# Interesting Esoterica Summation, volume 9

Oof! It’s been nearly a year since I last shared my findings in the field of interesting esoterica. I fear this may be quite a long post.

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.

### Lone Axes in Outer Space

Handel and Mosher define the axis bundle for a fully irreducible outer automorphism in “Axes in Outer Space.” In this paper we give a necessary and sufficient condition for the axis bundle to consist of a unique (geodesic) axis. As a consequence, we give a setting, and means for identifying in this setting, when two elements of an outer automorphism group $Out(F_r)$ are conjugate.

### Analyse algébrique d’un scrutin

I must have found this while Paul was writing his article about Eurovision voting. This terse, typewritten French essay goes way, way overboard in analysing methods of picking a winner from a set of judges’ rankings. Even if you can’t read French, there are some nice diagrams, in particular a net of the permutahedron on page 20.

### A knowledge-based approach of connect-four

Subtitle – “The Game is Solved: White Wins”

A Shannon C-type strategy program, VICTOR, is written for Connect-Four, based on nine strategic rules. Each of these rules is proven to be correct, implying that conclusions made by VICTOR are correct.

Using VICTOR, strategic rules where found which can be used by Black to at least draw the game, on each $7 \times (2n)$ board, provided that White does not start at the middle column, as well as on any $6 \times (2n)$ board.

In combination with conspiracy-number search, search tables and depth-first search, VICTOR was able to show that White can win on the standard $7 \times 6$ board. Using a database of approximately half a million positions, VICTOR can play real time against opponents on the $7 \times 6$ board, always winning with White.

Forget those guys who tried not to use any knowledge!

### WHAT IS Lehmer’s Number?

Lehmer’s number $\lambda \approx 1.17628$ is the largest real root of the polynomial

$f_\lambda(x) = x^{10} + x^9 – x^7 – x^6 -x^5 – x^4 – x^3 + x +1.$

2 pages; explains what Lehmer’s Number is.

### Solving Differential Equations by Symmetry Groups

Apparently step 1 is “make a guess”.

### Fair but irregular polyhedral dice

A MathOverflow question about whether an irregular $n$-sided polyhedron can work as a fair $n$-sided die. The accepted answer is a by the late Bill Thurston says it might be possible, by appealing to Brouwer’s fixed point theorem.

### Fair dice

Dice are usually cubes of a homogeneous material. Symmetry suggests that a homogeneous cube has the same chance of landing on each of its six faces after a vigorous roll, so it is said to be fair. Similarly the four other regular solids – the tetrahedron, octahedron, dodecahedron and icosahedron-are fair. Are there any other fair polyhedra?

To answer this question we must first define what we mean by fair. We shall say that a convex polyhedron is fair by symmetry if and only if it is symmetric with respect to all its faces. This means that any face can be transformed into any other face by a rotation, a reflection, or a combined rotation and reflection, which takes the polyhedron into itself. The collection of all these transformations of a given polyhedron is called its symmetry group. The fact that some transformation in the group takes any given face into any other given face is expressed by saying that the group acts transitively on the faces. Thus we can say that a convex polyhedron is fair by symmetry if and only if its symmetry group acts transitively on its faces.

### History-dependent random processes

Ulam has defined a history-dependent random sequence by the recursion $X_{n+1}=X_n+X_{U(n)}$, where $(U(n); n \geq 1)$ is a sequence of independent random variables with $U(n)$ uniformly distributed on $\{1,\dots, n\}$ and $X_1=1$. We introduce a new class of continuous-time history-dependent random processes regulated by Poisson processes. The simplest of these, a univariate process regulated by a homogeneous Poisson process, replicates in continuous time the essential properties of Ulam’s sequence, and greatly facilitates its analysis. We consider several generalizations and extensions of this, including bivariate and multivariate coupled history-dependent processes, and cases when the dependence on the past is not uniform. The analysis of the discrete-time formulations of these models would be at the very least an extremely formidable project, but we determine the asymptotic growth rates of their means and higher moments with relative ease.

What happens if you start the Fibonacci sequence, but pick a random previous term to add to the last one?

### An arctic circle theorem for groves

In earlier work, Jockusch, Propp, and Shor proved a theorem describing the limiting shape of the boundary between the uniformly tiled corners of a random tiling of an Aztec diamond and the more unpredictable `temperate zone’ in the interior of the region. The so-called arctic circle theorem made precise a phenomenon observed in random tilings of large Aztec diamonds.
Here we examine a related combinatorial model called groves. Created by Carroll and Speyer as combinatorial interpretations for Laurent polynomials given by the cube recurrence, groves have observable frozen regions which we describe precisely via asymptotic analysis of a generating function. Our approach also provides another way to prove the arctic circle theorem for Aztec diamonds.

This paper gets in for its title, but the maths is interesting too. Aztec diamonds are fun!

### The nesting and roosting habits of the laddered parenthesis

We refer to \begin{array}{} &&&&a \\ &&& .\\ && . \\ & a \\ a \end{array}

A rumination on the number of ways of parenthesising a stupid expression. There’s a nice page of diagrams in the middle, which would go nicely on a poster or a t-shirt.