's funny to see the term "blindingly obvious" followed by a paragraph of formal logic.
Fair enough. I expressed it that because I was asking a genuine question, not being rhetorical. That was the argument I used to convince myself that my (unstated) intuition was indeed true and wanted to know if there was something I was missing.
For sure, proving things even just as a sanity check is definitely interesting (and sometimes things we think are "intuitive" turn out to be false or at least intractable to prove).
^blindingly^mind-numbingly^
Continuous is not necessary - a break in the area can still have a halfway point.
Um, sure, but my argument is that the function will always be continuous, as long as you're slicing a single shape.
not every orientation will be
For example?
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.
> 3. Why the Obvious Answer Fails: The Center-of-Mass Trap
I'm pretty sure that's incorrect. The whole point of the center of mass is that it's at the intersection of all the bisecting lines. If you slice a shape in two and you don't get two parts of equal area then you didn't pass through the center of mass.
The vertial bisection of this:
is the | symbol, but the center of mass is near the o in the next graphic.XX XX XX|XXXXXXXXXX XX XXXX XX XXXXoXXXXXXXX XX XXYou're just asserting that. Why would the center of mass be further to the right from the vertical bisection? That would make it so the left side has a mass of 12.5 versus 8.5 on the right. If you were to pivot that shape freely from that point, it wouldn't stay balanced, it would swing until it ends up as an upside-down T.
You must consider how far away from the point they are in the horizontal direction. https://en.wikipedia.org/wiki/Center_of_mass#Definition
This is very unbalanced: 30 to the right and 55 to the left.
This is almost balanced (my graphic is not perfect). 38 the right and 36 to the left.21 21 21|123456789A 21 2143 43 4321o12345678 43 43Fair enough. I was thinking of center of mass as equivalent to the centroid and only taking into account geometric shape, not leverage.
Ah, thanks. Now that I think of it, the "center of mass" is the center of equal torque.
I was thinking the exact same thing. There is no way it’s not continuous (except maybe inside a black hole where space-time rules break down?)
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?
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.
What about chiral molecules, like say D-phenylalanine?
Why on Earth is this flagged?
Slop article generated to promote their game that they also posted yesterday.