Season 11 Episode 0

Constructing square roots of integers

Here’s another way to make square roots (if you just want roots of whole numbers); the Spiral of Theodorus, also known as Pythagoras's snail. You construct right-angled triangles, one at a time, and by Pythagoras’s theorem, the hypotenuses have length $\sqrt{2}$, $\sqrt{3}$, $\sqrt{4}$, and so on.

Rowan’s method is better than that though, because $x$ can be anything that we’ve already constructed (either by adding line segments together, or by taking a square root, or some combination of those steps). So Rowan can construct things like $\sqrt{1+\sqrt{3}}$ that don't appear in the Spiral of Theodorus.

Pentagons

Once you’ve got the ability to construct $\sqrt{5}$, you can think about drawing a perfect regular pentagon. Pentagons involve $\sqrt{5}$. If you didn't already know that, here's a geometry challenge to get started with; suppose the side lengths of a regular pentagon are 1, and all the angles are equal. Use the cosine rule to find the length of a diagonal (a line segment between two non-adjacent corners). We had an episode of OOMC on pentagon facts a few years ago; OOMC Season 5 Episode 3.

Here's one way to draw the pentagon! Start with a line segment of length 1. That’s going to be one of the sides of the pentagon, and I’m going to tell you the exact location of the opposite corner of the pentagon. Pentagons have reflectional symmetry, so this is somewhere on the line through the midpoint of your line segment that’s at right angles to your line segment. Draw that line. The distance along that line (the height of the pentagon, if you think of the first line segment as the base), is supposed to be $\sqrt{5+2\sqrt{5}}$ (this is not obvious!). So construct that length, and mark the corner the correct distance along your line. Now to fill in the remaining two corners, we know that each is distance one from two of the corners we’ve already found. So if we draw some circles of radius 1 around each of the three corners we’ve already got, these will intersect at the right places (plus a couple of extra points inside the pentagon).

Reading that, the magic step is the bit where I know exactly how tall a pentagon is. That’s not an obvious fact, at all! To work out the height, I need lots of facts like $\sin(72^\circ)$. I like the calculation of that value so much that I once put a bit of it on the Mathematics Admissions Test, and it got picked up by the blackpenredpen YouTube channel; YouTube | Oxford MAT asks: sin(72 degrees). 

Here’s an alternative way to do the construction of a regular pentagon, on the NRICH website.

Once you’ve thought about that activity, there’s a section of the Wikipedia article for a Pentagon with an animation, and a link to find out about Carlyle circles, which give a way to solve quadratic equations with compass and straightedge.

Compound angle formulas

There’s a way to use complex numbers to get compound angle facts like the $\sin(3\theta)$ fact that we hinted at on the livestream. Toby Lam has the details here; Toby Lam's Blog | Euler's formula and compound angle formulae.

Cubic Formula

You’ve met the quadratic formula for finding roots of $ax^2+bx+c=0$. There’s also a cubic formula to find roots of $ax^3+bx^2+cx+d=0$. It's messy! There is a quartic formula for $ax^4+bx^3+cx^2+dx+e=0$, but it’s such a mess that no-one ever uses it in its full form. 

What about a quintic formula for $ax^5+bx^4+cx^3+dx^2+ex+f=0$? There is no such formula using any number of operations (multiplication, addition, n-th roots), and there is a mathematical proof that there cannot be such a formula. I think it’s amazing that this can be known! This is a result from Galois Theory. For Galois Theory, I'm just going to link you to a biography of Galois, an article on the NRICH website, and the third-year Oxford Maths course where this is taught. Lecture notes for all our undergraduate courses are freely available to everyone; I'm not recommending that you read these or try to self-study Galois Theory, but I've linked to the 2020-21 iteration of the course because I thought you might be amused to see that on the page before last, we see

Corollary.
(1) It is not possible to divide a general angle into three equal parts using a lines and circles construction.

(2) (Gauss) It is possible to construct a regular 17-gon.

(a "corollary" is a mathematical result that follows from a previous, larger or more important, result.) In my opinion, that's a really cool place to get to after about three years of studying Mathematics at university.

Origami

[to follow – check back on Monday!]

Natural Number Game 

The Natural Number Game is hosted here.

The aim of the game is to prove mathematical statements (like $x+y=y+x$), starting from first principles. It is probably not a good idea to try the later levels until you’ve learnt about mathematical induction in the school or elsewhere! I'm not certain that this game is a good way to learn about that for the first time. But maybe I’m wrong, and one of the keen Y11 students will try it out!