Topology is weird: The Ham Sandwich Theorem 

Even if the colors are discontinuous?

Like water ice and land on earth but we have 2 world islands

@@manavnaik1607 Yeah, disconnected regions are fine

@@columbus8myhw wow yeah I thought so but I wasn’t sure how u meant it. Draw a line through the sphere. 2 points sit on the line. You have 3 colors. So one is always left out of the line and makes a new line that goes through itself. Imagine a knife cut through the same ball. It always circumscribes a triangle so it takes 4 colors to have the same effect right?

@@columbus8myhw or I’m misunderstanding?

Higher dimensional food could appear directly inside your stomach. No swallowing required!

it would be a really sus comment if u didn't specify the corpse sandwich

haha ham math > exams:D

SAME You're not in my French class, are you?

Hehe to help lol

ha, true!

I was wondering the same thing, chances are the theorem only applies in measurable spaces. It seems very intuitive for real-numbers but what about Borel ham sandwiches?

@@XENOGOD The theorem is very measure theoretic. It somewhat requires you to have the objects in R^n so you can sensibly talk about hyperplanes, but there are some generalizations. This works wrt any measure as long as the objects are measurable.

Wouldn't a non-measurable ham sandwich require an infinitely sized pig? 😉

@@EdKolis Or an infinitely fine cutting knife and whatever sized slice of ham you like 😉

Thank you! That is all MATLAB:)

ha thank you! Love that one:D

Ha, indeed! Although actually the arguments are rather different from that!

@@DrTrefor Mhmm! It just reminded me of the name

There are extensions of this theorem to basically different measure-theoretic notions

It's a nice idea, but you can't have a center of volume in this sense. For example, if you consider an equilateral triangle, the only possible place the center of volume could be is its center, by symmetry. However, a line parallel to one of its sides will divide the volume in ratio 4/9 : 5/9, as you can check. Maybe there's a related notion that works though!

In addition to the equilateral triangle counterexample, consider three small disjoint circles. (If it bothers you that this isn't a connected shape, you can add a thin curve connecting them of negligible area.) Then for any proposed "center of volume" you can find a line that splits the circles 2:1. (Either that or the center of volume is _in_ one of the circles, in which you can find a line that splits them ~2.5:~0.5 or so.)

@@nicholassullivan6105 oh yeah you're right. Thank you!

Yes there is. Just compute the center of mass assuming that the density is constant. There point you get is your "center of volume".

I don't think that's how center of masses work. Imagine a L shape with very skinny equally long legs, then the COM is outside the L. But if you slice from that point parallel to one of the L's legs, you don't get the whole leg so you get less than half the volume.

For spheres that is indeed fine, actually it is how I animated the planes. The much more interesting part is that the same argument works for nonspheres.

yes there is also related theorems like this

That's exactly right, for spheres it is obvious any slice through the centroid works. Is it obvious for any shape?

@@DrTrefor I didn’t go though an analytic proof, so heuristically speaking, the definition of centroid is the center of mass. If there’s a cut passing through the centroid and ended up with different volume(mass) on each side, it kind of violates the definition we started with. A rigid proof I’m afraid I have to invite the best friend of humans, fundamental theorem of calculus, to the discussion.

Indeed, there are a few different measure-theoretic versions of this theorem actually. Didn’t want to go there for the video itself:D

Nope all these are equally good for the theorem’s arguments I believe

Haha I didn’t mean it but definitely Ken

Yes indeed this works for anything you can give a "measure" to in some sense.

hahah true!

ooh I have been thinking about doing a video related to this!

The derivation with intuition can be found in "Classical mechanics" by John taylor

Thank you!

I had the exact same question throughout the video. I’m wondering what “slicing in half” means for objects that aren’t spherically symmetric.

Thank you!

This is just geogebra!

@@DrTrefor Thank you.

Exactly the same! The visualizing was with spheres but the arguments are general

@@DrTrefor but if you rotated the plane about the center we wouldn't always have the volume cut in half, if its an asymmetrical object?

@@replicaacliper there would always be some plane with that particular normal vector which cuts the first object in half (due to the intermediate value theorem).

@@replicaacliper what you need, however, is for the sets to be compact, otherwise the theorem might fail. So one of the objects can’t go off infinitely far in one direction, for example. This ensures that there is some plane with a given normal vector where all of the object is on the positive side of the plane, and some other plane with the same normal vector where all of the object is on the negative side of the plane. Then we can use the intermediate value theorem.

It sadly is not a constructive proof, so we know it must exist but the proof sketch of the video gives no mechanism by which to find it.

So in a way its similar to other impossible things like trisecting an angle or squaring the circle? Because using the same argument you can make corresponding claims about the existence of a trisected angle, squared circle or doubled volume of a cube. You just keep going until you have 1/3 which is somewhere between 1 and 0. And when you're there, you just start going from 1 to 0 until you get to 2/3 and this will be a trisected angle. But there is actually no way to stop, when you're there. Similar to how there is no way to contineously construct shapes like square of a circle or doubled volume cube which will be somewhere if you go from 0 area/volume until infinity, but because constructive things leave out an uncountably infinite amount of things out (the most primitive case being numbers and how you cant get past irrational numbers), there is always a chance, that you have missed and in those cases the chance is actually 100%. Does it mean, that constructively we and everything else are confined to a countably infinite set of things and we have no access to an uncountably many infinite things, because it all gets projected onto a some sort of a number line which goes only up to irrational numbers, because there is nothing connected with us, not even in a mathematical sense, beyond that. Like our universe might be mathematically unable to interact with any other universe in uncountably many different ways and only some ways actually will have a mathematical connection, like some objects in math do? Like the center of a length which must have some sort of connection to the end points, because there is a way to reach it with things arbitrary in size? Thinking of it, it might not even be true, because you'd need to lift up the compass and straight line and hit a point pin point from a higher dimension, and you might miss. So the actually constructive things are things that you can construct with a line and a compas without ever lifting it up and going to a point that was constructed and everytning else is literally a leap of faith into a higher dimension where things might work out. And this is why we have time. Because 3D alone wont give you anything even remotely resembeling quantum particles...

Basically the argument is sudden drops like this can't happen if we assume it is "continuous".

@@DrTrefor Assuming that something is continuous doesn't make it so (if 'continuous' even makes sense at all).

@@peterjansen7929 but the underlying assumption for this idea is that the function is continuous. Whether or not this assumption true in every physical system is not discussed.

@@lanog40 Point taken! The original reply wasn't very clear, but ought to have been clear enough, even though temperature and pressure are misleading examples, because strictly the question of continuity can't even be reached, as they are statistical quantities that aren't even defined in a point.

Should be there right now!!:) www.beautifulequation.com/collections/dr-trefor

@@DrTrefor I ordered one!

Like krista king

Note, I think this theorem is right if all the objects are compact. I'm just not sure how to prove there's a continuous function from the sphere to the space of planes cutting the first object in half.

Perhaps you can add an additional constraint, such as the plane always being the closest one to p? Something like this should mean you pick "similar" planes for nearby points on the sphere, preserving continuity. I can't think of an example where the closest plane jumps.

That's totally an issue, and one I chose to not expand on. Basically such multiple plans occur in an interval, so choose the midpoint of that interval to be the ONE plane

@@clahey Well, the volume here is just the measure of the object. You can get the measure of A by taking the integral of the characteristic function of A with respect to your measure. And integrals are continuous with respect to the set you're integrating over. So when you increase/decrease the set you integrate over (move the cutting line) the volume will change continuously. The sets don't have to be compact, but they do have to be measurable.

@@MK-13337 Yeah, I agree that the from the space of planar cuts to the volume above the plane is continuous and surjective. My objection was that it's possible that multiple planes give the same value, thus the inverse function isn't well defined and if you pick one arbitrarily, it may not be continuous. However, as was pointed out, the points where the volumes are the same is always an interval so you can choose the midpoint.

It is the potential actions that could not occur through the causality of entanglement that loop around the hyperbolic sphere or structure of the universe that allow potential actions to occur because they collide and shed impractical actions into hyperbolic structures that in turn become a part of the lower wavelength fields that increase in density over time and make the universe appear to be expanding. It’s in that potential action is the radiation that is emitted when potential actions in the future pop into the past through virtual particle interactions and cause fluctuations that create the present, in that the present is simply the most likely place for us to exist because all that exist are points of turbulence that correspond to the present of any given vector of observation. It is through the density of potential energy that is lost that these waves gain their amplitude.

This is it, this is the universe in it’s entirety. In that the multiverse, exist as a concept that holds action potential that propagates into the path of least resistance.