The Arte.tv channel has put online ten videos of ten minutes each, proposing ten “trips to the land of maths”, directed by Denis van Waerebeke. Of course, they are not part of the live program, but they will remain available until the end of 2026, which is even better. Ten very successful trips, both aesthetically and conceptually. In ten minutes, “you encounter epic landscapes, dizzying ideas, and sometimes even useful things,” as explained in the preamble.
The videos are all worth watching, but here’s a teaser for one of them, titled “The Game of Life”. The question of whether a virus is alive is a bit artificial as long as there is no definition of the word “alive”. Without clear definitions, there can be no science. In 1940, John von Neumann proposed a very theoretical definition. First of all, a living being must be able to reproduce. But this is not enough, because we can easily imagine a robot with its own assembly plan that moves around looking for the parts necessary for its replication. It can then make a copy of itself. However, nobody would call such a robot alive.
A living being must do something else than reproduce itself: Neumann asks that it can simulate a Turing machine, in other words that it can do what our computers do. This is a very abstract definition of life! In 1944, the physicist Erwin Schrödinger, one of the founding fathers of quantum physics, published a book entitled What is life? Today, this book is obsolete because the structure and functioning of DNA was not known at that time. But it contained the essential idea that a living cell must contain some sort of reproducible code.
Create infinite shapes
Around 1968, the mathematician John Conway invented a very simple game that tried to imitate life. On a (infinite) board divided into squares, like a huge checkerboard, one places a few tokens that draw a certain shape. The rule of the game is as follows. For each token, you count the number of tokens that are adjacent to it. If this number is 2 or 3, we leave it in place. If not, it is removed from the board. On each empty square surrounded by exactly three tokens, you place a new token. The initial shape then becomes a new shape, and the operation is repeated… We thus see a succession of shapes. At the beginning, Conway worked with real chips on a go board, but very quickly computers allowed to simplify the work. By dint of testing, he discovered a number of configurations that seemed to oscillate and periodically return to their initial position.
In 1970, he offered a $50 reward to the person who would discover a configuration whose size would tend to infinity during its development. He had to keep his promise because a “cannon” shape was found that regularly sends out “cannonballs”. Afterwards, there was a real craze for this game because it is very easy to program on a computer and anyone can play it. Download the free Golly application to have fun. In 1982, Conway demonstrated that it is indeed “life” as defined by Neumann. Today, the progress is incredible. Some configurations are made of a “membrane” that contains a filament of “DNA”. These virtual and abstract “living beings” can have a “sexual” reproduction in which the filaments mix. They can mutate and evolve, as in real life. You can find nine other mathematical journeys on Arte.tv.