Dust off your thinking hat and do some mind-stretches because here’s another course of Interesting Maths Esoterica! It’s been several months since the last volume so this is quite a big post. I won’t mind if you skim it.

In case you’re new to this: every now and then I encounter a paper or a book or an article that grabs my interest but isn’t directly useful for anything. It might be about some niche sub-sub-subtopic I’ve never heard of, or it might talk about something old from a new angle, or it might just have a funny title. I put these things in my Interesting Esoterica collection on Mendeley. And then when I’ve gathered up enough, I collect them here.

In this post the titles are links to the original sources, and I try to add some interpretation or explanation of why I think each thing is interesting below the abstract.

Some things might not be freely available, or even available for a reasonable price. Sorry.

Because there are so many entries this time round, I’ve picked a few particularly interesting or esoteric items that you should definitely have a look at. The rest might or might not interest you.

## Definitely have a look

### Light reflecting off Christmas-tree balls

‘Twas the night before Christmas and under the tree

Was a heap of new balls, stacked tight as can be.

The balls so gleaming, they reflect all light rays,

Which bounce in the stack every which way.

When, what to my wondering mind does occur:

A question of interest; I hope you concur!

From each point outside, I wondered if light

Could reach deep inside through gaps so tight?

Somebody asked a question on MathOverflow and, since it was about Christmas-tree baubles, posed it with a rhyme inspired by *The Night Before Christmas*.

And this is a good world we live in, because some wonderful person managed to give a beautifully concise, clear answer* *also in rhyme! (It doesn’t quite scan, but you can’t have everything)

### A mathematician’s survival guide

A career’s-worth of advice from Peter Castazza for new mathematicians on the subject of being a mathematician and taking part in maths culture, from dealing with “the public” to having the confidence to continue in research to seating mathematicians at dinner. Lots of good quotes and anecdotes.

### Topology Explains Why Automobile Sunshades Fold Oddly

We use braids and linking number to explain why automobile shades fold into an odd number of loops.

I like to think I’m the kind of man who understands things, but those foldy-outy windscreen shades always bamboozle me. The result in this paper might actually help me next time I need to fold one up. It’s a really interesting fact, and explained very well by the authors. I think this is in the same category of “maths you can explain to your parents while on a day trip” as the proof that use the intermediate value theorem to show that you can make a wobbly table stable just by rotating it.

### The wobbly garden table

And here’s that proof. It’s presented as a satirical comparison of the different ways engineers, physicists and mathematicians approach problems. Of course, the mathematician comes out looking considerably more rational than the other two. If all you want is the proof without the snarky tone, just read the mathematician’s bit.

### A cohomological viewpoint on elementary school arithmetic

From finite group theory to algebraic geometry to complex analysis, cohomological methods play a major role in modern mathematics. The subject has a long history throughout much of the twentieth century and strongly influenced the development of modern mathematics. Mathematicians view such techniques as powerful but sophisticated tools applicable to a remarkably wide field of study, but they usually react with surprise to learn that the ubiquity of cohomology in mathematics extends even to arithmetic at the primary school level.

It’s always fun when someone deliberately applies a sledgehammer to crack a very tiny nut. Somebody on Google+ mentioned that someone had told them that “carrying is a cocycle”, and wanted to know more. A commentor obliged with a link to this surprisingly not completely pointless article. Closed access on JSTOR, $12.

### Train sets

Suppose we have a train set — large stocks of straight and curved track, bridge-building materials, different kinds of sets of points, and a single engine, say clockwork for the sake of nostalgia. What can we do?

I think this was the first bit of interesting maths esoterica I ever encountered. When I was in sixth form my maths teacher read about this in the paper and brought it in to show us. Adam Chalcraft and Michael Greene show how to construct logic gates, and hence a Turing machine, using just train tracks and lazy (unmanned, uncontrolled) points. For years I’d forgotten who the authors were and tried in vain to find it again, so I’m very happy to finally have it in my collection.

### The top ten prime numbers

This isn’t a list of *the* top 10 prime numbers. It’s a list of lists of the top 10 prime numbers so far found of various types. For what is effectively just a big list of numbers, it’s a surprisingly gripping read. There are all sorts of kinds of prime numbers I’d never heard about, like Anti-Yarborough primes or strobogrammatic primes. I think I found this via the entry for the undulating palindromic primes on the OEIS.

### Seven staggering sequences

Neil Sloane, the man in charge of the Online Encyclopedia of Integer Sequences, wrote this paper for G4G7 about seven sequences that he found especially interesting. If anyone knows about really interesting integer sequences, it’s Sloane!

## The rest

### The Euler spiral: a mathematical history

The beautiful Euler spiral, deﬁned by the linear relationship between curvature and arclength, was ﬁrst proposed as a problem of elasticity by James Bernoulli, then solved accurately by Leonhard Euler. Since then, it has been independently reinvented twice, ﬁrst by Augustin Fresnel to compute diffraction of light through a slit, and again by Arthur Talbot to produce an ideal shape for a railway transition curve connecting a straight section with a section of given curvature. Though it has gathered many names throughout its history, the curve retains its aesthetic and mathematical beauty as Euler had clearly visualized. Its equation is related to the Gamma function, the Gauss error function (erf), and is a special case of the conﬂuent hypergeometric function.

I love the Euler spiral. I don’t know why. Maybe it’s because I first learnt of it as the “clothoid”, which is an excellent name, or maybe it’s because it gives me something to think about when I’m driving.

This shortish essay by Raph Levien gives a readable potted history of the spiral’s multiple discoveries and applications, illustrated with some lovely sparse diagrams of the sort that maths-illiterate Etsy craftspeople love.

### High Primes and Misdemeanours: Lectures in Honour of the 60th Birthday of Hugh Cowie Williams

This volume consists of a selection of papers based on presentations made at the international conference on number theory held in honor of Hugh Williams’ sixtieth birthday. The papers address topics in the areas of computational and explicit number theory and its applications. The material is suitable for graduate students and researchers interested in number theory.

I *think* I just kept this because of the title. That’s a good reason because it’s a good title, but I can’t remember why I found it. Maybe it contains something interesting.

### Statistical Modeling of Gang Violence in Los Angeles

Gang violence has plagued the Los Angeles policing district of Hollenbeck for over half a century. With sophisticated models, police may better understand and predict the region’s frequent gang crimes. The purpose of this paper is to model Hollenbeck’s gang rivalries. A self-exciting point process called a Hawkes process is used to model rivalries over time. While this is shown to ﬁt the data well, an agent based model is presented which is able to accurately simulate gang crimes not only temporally but also spatially. Finally, we compare random graphs generated by the agent model to existing models developed to incorporate geography into random graphs.

This is really interesting. I’ve had it in my folder since April, but didn’t get round to writing a post about it here before Peter scooped me thanks to UCLA’s hard-working press officer.

### Theory and History of Geometric Models

Irene Polo-Blanco’s PhD thesis about the history of mathematical models: physical sculptures of surfaces of the sort Felix Klein liked to make. It’s very long, but the first chapter gives a decent history. The rest is a pretty uninspiring exposition of the maths behind the models.

### Robust Soldier Crab Ball Gate

Based on the field observation of soldier crabs, we previously proposed a model for a swarm of soldier crabs. Here, we describe the interaction of coherent swarms in the simulation model, which is implemented in a logical gate. Because a swarm is generated by inherent perturbation, a swarm can be generated and maintained under highly perturbed conditions. Thus, the model reveals a robust logical gate rather than stable one. In addition, we show that the logical gate of swarms is also implemented by real soldier crabs (

Mictyris guinotae).

The title of this paper sounded very promising, and the content did not disappoint. It does exactly what it says on the tin: they’ve made a logic gate using balls of swarming soldier crabs, and it’s robust. Splendid. Not so splendidly, it’s closed access and a download from the AIP costs $28.

### Computer analysis of Sprouts with nimbers

Sprouts is a two-player topological game, invented in 1967 in the University of Cambridge by John Conway and Michael Paterson. The game starts with $p$ spots, and ends in at most $3p-1$ moves. The first player who cannot play loses. The complexity of the $p$-spot game is very high, so that the best hand-checked proof only shows who the winner is for the $7$-spot game, and the best previous computer analysis reached $p=11$. We have written a computer program, using mainly two new ideas. The nimber (also known as Sprague-Grundy number) allows us to compute separately independent subgames; and when the exploration of a part of the game tree seems to be too difficult, we can manually force the program to search elsewhere. Thanks to these improvements, we reached up to $p=32$. The outcome of the $33$-spot game is still unknown, but the biggest computed value is the $47$-spot game ! All the computed values support the Sprouts conjecture: the first player has a winning strategy if and only if $p$ is $3$, $4$ or $5$ modulo $6$. We have also used a check algorithm to reduce the number of positions needed to prove which player is the winner. It is now possible to hand-check all the games until $p=11$ in a reasonable amount of time.

I had always thought that Sprouts had been completely solved. I tried to find a reference which would persuade someone else, but I found this instead. Nimbers are a fun topic; see also this typewritten paper on multiplying games of Nim.

### Long finite sequences

Let $k$ be a positive integer. There is a longest finite sequence $x_1,\dots,x_n$ in $k$ letters in which no consecutive block $x_i,\dots,x_{2i}$ is a subsequence of any other consecutive block $x_j,\dots,x_{2j}$. Let $n(k)$ be this longest length. We prove that $n(1) = 3$, $n(2) = 11$, and $n(3)$ is incomprehensibly large. We give a lower bound for $n(3)$ in terms of the familiar Ackerman hierarchy. We also give asymptotic upper and lower bounds for $n(k)$. We view $n(3)$ as a particularly elemental description of an incomprehensibly large integer. Related problems involving binary sequences (two letters) are also addressed. We also report on some recent computer explorations of R. Dougherty which we use to raise the lower bound for $n(3)$.

First of all: this paper is set in fixed-space type and yet, inexplicably, it was published in 1998. The mind boggles.

Anyway, I like this sequence because it starts innocently enough and gets stupid very quickly.

### Calculus Made Easy

Being a very simplest introduction to

those beautiful methods of reckoning

which are generally called by the

terrifying names of the

DIFFERENTIAL CALCULUS

and the

INTEGRAL CALCULUSConsidering how many fools can calculate, it is surprising that it should be thought either a difficult or a tedious task for any other fool to learn how to master the same tricks.

Some calculus-tricks are quite easy. Some are enormously difficult. The fools who write the textbooks of advanced mathematics — and they are mostly clever fools — seldom take the trouble to show you how easy the easy calculations are. On the contrary, they seem to desire to impress you with their tremendous cleverness by going about it in the most difficult way.

Being myself a remarkably stupid fellow, I have had to unteach myself the difficulties, and now beg to present to my fellow fools the

parts that are not hard. Master these thoroughly, and the rest will follow. What one fool can do, another can.

I came across this when I noticed that Project Gutenberg were making TeX versions of old books. It’s supposed to be the one calculus book that gets it right. A hundred years on, every calculus book I’ve seen is still obtuse and long-winded and hard to understand, so maybe we should stick with this one.

### Cardinal arithmetic for skeptics

When modern set theory is applied to conventional mathematical problems, it has a disconcerting tendency to produce independence results rather than theorems in the usual sense. The resulting preoccupation with “consistency” rather than “truth” may be felt to give the subject an air of unreality. Even elementary questions about the basic arithmetical operations of exponentiation in the context of infinite cardinalities, like the value of $2^{\aleph_0}$, cannot be settled on the basis of the usual axioms of set theory (ZFC).

Although much can be said in favor of such independence results, rather than undertaking to challenge such prejudices, we have a more modest goal; we wish to point out an area of contemporary set theory in which theorems are abundant, although the conventional wisdom views the subject as dominated by independence results, namely, cardinal arithmetic.

Pretty sure I kept this just for the title.

### Survey on fusible numbers

We point out that the recursive formula that appears in Erickson’s presentation “Fusible Numbers” is incorrect, and pose an alternate conjecture about the structure of fusible numbers. Although we are unable to solve the conjecture, we succeed in establishing some basic properties of fusible numbers. We suggest some possible approaches to the conjecture, and list further problems in the final chapter.

Jeff Erickon’s fusible numbers are a fun class of fractions. However, Junyan Xu reckons he got his sums con*fused*(!)

### A categorical foundation for Bayesian probability

Given two measurable spaces $H$ and $D$ with countably generated $\sigma$-algebras, a perfect prior probability measure $P_H$ on $H$ and a sampling distribution $\mathcal{S}\colon H \rightarrow D$, there is a corresponding inference map $\mathcal{I}\colon D \rightarrow H$ which is unique up to a set of measure zero. Thus, given a data measurement $\mu\colon 1 \rightarrow D$, a posterior probability $\hat{P_H}=\mathcal{I} \circ \mu$ can be computed. This procedure is iterative: with each updated probability $P_H$, we obtain a new joint distribution which in turn yields a new inference map $\mathcal{I}$ and the process repeats with each additional measurement. The main result uses an existence theorem for regular conditional probabilities by Faden, which holds in more generality than the setting of Polish spaces. This less stringent setting then allows for non-trivial decision rules (Eilenberg–Moore algebras) on finite (as well as non finite) spaces, and also provides for a common framework for decision theory and Bayesian probability.

Category theory, yeah!

### Estimating the Effect of the Red Card in Soccer

We study the effect of the red card in a soccer game. A red card is given by a referee to signify that a player has been sent off following a serious misconduct. The player who has been sent off must leave the game immediately and cannot be replaced during the game. His team must continue the game with one player fewer. We estimate the effect of the red card from betting data on the FIFA World Cup 2006 and Euro 2008, showing that the scoring intensity of the penalized team drops significantly, while the scoring intensity of the opposing team increases slightly. We show that a red card typically leads to a smaller number of goals scored during the game when a stronger team is penalized, but it can lead to an increased number of goals when a weaker team is punished. We also show when it is better to commit a red card offense in exchange for the prevention of a goal opportunity.

Proper real football statistics.

### Lectures on lost mathematics

Notes from a series of lectures given in 1975 about “lost mathematics”, which might more properly be called “outsider maths” – discoveries by people in other disciplines, and which weren’t known by mathematicians at the time. Lots of nice, although poorly photocopied, pictures.

### On an error in the star puzzle by Henry E. Dudeney

We found a solution of the star puzzle (a path on a chessboard from c5 to d4 in 14 straight strokes) in 14 queen moves, which has been claimed by the author as impossible.

The star puzzle is very old. This dude decided he wouldn’t trust the author that a short solution was impossible, so worked one out. That’s all there is to this paper.

### To Dissect a Mockingbird: A Graphical Notation for the Lambda Calculus with Animated Reduction

The lambda calculus, and the closely related theory of combinators, are important in the foundations of mathematics, logic and computer science. This paper provides an informal and entertaining introduction by means of an animated graphical notation.

I love lambda calculus, me. I couldn’t follow this, though. I think the author’s made the classic error of picking a really weird analogy to to try to make it more accessible, and it just hasn’t worked for me. Though it says the bird terminology comes from a book by Smullyan; maybe I should read that instead.

### On distributions computable by random walks on graphs

We answer a question raised by Donald E. Knuth and Andrew C. Yao, concerning the class of polynomials on $[0,1]$ that can be realized as the distribution function of a random variable, whose binary expansion is the output of a finite state automaton driven by unbiased coin tosses. The polynomial distribution functions which can be obtained in this way are precisely those with rational coefficients, whose derivative has no irrational roots on $[0,1]$. We also show, strengthening a result of Knuth and Yao, that all smooth distribution functions which can be obtained by such automata are polynomials.

I was thinking about extending the princess on a graph puzzle to use finite state automata. This looks like a good place to start. Sadly, it’s closed access – a download costs $15 to the punter on the street.

### Equilibrium Solution to the Lowest Unique Positive Integer Game

We address the equilibrium concept of a reverse auction game so that no one can enhance the individual payoff by a unilateral change when all the others follow a certain strategy. In this approach the combinatorial possibilities to consider become very much involved even for a small number of players, which has hindered a precise analysis in previous works. We here present a systematic way to reach the solution for a general number of players, and show that this game is an example of conflict between the group and the individual interests.

I looked this up when I read here that an author was giving away copies of his book to fans who picked the lowest unique prime numbers. It’s odd that this game has an equilibrium solution, though I think the authors have assumed the players’ strategies have a certain form.