I've been watching a number of your videos over the past few weeks, I must say thank you for releasing these into the public domain, I've learnt a lot.
I totally agree, we need all this information to be shared, not just available through expensive books or courses in big universities, for which people would need to pay tens of thousands of dollars in tuition.
Given the copyright notice on the video, I'm pretty sure it is _not_ in the public domain. Edit: And, unless there's another notice somewhere, isn't even freely shareable, e.g., under CC BY-SA 4.0.
Honestly, this is great from to watch from a math major perspective and to a laymen as well. The ability to speak to both and keep everyone interested with such complex ideas is fantastic!
Whoops, you are right. Actually Descartes wrote a short paper on this which has been lost, except for a copy of it made by Leibniz in 1676. That was subsequently only found among Leibniz's papers in 1860.
Very good introduction in 55 min. Excellent comprehension. Thank you for the video. I had this class fall semester in 2012 and to tell you the truth I found this 55 min more interesting than the 14 week lecture. You know I wish professors would give something like this on day one of every advanced mathematics class instead of going over syllabus's and attendance which was a total waste of time.
+Thomas Platt And they just love wasting time dont they? the first day package includes.."describe yourslef to rest of the class" "what is your opinion about the subject" and punnishment for plajorism
tks NJ a wonderful lecture. I'am on a mission to understand "the road to reality" by roger penrose. roger is a bit brief in his discussion on the construction of Riemann surfaces and the geometric interpretation of complex functions.
≈ 14:30-15 min - - The "little bit of details to work out" when matching two different maps on S², seem to me to be principally, showing that those operations you cited from Poincaré, actually *can* get the two maps to match. It's intuitively kind of obvious; I trust that was something Poincaré carried out? Incidentally, you can connect an existing vertex to itself with a simple closed loop that doesn't cross any edge, and the effect is the same as if two distinct vertices were connected across some face. ≈ 42:30 (question from audience) The point about not allowing cuts and joins is, you're not allowed to do those things and then claim you have a topologically equivalent (homeomorphic) surface. And you're OK doing cuts & joins here, *because* you're *not* claiming homeomorphism; you're constructing a new surface that corresponds to what the square root (or other function) looks like.
Excellent lecture with an equally excellent lecturer. Thank you. One of the demoralizing aspects for an excellent lecturer is to have zero feedback from his numb students.
I have been very impressed by your lectures. I feel as if I am wrapping my head around abstract ideas I've wanted to learn about but couldn't get there myself.
Very good introduction to the historyof topology .Your brief one video englighten my mindset : Topology limitation shows that there are 'BigRoom' for future development/improvement ! Initially I thought Topology is just a 'Rubber-Sheet Maths Transformation' .Maths without any number/algebra/symbols. but GraphicPicture . Thanks for your knowledge sharing in Concepts until 21st century .
Thank you for the wonderful lesson. I am a little confused, however. If the curvature of a non-pointy flat plane is 1, then wouldn't the cube be pointyer than the tetrahedron? cube 1/4 tetrahedron 1/2? Thanks again.
This has been an absolutely amazing introduction to a lot of advanced mathematics. I cannot think of a better way to rapidly develop mathematical literacy in a short time with any less pain than this. Thank you for this course!
Perhaps the Constitution should have the phrase "all men are created homeomorphic" instead of equal. Then we could realize that we are all different, yet equal in the logical sense.
it's so sad that most universities simply teach ideas "handed down from the gods" rather than explore the fundamentals from which one can derive the rest and particularly within a constructive/computational theoretical foundation (where is type theory in most mathematics courses for example? despite being another candidate for foundations of mathematics and subsuming the mysterious (axiomatic sets) with the concrete and simple (types, homotopy type theories where there is a topological interpretation of types)
Thanks for this. I did my topology last year but the bit on Descarte's curvature was new to me and very enlightening. As a side note, it was "cool shapes" that originally attracted me to topology, but I unexpectedly ended up really getting into the point-set topology.
I'm trying to learn Point set topology with a minimal background in real analysis, and boy oh boy does it hurt. Not the lack of experience in real analysis, but the set theory part. So... much... set theory. If I see a capital letter after reading the textbook I'm going to have traumatic flashbacks to this subject (though I do really like it).
No the tetrahedron is pointyer (at a vertex) than a cube. The total curvatures are the same: for the tetrahedron this is spread between 4 vertices, for the cube it is spread between 8. You can see this geometrically: using a corner of a tetrahedron to crack a nut would be more effective than using the blunter corner of a cube.
[NB: I haven't watched this video yet, and can't until tomorrow. So I don't know the scope of what you cover in here. After watching it, I may have to revise some of what follows.] Yes, the "angular deficit" at each corner is a good concept to latch onto - it's a good analog to the exterior angle of a polygon. If you take the faces surrounding a vertex, split that along one edge, and open it out into a plane, for the regular tetrahedron, the gap-angle, indicating 'failure to close,' is 360º - 3·60º = 180º = π while for the cube, it's 360º - 3·90º = 90º = ½π Summing each over all the vertices gives: tetrahedron: 4·π = 4π cube: 8·½π = 4π - in both cases, equal to the total (surface-integrated) curvature of a sphere, S: ∫ κ dA = A·κ = 4πR²·(1/R²) = 4π S The same technique can be applied to each of the other 3 regular solids; or, for that matter, to any solid that's topologically equivalent to a sphere. For Kepler's stella octangula, e.g., there are 6 vertices, each with eight 60º angles, which make for a negative angular deficit = 360º - 8·60º = -120º = -⅔π ... and 8 vertices, each identical to the tetrahedron's, with angular deficit = 360º - 3·60º = 180º = π ... so the sum is 6(-⅔π) + 8·π = -4π + 8π = 4π All these results fall out of the Gauss-Bonnet theorem, one of the most beautiful in all of mathematics, IMHO.
Coincidentally watched a video right before this by Mathologer called "The Iron Man hyperspace formula really works". I would recommend it to anyone who enjoyed this lecture.
As a simple computer scientist, I am happy to understand your video, for a full beginner like me, you made me feel what Topology is
I've been watching a number of your videos over the past few weeks, I must say thank you for releasing these into the public domain, I've learnt a lot.
+SerBallister You're welcome. I hope you continue to learn a lot from these videos.
I totally agree, we need all this information to be shared, not just available through expensive books or courses in big universities, for which people would need to pay tens of thousands of dollars in tuition.
Given the copyright notice on the video, I'm pretty sure it is _not_ in the public domain. Edit: And, unless there's another notice somewhere, isn't even freely shareable, e.g., under CC BY-SA 4.0.
@NateROCKS112 I didn't mean legally in the public domain, like knowledge obtained from a library book, it's available to all.
I found your lectures only recently, and they're so intuitively addicting.
Honestly, this is great from to watch from a math major perspective and to a laymen as well. The ability to speak to both and keep everyone interested with such complex ideas is fantastic!
Thanks!
Whoops, you are right. Actually Descartes wrote a short paper on this which has been lost, except for a copy of it made by Leibniz in 1676. That was subsequently only found among Leibniz's papers in 1860.
What an incredible lecture. Neat, organized, clear, and a fair amount of detail. Thanks for the lecture!
Very good introduction in 55 min. Excellent comprehension. Thank you for the video. I had this class fall semester in 2012 and to tell you the truth I found this 55 min more interesting than the 14 week lecture. You know I wish professors would give something like this on day one of every advanced mathematics class instead of going over syllabus's and attendance which was a total waste of time.
Thanks for the nice remarks Thomas. I hope you will watch the other videos in the series too.
+Thomas Platt And they just love wasting time dont they? the first day package includes.."describe yourslef to rest of the class" "what is your opinion about the subject" and punnishment for plajorism
A "manifold" is an attempt to generalize the idea of a surface to higher dimensions. So the two terms are pretty close.
Thank you so much for taking the time and effort to upload these videos! This content is so enlightening.
no other word but thank u for that level of accessibility and simlicity
Your lectures are amazing!
tks NJ a wonderful lecture. I'am on a mission to understand "the road to reality" by roger penrose. roger is a bit brief in his discussion on the construction of Riemann surfaces and the geometric interpretation of complex functions.
Paul Vivers Thanks. You might also be interested in the MathHistory19: Complex numbers and curves lecture, which says more about Riemann surfaces.
Thank you for the physical insight. The idea of total curvature being spread over a number vertices helps me.
Thank you.
Thank you, Prof. Wildberger, for your seventeen videos on the subject that teachers and students here in my place are not very well interested in...
i have been watching your videos and they have made me understand even the most complex concepts in math
≈ 14:30-15 min - - The "little bit of details to work out" when matching two different maps on S², seem to me to be principally, showing that those operations you cited from Poincaré, actually *can* get the two maps to match.
It's intuitively kind of obvious; I trust that was something Poincaré carried out?
Incidentally, you can connect an existing vertex to itself with a simple closed loop that doesn't cross any edge, and the effect is the same as if two distinct vertices were connected across some face.
≈ 42:30 (question from audience)
The point about not allowing cuts and joins is, you're not allowed to do those things and then claim you have a topologically equivalent (homeomorphic) surface.
And you're OK doing cuts & joins here, *because* you're *not* claiming homeomorphism; you're constructing a new surface that corresponds to what the square root (or other function) looks like.
Thanks a lot. I studied topology without known the purpuse. The Professor gave définitions and demonstrasted theorems. A few people could pursue him.
He is my favorite mathematician. The best; for me; of course.
Excellent lecture with an equally excellent lecturer. Thank you. One of the demoralizing aspects for an excellent lecturer is to have zero feedback from his numb students.
Descartes died in 1650, when Leibniz was four... so there must be some confusion at 15:28.
I was wondering, thank you.
I have been very impressed by your lectures. I feel as if I am wrapping my head around abstract ideas I've wanted to learn about but couldn't get there myself.
A fabulous lesson, thank you very much, Prof. Wildberger!!
Absolutely amazing and fantastic. Thank you so much.
34:24: Basically, Quantum Mechanical spin. So QM spin, topology, and complex functions have common properties.
Very good introduction to the historyof topology .Your brief one video englighten my mindset : Topology limitation shows that there are 'BigRoom' for future development/improvement ! Initially I thought Topology is just a 'Rubber-Sheet Maths Transformation' .Maths without any number/algebra/symbols. but GraphicPicture . Thanks for your knowledge sharing in Concepts until 21st century .
If a topologist is drowning and asks for a rubber tire, just throw him a tea-cup! he can't tell the difference.
Very good job .. Thanks a lot professor Wildberger
Thank you!
Such enjoyable lectures. Thank you!
Excellent course.
Thank you for the wonderful lesson. I am a little confused, however. If the curvature of a non-pointy flat plane is 1, then wouldn't the cube be pointyer than the tetrahedron?
cube 1/4 tetrahedron 1/2?
Thanks again.
The Euler of UA-cam has done it again.
great video, i was hoping if you could maybe explain topology with a more conceptual and intuitive approach
thank you, great lecture
Thank you so much professor. It is really helpful lesson. But could you explain why you told s^3 in 3 dimension at the end of lesson ? Please.
thank you so much
Very nice explanation thank you ❤️sir.
Thank you! This helped me with my paper on the Euler characteristic.
Great overview👍
amazing series
Thank you sir
This is soooo cool!
thanks
Thank you
thank you very much but make substitutes plz
I love his lecture 😍🐈
Thank you, you are great - you make me great!
perfect
This has been an absolutely amazing introduction to a lot of advanced mathematics. I cannot think of a better way to rapidly develop mathematical literacy in a short time with any less pain than this. Thank you for this course!
Correction: the projective plane has Euler number 1, while the sphere has Euler number 2; but the sphere with a crosscap has Euler number 1.
Perhaps the Constitution should have the phrase "all men are created homeomorphic" instead of equal. Then we could realize that we are all different, yet equal in the logical sense.
it's so sad that most universities simply teach ideas "handed down from the gods" rather than explore the fundamentals from which one can derive the rest and particularly within a constructive/computational theoretical foundation (where is type theory in most mathematics courses for example? despite being another candidate for foundations of mathematics and subsuming the mysterious (axiomatic sets) with the concrete and simple (types, homotopy type theories where there is a topological interpretation of types)
Thanks for this. I did my topology last year but the bit on Descarte's curvature was new to me and very enlightening. As a side note, it was "cool shapes" that originally attracted me to topology, but I unexpectedly ended up really getting into the point-set topology.
I'm trying to learn Point set topology with a minimal background in real analysis, and boy oh boy does it hurt. Not the lack of experience in real analysis, but the set theory part. So... much... set theory. If I see a capital letter after reading the textbook I'm going to have traumatic flashbacks to this subject (though I do really like it).
This is so intuitive. Thanks for uploading!
Hobin, you look familiar. Did you teach a physics class? LOL
Insightful lecture. This is what history is all about - getting insights
No the tetrahedron is pointyer (at a vertex) than a cube. The total curvatures are the same: for the tetrahedron this is spread between 4 vertices, for the cube it is spread between 8. You can see this geometrically: using a corner of a tetrahedron to crack a nut would be more effective than using the blunter corner of a cube.
[NB: I haven't watched this video yet, and can't until tomorrow. So I don't know the scope of what you cover in here. After watching it, I may have to revise some of what follows.]
Yes, the "angular deficit" at each corner is a good concept to latch onto - it's a good analog to the exterior angle of a polygon.
If you take the faces surrounding a vertex, split that along one edge, and open it out into a plane, for the regular tetrahedron, the gap-angle, indicating 'failure to close,' is
360º - 3·60º = 180º = π
while for the cube, it's
360º - 3·90º = 90º = ½π
Summing each over all the vertices gives:
tetrahedron: 4·π = 4π
cube: 8·½π = 4π
- in both cases, equal to the total (surface-integrated) curvature of a sphere, S:
∫ κ dA = A·κ = 4πR²·(1/R²) = 4π
S
The same technique can be applied to each of the other 3 regular solids; or, for that matter, to any solid that's topologically equivalent to a sphere.
For Kepler's stella octangula, e.g., there are 6 vertices, each with eight 60º angles, which make for a negative
angular deficit = 360º - 8·60º = -120º = -⅔π
... and 8 vertices, each identical to the tetrahedron's, with
angular deficit = 360º - 3·60º = 180º = π
... so the sum is
6(-⅔π) + 8·π = -4π + 8π = 4π
All these results fall out of the Gauss-Bonnet theorem, one of the most beautiful in all of mathematics, IMHO.
njwildberger like your answer, professor. Thank you so much.
cool lecture :), thanks!
Thank you for the insights
Amazing. Thanks.
like this course!
Thankyou.
Coincidentally watched a video right before this by Mathologer called "The Iron Man hyperspace formula really works". I would recommend it to anyone who enjoyed this lecture.
I recently learned about the Dirac equation and spinors. The Riemann square root provides a lot of insight!
Great Lecture! Thanx !
Elegant lecture!
Great lecturer!
thanks a lot