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Homeomorphism and the group structure on a circle | Algebraic Topology 2 | NJ Wildberger

Insights into Mathematics

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This is the full second lecture in this beginner's course on Algebraic Topology. We give the basic definition of homeomorphism between two topological spaces, and explain why the line and circle are not homeomorphic.

Then we introduce the group structure on a circle, or in fact a general conic, in a novel way. This gives a gentle intro to the definition of a group. It also uses Pascal's theorem in an interesting way, so we give some background on projective geometry.

This course is given by N J Wildberger at UNSW, also the discoverer of Rational Trigonometry, and a leading advocate of logically correct thinking in mathematics.

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Research Gate page: https://www.researchgate.net/profile/...

Blog: http://njwildberger.com/

Online courses at openlearning.com (currently Algebraic Calculus One): https://www.openlearning.com/courses/... Please join us for an exciting new approach to one of mathematics' most important subjects!

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Wild Egg Maths YT channel: https://www.youtube.com/channel/UCriF...

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20 Comments

@Hbdhhkkj 7 years ago

Finally a good camera man! BRAVO, can you have him instruct other camera people?

@baptisteferrand8074 5 years ago

Very didactic, rich in examples, clear and interesting to follow (which is not so common among math teachers). Thank you for sharing sir.

@hanancohen8637 13 years ago

Spectacular classes, very, very nice job. Please, let the classes going on. Greetings from Brazil.

@solidstatejake 6 years ago

Thank you for making your wonderful lectures available, Dr. Wildberger.

@water0heaven 14 years ago

For the latter two problems, I get the group multiplication rule as alpha(x1)*alpha(x2)=alpha(x1+x2) for the parabola and beta(x1)*beta(x2)=beta(x1x2) for the hyperbola, the fitness of which are not difficult to check. Together with the case of the circle, all the apparently distinct formulas seem to result from the same geometrical interpretation as drawing the parallel intersects, which is really amazing.

@Igdrazil 8 years ago

Excellent. A similar group structure works on a Elliptic Curve giving the most efficient cryptographic protection needed for small devices like smartphones etc that can't compute to fast but need nevertheless to be secured.
A kind of "asymmetry" pin's out in your excellent presentation. Indeed through many aspects, Projective Geometry shows to be an unifying geometry. But that makes the "line at infinity" and thus "infinity" a central concept. A "miror" of the CENTER. And thus in this regard A and S1 are HOMEOMORPHIC because in projective sens, both ends are well connected...

@carolinafraga7872 6 years ago

I wish I had a teacher like you

@JustNow42 3 years ago

Wow. Excellent.

@durgaaratikatla8729 5 years ago

Really nice explanation 👍👌👌👌

@dkwc4019 5 years ago

wonderful presentation!! thanks alots !

@scottvska 15 years ago

Thanks for uploading these.

@francescoapg 4 years ago

thanks for sharing!

@TomPps 6 years ago

Thank you so much for your videos they're really helpful

@ThatHippyPerson 4 years ago

Thankyou. Dr. Wuldberger

@RichardSouthwell 13 years ago

Its very good to present math like this on youtube, its great, it makes other people want to upload their mathematics on to Youtube too

@sguzzygang 7 years ago

Good vid!

@SatishJadhav-rr1oq 7 years ago

It's really interesting ...and the way you teach the subject is very nice
And nice to here 16:30 Indian name...(highschool teacher).

@mohdibrahimsheikh5181 9 years ago

circle group The group structure on a circle is the work of F. Lemmermeyer, S. Shirali found in the Mathematical Gazette, Vol 93 #526, March 2009. Given points A, B, and C on the circle their products are also found on the circle giving the group property of closure.

Group properties are the following:
1) Closure property as mentioned above.2) The point O acts as the group identity such that O*A=A*O=A.3) For every point A there is a point B such that A*B=B*A=O existence of inverse.4) (A*B)C=A(B*C), associative law.

@ThatHippyPerson 5 years ago

Thankyou so much.

@CranCowan 1 month ago

Interesting video. A cool property of these conic group structures is how they lead to Fibonacci sequences using the recursive relationship C=A*B, D=B*C ... etc. where A,B,C... are points along the conic. For a circle, you get a Fibonacci sequence of the circular angles, for the hyperbola you get a Fibonacci sequence of hyperbolic angles and for the parabola you get a Fibonacci sequence along the x-axis.