# 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.

(JSTOR, \$12. By the way, I find JSTOR’s policy of logging me out each time I leave the site rather annoying.) ### Mathematics and group theory in music The purpose of this paper is to show through particular examples how group theory is used in music. The examples are chosen from the theoretical work and from the compositions of Olivier Messiaen (1908-1992), one of the most influential twentieth century composers and pedagogues. Messiaen consciously used mathematical concepts derived from symmetry and groups, in his teaching and in his compositions. Before dwelling on this, I will give a quick overview of the relation between mathematics and music. This will put the discussion on symmetry and group theory in music in a broader context and it will provide the reader of this handbook some background and some motivation for the subject. The relation between mathematics and music, during more than two millennia, was lively, widespread, and extremely enriching for both domains. This paper will appear in the Handbook of Group actions, vol. II (ed. L. Ji, A. Papadopoulos and S.-T. Yau), Higher Eucation Press and International Press. I was sure I already had a paper on music and group theory in my collection, but I can’t find it. Oh well! There are a few on the arXiv, anyway. ### Foldings and meanders We review the stamp folding problem, the number of ways to fold a strip of n stamps, and the related problem of enumerating meander configurations. The study of equivalence classes of foldings and meanders under symmetries allows to characterize and enumerate folding and meander shapes. Symmetric foldings and meanders are described, and relations between folding and meandric sequences are given. Extended tables for these sequences are provided. I love a paper that provides extended tables! I will apply this paper’s findings to the stamps in my wallet. ### The shape of a Mobius Band The shape of a Mobius band made of a flexible material, such as paper, is determined. The band is represented as a bent, twisted elastic rod with a rectangular crosssection. Its mechanical equilibrium is governed by the Kirchhoff-Love equations for the large deflections of elastic rods. These are solved numerically for various values of the aspect ratio of the cross-section, and an asymptotic solution is found for large values of this ratio. The resulting shape is shown to agree well with that of a band made from a strip of plastic. Reminiscent of the classic “can one hear the shape of a drum?”, but almost entirely different. My immediate question is: can one hear the shape of a Mobius band? ### A Fresh Look at Peg Solitaire Peg solitaire is a one-person game that is over 300 years old; most people are familiar with the puzzle on the “standard 33-hole board”. When I first saw this game, what struck me was the unusual shape of the board. How was this strange cross-shaped board discovered and what is so special about it? While the history of the game is too fragmented to answer the question of the origin of this board, this paper will demonstrate that the special shape of the standard board can be derived from first principles. This board arises as a consequence of two very natural requirements: that of symmetry, and the ability to play from a board position with one peg missing to a single peg at the same location. We’ll show that in a certain well-defined sense, the shape of this board is unique. George Bell makes another appearance in this volume of the summation. Figure 9 is nice! ### Pondering an Artist’s Perplexing Tribute to the Pythagorean Theorem On the subject of a photo featured on the cover of The College Mathematics Journal, which looks like it demonstrates Pythagoras’ theorem on a 3-4-5 triangle but doesn’t. In short: everyone is wrong about everything to do with it. ### How often should you clean your room? We introduce and study a combinatorial optimization problem motivated by the question in the title. In the simple case where you use all objects in your room equally often, we investigate asymptotics of the optimal time to clean up in terms of the number of objects in your room. In particular, we prove a logarithmic upper bound, solve an approximate version of this problem, and conjecture a precise logarithmic asymptotic. I suppose this should really be “how often should you tidy your room?”, because it doesn’t talk about the accumulation of dusty nonsense on your things and furniture. Anyway, the authors reckon you should only start putting books back on a bookshelf after you’ve removed about$4 \log_2(n)\$ of them.

• #### Christian Lawson-Perfect

Mathematician, koala fan, Aperiodical editor. Usually found paddling in the North Sea, or fiddling with computers.