Something Mathly

Consider the following recursively defined sequence:

Innocent question: is this sequence unbounded? Surprisingly, the answer to this is unknown—at least according to the source article dating from 2000, Unbounded orbits and binary digits by M. Chamberland and M. Martelli.

Yep! Here’s the figure you’re given (roughly):

The key to this question is knowing that when a + b = 90, sin a = cos b. We’ll come right back to that.

If we draw a point W somewhere on segment RT, like the question says, we can see that the two angles they’re asking about, RSW and WST, are complimentary (that is, they add up to 90°). Look, there they are in green and red below:

The curvature of curves.

x²

x³

sin(x)

exp(x)

Normal distribution (y=exp(-x²/2))

Ellipse

r=5/2+cos(3τθ)

x=(t-1)(t+1), y=t(t-1)(t+1)

Archimedes’ Spiral

Logarithmic spiral

If you want to try your own curve, try on Desmos graphing calculator!

affine

nonabelian

Lie group

matrices mapping from polynomial to polynomial space

dynamical systems (repeating a map over and over)

This semester I am teaching a class called “Linear Algebra and Differential Equations”, so I thought I’d spend a couple minutes at the beginning of classes talking about linear algebra. (The full backstory can be found here)

For those of you coming from the linear algebra sequence, we’re going to focus primarily on vector spaces instead of the full force of modules; Most things I’m going to say are also true at least for free modules, but definitely not everything and I’ll do my best to highlight where those differences arise.

——

So at some point in your life, you’ve probably heard about function composition. Probably for the first time in precalc and then you got a lot of practice with it in calc from the chain rule, u-substitutions, change of variables, and so on. In any case, I’ll remind you how it works: we have two functions $f$ and $g$ and we want to make a third function $f\circ g$:

$$ (f\circ g)(x) = f(g(x)) $$

so you take the input, put it into $g$, and whatever it gives out you put that immediately into $f$, and whatever comes out of $f$ is the number you want.

Not often covered in a precalc class is the following issue:

$$ \begin{align*} f(x) &= 3\log(x) \ g(x) &= \begin{bmatrix} x \ 2x-1 \end{bmatrix} \end{align*} $$

So I’ve defined two perfectly good functions, but if I try to take $f\circ g$, we have to take the logarithm of a vector— of course that doesn’t make any sense. We say that $f$ and $g$ aren’t compatible. Why not? Well, the key here is that the sorts of objects that $f$ takes in need to be the same types of objects that $g$ spits out.

But now we know— or at least I’ve said it enough times that maybe you’re starting to believe me— that there is this relationship between matrices and linear maps. So if $F$ and $G$ are not just functions, but linear maps, you can’t stop me from taking the matrix associated to $F$ and the matrix associated to $G$ and the matrix associated to $F\circ G$.

But of course the three maps weren’t just random, the last one has a very strong relationship to the first two. We’d like to be able to say the same for the matrices. In other words, we want to fill in the “missing” arrow on the right-hand side.

Of course, we can draw that arrow, which is given precisely by matrix multiplication: $C=AB$. So if I were in the mood to make grandiose statements, I could say “Therefore, in order to understand matrix multiplication, we must first understand the relationship between matrices and linear maps.”

But of course that’s completely false…

I mean, it has to be, right? Because we did talk about matrix multiplication in lecture yesterday and we didn’t talk about linear maps. And really, that’s the whole point of matrix multiplication: we don’t want to go the long way around the square, and matrix multiplication says we don’t have to. We can stay on the right side of the picture without ever even realizing there is a left side.

Recently I heard about this surprising theorem. I already knew about Fáry’s theorem: if a simple graph can be drawn on the plane without crossing edges (i.e. if the graph is planar), then it can always be drawn in such a way that all edges are non-crossing straight lines. Hence, restricting to straight-line edges doesn’t give a strictly smaller class of planar graphs. I found this to be quite surprising already.

However, much more is true: you can always draw a simple planar graph in the plane in such a way that you can place non-overlapping circles on the vertices, such that two circles touch if and only if the corresponding vertices are joined by an edge. In particular, the drawing clearly satisfies the result of Fáry’s theorem. This result is called the circle packing theorem.

There are 2 days to Christmas and I thought I would do something a little bit different. So, everyday till Christmas I will post a book recommendation. It will be a math or math related book of course ^_^ Enjoy

The Golden Ratio Coloring Book by Rafael Araujo

Rafael Araujo is a Venezuelan architect and illustrator living in Caracas. For over 40 years he’s been drawing the most beautiful illustrations of nature, entirely by hand. At an old drafting table he adeptly renders the mathematical brilliance of nature with just a pencil, compass, ruler and protractor.

Araujo’s illustrations revolve around intelligent patterns of growth that are ruled by the Golden Ratio. This special number, commonly annotated with the Greek letter Phi (ϕ) can be seen in all sorts of natural spirals, sequences, and proportions. “Phyllotaxis” is the name given for the tendency of organic things to grow in spiral patterns and this number pattern reoccurs so often in nature that some researchers have deemed it a universal law for the perfection of structures, forms, and proportions. From sea shells, leaves, crystals, and even butterfly wings, Phi can be traced throughout our environment, time and time again.

also this should entirely be accessible to anyone who’s done just enough calculus to talk about maxima and minima of continuously differentiable functions, which means either high school or undergrad calc, depending on your educational path; don’t be so worried ;p

Complex analysis - the perfect subject for combining visual and analytic thinking. I just read “Visual Complex Analysis” by Tristan Needham, a basic book on complex analysis: lots of historical references, uncompromising explanations, lots of problem solving and plenty of beautiful illustrations!

So this is great :)

This image is the summary (well, it’s at least a summary) of the longest human-readable proof ever written— by far. This monster of “well over 10000 pages” goes by many names, including The Classification Theorem, a sweepingly grand moniker that any lesser theorem would be scolded for taking. The things that this result classifies are the so-called “finite simple groups”; for an explanation of what those are, I would direct you to this excellent article which attempts to explain the matter.