Topology For Beginners: Brouwer Fixed Point Theorem

The Brouwer Fixed Point Theorem is one of the most elegant results in topology, for it implies that a large number of real and abstract processes have fixed points without referring to quantitative details.
sites.google.c...
See how to design beautifully complex shapes, and 3D print them easily
• How to Create Beautifu...
Check out my 3D printed jewelry inspired by topology and complex networks
www.shapeways.c...
Topology is one the most important and interesting subjects in mathematics. Since topology is is hardly taught at school, I thought I'd make some lessons about it. The material is accessible to everybody, and will hopefully be educational for children and adults alike. Some of these problems are great for expanding the mind and practicing visualization.

[@ABHISHEKMATHEWM]

What a legend! I watched dozens of videos on this topic, and finally this guy got the theorem across to me.

[@Kujiranoai]

If the tea includes a teabag does the theorem still apply? Might not the tea bag promote a discontinuous mapping of the tea onto itself, due to permeability or impacts on liquid flow?

[@3asfour101]

Is translation of the disk considered a mapping from the disk to itself?
if no then why? if yes then I can translate all points on the disk far enough such that I can guarantee no common points

[@johnwolf7073]

VSauce

[@johnnyplatis]

Little theorem?????

[@redada1900]

that's pretty much look's like the hairy ball theorem just there instead of a fixed point we have a zero tangent vector mathematics repeat itself greating to who CREATED this universe

[@RichardSouthwell]

redada1900 Yep, its crazy how many places these kind of results find uses

[@akhilgandu5279]

VSauce

[@Wumbolo]

Vsauce brought me here

[@rauleduardomartinez-campos955]

Unearthed gem by Vsauce.

[@omegatheta9058]

T99

[@hansmatter4705]

in 1 dimension the theorem is not suprprising by stretching a line

[@memeolicious7949]

I know I’m late to the video, but does the fixed points theorem apply on a deck of cards

[@TheZurul]

these videos are great thank you so much. I have a question though: can you recommend any books like this on topology which are this specific but straightforward and formula-free (for people like me who barely remember their high school math)

[@RichardSouthwell]

Glad you like them. I got interested in this theorem because of the book:
'Five Golden Rules: Great Theories Of 20th Century Mathematics And Why They Matter by John L. Casti'.
It claims to be for the laymen, although there are a few formulas. My top recommendation would have to be:
'The Shape of Space: How to Visualize Surfaces and Three-Dimensional Manifolds by Jeffrey R. Weeks'.
This is an amazing book, that teaches really interesting topics in topology and geometry with no formulas at all. I highly recommend it.
Probably the easiest way to learn is to watch Wildberger's lecture series, he explains things in a very nice and simple way:
AlgTop0: Introduction to Algebraic Topology
AlgTop0: Introduction to Algebraic Topology

[@TheZurul]

Richard Southwell thank you! I study the philosophy of Gilles Deleuze who uses a lot of ideas from topology as well as differential geometry but doesn't really explain them! It's great to get the background on these topics

[@RichardSouthwell]

TheZurul Philosophy+topology huh, sounds like an interesting combination. I did not know about Gilles Deleuze so I just watched this lecture:
Manuel DeLanda - The Philosophy of Gilles Deleuze. 2007 1/5
Manuel DeLanda - The Philosophy of Gilles Deleuze. 2007 1/5
but so far I have not found any topology. Are there any youtube videos about the connection?

[@knowledgeisfree1]

I understand the theorem much better now. Brower's fixed point theorem came up as part of my general equilibrium studies in advanced microeconomics.

[@ryanjbuchanan]

Dude, you are so fucking cool! I love the passion you have for explaining this stuff and the devotion you have to these topics and the treatment you give them, allowing anyone to learn from start to finish, while showing off cool projects like cellular automata and stuff!

[@Hythloday71]

Can't comb a hair sphere flat ? = Brouwer Thm ? Is it related to black hole no hair theorem ?

[@RichardSouthwell]

Glad to be of help :-)

[@jacoboribilik3253]

A very clear explanation, you look like that guy from 21

[@wesen1000]

your approach is nice to present this abstract concept by supportive real world example

[@StephenPaulKing]

Isn't a mapping equivalent to a function from one set or class or category or collection or whatever to another?

[@mohammadalanazi2274]

the most understanding that i got from ur explanation is state the theorem

[@RichardSouthwell]

[@Patriciovaldivieso]

Great for intuition, thanks for sharing.

[@sampadlahireee3036]

Great Effort

[@salembenahmed6208]

You've said that the course is for anyone ? how can a high school kid learn or understand a complex topic ?