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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?

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>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:

  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

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.

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.

  21
  21
  21|123456789A
  21
  21

This is almost balanced (my graphic is not perfect). 38 the right and 36 to the left.

  43
  43
  4321o12345678
  43
  43

Fair 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?)

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