Short Math Facts

This one will be environmental. Suppose you have a reusable bag (yes, one of those from the conferences). Then inside your bag you can put a lot of things, like apples, cucumbers, potatoes… or other reusable bags with other things inside. Okay, now let’s assume that your fabric bag is so big that you can fit any reusable bag with anything inside, including other bags (all of them made of fabric, of course), and so on. Now the question is: is it possible to have a fabric bag that contains all the existent reusable bags?

This question (well, without all of the environmental setting) is known as Russell’s Paradox. Bertrand Russell, born in May of 1872, was a British mathematician, philosopher and writer, who received the Nobel prize of literature in 1950.

Imagine you want to install ceramics in your house. Suppose that the price is the same, independent on the shape (and the size in terms of area is the same as well) but that the glue is extremely expensive. Then, you want to buy the pack of ceramics that use the least amount of glue to put them together, and that the shape is an admissible shape to cover the floor with no need of cutting them into pieces (like squares, rectangles, triangles… what else?). Pappus of Alexandria, who was born in 350 A.C., suggested around the year 400 that regular hexagons minimize the glue you will need (in terms of perimeter per area, to fill up the plane), but this wasn’t proved until 1999 by Thomas Hales.

What is curious is that bees knew it even before Pappus of Alexandria; they construct their hives using hexagons! (Nature is amazing, isn’t it?)

Recommended video (Talk of Eduardo De Cabezón in TEDx)

Do you remember the scene where Mr. Burns tells Homer that he has a thousand monkeys with a thousand typewriters, and that soon they will finish writing the greatest novel of history? The idea of this scene is not original; it comes from the “Theorem of Infinite Monkeys” by Borel, in which the claim is that a monkey typing random characters during an infinite time will write any given text (the most common example is Hamlet by Shakespeare). Émile Borel, French mathematician and politician, was born in January of 1871, and for him are named the borelean algebras, Heine-Borel theorem, and the lunar crater Borel.

Imagine you are drinking a glass of wine and a drop falls in the tablecloths (dishonor). The tableclothes, which we’ll call “complex plane”, now has a mark. The Conformal Map Theorem of Riemann says that if the mark has no holes, then it is “the same” as a circular mark. This “the same” is defined in terms of smoothness of how to move one mark into the other (you can imagine the tableclothes were plastic and you could actually deform the drop to get it circular). The smoothness and inversibility of thee movement ensures that you also respect angles in the process.

So, once you drop some wine, just put the glass back on top of the circle and nobody will notice!

Bernhard Riemann, German mathematician, born in September, 1826, had a huge influence in analysis, number theory and differential geometry. PS: please, don’t spill wine.

Let’s do an experiment, in which particles move in some space thru time, let’s say in a cube of 1 m3, and you observe the changes of their positions every minute. Observe now a smaller cube inside the first one. Particles may get out of that smaller cube, travel around and may or may not come back. The question is: how do I know if they will come back?

Henri Poncaré, French mathematician born in April, 1854, proved that if the particles are moving in a nice way, most of them will come back. Actually, they return infinitely many times!

Poincaré’s Recurrence Theorem must be one of my favorites theorems, not only for the beauty of the result and how useful it is, but also because the proof is literally three lines that are easy to read.

What about women in math? Almost 250 years ago, in April, was born a French female, Marie-Sophie Germain. Since mathematics wasn’t a carrier for a woman, she got herself the notes of the university to study without being an actual student. She used to send mail to Lagrange, signing with the name of a man: Monsieur Antoine-August Le Blanc. When Lagrange was surprised enough about this student, he invited him to a meeting. In that moment, Germain had to let him know her true identity.

Germain did great in her research in number theory (actually, there are numbers that are named after her) and she was very admired by Gauss, who tried to giver her an award, but it wasn’t known until she died.

Well, there is a lot more about women in math… Some information can be found here

Did you know that a famous mathematician was the scientist in charge of the Cold War in the US? The Hungarian-American John Von Neumann, well known for his progress in science and computer science, was born in December of 1903 and was the scientist that reached the highest political power in the US. Together with his collaboration in the elaboration of explosives, he was the one that computed the precise moment they needed to detonate the bombs of Hiroshima and Nagasaki on the 6th and 9th of August, 1945, so they could destroy the most possible.

In mathematics, this man is famous for the formalization of the modern axiomatic and by his work in Ergodic Theory. He also worked in computer science, physics and statistic. There is a lunar crater and an asteroide named after him.

(For the full gossip, read Wikipedia)

This is going to challenge your intuition. There is a theorem that claims that you can cut a ball in finitely many pieces and glue them together to get two balls of the same size as the first one. This is known as the Banach-Tarski Paradox, even though it is not a paradox at all, but a counter-intuitive consequence of the Axiom of Choice (the movement or gluing the pieces back together doesn’t change the shape of the pieces, but the pieces are not “solid” in the usual way). Alfred Tarski, Polish mathematician and philosopher, was born in January of 1902.

My friend Claudio Valdivia commented on the instagram post how to get an anagram of Banach-Tarski: Banach-Tarski Banach-Tarski.

If you know how to differentiate, you have probably heard of L’Hôpital’s Rule for limits of indeterminate forms. Guillaume de L’Hôpital was a French mathematician who was born in February of 1661, and after he died, Bernoulli claimed he had an agreement with L’Hôpital in which Bernoulli would receive 300 francs per year in exchange of his discoveries, and he claimed as well that most of what was published in L’Hôpital’s belonged to him. Later, some people found documentation that support his claims. Either way, the rule that we all know as L’Hôpital’s rule, does not have his name in his book. Moreover, he published the book as anonymous and thanked Bernoulli for his help.

Math is a game about truth discovery. As well as any game, it has some rules. This rules are called axioms, and they are the base to decide when something is true or false. Within the 23 problems that Hilbert published in 1900, he ask to find a list of rules that are inductive (like the rules of natural numbers) and consistent (with no contradictions), and so that any phrase can be decided to be true or false.

Kurt Gödel, Austro-Hungarian mathematician, born in April of 1906, proved that what Hilbert asked for was impossible. In particular, he said that in every inductive system that is consistent, it is impossible to decide if the claim “this system is consistent” is true or false.

Gödel was convinced that people wanted to kill him, so he wouldn’t eat anything besides what his wife would cook. When she got sick and stayed in the hospital, he died of hunger, with only 29.5 kilograms of weight.

Ruler-Compass constructions are very exciting to play with (http://www.euclidthegame.com). The famous mathematician Carl Friedrich Gauss, born in April of 1777, proved that the heptadecagon (regular polygon of 17 edges) is constructible using only ruler and compass. He’s known, however, for many other contributions.

https://www.youtube.com/watch?v=87uo2TPrsl8