Every shape can be cut exactly in half with one straight line

fluoridation 28 days ago |

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?

happytoexplain 28 days ago | |

>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...

fluoridation 28 days ago | | |

Well, I'm listening.

happytoexplain 28 days ago | | |

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.

fluoridation 27 days ago | | |

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.

happytoexplain 26 days ago | | |

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).

happytoexplain 27 days ago | | |

^blindingly^mind-numbingly^

IAmBroom 28 days ago | | |

Continuous is not necessary - a break in the area can still have a halfway point.

fluoridation 28 days ago | | |

Um, sure, but my argument is that the function will always be continuous, as long as you're slicing a single shape.

WithinReason 28 days ago | | |

not every orientation will be

analog31 28 days ago | |

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.

asplake 28 days ago | | |

> 3. Why the Obvious Answer Fails: The Center-of-Mass Trap

fluoridation 28 days ago | | |

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.

gus_massa 27 days ago | | |

The vertial bisection of this:

  XX
  XX
  XX|XXXXXXXXXX
  XX
  XX

is the | symbol, but the center of mass is near the o in the next graphic.

  XX
  XX
  XXXXoXXXXXXXX
  XX
  XX

fluoridation 27 days ago | | |

You'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.

analog31 28 days ago | | |

Ah, thanks. Now that I think of it, the "center of mass" is the center of equal torque.

DougN7 27 days ago | |

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?)

margalabargala 28 days ago |

The math behind the interactive bits is terribly broken. You may want to add testing to the prompt that wrote this article.

bpx51 28 days ago | |

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?

margalabargala 28 days ago | | |

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.

bpx51 28 days ago | | |

Yes, it was a floating point rounding issue when slicing holes. Just pushed a fix, thank you for pointing it out

Intermernet 28 days ago |

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?

VCFundedGenYer 28 days ago |

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.

Lerc 28 days ago |

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.

bpx51 28 days ago | |

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.

bpx51 28 days ago |

The math behind why you can always find a perfect 50/50 cut, even for weird shapes.

Obscurity4340 28 days ago | |

What about chiral molecules, like say D-phenylalanine?

ano-ther 28 days ago |

The game is very nice: https://bisecto.com/

bpx51 28 days ago | |

Thank you

josefritzishere 28 days ago |

Why on Earth is this flagged?

minitech 28 days ago | |

Slop article generated to promote their game that they also posted yesterday.