Nine chapters, two hours of math, to gradually discover the fourth dimension.

How to use this movie?

The film is designed so that all audiences (who want it!) can enjoy it, provided they choose the chapters well. It consists of nine chapters, each lasting thirteen minutes. In fact, chapters 3-4, 5-6 and 7-8 are double chapters, but other than that, they are pretty much independent.

Of course, you could sit down in front of a TV or computer and watch all 117 minutes in one go! But there may be passages that seem too fast, or on the contrary, too elementary. It all depends on what you're interested in, what you already know, or simply what mood you're in at the time... It's also possible to make do with a few well-chosen chapters. Here are a few pointers to help you get the most out of the film. If you'd like to know a little more, you can consult the explanatory pages, where you'll also find short extracts from the film...

Roughly speaking, the mathematical level of the chapters gradually increases:

Chapter 1, Dimension 2, is very elementary. It should be enjoyable for middle-school students, but we think that even if you already know what meridians and parallels are, you might enjoy seeing the spectacle of the Earth rolling like a ball!

Chapter 2, Dimension 3, is still elementary, but requires a little imagination. It can also be viewed as a show, with a touch of philosophy thrown in... It even includes exercises to make sure you've understood. For explanations, supplements and additional references, please consult this page on this site.

Chapters 3 and 4 introduce us to the fourth dimension. This is of course more difficult, and you may feel a little dizzy? If you want to understand everything, don't hesitate to use the pause button on your remote control, to review these chapters several times, and to consult this page, where you'll find references for additional information. And if you don't want to understand everything, you can simply admire the pictures!

Chapters 5 and 6, Complex Numbers, provide an introduction to the complex numbers taught in the French final year of high school. It's not intended as a substitute for a standard course, but we think these chapters could be a welcome addition. If you learned about complex numbers a long time ago, and have forgotten..., this might refresh your memory. If you don't know anything about complex numbers, you'll need to press the pause key several times and try to understand using the references we provide. These chapters are the most “academic” in the film. To thank you for your effort, the end of chapter 6 features a breathtaking diving scene.

Chapters 7 and 8 contain an introduction to Hopf fibration, which is not discussed in high school or even undergraduate courses. So it's clearly not for real beginners. But it's all very well, and the game is well worth the effort of understanding. In principle, everything is explained, even if, of course, things sometimes go a bit too fast. Here again, the references given may be useful in case of difficulties... Good luck and enjoy the show!

Finally, Chapter 9 has a special status. It offers a proof of a geometric theorem. There's nothing inaccessible at middle-school level about this demonstration, and this chapter could have been placed after chapter 1. Mathematics wouldn't exist if theorems weren't proved. We wanted to make this clear at the end of a film whose main purpose is to show mathematical objects.

Here are some possible routes:

Middle school: 1 or 1-2 or 1-2-9

High school: 1-2-(3-4)-9

High school science 5-6

Undergraduate science: 2-3-4-5-6 or 5-6-(7-8-9)

Graduate science: 7-8-(9)

General public: 1-2-3-4-(9)

Commentary in Arabic, English, French, German, Italian, Japanese, Russian and Spanish. Subtitles in German, English, Arabic, Bosnian, Chinese, Spanish, French, Greek, Hebrew, Italian, Japanese, Dutch, Persian, Polish, Portuguese, Russian, Serbian (Latin and Cyrillic), Slovenian, Czech, Turkish.