Fixed that for ya.
While the two pieces of bread and the piece of ham might have centroids, the problem here is to slice them simultaneously so that all three are bisected with a single slice, without moving them.
When I teach the Ham Sandwich Theorem I usually emphasize that the three volumes can be anywhere, for example one piece of bread on your kitchen counter, the second piece of bread in orbit around the moon, and the ham a chunk the size, shape, and location of the star Betelgeuse. There exists a single plane that slices all of them simultaneously.
By the way, I’m pretty sure I saw a video (we called it a “movie” back then) on the Ham Sandwich and the Hairy Billiard Ball Theorems when I was an undergraduate in the mid-1970s. I don’t know where that video is today. Funny theorem names seem to attract popularizations now as well as they did then.
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Unfortunately, Orwell’s Corollary tells us that one half of a ham sandwich is always more equal than the other half.
And if you think that the labor conditions are bad; have fun elaborating a system of mathematics where the equivalence relation is subject to those constraints…
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Right. And if you’ve got three centroids, you’ve got three points that uniquely define a plane. Since this plane ipso facto passes through the centroid of each shape, the plane bisects each volume! No calculus, topology, or higher geometry required. So why does Intermediate Value Theorem / Borsuk-Ulam come into it at all?
I suppose that at some point in proving the existence of centroids, you would probably use I.V.T., but it seems sloppy to involve I.V.T. in the traditional proof of Ham Sandwich Theorem when you could just offload it onto the existence of centroids as a lemma. As for higher dimensional cases, is it possible that centroids are not generally well-defined?
No, that is not correct. The centroid is the effective center of volume, but it is not the case that every plane through it bisects the volume. Consider, for example, a single object consisting of two spheres of unequal size separated by a long line segment. The centroid will be on that segment between the two spheres, and a plane through that centroid orthogonal to the line will not bisect the volumes.
I’m not sure I believe you. I don’t think you can have a center of volume whereby a separating hyperplane through that point doesn’t bisect the volume. In the example you give, my intuition tells me that the centroid would have to be within the larger sphere (near its perimeter, I guess).
Maybe I’m just assuming that “centroid” always means “center of volume” when it doesn’t?
Maybe I’m just assuming that “centroid” always means “center of volume” when it doesn’t?
Well, it depends on what “center of volume” means. If it helps, recall that the centroid is an arithmetic mean, not a median. With my example, imagine that there is a uniform mass density, so the centroid is the balancing point. If the two balls are very nearly (but not exactly) the same volume/mass, and the distance between them is relatively great, then you wouldn’t expect to be able to balance them by putting your finger adjacent to (or within) the larger sphere.
[Edited for I need to go to bed and this is giving me a headache.]
OK, I think my problem is that I never internalized the fact that the calculation of center of mass calculation is weighted by the relative moment of each part of the mass. I don’t know how I got through a math degree without really being cognizant of this.
So I guess my question is: does an arbitrary volume contain a uniquely defined (point) center such that any dividing hyperplane through that point bisects the volume into two equal volumes? I had always assumed yes, and that it would be easy to find, but I guess maybe not. And now that I think about it there are plenty of trivial counterexamples (e.g., a three-pointed star shape).
Never change, none pizza with left beef.
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actually, I kind of wonder if that approximates the spiky spheres bit that Matt Parker alluded to.
Do you have kids?
Just one, a spoiled little ape, now bigger than me.
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I’ve only got one too, but she is very little. I do remember vividly fighting with my brothers over how each piece had to be exactly equal.
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