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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.

What a sad title :(

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.

Of course Persi Diaconis has thought about fair dice. (\$12, or free-ish if you haven’t read too many other JSTOR articles recently)

The mathematics of Septoku

Septoku is a Sudoku variant invented by Bruce Oberg, played on a hexagonal grid of 37 cells. We show that up to rotations, reflections, and symbol permutations, there are only six valid Septoku boards. In order to have a unique solution, we show that the minimum number of given values is six. We generalize the puzzle to other board shapes, and devise a puzzle on a star-shaped board with 73 cells with six givens which has a unique solution. We show how this puzzle relates to the unsolved Hadwiger-Nelson problem in combinatorial geometry.

The competitive Septoku-solving circuit is a non-starter.

The topology of competitively constructed graphs

We consider a simple game, the $k$-regular graph game, in which players take turns adding edges to an initially empty graph subject to the constraint that the degrees of vertices cannot exceed $k$. We show a sharp topological threshold for this game: for the case $k=3$ a player can ensure the resulting graph is planar, while for the case $k=4$, a player can force the appearance of arbitrarily large clique minors.

I love to force the appearance of arbitrarily large clique minors. It’s sort of my signature move.

Circular orbits on a warped spandex fabric

We present a theoretical and experimental analysis of circular-like orbits made by a marble rolling on a warped spandex fabric. We show that the mass of the fabric interior to the orbital path influences the motion of the marble in a nontrivial way, and can even dominate the orbital characteristics. We also compare a Kepler-like expression for such orbits to similar expressions for orbits about a spherically-symmetric massive object in the presence of a constant vacuum energy, as described by general relativity.

People sometimes try to explain hoe relativity works by rolling balls on a stretched tablecloth. They shouldn’t, though: you get the wrong shape of orbit!

The Stick Problem

Given sticks of possible sizes one through six, what is the smallest number of sticks you can have to ensure that you are able to form a perfect square? The Pigeonhole Principle tells us that if we have nineteen sticks we would have at least four of one of the sizes, but can we do better if we take partitions into account? This is one case of the stick problem which, though simple in statement, proves to be not so simple in solution. In this paper, we define the stick problem clearly, discuss our methods for approaching and simplifying the problem, provide an algorithm for generating solutions, and present some computer generated solutions for specific cases.

Not ‘a’ stick problem: the stick problem. That’s a bold terminological land-grab! Gives special thanks to Alabama Supercomputer Authority – I really liked their debut album, but their recent work is rather lacklustre.

Linear recurrences through tilings and Markov chains

We present a tiling interpretation for $k$-th order linear recurrences, which yields new combinatorial proofs for recurrence identities. Moreover, viewing the tiling process as a Markov chain also yields closed form Binet-like expressions for these recurrences

3-bonacci numbers!

Mathematical Games

Slides about mathematical games, going back to Ur and through Boethius. Includes a set of rules for Rhythmomachy.


A more readable set of rules for r(hy|i)thmomach(y|ia).

More ties than we thought

We extend the existing enumeration of neck tie knots to include tie knots with a textured front, tied with the narrow end of a tie. These tie knots have gained popularity in recent years, based on reconstructions of a costume detail from The Matrix Reloaded, and are explicitly ruled out in the enumeration by Fink and Mao (2000).
We show that the relaxed tie knot description language that comprehensively describes these extended tie knot classes is either context sensitive or context free. It has a sub-language that covers all the knots that inspired the work, and that is regular. From this regular sub-language we enumerate 177 147 distinct tie knots that seem tieable with a normal necktie. These are found through an enumeration of 2 046 winding patterns that can be varied by tucking the tie under itself at various points along the winding.

There are way more than 85 ways to tie a tie. I wrote a whole post about this.

Table for Fundamentals of Series : Part I : Basic Properties of Series and Products

It’s a big huge list of identities.

Eponymy in Mathematical Nomenclature: What’s in a Name, and What Should Be?

Makes the case (which I disagree with) that naming mathematical things after people is less helpful than describing what they do. We’re implored to “exploit the riches of the vernacular”. Because it’s immediately obvious how categories, groups, and monoids differ from each other. (\$45/£35, Springer)

A number system with an irrational base

Like, what would happen if, like, you wrote a number down in base $\pi$? Whooahhhh. (\$12, JSTOR)

On the diagram of 132-avoiding permutations

The diagram of a 132-avoiding permutation can easily be characterized: it is simply the diagram of a partition. Based on this fact, we present a new bijection between 132-avoiding and 321-avoiding permutations. We will show that this bijection translates the correspondences between these permutations and Dyck paths given by Krattenthaler and by Billey–Jockusch–Stanley, respectively, to each other. Moreover, the diagram approach yields simple proofs for some enumerative results concerning forbidden patterns in 132-avoiding permutations.

I think I just like papers with diagrams.

A Mathematical Coloring Book

Line drawings of mathematical objects. Keep a copy handy in case a tiny person visits. There are some notes on each of the pictures at the end of the book.

Useful inequalities cheat sheet

This is a collection of some of the most important mathematical inequalities. I tried to include non-trivial inequalities that can be useful in solving problems or proving theorems. I omitted many details, in some cases even necessary conditions (hopefully only when they were obvious). If you are not sure whether an inequality can be applied in some context, try to find a more detailed source for the exact definition. For lack of space I omitted proofs and discussions on when equality holds.

I didn’t include inequalities which require lengthy definitions, inequalities involving complex functions, number theory, advanced calculus (most integral inequalities) or inequalities with a pure geometric character. Some of the inequalities are special cases of others, but I tried to resist the temptation of including only the most general form (which may not be the most easily applicable).

Much like the big huge list of identities mentioned earlier, this is a big huge list of inequalities. I assume it’s only useful if you already know what they’re for and just need a reminder.

Generalizing Zeckendorf’s Theorem to f-decompositions

A beautiful theorem of Zeckendorf states that every positive integer can be uniquely decomposed as a sum of non-consecutive Fibonacci numbers $\{F_n\}$, where $F_1=1$, $F_2=2$ and $F_{n+1}=F_n+F_{n−1}$. For general recurrences $\{G_n\}$ with non-negative coefficients, there is a notion of a legal decomposition which again leads to a unique representation, and the number of summands in the representations of uniformly randomly chosen $m \in [G_n,G_{n+1})$ converges to a normal distribution as $n \to \infty$.
We consider the converse question: given a notion of legal decomposition, is it possible to construct a sequence $\{a_n\}$ such that every positive integer can be decomposed as a sum of terms from the sequence? We encode a notion of legal decomposition as a function $f:\mathbb{N}_0\to\mathbb{N}_0$ and say that if $a_n$ is in an “$f$-decomposition”, then the decomposition cannot contain the $f(n)$ terms immediately before an in the sequence; special choices of f yield many well known decompositions (including base-$b$, Zeckendorf and factorial). We prove that for any $f:\mathbb{N}_0\to\mathbb{N}_0$, there exists a sequence $\{a_n\}_{n=0}^\infty$ such that every positive integer has a unique $f$-decomposition using $\{a_n\}$. Further, if $f$ is periodic, then the unique increasing sequence $\{a_n\}$ that corresponds to $f$ satisfies a linear recurrence relation. Previous research only handled recurrence relations with no negative coefficients. We find a function $f$ that yields a sequence that cannot be described by such a recurrence relation. Finally, for a class of functions $f$, we prove that the number of summands in the $f$-decomposition of integers between two consecutive terms of the sequence converges to a normal distribution.

I love Zeckendorf’s theorem, so I love this paper too.

Nim Fractals

We enumerate P-positions in the game of Nim in two different ways. In one series of sequences we enumerate them by the maximum number of counters in a pile. In another series of sequences we enumerate them by the total number of counters.
We show that the game of Nim can be viewed as a cellular automaton, where the total number of counters divided by 2 can be considered as a generation in which P-positions are born. We prove that the three-pile Nim sequence enumerated by the total number of counters is a famous toothpick sequence based on the Ulam-Warburton cellular automaton. We introduce 10 new sequences.

Nim! Fractals! Integer sequences!

The Number-Pad Game

Problem 3 in the 1st Mathematical Olympiad of Central America and the Carribean,
held in Costa Rica in 1999 [2] was the following:

In a calculator the number keys (except 0) are arranged as shown. Player $A$ turns on the calculator, presses a digit key and then presses the $+$ key. Then a second player $B$ presses a digit key in the same row or column of the last digit key pressed by $A$, then presses the $+$ key. The games proceeds with the two players taking turns alternately. The first player who reaches a sum greater than 30 loses. Which player has a winning strategy? Describe the strategy.

LIM is not slim

In this paper LIM, a recently proposed impartial combinatorial ruleset, is analyzed. A formula to describe the $\mathcal{G}$-values of LIM positions is given, by way of analyzing an equivalent combinatorial ruleset LIM’, closely related to the classical NIM. Also, an enumeration of $\mathcal{P}$-positions of LIM with $n$ stones, and its relation to the Ulam-Warburton cellular automaton, is presented.

Not particularly readable, but it starts at nim sums and ends up at cellular automata. (Springer, \$45/£35)

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.

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