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Showing posts from July, 2018

### The hairy ball theorem and why there is no wind (somewhere) on Earth

PtEn< change language What if I told you that right now there is a place on Earth where there is no wind blowing to the sides? None at all! How can I know that? All we need is what is usually called the Hairy Ball Theorem.

In less rigorous contexts, one can phrase the Hairy Ball Theorem as such:
Theorem (Hairy Ball I): if you have a hairy ball, regardless of the way you comb its hair there will always be a spot where the hair points right up.
In this particular image, the hair is pointing up both on the top and on the bottom.

More formally, the Hairy Ball Theorem can be formulated like so:
Theorem (Hairy Ball II): every continuous vector field over $S^2$ has at least a point where the tangential component is $0$.

From this theorem it is actually quite easy to establish our interesting fact!

If we think of the wind at the Earth's surface as a continuous vector field, the Hairy Ball Theorem says that there must be a point where the wind isn't blowing to the sides!

Now all tha…

### Problem #11 - the salvation of the monks

PtEn< change language Today's problem is a riddle of the same style as problems #09 and #10.

Problem statement: the island depicted above used to be home to $2018$ monks that led a simple life, far from the bad, vain and consumerist habits of today's societies. Between many other deprivations, the monks never saw themselves on the mirror nor on any other reflecting surface. One day the volcano in the centre of the island - slowly but steadily - started hinting at an incoming eruption and the monks grew worried as the days passed. Some days later, at exactly 23h59, a divine being showed itself to the monks in their dreams and offered them a painless death - thus avoiding the wrath of the volcano - as long as every monk found out the colour of his eyes. The divine being also told them that every monk in the island had either blue or green eyes and that there was at least one monk with blue eyes. The divine being left with its final condition: every day, at 23h59, all monks wh…

### To measure or not to measure... a real problem!

PtEn This post will be more theoretical than usual. If you are afraid of mathematics, go away now before it is too late!

I will write about a set, the Vitali set, which arises in measure theory, to show that there is no coherent way of assigning a size to every subset of the real line. Because of that I will try to define size for subsets of the real line and then verify that such task is impossible.
I will also enforce a couple of (seemingly reasonable) restrictions on my definition.

I challenge you to read this post and to calmly follow my reasoning. Whenever something doesn't seem obvious, try to make it clear by yourself with a piece of paper and pen/pencil. If any doubts persist, drop me your question(s) in the comments and I will answer gladly!

Let us call $m$ to the function that, given a subset of the real line, returns its size; that is, let us try to define $m: \mathcal{P}(\mathbb{R}) \to [0, \infty]$.

Let us also suppose that the function $m$ satisfies the following re…