Real Mathematics – Strange Worlds #17

Topology On Your Head

Often you see me writing about changing our perspective. For example, when you encounter a baby first thing you do is to make baby sounds and try to make the baby laugh. Whereas if you’d looked carefully at baby’s hair, you could have seen a very valuable mathematical knowledge hidden on the baby’s head:

As shown above, there is a point on each baby’s head. You can see that the hair besides that point is growing in different directions. Can you tell me which direction the hair grows at that exact point?

Hairy ball theorem can give us the answer.

Hairy Ball Theorem

Hairy ball theorem asks you to comb a hairy ball towards a specific direction. The theorem states that there is always at least one point (or one hair) that doesn’t move into that direction.

You can try yourself and see it: Each time at least one hair stands high. This hair (or point) is a sort of singularity. That hair is too stubborn to bend.

Baby’s hair is some kind of a hairy ball example. (I use the expression “some kind of”because the hairy ball has hair all over its surface. Though the baby’s head is not covered with hair completely.) This is why the point on the baby’s head is a singularity. It is the hair that gives a cowlick no matter how hard you comb the baby’s hair.

Torus

Hairy ball theorem doesn’t work on a torus that is covered with hair. In other words, it is possible to comb the hair on a torus towards a single direction.

No Wind

Hairy ball theorem can be used in meteorology. The theorem states that there is a point on earth where there is no wind whatsoever.

To prove that, you can use a hairy ball. Let’s assume that there is wind all over the earth from east to west. If you comb the ball like that, you will realize that north and south poles will have no wind at all.

On Maps

The hairy ball theorem is a kind of a fixed point theorem. Actually, it is also proven by L. E. J. Brouwer in 1912.

One of the real-life examples of the fixed point theorem uses maps. For example, print the map of the country you live in, and place it on the ground:

You could use a smaller map though.

There is a point on the printed map that is exactly the same as the map’s geographical location.

“You are here” maps in malls or bus stops can be seen as an example of this fact.

One wonders…

Assume that all the objects below are covered with hair. Which one(s) can be combed towards the same direction at all its points? Why is that?

M. Serkan Kalaycıoğlu

Real Mathematics – Strange Worlds #16

The Walk

  • Select two points in the classroom.
  • Draw a line between them.
  • Send a student to one of those points.
  • Once the student starts his/her walk, he/she should arrive at the other point exactly 10 seconds later.
  • Everybody in the classroom would count to 10 to help the walker.
    Ask the student to do the same walk twice while recording the walk using a camera.

The goal of the experiment

After the experiment is done, the following question is asked to the classroom:
“Is there a moment during both walks when the student stands at the exact point?”
In other words, the student walks the same distance in the same amount of time at different speeds. The goal is to find if there is a moment in both walks when the student passes the exact point on the line.
First of all, we should give time to the students for them to think and brainstorm on the problem. Then, using the video shots, the answer is given.
The most important question comes at last: Why so?

Weeding out the stone

In my childhood, one of my duties involved weeding out the stones inside a pile of rice. To be honest, I loved weeding out. Because I was having fun with the rice as I was making different shapes with it.

Years later when I was an undergrad mathematics student I heard of a theorem that made me think of my weed out days. This theorem stated that after I finish the weed out, there should be at least one rice particle that sits in the exact point where it was before the weed out started. (Assuming that the rice particles are covering the surface completely.) In other words; no matter how hard to stir the rice particles, there should be at least one rice particle that has the exact spot where it was before stirring.

This astonishing situation was explained by a Dutch mathematician named L.E.J. Brouwer. Brouwer’s fixed point theorem is a topology subject and it is known as one of the most important theorems in mathematics.

The answer to the walking problem,

The walking problem is an example of Brouwer’s fixed point theorem. This is why the answer to the question is “yes”: There is a moment in both walks when the student stands at the exact point on the line.

I will be talking about Brouwer’s fixed point in the next article.

One wonders…

A man leaves his home at 08:00 and arrives at another city at 14:00. Next morning at 08:00 he leaves that city and arrives at his home at 14:00, using the exact roads.

Conditions

  • Starting and finishing points are the same, as well as the time intervals of both trips.
  • The first condition means that the man could travel in his choice of speed as long as he sticks to the first condition.

Is there a point on these trips where the man passes at the exact time during both trips?

Hint: You could assume that the distance is 600 km and the man must finish that in 6 hours. For instance, he could have been traveling 100 km/h the first day, and the next day 80 km/h in the first 2 hours; 100 km/h in the next 2 hours, and 120 km/h in the last 2 hours of the trip.

M. Serkan Kalaycıoğlu