#borsuk-ulam theorem

[trendydigests]

How to Cut a Ham Sandwich and Other Math Tricks

Have you ever wondered how to cut a ham sandwich into two equal parts with one slice? Or how to find a point on a sphere that has the same temperature as its opposite point? Or how to fold a protein or control a robot using polygons? If you are curious about these questions, you might be interested in learning about the Borsuk-Ulam theorem, a mathematical theorem that has many surprising and…

[jonathankatwhatever]

This AMT was intense. I made a ton of mental notes to type out and can’t remember any of them.

The typing appears back to normal, which is a relief. Had to restart the app a few times.

I saw my fear in light of something called the Borsuk-Ulam Theorem, which relates pairs over a dimensional enclosure, meaning over a Euclidian space within an n-sphere.

My intense fear that somehow I’m wrong, that this can’t be happening, that there must be something crucial I’ve missed, that I’ve either been misled or have misled myself, that I’m delusional in the bad sense, that I’m being betrayed, that maybe I don’t know the real mission and I’m being used. Those all align on the fear axis. That axis can’t go away because it contains its own positives, meaning it pairs in 1-0Segments over the 1-0Space (which may be a better name than 0Space). Fear that I’m not doing enough is an obvious example. Fear for everything that happens with regard to you. Fear of unknowns.

And it isn’t that there’s a single voice which stretches 0-1-0 over the enclosed dimensional space. No, it’s more that there are baubles, those dimensional bubbles which connect on a counting string, which connect over the 1-0Space. They lie on the counting string which connects the polar Ends. And the string they connect across is the 1Space counting, meaning the gs processes that say this is the same over this extent.

I’m not sure how best to say that all the mathematics has come together. All the visions seem to have clear, proven analogues. e eyboard acting up again .

——

Trying again. Nope. Need to use onscreen keyboard. Yuck.

Just looked at L-functions again. Made sense with the injection model. I looked at Dirichlet characters and that connects injection to the modularity of the identity functions we describe.

[auroral-melody]

bonus ask: infodump about the most obscure math thing you can think of off the top of ur head

this isn’t obscure but it IS really cool

borsuk-ulam theorem, or “meterologist’s theorem”: if you look at earth, there are always two points across from each other (”antipodal”, think north and south poles) that have the same temperature/pressure/humidity/whatever you’re measuring, as long as whatever you’re measuring is continuous.

it has to do with the Intermediate Value Theorem. if you don’t know anything about calculus, it basically means “if i have a line between 1 and 5, somewhere on that line is a 4″. it’s one of those math theorems that’s feels A Bit Obvious but it’s important to establish ok. basically, if things don’t suddenly jump/teleport, we can say this.

okay, let’s get into the why! pick two points on earth, call em a and b. they’re opposite each other.

now move them along any path you like (as long as they stay antipodal) so that they get to the other point. the temperatures HAVE to cross, and that’s where they’re equal!

to put it another way, more mathematically: make a function f, where

f = the temp at a - the temp at b.

ignore my ms paint drawing of the earth and the fact that those AREN”T the north and south poles

the order here matters which you subtract. it flips to the negative when you’ve moved to the other point. now, f is a continuous function (we’re assuming the earth’s temperature doesn’t Suddenly Jump, so we can apply the IVT), so f hits every point in between 30 and -30 if you go between them like in the gif above.

so, when you look at f on one point, you’re positive. then you travel t the other point and check, and it’s negative. at some point there, it crossed zero! at that point, the temperatures were equal.

0 = temp of a - temp of b

temp of b = temp of a

now, if you do this infinitely many times, you end up with a band of points that have antipodal equal temperatures, because blah blah continuity, basically. if you do this same process with pressure, but this time only measuring on those points / following that band --

you’ll find a point with the same temp and pressure :)

references i stole these explanations and gifs from:

https://brilliant.org/discussions/thread/borsuk-ulam-theorem-the-global-version/

http://mathforum.org/library/drmath/view/54746.html

[bsdndprplplld]

8 V 2022

I am on my way home from a math conference, the first one in which I participated actively – I prepaired the talk about the Borsuk-Ulam theorem

my lecture was centered around the connection between the classic "continuous" BUT and its combinatorial analog: Tucker's lemma

I wanted to talk about this because I was amazed at how cool and "versatile" this theorem is. there is a whole book about its applications and generalizations, which is btw very well-written, I highly encourage everyone to read it:

my presentation went well, although after practicing it for about a week the topic seemed really fucking boring to me, no wonder

other than that I have another recommendation to make. do you also hate how messy multivariable calculus is? I do. calculations and technical definitions everywhere, and at the end everything comes down to calculating the determinant of some jacobian. bluh. I stumbled upon a book that describes everything from a sort of algebraic perspective, smells a little bit like category theory too. very clean, very satisfying to read:

I have been studying covering spaces recently and I can give some dope motivation for learning about the structure induced by the covering mapping:

I will never forget that the homomorphism induced by the covering projection is injective

that would be it for my mathemathical life. my personal life, which is still closely connected to math, brings me some psychological progress. I no longer get stuck in loops of "oh I'm so bad at math. maybe I'm not? I got a good grade from X. ah but I got a shit grade afterwards". it might be because I didn't fall on my face for a while now, only decent grades, good ideas, a good presentation, this is correct. but I also do not negotiate with myself that this is supposed to be proof that I'm good enough, I just stopped paying attention to these and focused on math instead. and paradoxically when I stopped caring about being good at math I was rewarded with getting better at math???

a coincidence,

a pleasant one, nonetheless.

anyway I will have to take a fall at some point, unavoidable. and it will be the final test of my progress, becauase I used to get very elevated in my sense of self-worth after receiving a single good grade among trash ones and now I'm just ok. not the god, just ok. but back then, at some point I would no longer be god, I would get smacked in the face by some "proof that I'm actually trash" and that would be a fall from a significant altitude. so I'm hoping that the fall will also be less painful now

I think the biggest change I made was giving up, I abandoned all hope. nooow here is the moment when people interrupt me with "nooo that's horrible don't give up you're a great person you just have to notice that"

fuck off you don't understand shit

I'm doing better now precisely because I stopped hoping that one day I'll stop feeling worthless, that one day something great will happen that will prove once and for all that I'm meant for something great. I can't stand this anymore, I am disgusted by the fact that deep down I still believe that I'm supposed to be the best and that I can't enjoy anything unless I am winning. I want to puke when I'm reminded that everything I do serves the purpose of winning the negotiations I have with myself about what my actual value is

my self-hatred runs much deeper now than ever before and I have no more patience for self-victimization, no more room for "allowing myself to feel". fuck off, all I feel is rage. I want to be able to do things without the prospect of a reward, my goal is to enjoy things, not the sense of being good at doing things

so that's what I'm doing, I made peace with the fact that I will probably never feel good about myself and that I have no chance at achieving the greatness I crave. and I must say I started respecting myself more, turns out I am actually able to do things without the promise of being the best at them, the vision of bringing value to the world motivates me. and fuck the western culture with its oh you must love yourself you are a great person. no, you don't have to do that and you have no way of knowing what kind of person you are, nobody has ever defined it in a strict formal sense, people just use this phrase to trigger the feel-good in others

I am aware that all of this sounds really bad, but I don't care, it works. and my math will be better like that because now that I stopped crying over being trash I have more time to study

I just hope that the fall won't be as painful

[lolt64]

firmly dont believe the borsuk-ulam theorem. wont even capitalize it