I discovered a wonderful tutorial movie and web site on Topology and Geometry

http://www.dimensions-math.org/Dim_regarder_E.htm (watch online)

I recommend that you watch the movie first before you read the text on the web pages

http://www.dimensions-math.org/Dim_tour_E.htm (tour/guide)

http://www.dimensions-math.org/Dim_chap_E.htm (details)

I congratulate Jos Leys, Étienne Ghys and Aurélien Alvarez.

Short descriptions of the chapters of the movie from dimensions-math.org:

Chapter 1dimension two, is very elementary. Secondary school students should be able to appreciate it, but we think that, even if you know already what meridians and parallels are, you will enjoy the spectacle of the Earth rolling like a ball ! (Look here).

Chapter 2dimension three, is still elementary, but requires a bit of imagination, and it has some philosophical elements… There are even some exercises to check that you have understood. For explanations, additional information and references, one can consult this page

Chaptres 3 and 4 get us into the fourth dimension. This is of course more difficult, and maybe it will make your head spin ! In order to understand everything, don’t hesitate to push the pause button on your remote, to watch these chapters several times, and to consult this page where you will find references to additional information. But even if you do not feel like making the effort to understand it all, you can always sit back and enjoy the pictures !

Chaptres 5 and 6complex numbers, contain an introduction to, well… complex numbers. In France, complex numbers are taught in the final year of secondary school. We don’t see this as a replacement for a classic course, but we think that these chapters could accompany such a course in a pleasant way. If you learned about complex numbers a long time ago, and you forgot most of it…, this could refresh your memory. If you know nothing about complex numbers, you should push the pause button as often as you like, and try to understand using the references that we propose. These chapters are the most “school-like” of the film. To thank you for your efforts, chapter 6 ends with an amazing deep zoom scene.

Chaptres 7 and 8 give you an introduction to the Hopf fibration, which is not taught in secondary school, and not even in the first years at university. This is certainly not beginner’s stuff ! On the other hand, it is quite pretty and deserves to be understood. Everything is explained in the film, but of course, things may go a bit fast. Here also, the references that we provide can be useful in case you have trouble understanding… Good luck, and enjoy the show !

Finally, chapter 9 is a special one. It shows the proof of a theorem of geometry. This proof uses nothing above the level of secondary school, and we could very well have put this chapter right after chapter 1. Without proofs for theorems mathematics would not exist, and we wanted to make this very clear at the end of a film that is essentially about mathematical objects. (Look here).

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