## All scientific knowledge

If, in some cataclysm, all scientific knowledge were to be destroyed, and only one sentence passed on to the next generation of creatures, what statement would contain the most information in the fewest words? I believe it is the atomic hypothesis (or atomic fact, or whatever you wish to call it) that all things are made of atoms — little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another. In that one sentence you will see an enormous amount of information about the world, if just a little imagination and thinking are applied.

The Feynman Lectures on Physics, Vol. I (1964)
Richard Feynman

## This is not yet a scientific age

Is no one inspired by our present picture of the universe? This value of science remains unsung by singers, you are reduced to hearing not a song or poem, but an evening lecture about it.

This is not yet a scientific age.

What Do You Care What Other People Think? (1988)
Richard Feynman

## The Pleasure of Finding Things Out

I can live with doubt and uncertainty and not knowing. I think it’s much more interesting to live not knowing than to have answers that might be wrong. [...] I don’t feel frightened by not knowing things, by being lost in the mysterious universe without having any purpose.

The Pleasure of Finding Things Out (1999)
Richard Feynman

(HT @mattischrome for the quote)

## Feynman, Bethe and the beauty of mathematics

To those who do not know mathematics, it is difficult to get across a real feeling as to the beauty, the deepest beauty, of nature…If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in.

The Character of Physical Law (1965)
Richard Feynman

This term I have been teaching a new first year undergraduate module, Mathematics for Computing, in which I have been trying to impart a little bit of love for mathematics. While we have covered a number of underpinning topics relevant to computer science, such as propositional logic, set theory and number theory, I have also tried to show that there are a multitude of clever little tricks that can make arithmetic and performing seemingly complex calculations that little bit easier. And in doing so, I was reminded of the mathematical prowess of Richard Feynman as well as Hans Bethe, Nobel laureate in physics and Feynman’s mentor during the Manhattan Project. Bethe is one of the few scientists who can make the claim of publishing a major paper in his field every decade of his career, which spanned nearly 70 years; Freeman Dyson called Bethe the “supreme problem solver of the 20th century.

An example of Bethe’s mastery of mental arithmetic was the squares-near-fifty trick (taken from Genius: The Life and Science of Richard Feynman by James Gleick):

When Bethe and Feynman went up against each other in games of calculating, they competed with special pleasure. Onlookers were often surprised, and not because the upstart Feynman bested his famous elder. On the contrary, more often the slow-speaking Bethe tended to outcompute Feynman. Early in the project they were working together on a formula that required the square of 48. Feymnan reached across his desk for the Marchant mechanical calculator.

Bethe said, “It’s twenty-three hundred.”

Feynman started to punch the keys anyway. “You want to know exactly?” Bethe said. “It’s twenty-three hundred and four. Don’t you know how to take squares of numbers near fifty?” He explained the trick. Fifty squared is 2,500 (no thinking needed). For numbers a few more or less than 50, the approximate square is that many hundreds more or less than 2,500. Because 48 is 2 less than 50, 48 squared is 200 less than 2,500 — thus 2,300. To make a final tiny correction to the precise answer, just take that difference again — 2 — and square it. Thus 2,304.

Bethe’s trick is based on the following identity:

$(50 + x)^2 = 2500 + 100x + x^2$

For a more intuitive geometric proof of this formula, imagine a square patch of land that measures $50 + x$ on each side:

Its area is $(50 + x)^2$, which is the value we are looking for. As you can see in the diagram above, this area consists of a 50 by 50 square (which contributes the $2500$ to the formula), two rectangles of dimensions 50 by x (each contributing an area of $50x$, for a combined total of $100x$), and finally the small x by x square, which gives an area of $x^2$, the final term in Bethe’s formula.

While Feynman had internalised an apparatus for handling far more difficult calculations (for which he became famous for at Los Alamos, such as summing the terms of infinite series or inventing a new and general method for solving third-order differential equations), Bethe impressed him with a mastery of mental arithmetic that showed he had built up a huge repertoire of these easy tricks, enough to cover the whole landscape of small numbers. Bethe knew instinctively that the difference between two successive squares is always an odd number (the sum of the numbers being squared); that fact, and the fact that 50 is half of 100, gave rise to the squares-near-fifty trick.

Unfortunately, the skill of mental arithmetic that did so much to establish Bethe’s (as well as Feynman’s) legend was doomed to a quick obsolescence in the age of machine computation — it is very much a dead skill today.

## The importance of playing

As you may already be aware, Richard Feynman is a hero of mine. I highly recommend his book, Surely You’re Joking, Mr. Feynman!, an edited collection of reminiscences published in 1985. While there are light-hearted anecdotes about safe-cracking, art, languages and samba, what he says about playing and actually doing the things you love has always resonated with me:

But when it came time to do some research, I couldn’t get to work. I was a little tired; I was not interested; I couldn’t do research!

Then I had another thought: Physics disgusts me a little bit now, but I used to enjoy doing physics. Why did I enjoy it? I used to play with it. I used to do whatever I felt like doing — it didn’t have to do with whether it was important for the development of nuclear physics, but whether it was interesting and amusing for me to play with. When I was in high school, I’d see water running out of a faucet growing narrower, and wonder if I could figure out what determines that curve. I found it was rather easy to do. I didn’t have to do it; it wasn’t important for the future of science; somebody else had already done it. That didn’t make any difference. I’d invent things and play with things for my own entertainment.

So I got this new attitude. Now that I am burned out and I’ll never accomplish anything, I’ve got this nice position at the university teaching classes which I rather enjoy, and just like I read the Arabian Nights for pleasure, I’m going to play with physics, whenever I want to, without worrying about any importance whatsoever.

Within a week I was in the cafeteria and some guy, fooling around, throws a plate in the air. As the plate went up in the air I saw it wobble, and I noticed the red medallion of Cornell on the plate going around. It was pretty obvious to me that the medallion went around faster than the wobbling.

I had nothing to do, so I start to figure out the motion of the rotating plate. I discover that when the angle is very slight, the medallion rotates twice as fast as the wobble rate — two to one. It came out of a complicated equation! Then I thought, “Is there some way I can see in a more fundamental way, by looking at the forces or the dynamics, why it’s two to one?”

I don’t remember how I did it, but I ultimately worked out what the motion of the mass particles is, and how all the accelerations balance to make it come out two to one.

I still remember going to Hans Bethe and saying, “Hey, Hans! I noticed something interesting. Here the plate goes around so, and the reason it’s two to one is…” and I showed him the accelerations.

He says, “Feynman, that’s pretty interesting, but what’s the importance of it? Why are you doing it?”

“Hah!” I say. “There’s no importance whatsoever. I’m just doing it for the fun of it.” His reaction didn’t discourage me; I had made up my mind I was going to enjoy physics and do whatever I liked.

I went on to work out equations of wobbles. Then I thought about how electron orbits start to move in relativity. Then there’s the Dirac Equation in electrodynamics. And then quantum electrodynamics. And before I knew it (it was a very short time) I was “playing” — working, really — with the same old problem that I loved so much, that I had stopped working on when I went to Los Alamos: my thesis-type problems; all those old-fashioned, wonderful things.

It was effortless. It was easy to play with these things. It was like uncorking a bottle: Everything flowed out effortlessly. I almost tried to resist it! There was no importance to what I was doing, but ultimately there was. The diagrams and the whole business that I got the Nobel Prize for came from that piddling around with the wobbling plate.

I truly believe in the importance of playing and solving problems that you find interesting; in fact, that’s why I do research. While I strive to adhere to this philosophy, it is not always possible — especially considering the academic research environment in which we currently reside, with its minimum publishable unit (as well as the shadow of the Research Excellence Framework in 2014).

But the main point I wanted to draw from this extended Feynman quote is how important it is in education to stimulate curiosity by playing: tinkering, fiddling and finding interesting real-world problems to solve. And while Feynman was talking about physics, I think this is especially relevant for computing: it is crucial that we give students something to play with! It should be trivial to engage students in computing and technology, but I think this is something that is (in general) missing from UK schools.

However, at a TeachMeet I attended in Reading last night as part of the 2011 Microsoft UK Partners in Learning Forum (where I am giving a talk today), I met teachers who were showcasing incredible examples of innovative teaching to engage students in computing and technology. But as with talking to members of the Computing at School (CAS) group, it is very easy to preach to the converted; it is therefore crucial that this “network of excellence” interacts with people who are currently outside of the network who need help and support.

So let’s try and get a Raspberry Pi, Arduino, LEGO Mindstorms, .NET Gadgeteer, Kinect et al. into the hands of kids at school (as well as exposing them to Scratch, Kodu, Alice, Greenfoot, etc) so they can program, hack and solve interesting problems.

But more importantly, play.

## The most remarkable formula in all of mathematics

I have been re-reading Genius: The Life and Science of Richard Feynman by James Gleick (hence the recent Feynman-themed post), which reminded me of a very special formula in mathematics; one that Feynman himself described as follows in his famous Feynman Lectures on Physics:

In our study of oscillating systems, we shall have occasion to use one of the most remarkable, almost astounding, formulas in all of mathematics. From the physicists’ point of view, we could bring forth this formula in two minutes or so and be done with it. But science is as much for intellectual enjoyment as for practical utility, so instead of just spending a few minutes, we shall surround the jewel by its proper setting in the grand design of that branch of mathematics called elementary algebra.

This remarkable formula? Euler’s Identity:

$e^{i\pi} + 1 = 0$

In analytical mathematics, Euler’s identity (named for the pioneering Swiss-German mathematician, Leonhard Euler), is an equality renowned for its mathematical beauty, linking five fundamental mathematical constants:

• The number 0, the additive identity.
• The number 1, the multiplicative identity.
• The number $\pi$, which is ubiquitous in trigonometry, the geometry of Euclidean space, and analytical mathematics ($\pi$ = 3.14159265…)
• The number $e$, the base of natural logarithms, which occurs widely in mathematical and scientific analysis ($e$ = 2.718281828…); both $\pi$ and $e$ are transcendental numbers.
• The number $i$, the imaginary unit of the complex numbers, whose study leads to deeper insights into many areas of algebra and calculus.

The identity is a special case of Euler’s Formula from complex analysis, which states that:

$e^{ix} = \cos x + i \sin x$

for any real number $x$. The derivation to the identity follows, as $\cos \pi = -1$ and $\sin \pi = 0$.

For me, Euler’s Identity reinforces the underlying beauty and interconnectedness of mathematics, pulling together three seemingly disparate fields into one simple formula; it certainly deserves being known as “the most remarkable formula in all of mathematics“.

## Feynman Problem-Solving Algorithm

When I am doing research, I often think of the Feynman Problem-Solving Algorithm, supposedly coined in jest by another Nobel Prize-winning physicist, Murray Gell-Mann, about Richard Feynman‘s innate problem-solving ability:

1. Write down the problem.
2. Think very hard.