Let me count the ways…

Alice was so bored, waiting for a message from Bob, that she started to play with the five white rabbits she had got from the Queen of Hearts. She tried to figure out in how many ways she could split her rabbits in groups, like 5 = 4+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1, so, for 5 rabbits, 7 ways.

She decided to count the partitions in which no group contained more than 3 rabbits… in our example, there are 5. And then, she counted the partitions with no more than 3 groups. Amazingly, although the groups were not the same, the two numbers coincided, also 5.

Alice wondered… She wonders all the time (why?). Is that a coincidence?

One plus one plus one plus one...

π=80

A few unrelated questions around π…

  • Why is it true that  π=80?
  • Why on Earth did we define π as we did, instead of giving a nice symbol to 2π? Life would be much easier… So many less factors 2 in our books… A quadrant would be just  π/4, not the nonsensical π/2… Can you see any notational advantage? Read this for more info.
  • Do you recognize this sequence: 3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, …?
  • Why should I have posted this yesterday?

OK, let’s keep it short. And thanks to S.N. Santalla…

Update (March 17) My birthday date appears at position 45,260,128 of π, not counting the initial 3. When was I born? ;) Hint. (Via Pepe Aranda) Moreover: possession of all digits of π makes you infringe all known copyright laws… Do you know why?

Why is g so close to π squared?

The hard facts: (a) The acceleration of gravity on Earth is g ~ 9.8 m/s2; (b) π2 ~ 9.87.

The question: Is that pure chance?

The naive answer: Sure. Just change the units, the similarity is gone. Just change the planet, the similarity is gone.

Yet… a little bit of historical research tells us that it is not pure chance. How come?

Of course, if there is a connection between the two values, it must be historical, not physical. The similarity between the two values is just on Earth, and with our units. But how is the meter defined? The definition has evolved with time (and in the US they still use units related to the lengths of their extremities… ains…). For a long time, it was one ten-millionth of the length of the Earth’s meridian. So the relation to the Earth is ensured in the definition, no doubt.

No magic involved, just history. It was the French National Assembly, during the Revolution, defining the meter. They wanted a universal definition, and they came up with that one. But it was not the first one… Before, there were others.

As far as we know, it was the marvellous mind of John Wilkins the first to conceive the idea of meter. And what was his definition? No wonder, the length of a seconds pendulum. That means: a pendulum whose period is two seconds. Now, for a bit of physics, remember that, within the small angles approximation, the period of a pendulum is

T=2\pi \sqrt{L\over g}

Now, imagine that we were using Wilkins’ meter. Then with a pendulum of length 1 length-units, we would have a period 2 time-units. Just solve for g and… hey! You get… π2.

Wilkins’ idea went all the way down to Huygens, and to Talleyrand, who proposed it to the French revolutionaries. Technical difficulties, mostly the fluctuations of length with temperature, made them change the choice, but nonetheless picking up a close value.

Le jour du mètre est arrivé!

 

Ferrofluid monsters

Imagine you put a lot of magnetic particles inside an oily substance, so that they get “stuck”, but they can flow. Then you got a ferrofluid. These fluids have very strange behaviour. Look at the almost alive forms in these videos…

Why do they behave like that? The magnetic particles get attracted by the magnetic field, and they “drag” the oil. Of course, a special kind of mixing is necessary here, but we won’t go into that. But, why the spikes? Because of the surface tension. As in any liquid, ferrofluids want to minimise their free-surface area, so the optimum shape is this array of conical shapes.

They do have practical applications. For example, in your hard drive, they are a good lubricant to the magnetic shafts. They are normally used as seals, since they will attach to any magnetized surface very tightly.

But, despite their name, ferrofluids are not ferromagnetic! The magnetic particles inside are not interacting, so it does not acquire a net magnetization in absence of external fields. I am developing myself some theory regarding possible “truly” ferromagnetic fluids, if they might exist… you know how theorists work, we have the idea, and then we pray the experimentalists to find it out…

With thanks to Rodolfo

Robots, ho!

May I introduce you to Paquito? Less shiny than C3PO and less obnoxious than R2D2… yet a great basketball player! Poor Paquito didn’t win the Galapabot’10 competition, which took place last weekend. For some strange reason, I was appointed judge in the event. In the pic, my face is strategically covered by one of Paquito’s gears.

So the game, organized by one of my little padawans (recently upgraded to young jedi), Irene by name, was basically to clean your ground half of balls, by dumping them to your opponents’ half.

In the pic you can see BEAST III in action, aka Terminator, it was really a great piece of work! As a remarkable side issue: you can see a girl driving a robot. There are geek girls in robotics!!  BEAST III was created and handled by a Swedish-non Swedish team. OK, I guess this last sentence needs explaining. The Svartmetall team came from the Intl. school of Stockholm, yet none of the members were Swedish! Also worth mentioning the other robots: Chucky, Franky, Smurf (made on the spot!) and BEAST 3.5.

The competition had an autonomous stage, where the robots had to act on their own, following their programs only. After that, the human drivers take control. I guess it is the first stage when you have the feeling that you have created something… The soul of Isaac Asimov was looking upon us, I am pretty sure.

Merry Newtonmas!

Nature and Nature’s laws lay hid in night: God said, let Newton be! and all was light.

Alexander Pope


I know it is just a myth, but who cares? Dec 25, 1642, Isaac Newton was born. Not a nice person, for sure, but maybe the most brilliant genius humanity has known, it is only fair to offer an alternative to all those jingles and Santa Claus… We need new myths, let us try this one out! :)

BTW, Wikipedia had an article on Newtonmas and had it deleted. The alleged reason was irrelevance of the term. If you check google, you’ll get more than 2.5e6 hits… is that irrelevant? I have asked the editor for undeletion. Please, do also the same.

Now, for some cheering up, a Newtonmas carol… Sing it to the music of “Jingle bells” (Credit should be given to Alvin Lee.)

A comet hit the earth;
It’s made of methane ice.
It makes a giant force.
Now, isn’t that so nice?
What made it come here thus?
What made it hit the earth?
The answer’s very clear my friend.
It fills you up with mirth.
O-o-o-o-oh!
Gravity, gravity,
Keeps us on the ground.
An apple fell on Newton’s head,
What goes up must come down.
O-o-o-o-oh
Gravity, gravity,
Mass times nine point eight
Remember travel very fast
If earth you must escape.
Walk around the earth.
Keep a steady pace.
If gravity weren’t here
You’d float away in space.
Call it what you want,
Call it any name.
This force is sure a heavy weight.
Attraction is its game.
O-o-o-o-oh!
Gravity, gravity,
Keeps us on the ground.
An apple fell on Newton’s head,
What goes up must comes down.
O-o-o-o-oh
Gravity, gravity,
Mass times nine point eight
Remember travel very fast
If earth you must escape.

Also check this nice explanation about Newtonmas published in the 500th anniversary of Newton’s birth, so 132 years in our future light cone…

Superhighway (or funny minima)

I proposed this problem to my calculus students. It turned out to be more interesting than I thought (thanks, Ignacio and Noema).

The government intends to build a superhighway without speed limit so, therefore, without curves. It will start from city A, and should pass as close as possible to cities B and C.

In order to solve it you should start by stating what you mean by “as close as possible”. An option is to minimize the sum of the distances. But then, you get the following funny configuration:

Cities B and C are at the same distance from A, and make up a right angle. Intuition dicatates that the best route for the highway would be the bisector. Let d(A,C)=d(A,B)=1. Then, the sum of the distances from the cities to the highway is \sqrt{2}. But there is a better highway! You can just break the symmetry between B and C and make the road pass through C exactly. Then, the sum of the distances is just… 1.

If this was a real highway, the politician in charge would tell us that symmetry is also worth. Citizens of B may riot with un asymmetric solution… So, now let us change our target function. Let us minimize the sum of the squares of the distances. In that case, the bisector gives a square total distance of 1, same as the asymmetric road… Can you explain it?

Which is the best choice? Of course, it depends on the reason for which you’re fighting with the problem. If it is just to pass an exam, any of them will do…:) [Of course, there are many other alternatives. A student minimized the sum of the squares of the distances from the points to the straight line in the y direction. This makes sense in some cases, e.g.: when you’re fitting experimental points to a line.]

So, choosing the right function to minimize is crucial in practice… Kadanoff once explained in a talk that the government of the city of London has ordered a huge global study of traffic, taking into account both public and private transport, energetic and economic issues, taxes and prices, eeeeeverything. They made an enormous computer program that was running for days and, finally, told them the answer, how to optimize traffic in London. They had to remove all traffic lights. Why???? The program had many things into account… also the fact that in street car accidents, it’s normally old people who die. And old people do not pay taxes, they receive their pension money from the government. So it was convenient to remove the traffic lights… So, you see: garbage-in, garbage-out. Yes, maths is a nice girlfriend, she gives you more than you put in the relationship… but she can do no miracles. If you’re stupid, she can’t fix that…