4

Googol and Google

10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000

In 1938 American mathematician Edward Kasner (pictured here) asked his two young nephews for the name of a huge arbitrary number, which Kasner set at one followed by one hundred zeros. Edward Sirotta, then nine years old, suggested googol which Kasner subsequently described in his book Mathematics and the Imagination two years later. As Kasner told the story in his book, co-written with James R. Newman:

Words of wisdom are spoken by children as least as often by scientists. The name “googol“ was invented by a child (Dr. Kasner’s nine-year-old nephew: Milton Sirotta) who was asked to think up a name for a very big number, namely, 1 with a hundred zeros after it. He was very certain that this number was not infinite, and therefore equally certain that it had to have a name. At the same time that he suggested “googol” he gave a name for a still larger number: “Googolplex.” A googolplex is much larger than a googol, but is still finite, as the inventor of the name was quick to point out. It was suggested that a googolplex should be 1, followed by writing zeros until you get tired. This is a description of what would happen if one actually tried to write a googolplex, but different people get tired at different times and it would never do to have Carnera a better mathematician than Dr. Einstein, simply because he had more endurance. The googolplex then, is a specific finite number, with so many zeros after the 1 that the number is a googol. A googolplex is much bigger than a googol. You will get some idea of the size of this very large but finite number from the fact that there would not be enough room to write it, if you went to the farthest star, touring all the nebulae and putting down zeros every inch of the way.

The word has no etymology or history-it came straight from the imaginative brain of a 9 year old boy and into history! The word google was already in the American lexicon as the title of children’s cartoon, Barney Google, which came from a song called The Goo-Goo Song (1900) in Vincent Cartwright Vicker’s The Google Book, a childrens book about all the unusual creatures who live in Googleland.

Other names for the number googol include ten duotrigintillion (short scale), ten thousand sexdecillion (long scale) or ten sexdecilliard (Peletier scale).

It has notably been used in the last eighteen years in a slightly different spelling as the name of the world’s largest search engine, when on September 27, 1998, Sergey Brin and Larry Page launched as small company they named Google.

Happy Birthday, Google!

Google doodle courtesy google; Edward Kasner, Barney Google and creatures from Googleland all in the public domain.

2

What I learned this week:

- Don’t call ρ (greek letter rho) p. Your math prof will hunt you down.

- Stay hydrated. Not only is tumblr very enthusiastic about this, but it will force you to get up and walk a few feet every so often. Or however far away your bathroom is.

- If you use the back of your chemistry notebook as scratch paper for math your chemistry teacher will think you’re a very good student. Even if it’s mostly scribbles.

- This will also save trees. Or at least not kill more trees.

- Don’t try to use the built-in graphing program on the above laptop. It will only graph small blue spheres. Piles and piles of small blue spheres. Instead stare the screensaver in despair.

Rank Two Cluster Algebras from Very Nearly Nothing

the greedy basis

equals the theta basis

a rank-two haiku

Gregg Musiker (with respect to Cheung, Gross, Muller, Rupel, Stella, and Williams), UMN Combinatorics Seminar. [**]

A warning to the experts: I really will be doing this “from very nearly nothing”, which in this case means that I’m assuming only knowledge of high-school algebra… and persistence. (Of course, this means I will use imprecise language in some places— but if you know enough that you can find these holes, you know enough that you can fill them.)

——

We begin by defining the rational function field (in two variables), which you can construct as follows. 

  1. First take all numbers, and throw them into a box. 
  2. Then, take $x_1$ and throw it in there, and $x_2$ as well. (These two objects aren’t related in any way, they’re just variables.) 
  3. Then, enlarge the collection of things in the box as follows: for any pair of elements in the box, take them out, copy them, and then add, subtract, multiply, or divide the copies (with only one exception: you still can’t divide by zero!).
  4. Put the resulting object, as well as the originals in the box.
  5. Repeat steps 3 and 4 forever.

At the end of all that, what you have in the box is an enormous collection of objects that look something like these:

$$0.25+x_2 \qquad x_2(x_1)^2-3x_2 \qquad \frac{8+(x_1)^3-0.4x_2}{3+8x_2x_1} \qquad \frac{x_1^2+(x_2)^2+1}{x_1x_2} \qquad\cdots$$

This box is sort of our “raw materials” that we can play with. By that, I mean that we won’t care much about the box per se, but rather we will use the contents of that box to make more interesting things. Namely choose a (small) subcollection $S$ of things in that box, and put them in a new container— maybe a bin, this time. We obtain the algebra generated by S in a very similar way as we obtained the rational function field:

  1. First take all numbers and throw them into the bin.
  2. Put all the elements of $S$ in the bin.
  3. Then, enlarge the collection of things in the bin by performing almost the same Step 3 as before, except this time, division of any kind is forbidden.
  4. Put the resulting object, as well as the originals, into the bin.

Note that the lack of division means that unless $S$ is very big, you’ll get much less in the bin than in the box. For instance, if $S$ consists of $x_1$ and $x_2$ you’ll never get anything with a fraction bar. Or, in a somewhat mind-bendier way, if $S$ consists of $x_1-x_2$ and $1/(x_2)^3$, you’ll get things like

$$0.25+1.3x_1-1.3x_2 \qquad \frac{x_1-x_2+\frac{7}{(x_2)^3}}{4(x_2)^3}\qquad \cdots$$

but you’ll never, ever get an $\frac{x_1}{x_2}$, or even an $x_1+x_2$. If you have some time, play around with it and convince yourself :)

[ A brief bit of broader perspective: this sort of construction is ubiquitous in the field of math called commutative algebra: because an algebra is a setting in which we can add, subtract, and multiply, we think of it as an abstract version of the whole numbers. ]

——

Anyway, the reason why I’m calling our variables $x_1$ and $x_2$ instead of something easier like $x$ and $y$ is because we’re about to introduce some more $x_n$’s, so that we can generate an algebra (like we just did in the step above)

Let $b$ and $c$ be positive whole numbers, and suppose we’ve already defined $x_1, x_2, x_3,x_4$, and so on up to $x_n$. We then define $x_{n+1}$ in terms of the previous two:

$$x_{n+1} = \left\{ \begin{array}{ll} \displaystyle \frac{x_n^b+1}{x_{n-1}} & \text{ when $n$ is even,} \\ \\ \displaystyle\frac{x_n^c+1}{x_{n-1}} & \text{ when $n$ is odd.} \\ ~ \end{array} \right. $$

We are at last ready to define our object of interest: the cluster algebra (of rank two) (with parameters b and c) is the algebra generated by the infinite set $ \{x_1,x_2,x_3,x_4, x_5\cdots\}$.

—–

You may wonder why we’d ever care about such a thing. A quick, dirty, and slightly misleading answer is that cluster algebras are a vast generalization of certain sequences that you (might) know and (should) love. For instance, if $b=c=2$, and we perform the substitution $x_1=x_2=1$, then we get the even terms of the Fibonacci sequence:

$$\begin{align*} x_2 &= 1 \\ x_3 &=2 \\ x_4 &= 5 \\ x_5 &=13 \\ x_6 &=34 \\ x_7 &= 89 \end{align*}$$

In other words, somehow the functions that these terms come from convey more information than the numbers do themselves. Indeed, if you learn how to decode the functions properly, they shed some light on patterns that arise when we count certain types of objects.

There are more intrinsic reasons to care, but they take more time to explain.

You might imagine that the expressions become very difficult once you get to $x_4$ or $x_5$. In general this is true, but actually there is a surprising glob of structure in the madness: any of the $x_n$’s and (hence) any element in the cluster algebra, is of the form

$$\frac{P(x_1,x_2)}{(x_1)^k(x_2)^\ell}.$$

The notation $P$ means that the numerator is a polynomial, which in our context is just a fancy way of saying there are no fraction bars in the numerator. The denominator is even better: not only are there no fraction bars, there are also no plus or minus signs: it’s just the single term $(x_1)^k(x_2)^\ell$. This fact is not at all obvious, which you will see if you try to do some calculations. For instance, even in the nice “Fibonacci case” where if $b=c=2$, we have

$$x_5 = \frac{\displaystyle\left(\frac{x_2^4+2x_2^2+x_1^2+1}{x_1x_2}\right)^2+1}{\displaystyle\frac{x_2^2+1}{x_1}}$$

which, despite being a heaping mess in the numerator and having a plus sign in the denominator, ends up magically cancelling to have the desired form.

The fact that this happens is known as the Laurent phenomenon, because expressions of this form are called Laurent polynomials.

An even more remarkable thing happens: when you write out an $x_n$ in this way, there will never be any minus signs. This fact was conjectured by the people who first described cluster algebras (Fomin and Zelavinsky), but this positivity conjecture was only proven in 2013.

[ In his talk, Musiker remarked that there were two proofs of positivity that came out very nearly at the same time. The first one, by Lee and Schiffler, used “only algebra and patience”. This was very surprising: people were using all sorts of fancy machinery to attack the problem, yet the first proof was ultimately rather elementary. The second proof, by Gross, Hacking, Keel, and Kontsevich, was more along the lines of what people were expecting: “via scattering diagrams, mirror symmetry, and theta functions”. In other words, their proof uses extremely sophisticated and subtle, modern machinery from algebraic geometry. ]

—–

[ ** To clarify this citation somewhat:

This post is inspired by a talk (whose title is the haiku at the top of the post), but as you can tell by the fact that I never mentioned bases at all, it is quite far away from the actual subject of the talk. There are two reasons for this choice. One is because the definitions of the bases in the title were way too involved for me to effectively convey here. If I can figure out how to package the rest of my notes in a digestible format, I’ll write a more technical follow-up. The other is that I’m sick of sitting on my hands trying to figure out how to explain what a cluster algebra is, and I’m hoping that having some documentation for the rank two case will grease the wheels a little bit.

Credit where it’s due: I stole Musiker’s explanation of how to define a rank-two cluster algebra. ]

  • Me:*works really hard on lengthy math problem*
  • Me:*did one part wrong, over and over, must restart*
  • Me:I am a failure. What is wrong with me.
  • Me:*tries again, finishes it flawlessly*
  • Me:I did it. Me. I did it all by myself. I fucking did it. Me. I am amazing. Look at me go. I am the best. The master. You can't stop me now

Can you flatten a sphere?

The answer is NO, you can not. This is why all map projections are innacurate and distorted, requiring some form of compromise between how accurate the angles, distances and areas in a globe are represented.

This is all due to Gauss’s Theorema Egregium, which dictates that you can only bend surfaces without distortion/stretching if you don’t change their Gaussian curvature.

The Gaussian curvature is an intrinsic and important property of a surface. Planes, cylinders and cones all have zero Gaussian curvature, and this is why you can make a tube or a party hat out of a flat piece of paper. A sphere has a positive Gaussian curvature, and a saddle shape has a negative one, so you cannot make those starting out with something flat.

If you like pizza then you are probably intimately familiar with this theorem. That universal trick of bending a pizza slice so it stiffens up is a direct result of the theorem, as the bend forces the other direction to stay flat as to maintain zero Gaussian curvature on the slice. Here’s a Numberphile video explaining it in more detail.

However, there are several ways to approximate a sphere as a collection of shapes you can flatten. For instance, you can project the surface of the sphere onto an icosahedron, a solid with 20 equal triangular faces, giving you what it is called the Dymaxion projection.

External image

The Dymaxion map projection.

The problem with this technique is that you still have a sphere approximated by flat shapes, and not curved ones.

One of the earliest proofs of the surface area of the sphere (4πr2) came from the great Greek mathematician Archimedes. He realized that he could approximate the surface of the sphere arbitrarily close by stacks of truncated cones. The animation below shows this construction.

External image

The great thing about cones is that not only they are curved surfaces, they also have zero curvature! This means we can flatten each of those conical strips onto a flat sheet of paper, which will then be a good approximation of a sphere.

So what does this flattened sphere approximated by conical strips look like? Check the image below.

External image

But this is not the only way to distribute the strips. We could also align them by a corner, like this:

External image

All of this is not exactly new, of course. In the limit, what you have is called a American polyconic projection, which does require stretching in order to fill the gaps between the ending of the strips. Gauss’s Theorema Egregium demands this.

But I never saw anyone assembling one of these polyconic approximations. I wanted to try it out with paper, and that photo above is the result.

It’s really hard to put together and it doesn’t hold itself up too well, but it’s a nice little reminder that math works after all!

Here’s the PDF to print it out, if you want to try it yourself. Send me a picture if you do!

DROP THE BASS

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