If you get stuck, you can read the Mathematica code on his website, or check out the comments on my G+ post. But it's probably more fun to watch it carefully and see for yourself!
The British mathematician James Joseph Sylvester, who lived from 1814 to 1897, was one of the first to dig deeply into the beautiful patterns you can form using finite sets. But he got into lots of trouble.
For example, he entered University College London at the age of 14. But after just five months, he was accused of threatening a fellow student with a knife in the dining hall! His parents took him out of college and waited for him to grow up a bit more.
Later, at the age of 27, he went to the United States and became the chair of mathematics at the University of Virginia in Charlottesville. After just a few months, a student reading a newspaper in one of Sylvester's lectures insulted him. Sylvester struck him with a sword stick. The student collapsed in shock. Sylvester thought he'd killed the guy! He fled to New York where one of his brothers was living.
Later he came back. According to the online biography I'm reading, "the abuse suffered by Sylvester from this student got worse after this". Soon he quit his job.
One thing I like about Sylvester is that he invented lots of terms for mathematical concepts. Some of them have caught on: matrix, discriminant, invariant, totient, and Jacobian! Others have not: cyclotheme, meicatecticizant, tamisage and dozens more.
Sylvester defined a 'duad' to be a way of choosing 2 things from a set. A set of 6 things has 15 duads. A hypercube has 16 corners. The picture by Greg Egan above shows a hypercube with 15 of its 16 corners labelled by duads. The bottom corner is different.
This may seem just cute, but in fact it can help you visualize a rather wonderful fact: the group of permutations of 6 things is isomorphic to the symmetry group of a 4-dimensional symplectic vector space over the field with 2 elements.
For details, read this:
This Friday I was hanging out and drinking beer with some philosophy professors. This is always fun, because they think sort of like me, but different. They seem more optimistic about our ability to solve all sorts of puzzles just by talking.
To annoy them a bit, I said that philosophers are great at verbal reasoning, but mathematicians should be good at three kinds of reasoning: verbal, symbolic and visual reasoning.
In response, one of them showed me this picture proof that \(\sqrt{2}\) is irrational.
We just need to show that it's impossible to have
$$ a^2 = b^2 + b^2 $$
for whole numbers \(a\) and \(b\). So let's do a proof by contradiction. We can assume \(a\) is the smallest whole number that obeys this equation for some whole number \(b\). We'll get a contradiction by finding an even smaller one.
We do it by drawing a picture.
The big square in this picture is an \(a \times a\) square. The two light blue squares, which overlap in the middle, are \(b \times b\) squares.
The area of the big square is the sum of the areas of the light blue squares. But there are two problems. First, the light blue squares overlap. Second, they don't cover the whole big square! These two problems must exactly cancel out.
So, the area of the overlap — the dark blue squares — must exactly equal the area that's not covered — the two pink squares.
So, the area of the dark blue square is the sum of the areas of the pink squares! But the lengths of the sides of these must be whole numbers, say \(c\) and \(d\). So we have
$$ c^2 = d^2 + d^2 $$
But \(c\) is smaller than \(a\). So, we get a contradiction!
Actually this proof uses a mix of verbal and visual reasoning, with just a tiny touch of symbolic reasoning. I wrote the formulas like \(a^2 = b^2 + b^2\) just to speed things up a bit and reassure you that this was math. I didn't really do anything with them.
The philosophers who told me about this are Mike Pelczar and Ben Blumson. The picture here comes from a website Mike pointed me to:
Richeson says:
Apparently the proof was discovered by Stanley Tennenbaum in the 1950's but was made widely known by John Conway around 1990. The proof appeared in Conway's chapter "The Power of Mathematics" of the book Power, which was edited by Alan F. Blackwell, David MacKay (2005).
On the other hand, Ben says John Bigelow published the proof in his book The Reality of Numbers in 1988, without citing anyone.
We wondered if it was known to the ancient Greeks.
You can do similar proofs of the irrationality of \(\sqrt{3}, \sqrt{5}, \sqrt{6}\) and \(\sqrt{10}\):
And this particular style of proof by contradiction is famous! It's called proof by infinite descent. You assume you have the smallest whole number that's a counterexample to something you want to prove, and then you cook up an even smaller one. It's really just mathematical induction in disguise, but it's more fun. It was developed by Pierre Fermat — who, by the way, was a lawyer.
If you want to take all the fun out of the proof I just gave, you can do it like this.
Assume \(a\) is the smallest whole number for which there's a whole number \(b\) with
$$ a^2 = b^2 + b^2 $$
Let
$$c = 2b - a$$
and
$$d = a - b$$
Then \(c\) and \(d\) are whole numbers and
$$ c^2 = d^2 + d^2 $$
(You can do some algebra to check this.) But \(c \lt a\), so we get a contradiction.
Wikipedia shows you how to prove by infinite descent that whenever \(n\) is a whole number, either \(\sqrt{n}\) is a whole number or it's irrational:
Fermat did a lot more interesting stuff with this method, too!
Right now physicists are struggling with the 'firewall paradox' — a problem in our theory of black holes. But this is far from the first time physicists have been stuck with an annoying 'paradox'.
Back in the late 1800s, physicists noticed that an electron should get mass from its electric field. Nowadays we'd say this is obvious. The electric field has energy, and \(E = mc^2\), so it contributes to the mass of the electron. But this was before special relativity!
How did they figure it out? They were very clever. They used Newton's \(F = ma\). When you push on something with a force, you can figure out its mass by seeing how much it accelerates!
So, they did some calculations. When you push on an electron with a force, you also affect its electric field. It's like the electron has a cloud around it, that follows wherever the electron goes. This makes it harder to accelerate the electron. So, it effectively increases the electron's mass. They calculated this extra mass.
They also did an easier calculation: how much energy this electric field has!
Say \(m\) is the extra mass due to the electric field surrounding the electron, and \(E\) is the energy of this electric field. Then they discovered that $$ E = \frac{3}{4} mc^2 $$ Whoops!
Had they made an algebra mistake? Not really.
Some really smart people all got the same answer! Oliver Heaviside got it in 1889 — he was one of the world's smartest electrical engineers. J.J. Thomson got it in 1893 — he's the guy who discovered the electron! Hendrik Lorentz kept getting the same answer, even as late as 1904 — and he's one of the people who paved the way for relativity!
But in 1905, Einstein wrote his paper showing that E = mc2 is the only possible answer that makes sense.
So what went wrong?
All those guys were assuming the electron was a little sphere of charge. Why? In their calculations, if was a point, the energy in its electric field would be infinite, because the electric field gets extremely strong near that point. The mass contributed by this field would also be infinite.
If the electron were a tiny sphere, they could avoid those infinite answers.
But then they ran into this \(E = \frac{3}{4}mc^2\) problem. Why? Because electrical charges of the same sign repel each other. So a tiny sphere of charge would explode if something weren't holding it together. And that something — whatever it is — might have energy. But their calculation ignored that extra energy.
In short, their picture of the electron as a tiny sphere of charge, with nothing holding it together, was incomplete. And their calculation showing \(E = \frac{3}{4}mc^2\), together with special relativity saying \(E = mc^2\), shows this incomplete picture is inconsistent.
So in the end, it's not a case of people being stupid. It's a case of people discovering something interesting... by taking a plausible idea and showing it can't work.
If you want the details, read what Feynman has to say:
Here's a bit of what he says:
The discrepancy between the two formulas for the electromagnetic mass is especially annoying, because we have carefully proved that the theory of electrodynamics is consistent with the principle of relativity. [...] So we are in some kind of trouble; we must have made a mistake. We did not make an algebraic mistake in our calculations, but we have left something out.In deriving our equations for energy and momentum, we assumed the conservation laws. We assumed that all forces were taken into account and that any work done and any momentum carried by other 'nonelectrical' machinery was included. Now if we have a sphere of charge, the electrical forces are all repulsive and an electron would tend to fly apart. Because the system has unbalanced forces, we can get all kinds of errors in the laws relating energy and momentum. To get a consistent picture, we must imagine that something holds the electron together. The charges must be held to the sphere by some kind of rubber bands.something that keeps the charges from flying off. It was first pointed out by Poincaré that the rubber bands — or whatever it is that holds the electron together — must be included in the energy and momentum calculations. For this reason the extra nonelectrical forces are also known by the more elegant name "the Poincaré stresses". If the extra forces are included in the calculations, the masses obtained in two ways are changed (in a way that depends on the detailed assumptions). And the results are consistent with relativity; i.e., the mass that comes out from the momentum calculation is the same as the one that comes from the energy calculation. However, both of them contain two contributions: an electromagnetic mass and contribution from the Poincaré stresses. Only when the two are added together do we get a consistent theory.
This was a bummer back around 1905, because people had actually hoped all the mass of the electron was due to its electric field. Note: this extra assumption is not required for the \(E = \frac{3}{4}mc^2\) problem to bite you in the butt. It's already a problem that the energy due to the electric field is \(\frac{3}{4}mc^2\) where \(m\) is the mass due to the electric field. But the solution to the problem — extra 'rubber bands' — killed the hope that the electron could be completely understood using electromagnetism.
It is therefore impossible to get all the mass to be electromagnetic in the way we hoped. It is not a legal theory if we have nothing but electrodynamics. Something else has to be added. Whatever you call them — 'rubber bands', or 'Poincaré stresses', or something else — there have to be other forces in nature to make a consistent theory of this kind. Wikipedia has a good article on the history of this problem:
and this paper is also good:
The spacecraft Dawn has gotten a closer look at the mysterious white spots on the asteroid Ceres. In the first photos, they were so bright they were overexposed!
Now Dawn is closer. Here's the biggest patch of white stuff, called Spot 5, in a crater called Occator.
What is it? The obvious guess is some sort of ice or salt, reflecting sunlight. But here's the cool part: you can sometimes see haze over Spot 5. This suggests that some sort of gas is coming up from beneath the surface! Or maybe the ice is sublimating, turning into vapor... perhaps explosively?
The mission director writes:
Dawn has transformed what was so recently a few bright dots into a complex and beautiful, gleaming landscape. Soon, the scientific analysis will reveal the geological and chemical nature of this mysterious and mesmerizing extraterrestrial scenery.
This picture is a composite of two images: one using a short exposure that captures the detail in the bright spots, and one where the background surface is captured at normal exposure. Each pixel here is a 140 meter × 140 meter square.
Right now Dawn is orbiting Ceres at a distance of 1450 kilometers. In December, it will descend to just 375 kilometers from the surface. Then we'll get even better images!
And when the mission ends? Then Dawn will remain as a permanent satellite of Ceres. A fitting end to a great mission — it's the first spacecraft to orbit two bodies, Vesta and Ceres.
You can watch a short video of what Dawn has been seeing, here:
This image came from here:
For more on the haze, read this:
This is based on a famous print by M. C. Escher called 'Ascending and Descending'. That in turn was based on an idea by the mathematician Penrose, called the Penrose stairs. But Penrose in turn was inspired by Escher, who was inspired by Penrose.... in an endless loop!
At least that's the story on Wikipedia:
At an Escher conference in Rome in 1985, Roger Penrose said that he had been greatly inspired by Escher's work when he and his father discovered the Penrose stairs.
Penrose said he'd first seen Escher's work at a conference in Amsterdam in 1954. He was "absolutely spellbound", and on his journey back to England he decided to produce something "impossible" on his own. After experimenting with various designs he finally arrived at the impossible Penrose triangle.
Roger Penrose then showed his drawings to his father Lionel, who immediately produced several variants, including the impossible stairs. They wanted to publish their findings but didn't know where to do it. Because Lionel Penrose knew the editor of British Journal of Psychology, the finding was finally presented as a paper there. After its publication in 1958, they sent a copy of the article to Escher as a token of their esteem.
But here's the weird part. While the Penroses credited Escher in their article, Escher himself noted in a letter to his son in January 1960 that he was:
... working on the design of a new picture, which featured a flight of stairs which only ever ascended or descended, depending on how you saw it. They form a closed, circular construction, rather like a snake biting its own tail. And yet they can be drawn in correct perspective: each step higher (or lower) than the previous one. I discovered the principle in an article which was sent to me, and in which I myself was named as the maker of various 'impossible objects'. But I was not familiar with the continuous steps of which the author had included a clear, if perfunctory, sketch..."
So it seems this impossible staircase willed itself into existence in a paradoxical loop of causality!
But now that I think about it, all good ideas come into being a bit like this. At first they exist only in vague form, in the random jostlings of thoughts and words. They exist just a little, merely because they can exist. But thanks to their power, the more people think and talk about them, the more they spiral into existence... until finally they are clear and undeniable. Like this idea here.
Puzzle 1: just by looking carefully at this animated gif, can you guess who made it?
Puzzle 2: can you see the slight mistake in the picture here?
Puzzle 3: does that mistake occur in Escher's original print? Here's the original
Here's the Wikipedia article where I got the story:
For answers to the puzzles, see the comments on my G+ post.
This is the McGee graph. You'll notice each vertex has 3 neighbors. Also, if you go around a cycle of edges, there will always be at least 7 edges in your path. The McGee graph's claim to fame is that it has the fewest vertices possible for a graph with these two properties.
This isn't so amazing, so the McGee graph is not as famous as some others I've discussed on Visual Insight. It's fairly symmetrical, though. Besides the obvious symmetries — rotating it 1/8 of a turn, or flipping it over — there are some sneaky symmetries, like the one shown in this animation by Greg Egan.
He drew four red 'bands', actually hexagons, because this sneaky symmetry mixes up the nodes in each 'band' in a clever way. Also, you'll notice that some vertices are red, and others blue. That's because all the symmetries send red vertices to red ones, and blue vertices to blue ones.
In trying to understand the McGee graph better, I wanted to start by understanding its symmetries. It has 8 rotational symmetries; if we throw in our ability to flip it over we get a total of 16, and if we throw in all the sneaky symmetries we get a total of 32.
There are lots of different groups with 32 elements. In fact, there are 51 of them.
(If you start listing finite groups, you'll soon discover that most of them have a power of 2 as their number of elements. Indeed, of the 50 billion or so groups of size at most 2000, more than 99% have 1024 elements!)
Luckily there's an online list of all 51 groups with 32 elements, and a fellow named Gordon Royle told me which of these was the symmetry group of the McGee Graph. So, I soon found out that the symmetries of the McGee graph are the affine transformations of the integers mod 8.
I should explain this. An affine transformation is something like this: $$ x \mapsto ax + b $$ where \(a\) is invertible. You may be used to calling them 'linear', but showoffs like me prefer to save that word for transformations of this sort: $$ x \mapsto ax $$ Anyway, you may be used to affine transformations when \(x, a,\) and \(b\) are real numbers. The real numbers form a line. The affine transformations just slide, stretch or squash, and maybe flip that line.
But here we are dealing with the case where \(x, a,\) and \(b\) are integers mod 8. What is the 'line' like then? Well, the integers mod 8 can be visualized as an octagon — that is, the 8 red nodes in the animation here. So now our 'line' looks like an octagon!
This may freak you out, but to mathematicians there's nothing more fun than taking intuitions from a familiar example and applying them to some weird other situation. So now a line looks like an octagon? — sure, we can handle that.
What are affine transformations like? Well, these are rotations of the octagon: $$ x \mapsto x + b $$ We can also flip the octagon like this: $$ x \mapsto -x $$ But there are other transformations of this sort: $$ x \mapsto ax $$ where \(a\) isn't \(1\) or \(-1\), and those are the 'sneaky symmetries'.
You can check that in the integers mod 8, only four numbers are invertible: \(1, 3, 5\) and \(7 = -1\). So, we get \(4 \times 8 = 32\) affine transformations. And those are the symmetries of the McGee graph!
So: the red vertices of the McGee graph are just integers mod 8. The symmetries are affine transformations. Then the question becomes: how can we understand the blue vertices, and the edges, in terms of the integers mod 8? If we answer this correctly, it will become obvious why affine transformations give symmetries of the McGee graph.
This is a fun puzzle for people who like Felix Klein's philosophy relating groups and geometry. Egan and I worked out the answer, and explained it — with lots of pictures — on my blog Visual Insight:
So go there if you want the full story.
Not coincidentally, all the groups of size \(\le 2000\) were listed in the year 2000. You can read more about them here: