Topology is weird: The Ham Sandwich Theorem

[@magic_cfw]

you may laugh at physicist's spherical cows, but here's mathematicians with their n-dimensional sandwiches and their n-1 knives.

[@columbus8myhw]

A neat theorem: say you color a sphere with three colors, so that every point has a color. (A point on the boundary of two colored regions is considered to have _both_ colors simultaneously.) Then there exists a pair of directly opposite points of the same color. Said another way, one of the colored regions contains a pair of opposite points.

[@manavnaik1607]

Even if the colors are discontinuous?

[@manavnaik1607]

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

[@columbus8myhw]

@@manavnaik1607 Yeah, disconnected regions are fine

[@manavnaik1607]

@@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?

[@manavnaik1607]

@@columbus8myhw or I’m misunderstanding?

[@GigaOtomia]

Without looking at the chapters, at about eight minutes in, I was thinking "Yeah, this can be extrapolated to however many dimensions you want." Then the "higher dimensions" chapter came around and I went "Oh, he went over that. Neat."

[@marcushendriksen8415]

"That's a bit of a mouthful" an n-dimensional ham sandwich would be too, I'm sure! It's sometimes hard to swallow in three dimensions; I'm not sure I could manage more 🙃

[@jessstuart7495]

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

[@agrajyadav2951]

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

[@TheTriggor]

On a slightly less mathematical note, I like how you turned the poor Borsuk into Rorschach. To your defense, both guys liked shapes.

[@fanrco766]

You really glossed over the part about how sweeping a plane over a shape and getting the total area on one side of it is a continuous function. That seems intuitively correct to me, but it isnt obvious that it would work for EVERY shape. Other than that, great video!

[@tollspiller2043]

I'm supposed to study for a french examn tomorrow in the first lesson. So here I am watching youtube about ham math, couldn't complain, there's always time tomorrow

[@DrTrefor]

haha ham math > exams:D

[@columbus8myhw]

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

[@Māṇikyavēṇā]

Sir how you make those diagrams in multivariable calculus videos?Just loving that playlist.❤

[@DrTrefor]

Thank you! That is all MATLAB:)

[@iaagoarielschwoelklobo1894]

Thanks man, I really hate when my n-dimentional ham is cut uneven

[@DrTrefor]

Hehe to help lol

[@MK-13337]

Well the objects have to be measurable, so if you have a non-measurable slice of ham you run into problems.

[@DrTrefor]

ha, true!

[@XENOGOD]

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?

[@MK-13337]

@@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.

[@EdKolis]

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

[@MK-13337]

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

[@General12th]

Hi Dr. Bazett! The animations were very useful!

[@JAzzWoods-ik4vv]

Does the Borsuk-Ulam theorem provide a constructor for the solution of problem with some arbitrary 3 bodies, or is the proof only of existence? I would think it does, since I'm pretty sure it does for the temperature and pressure problem. I guess I can rationalise that f:s^2-> R exists since it's just points on a circle, but really makes me wonder how the function looks, what would be the most "natural" way to describe it for spheres, and how hard is it to find a solution. Really good video! wish I knew enough math to solve those questions myself

[@Roy-K]

In curious; since you can define any plane with three points, why is it not sufficient to simply say that the plane which slices all of the spheres in half is defined as the plane containing the origin point of each of the spheres? Since any plane that cuts a sphere such that the origin of the sphere is on the that plane necessarily means the volume of the sphere is cut in half, a plane which travels through the center of all the spheres would cut all three in half

[@DrTrefor]

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.

[@eamon_concannon]

To clarify how to determine which side of plane is positive; It seems that we can take a vector with tail at origin and head at point p on reference sphere. Then we can translate this vector so its tail is on the plane which cuts chosen (middle) sphere in half. The side of the plane that the head of the vector is on is designated as the positive side of the plane. Is this correct? We note that volume of half of the chosen (middle) sphere is referred to as vol(A_2) which is constant for all points p on reference sphere. Vol(A_3) refers to volume of the part of a non-chosen sphere that is on positive side of plane

[@ntvonline9480]

Nice topology shirt.

[@DrTrefor]

ha thank you! Love that one:D

[@idontwantahandlethough]

Hey man, so I gotta ask: in the thumbnail, which famous TV scene were you modeling your pose off of: 1. Ken Jeong (playing Chang in _Community)_ saying "HAAMMMMM, girl!" 2. Sherri Shepherd (playing Angie Jordan in the fake reality TV show _Queen of Jordan,_ inside of another show, _30 Rock)_ saying "HAAAAMM!". People do like the way she says 'ham'. I refuse to believe it's neither of them. Thank you for your time, I will await your response by my personal computer!

[@DrTrefor]

Haha I didn’t mean it but definitely Ken

[@diskret7412]

You are so good in explaining, you have my sub! :D

[@santos3131]

is there anything special about the choice of 1/2, does the theorem apply for any number between 0 and 1?

[@DrTrefor]

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

[@philipbarthelma1903]

Does anyone know wheter any object has a "center of volume"(similar to a center of mass), at which you can slice through the object at any orientation and will always cut its volume in half? If that was the case i think one could just construct the the (n-1)d-hyperplane with the (n-1)d centers of volume and would have also proven the theorem.

[@DrTrefor]

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

[@nicholassullivan6105]

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!

[@columbus8myhw]

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.)

[@philipbarthelma1903]

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

[@jonathandawson3091]

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

[@Alex-jk2qy]

Reminds me of the sandwich theorem when calculating limits of complex sequences

[@DrTrefor]

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

[@Alex-jk2qy]

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

[@Damic_Damic]

Like that theorem. Still a Motive for a science theme tattoo I'm thinking of.

[@hugowong4436]

maybe this is a stupid question, and for the pancake cake. How can we ensure that during the rotation, the red one is always divided by half. The red pancake may not necessarily to be a circle, right. Thank you

[@rigoluna1491]

Is it not enough to state: (1) any plane or hyperplane that cuts/passes through the a sphere's origin will cut it in half (2) pick the centers of all the spheres, you will always be able to fit a plane or hyperplane to all the centers.

[@Edu888777]

For another topic, Whats the intuition behind the differential Wave Equation?

[@DrTrefor]

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

[@mastershooter64]

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

[@rkdeshdeepak4131]

Is height as a function of age necessarily continuous?

[@platykurtic5510]

Pronounciation note: antipodes nuts

[@flameendcyborgguy883]

My more physics brain though: There always exists a point, which is the center of mass for an object. If we assume the constant density of an object, no matter how we rotate the plane set in that object, we gonna cut it in half. As such there are 3 points, all of which can be used to always cut the object in half. As such, and a bit of geometrical understanding, there MUST exist a plane that connects those 3 points, and as such cuts all of the 3 objects in half.

[@pianoplayer1262]

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.

[@TheArmyofWin]

Am I missing some nuance, it feels like you can just take the center of volume (e.g. center of mass of a uniform density object) points to define your plane that way. 3 points define a plane, and any plane through a center of volume bisects each object. The only thing needed to be proven is that there exists a center of volume as a point for any given object, instead of multiple objects sharing a center of volume plane.

[@ilpreterosso]

My instant though after seeing the headline: three points make a plane in 3D, there are three objects to be evenly sliced… assume they have uniform density, so I guess the subproblem of this statement is to prove the even slice passing through the centroid is rotation invariant

[@DrTrefor]

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

[@ilpreterosso]

@@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.

[@tothm129]

I thought the ham sandwich theorem was different. I thought you had m points in n dimension, and your goal was to put an even number on both sides (for odd m, one point on your n-1 dimensional plane). my complex analysis professor would go on wild tangents and would almost never teach what he was supposed to.

[@DrTrefor]

yes there is also related theorems like this

[@dansheppard2965]

Better to use an n-2 dimensional knife to cut an n-1 dimensional plane in n dimensions, I think! 😀

[@manavnaik1607]

Wait what about objects of different compositions and densities tho. I’m assuming you still can because there exists a center point to each object then and u can make a plane through 3 points but I’m not sure

[@DrTrefor]

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

[@Grassmpl]

There had to be some notion of niceness with the shapes that we cut. For example, what if our shape is not Lebesque measurable?

[@DrTrefor]

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

[@svenwindpassinger2170]

What means the half, what equals? The shape, mass or something else which kind of volume? In which opinion of symmetry? And what means a function? A manyfold in which topology? I find it very interesting. But now I have more questions than before😅. I think that is normal for a Scientist? Great Video 👍 Greetings Sven

[@ogreeni]

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

[@anto2593]

Wonderful video! For a relatively light and digestible youtube video, the argumentation was quite rigorous. I also love your contagious enthusiasm. I have only one criticism: After all the time spent talking about cutting the ham sandwich, you never did! The unresolved buildup is unbearable, nooo

[@DrTrefor]

hahah true!

[@cernejr]

Nice video. I like the bread in the photo lot more than the bread in your hand :). Would you share what software you use to do your 3D images?

[@DrTrefor]

This is just geogebra!

[@cernejr]

@@DrTrefor Thank you.

[@stasglazkov8734]

I find it weird how you say "you can find" instead of "there exists" while talking about halving the contents of figures. Is there actually a way to find (construct) those? Or do they just exist when we are talking about any weird shapes, but you dont neccessary have the means to construct those?

[@DrTrefor]

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.

[@stasglazkov8734]

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...

[@Utesfan100]

Now, to use the 4th dimension to create a ham and cheese sandwich. I can see Buckaroo Bonzai's interesting the the 8th dimension now.

[@SebastianSchwank]

Generalizes this to solution wheights of neuonal Networks?

[@SaidVSMath]

Another fantastic video!

[@DrTrefor]

Thank you!

[@deleted-something]

So we got to a point in mathematics in which they are things like the ham sandiwch theorem

[@DanDart]

The real news is that this absolute chad eats red and blue pancakes.

[@bentpen2805]

This video appeared in my recommendations minutes after I made myself a ham sandwich. Should I be scared?

[@replicaacliper]

how would the proof change if we were working with non-spherical objects?

[@DrTrefor]

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

[@replicaacliper]

@@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?

[@kilian8250]

@@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).

[@kilian8250]

@@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.

[@KaliFissure]

Doesn't intermediate value theorem prove that R is continuous?

[@Blank00]

Who's here after their Calculus teacher mentioned that the Sandwich Theorem is not to be confused with the Ham Sandwich Theorem?

[@wjrasmussen666]

When will that Tshirt return to your shop?

[@DrTrefor]

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

[@wjrasmussen666]

@@DrTrefor I ordered one!

[@BraxtonMeyer]

Trefor, we're going to need to see your birth certificate to prove that you were less than 4 feet tall at that point..

[@gideonlapidus8996]

Love this

[@DrTrefor]

Thank you!

[@sicko_the_ew]

But what if you encountered a rain storm along the way? Then you might have a problem if it coincides with the spot where the antipodes would've been of equal temperature (because at the instant the first drop hits, the temperature is going to plunge to cloud top temperature). (I'm just trying to raise a silly out-of-illustration objection, but maybe someone can make it into something interesting?) (And maybe not.)

[@DrTrefor]

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

[@peterjansen7929]

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

[@lanog40]

@@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.

[@peterjansen7929]

@@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.

[@mohamedmouh3949]

and the kids says that thease theorems are obvious. thnak you 🥰🤩

[@incredulity]

0th

[@zefciu]

Bore-sook, not boar-shack :)

[@egycg3569]

Why don't you make udemy courses?

[@egycg3569]

Like krista king

[@deletefacebook8419]

The pancake theorem perfectly describes why we cannot assign a single vector to the location of the Big Bang. It’s that we evolved to increase the rate at which we can consume matter in order to continue to radiate potential actions so that we would not become consumed by dark energy. This is why we can only see specific wavelengths of light, and why we can only create structures within a specific amount of action potential to perform measurements, it’s because the potential actions that could have occurred in order to measure higher dimensional fields became consumed by dark energy which is just potential action that could not be performed with the constraints caused by the state of density in the past in relation to the atomic structure and subatomic structures that the systems in question use to communicate potential action in order to avoid consumption.

[@deletefacebook8419]

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.

[@deletefacebook8419]

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.

[@clahey]

What if there are multiple planes at the same angle that all split the first object in half? Say your first object is the ham and there's two equal pieces with a gap between them. In this case there's multiple horizontal planes that split the ham in half.

[@clahey]

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.

[@TheMadRyaner]

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.

[@DrTrefor]

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

[@MK-13337]

@@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.

[@clahey]

@@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.

[@Loots1]

Sounds like baloney to me! Kidding :)