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?
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?
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.
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.