A Cube of Resistors

Here is a puzzle for those of you with some knowledge of circuit analysis.

Imagine welding a cube of resistors, each resistor with the resistance of 1 Ohm. If you measured the resistance between opposite corners, what would it be?

Each black line represents a resistor of 1 Ohm. What is the equivalent resistance between the red points?
Each black line represents a resistor of 1 Ohm. What is the equivalent resistance between the red points?

Hints:

Do it the hard way: Distort the cube into a two-dimensional surface and use the full machinery of equivalent resistance (Y-Delta transformations, etc) to solve all the loops.

Do it the fun way: Think of the particular geometry of the cube, and its symmetries. Think of the fundamentals of circuit analysis. You really don’t need to calculate much if you know how to visualize the problem.

“The art of making love, muffled up in furs, in the open air, with the thermometer at Zero, is a Yankee invention.”
-President John Quincy Adams

Donate computing time for the environment

There is the environmental need and the political will to take solar energy seriously. Our group at Harvard is leading several theoretical and computational efforts to develop more efficient solar panels. One of our efforts (not my project) is to use computational chemistry tools to find novel materials that would lead to better solar technologies. This is mostly performed by trial and error. A lot of it.

A certain molecular arrangement is “proposed” randomly, and the computer calculates its molecular energy to see if it makes sense (if it is a realistic material) and then if it is useful for solar panels. This trial and error approach takes a long time, requires a lot of computational power, but it can be parallelized in a straight-forward manner.

How can you help us?

By downloading the Harvard Clean Energy Project software. With it, you can donate your unused computer cycles, when the computer is on but not using the processor much, to help perform the combinatorial calculations. With these small computing contributions from thousands of students it is expected that the calculations will be done ten times faster than in a supercomputer.

Right now it is only available for Windows, but in the next few weeks it should be compatible with Linux and Mac too.

The project has had a lot of visibility, the other day BBC called our office!

Check out all the news articles written about it.

The fixed point of the cow within the cow… and time travel.

Consider a matryoska, russian dolls that you open up and have another doll that you open up, etc etc.

A matryoshka is a doll with a matryoshka inside.
A matryoshka is a doll with a matryoshka inside.

Now, imagine this could go on forever, an infinite recursion of a doll within a doll within a doll…

This couldn’t happen in real life, after all, there will be a point that the doll will be made of so few atoms that it wouldn’t look like a doll anymore. In fact, this is sort of the idea that motivated Democritus, the greek philosopher from around 400BC, to propose the concept of atoms, atoms providing an end to the philosophical difficulties of making things smaller and smaller. But, time has passed, Democritus died, Newton was born, solved the mathematical difficulties of these kind of arguments, and then died too, and luckly now we feel empowered and gutsy enough to think about a doll within a doll within a doll within a doll… even if in real life this wouldn’t be possible.

In your minds eye imagine that point that is common to the center of all dolls, yes, all infinite number of them.  This point is called the fixed point.

A simpler, more visual example might be of help. Look at the label of a chocolate box, where a nun carries a tray with a chocolate box whose label has a nun that carries a tray…

A nun with a funky hat.
A nun with a funky hat.

All those chocolate boxes within chocolate boxes have one point in common. If you where going to imagine the point where the next chocolate box will come from, that would be the common point, the fixed point.

The fixed point is a property of maps, a mathematical transformation, that converts something into a something within itself. Maps are very general kinds of transformations.

Mathematically, an object in the original “Something” (the domain) could be called $$X$$. The transformation $$M$$ would take $$X$$ to $$Y$$, which is also inside the “Something”. These are all related by: $$X rightarrow Y=Mleft(Xright)$$.

Any map that takes you inside of yourself has a fixed point.
Any map that takes you inside of yourself has a fixed point.

The oval on the left gets mapped into another oval in the right, that is smaller and within the space occupied by the original oval. The red arrows indicate how a point at the top (bottom) of the oval gets mapped to another oval.

A fixed point would be a point $$F$$ such that $$F=Mleft(Fright)$$, that is, after the mapping nothing happens to it. In the diagram this is illustrated by the green arrow.

If you where going to map using $$M$$ the small oval on the right, $$Y$$, to another oval, $$Z$$, just like $$Y$$ was inside $$X$$, the oval $$Z$$ would be inside $$Y$$. Repeating this until you get sick of this (or reach infinity) would show that the fixed point $$F$$ is common to every single one of those ovals. There is a mathematical theorem that says that if a map maps something into itself, it will always have at least one fixed point.

Let me say that again to summarize: Maps have at least one fixed point.

One of my favorite applications of maps and fixed points is time travel. Imagine a person, let’s call him McFly. McFly’s person, life, memories, history, everything that he is and has been, will be labeled by $$X$$. McFly is friends with a mad scientist who gives him a time machine $$M$$.

Your friendly neighborhood mad scientist next to your average time traveller.
Your friendly neighborhood mad scientist next to your average time traveler.

With it, McFly can travel back in time, and change his own history, becoming a new self $$Y=Mleft(Xright)$$. He can for example go back in time, deposit a penny in a bank account, wait for many years collecting interests, take the money out in the present and become rich. The new McFly, $$Y$$, is now rich. McFly could also go back in time, prevent his parents from ever meeting, and he never been born. The new McFly $$Y$$ doesn’t even exists.

These paradoxes are what make imagining time travel so much fun. The problem is that it also makes it seem implausible, violations of cause and effect being most unphysical. Except for one kind of solution called closed-timelike curves.

Closed-timelike curves would be like traveling back in time but not affecting the future at all, therefore, not violating causality. Closed-timelike curves is time travel that is consistent with itself. For example, a closed-timelike curve would be that McFly travels back in time to deposit the money in the bank whose collected interests will allow the mad scientist to build the time travel machine that will allow McFly to travel back in time to deposit the money… etc etc.

A closed-timelike curve is a fixed point $$F$$ in time travel $$M$$, such that $$F=Mleft( F right)$$.

A closed-timelike curve would not be one when McFly changes history in such a way he is unable to travel back in time to change history. For example, if McFly lost the penny he woudn’t be able to deposit it in the bank and thus the mad scientist won’t have any money to build the time machine creating a paradox.

The theorem that there must be a fixed point in any map implies that there will also be closed-timelike curves. Time travel in one of those wouldn’t violate causality, but might not be to useful either.

I leave you with a picture of a MinusTwoFish-approved cheese brand.

The cow within the cow within the cow within the cow...
The cow within the cow within the cow within the cow…

Also, a very cool video of the effect of self maps and fixed points. Click it!
Flash Video of Droste Effect

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“A witty saying proves nothing.” –Voltaire

No greens for crimson

A few weeks ago, Nobel Laureate Al Gore came to visit our campus to officially inaugurate the new initiative to make Harvard University environmentally friendly. The following banner can now be seen everywhere around the university.

Crimson is a reference to the university colors and sports teams, The Crimsons.

Not too long ago we got an email from the president of the university announcing how, thanks to the economic disaster, things were going to get ugly here. Harvard’s endowment has been managed very conservatively and successfully through its history, making it the richest university, attracting people like me to do do research. About $$frac{1}{3}$$ of the employees here get paid from the profits generated from the investments of this endowment. With the market crash, there is now place to make money, no matter how safe you play it.

Yep, no greens for crimson.

The news hit the main media of how big the losses have been. And they have been big.

Harvard Endowment Fell 22 Percent in Four Months

Harvard’s endowment—the largest in higher education—fell 22 percent in four months from its June 30 value of $36.9 billion, marking the endowment’s largest decline in modern history, University officials announced yesterday.

The precipitous drop will require Harvard’s faculties to take a “hard look at hiring, staffing levels, and compensation,” wrote University President Drew G. Faust and Executive Vice President Edward C. Forst ’82 in a letter informing the deans of Harvard’s losses.

What is the impact that these news will have on my fellowship? I don’t know.

A Stochastic Goodbye to Ito

Kiyoshi Ito, a Stochastic Man of Longevity
Kiyoshi Ito, a Stochastic Man of Longevity

Mathematician Kiyoshi Ito died at the young age of 93 this past month. Ito was the inventor of calculus for stochastic processes, known as the Ito Calculus.

Calculus, as invented by Isaac Newton and Gottfried Leibniz, studied the rate of change of nice smooth variables, $$x$$ in terms of their differentials, infinitesimal quantities described by $$dx$$. To properly define a Leibniz differential, the variable $$x$$ must be nicely behaved. Words that are often associated with nice variables are smooth, differentiable and/or continuous.

This limited the scope of applications of calculus. In particular, it does not apply to a random process. A random process, such as rolling a dice, is not nicely behaved, each roll of the dice being very different from the one before, its values literally jumping around a lot. A processes given by probabilistic, random, rules is called a stochastic processes.

My favorite stochastic process is the random walk, and is defined as follows.

Imagine a drunk guy, who can either take a step forward or backwards. Each direction has an equal probability, so you can think of the drunk guy carrying out a random walk, where the direction of each step is determined by a coin toss, heads giving a step forward while tail signifying a step backwards.

This class of problems are very common in statistical physics, finance and biology. The difficulty with doing calculations of stochastic processes is that the variables are not nice, and thus their differentials are not well defined.

Ito invented his own type of differential for exactly this purpose. Although the rules he computed were inspired by traditional calculus, they are on a different class of their own. It’s impact is so broad that is difficult to think of a field with a component of applied math where Ito calculus does not play a role.

New York Times has the story.

$$langle mbox{Ito} rangle = 46.5 $$

A solution to a fishy affair

Previously, on Minus Two Fish:
P.A.M. Dirac was presented with a puzzle.

Three fishermen come back from the sea. Each collapses in their respective tent. Fisherman #1 wakes up and decides to get his share of the catch. He counts the fish, realizes it is not a number divisible by three, throws away one fish to the sea correcting the situation, and takes a third of the remaining fish into his tent. Fisherman #2 wakes up later and also decides he is going to get his share of the catch. Fisherman #2 wants a third of the fish he sees. It is not a multiple of three, but he throws away one fish and takes a third of the fish and goes to sleep. Fisherman #3 wakes up after, and does the same: he throws away one fish, takes a third of the fish, and goes into his tent.
What is the smallest number of fish for which this would happen?

The real solution was not revealed, until now.

[Dirac’s Fish Puzzle Solution Spoiler Warning]

The want to find what is the smallest number of fish $$f$$ at the beginning.

$$f-1$$ is the number of fish after Fisherman #1 throws away one. $$left(f-1right)frac{1}{3}$$ is the number of fish that Fisherman #1 takes for himself, leaving only $$left(f-1right)frac{2}{3}$$. $$(f-1)frac{2}{3}-1$$ is the number of fish after Fisherman #2 throws away one. $$left( left(f-1right)frac{2}{3}-1right)frac{2}{3}$$ is the number of fish that Fisherman #2 leaves after he takes his share. $$left( left( f-1 right) frac{2}{3} -1 right)frac{2}{3}-1 $$ is the number of fish after Fisherman #3 throws away one. $$left(left(left(f-1right)frac{2}{3}-1right)frac{2}{3}-1right)frac{2}{3}$$ is the number of fish that Fisherman #3 leaves after he takes his share.

The last number must be an integer $$i$$, as it is implicit that only a whole number fish are left.
$$i=(((f-1)frac{2}{3}-1)frac{2}{3}-1)frac{2}{3}=ffrac{8}{27}-frac{38}{27}$$. Solving for $$f$$,
$$f=ifrac{27}{8}+frac{38}{8}$$.

Now, $$i$$ must be a small integer. The easiest way to explain the solution is by trial and error. So, try $$i=1$$, and you will see $$f$$ is not an integer. Try $$i=2$$, $$i=3$$ … still $$f$$ is not an integer. Until you reach $$i=6$$, where $$f=25$$.
So, the smallest number of fish $$f$$ is 25.
Also, try in the equation to plug in $$f=-2$$, Dirac’s solution and you will see $$i=-2$$. That is why it is so weird, Dirac assumes that there are negative fish to start, and negative fish at the end.

This solution permits for infinite number of fishermen to come and throw away $$1$$ fish and take $$frac{1}{3}$$ of the remaining fish as the number of fish at the end is the same as the beginning.

Baby’s first physics book

Here are Paul Dirac and Enrico Fermi. They can do lots of things!
Here are Paul Dirac and Enrico Fermi. They can do lots of things!

How can I jump start my toddler’s career as a theoretical physicist?, I get asked often. I knew only one way how to answer, a way following the Talmudic tradition, answering with another question: Why?.

But, finally, the resource everyone has been looking for is available for sale, online.

Baby’s first physics book – Pat Schrodinger’s Kitty

This is the perfect gift for the nerdy babies in your life. With seven colorfully illustrated activities that will delight any infant who is interested in exploring theoretical physics.