### Carnival of Mathematics

The Aperiodical is described on its About’ page as “a meeting-place for people who already know they like maths and would like to know more”. The Aperiodical coordinates the Carnival of Mathematics (CoM), a monthly blogging roundup hosted on a different blog each month. Generally, the posts describe a collection of interesting recent items on mathematics from around the internet. This month, it is the turn of thatsmaths.com to host CoM. In this CoM contribution — Number 196 — I review five mathematical posts that have appeared during the past month or so. URL links to the originals are included. There is a great wealth of alternative posts from which to choose; what appears below is just a taster.

1. Animation about Gödel’s Incompleteness Theorem

We must all be familiar with the amazing results proved by Kurt Gödel. In July, a video written by Marcus du Sautoy was published on YouTube. Entitled The paradox at the heart of mathematics: Gödel’s Incompleteness Theorem, it has already attracted millions of viewers and thousands of comments.

Paradoxical self-referential sentences like This sentence is false’ are easy to construct. Gödel devised an ingenious coding and constructed a mathematical equation that refers to itself, effectively reading “This statement cannot be proved”. He had constructed a mathematical result that is true but unprovable.

Marcus du Sautoy (Twitter @MarcusduSautoy) has been working for the last year on this cartoon animation, which gives a nice clear description of some of the subtle and counter-intuitive ideas underlying Gödel’s results: Video here.

2. Generating Pairs of Co-prime Numbers

In an Aperiodical post, Colin Beveridge discussed “almost Pythagorean triplets”, e.g., three integers such that ${a^2 + b^2 = c^2+1}$. He quoted Martin Gardner: “You’d be surprised how much math you can learn by exploring some of the implications and ramifications of what may seem at first no more than a trivial brainteaser”.

This investigation led Colin to ask if there is a method to generate all of the pairs of co-prime integers. He found that there are several, including the Farey sequence and the Stern-Brocot tree. Both have been described on thatsmaths,com, at Farey sequence and Stern-Brocot sequence. He found a range of interesting results and noted that “there’s loads more to play with here”. If you would like to explore this topic, see the twitter thread here.

3. Fluids and Probability

Counting nothing as something, we can say that “something must happen” (even if it is nothing!). So, the probability of something happening is 1.

Kartik Chandra (blog at Comfortably Numbered) has written a piece with the attractive title “Fortune in Flux”, linking intuition about fluid motion and probability distributions. Kartik, who is fascinated by analogies between physical systems, asks “Can we get any mileage out of the probability-fluid analogy?” As argued in the post, the answer is “Yes”.

A simple example of a useful analogy is the equivalence between the equations for a damped spring, with mass, stiffness and damping, and an LCR circuit, with inductance, capacitance and resistance. It is often the case that one system is easier to work with than the other, and we can learn something about both systems by experiments on one. Simple mechanical oscillator (left) and its electric circuit analogue (right) [Image Wikimedia Commons].

For a closed fluid system, the total mass remains constant irrespective of how distorted or convoluted the flow may be. This is reminiscent of the requirement that the integral of a probability density must equal 1: “something must happen”. Kartik exploits this analogy to deduce the value of parameters in physical systems, something that can be difficult by conventional means.

Kartik concludes: `How wonderful it is, that we can start with the simplest axiom of probability — “something must happen” — and, reasoning only with physical intuition, reach a sampling algorithm that solves a real problem.’ Read the post Fortune in Flux.

4. Triangulations, Geometry and Knots In a YouTube video entitled Triangulations, geometry and knots, Prof Jessica Purcell (Monash University) discusses hyperbolic geometry with Prof Stephan Tillmann (Sydney University). They plan to collaborate on a project on the triangulation of 3-manifolds. Purcell explains why mathematicians are mesmerised by mathematical knots!

Hyperbolic geometry, first dreamt up independently by Gauss, Lobachevskii and Bolyai, is the geometry of manifolds with negative curvature. Purcell is investigating the triangulation of 3-manifolds in hyperbolic space. This has potential applications in computational topology.

In the video, Stephan Tillmann describes how a topological knot can be studied by considering its complement, the 3-space around it. Purcell concludes by speculating on the shape of the universe: “perhaps we are sitting inside the complement of a trefoil knot”.

5. Approximations for ${\pi}$ that beat ${22/7}$

There are numerous ways of remembering digits of ${\pi}$. A piem, or ${\pi}$-poem, has words with the number of letters corresponding to the digits. One well-known piem begins “How I want a drink, alcoholic of course  … ”. The letter count gives 3.1415926 … . John Beach (twitter account here) has tweeted a thread with several alternative approximations to ${\pi}$. The familiar estimate ${22/7}$ gives a poor result, ${3.14286}$, with only three digits correct. Beach notes that ${22/7}$ is often associated with Archimedes although, of course, Archimedes did better than this, squeezing ${\pi}$ between limits ${\textstyle{\frac{223}{71}<\pi<\frac{22}{7}}}$, or ${3.1408<\pi<3.1429}$. He also notes that ${4000/1273}$ is an improvement on ${22/7}$.

But the simple inverse piem for ${1/\pi}$, “Can I remember the reciprocal”, gives ${\pi\approx 3.14159}$, good for six digits. There are opportunities for exploring and discovering more interesting and accurate estimates of ${\pi}$. The twitter thread is here. * * * * *

That’s Maths II: A Ton of Wonders

by Peter Lynch now available.
Full details and links to suppliers at
http://logicpress.ie/2020-3/

>>  Review in The Irish Times  <<

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