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Interesting Esoterica Summation, volume 6

Cor, it’s been longer than I thought since I last did one of these. I’ve been happily collecting esoterica for months, thinking I didn’t have enough new stuff to do a summation. It turns out I’ve got 22 new things! Better get cracking with the interest and the summing.

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.

On sphere-filling ropes

What is the longest rope on the unit sphere? Intuition tells us that the answer to this packing problem depends on the rope’s thickness. For a countably infinite number of prescribed thickness values we construct and classify all solution curves. The simplest ones are similar to the seamlines of a tennis ball, others exhibit a striking resemblance to Turing patterns in chemistry, or to ordered phases of long elastic rods stuffed into spherical shells.

Spherical geometry makes the world go round.

The Lost Squares of Dr. Franklin: Ben Franklin’s Missing Squares and the Secret of the Magic Circle

If I mention Benjamin Franklin and mathematics in the same breath, your reaction will likely be:
1. “I didn’t know Franklin did mathematics,” or:
2. “I already know all about that subject.”

My counter-reply, as appropriate:
1. Yes, he did.
2. No, you don’t.

A lengthy piece from the American Mathematical Monthly about Benjamin Franklin’s adventures with magic squares. Closed access on JSTOR, $12.

Papy’s minicomputer

Georges Papy was gloriously batty, in that inimitably high-faluting way accessible only to French-speakers. Before I talk about the minicomputer, I have to direct you to this review of his book Groups, which is my absolute least favourite maths book, or the book I most love to hate, in New Scientist magazine. My eternal thanks to the reviewer for stepping up to the plate and smacking it out of the park with this snippet:

A final 32 plates in colour have all the discontinuity of a death-bed repentance. They leave disappointment at an opportunity wasted. Many are trivial, all are insufficiently captioned and – quite inexcusably – some exhibit colours other than those named in the accompanying text.

My thoughts exactly. Anyway, the minicomputer. Georges collaborated closely with his wife Frédérique, and she wrote this piece for the Bulletin of the Association of Teachers of Mathematics in 1970. It’s about a square piece of card which was meant to help children learn about numbers. I’m non convainçu.

Conway’s Wizards

I present and discuss a puzzle about wizards invented by John H. Conway.

Tanya Khovanova is doing an admirable job of collecting trickier puzzles and providing lucid discussion of their solutions, all on the arXiv for everyone to see. This one’s a logic puzzle involving two wizards on a bus.

Picture-hanging puzzles

We show how to hang a picture by wrapping rope around n nails, making a polynomial number of twists, such that the picture falls whenever any k out of the n nails get removed, and the picture remains hanging when fewer than k nails get removed. This construction makes for some fun mathematical magic performances. More generally, we characterize the possible Boolean functions characterizing when the picture falls in terms of which nails get removed as all monotone Boolean functions. This construction requires an exponential number of twists in the worst case, but exponential complexity is almost always necessary for general functions.

A wonderful puzzle. Good luck forgetting the definition of a commutator after working through this paper. This was so interesting that I made it the subject of our first recreational maths seminar.

A stratification of the space of all $k$-planes in $\mathbb{C}^n$

To each $k \times n$ matrix $M$ of rank $k$, we associate a juggling pattern of periodicity $n$ with $k$ balls. The juggling pattern actually only depends on the $k$-plane spanned by the rows, so gives a decomposition of the “Grassmannian” of all $k$-planes in $n$-space.
There are many connections between the geometry and the juggling. For example, the natural topology on the space of matrices induces a partial order on the space of juggling patterns, which indicates whether one pattern is “more excited” than another.
This same decomposition turns out to naturally arise from totally positive geometry [Lusztig 1994, Postnikov ∼2004], characteristic $p$ geometry [Knutson-Lam-Speyer 2011], and noncommutative geometry [Brown-Goodearl-Yakimov 2005]. It also arises by projection from the manifold of full flags in $n$-space, where there is no cyclic symmetry.

Allen Knutson has spent more time than probably anyone else thinking about the maths of juggling. These are some slides from a talk about the connection between juggling and some very abstract geometry.

How to eat 4/9 of a pizza

Given two players alternately picking pieces of a pizza sliced by radial cuts, in such a way that after the first piece is taken every subsequent chosen piece is adjacent to some previously taken piece, we provide a strategy for the starting player to get 4/9 of the pizza. This is best possible and settles a conjecture of Peter Winkler.

Biologically unavoidable sequences

A biologically unavoidable sequence is an infinite gender sequence which occurs in every gendered, infinite genealogical network satisfying certain tame conditions. We show that every eventually periodic sequence is biologically unavoidable (this generalizes Koenig’s Lemma), and we exhibit some biologically avoidable sequences. Finally we give an application of unavoidable sequences to cellular automata.

I saw the main result of this paper written very clearly somewhere, probably Google+, but I’ve forgotten it and I’m finding the paper quite hard to read.

Figures for “impossible fractals”

Take “impossible” shapes and turn them into fractals. Admire pretty pictures.

The paramagnetic and glass transitions in sudoku

We study the statistical mechanics of a model glassy system based on a familiar and popular mathematical puzzle. Sudoku puzzles provide a very rare example of a class of frustrated systems with a unique groundstate without symmetry. Here, the puzzle is recast as thermodynamic system where the number of violated rules defines the energy. We use Monte Carlo simulation to show that the “Sudoku Hamiltonian” exhibits two transitions as a function of temperature, a paramagnetic and a glass transition. Of these, the intermediate condensed phase is the only one which visits the ground state (i.e. it solves the puzzle, though this is not the purpose of the study). Both transitions are associated with an entropy change, paramagnetism measured from the dynamics of the Monte Carlo run, showing a peak in specific heat, while the residual glass entropy is determined by finding multiple instances of the glass by repeated annealing. There are relatively few such simple models for frustrated or glassy systems which exhibit both ordering and glass transitions, sudoku puzzles are unique for the ease with which they can be obtained with the proof of the existence of a unique ground state via the satisfiability of all constraints. Simulations suggest that in the glass phase there is an increase in information entropy with lowering temperature. In fact, we have shown that sudoku have the type of rugged energy landscape with multiple minima which typifies glasses in many physical systems, and this puzzling result is a manifestation of the paradox of the residual glass entropy. These readily-available puzzles can now be used as solvable model Hamiltonian systems for studying the glass transition.

Invited Commentary: The Perils of Birth Weight—A Lesson from Directed Acyclic Graphs

The strong association of birth weight with infant mortality is complicated by a paradoxical finding: Small babies in high-risk populations usually have lower risk than small babies in low-risk populations. In this issue of the Journal, Hernández-Díaz et al. (Am J Epidemiol 2006;164:1115–20) address this “birth weight paradox” using directed acyclic graphs (DAGs). They conclude that the paradox is the result of bias created by adjustment for a factor (birth weight) that is affected by the exposure of interest and at the same time shares causes with the outcome (mortality). While this bias has been discussed before, the DAGs presented by Hernández-Díaz et al. provide more firmly grounded criticism. The DAGs demonstrate (as do many other examples) that seemingly reasonable adjustments can distort epidemiologic results. In this commentary, the birth weight paradox is shown to be an illustration of Simpson’s Paradox. It is possible for a factor to be protective within every stratum of a variable and yet be damaging overall. Questions remain as to the causal role of birth weight.

Algorithmic Self-Assembly of DNA Sierpinski Triangles

Algorithms and information, fundamental to technological and biological organization, are also an essential aspect of many elementary physical phenomena, such as molecular self-assembly. Here we report the molecular realization, using two-dimensional self-assembly of DNA tiles, of a cellular automaton whose update rule computes the binary function XOR and thus fabricates a fractal pattern—a Sierpinski triangle—as it grows. To achieve this, abstract tiles were translated into DNA tiles based on double-crossover motifs. Serving as input for the computation, long single-stranded DNA molecules were used to nucleate growth of tiles into algorithmic crystals. For both of two independent molecular realizations, atomic force microscopy revealed recognizable Sierpinski triangles containing 100–200 correct tiles. Error rates during assembly appear to range from 1% to 10%. Although imperfect, the growth of Sierpinski triangles demonstrates all the necessary mechanisms for the molecular implementation of arbitrary cellular automata. This shows that engineered DNA self-assembly can be treated as a Turing-universal biomolecular system, capable of implementing any desired algorithm for computation or construction tasks.

Delay can stabilize: Love affairs dynamics

We discuss two models of interpersonal interactions with delay. The first model is linear, and allows the presentation of a rigorous mathematical analysis of stability, while the second is nonlinear and a typical local stability analysis is thus performed. The linear model is a direct extension of the classic Strogatz model. On the other hand, as interpersonal relations are nonlinear dynamical processes, the nonlinear model should better reflect real interactions. Both models involve immediate reaction on partner’s state and a correction of the reaction after some time.

The models we discuss belong to the class of two-variable systems with one delay for which appropriate delay stabilizes an unstable steady state. We formulate a theorem and prove that stabilization takes place in our case. We conclude that considerable (meaning large enough, but not too large) values of time delay involved in the model can stabilize love affairs dynamics.

Modelling romantic relationships as dynamical systems, with some resulting advice for couples going through a rough patch. Closed access, ScienceDirect, $27.95.

Incorporating Voice Permutations into the Theory of Neo-Riemannian Groups and Lewinian Duality

A familiar problem in neo-Riemannian theory is that the P, L, and R operations defined as contextual inversions on pitch-class segments do not produce parsimonious voice leading. We incorporate permutations into T/I-PLR-duality to resolve this issue and simultaneously broaden the applicability of this duality. More precisely, we construct the dual group to the permutation group acting on n-tuples with distinct entries, and prove that the dual group to permutations adjoined with a group G of invertible affine maps Z12 -> Z12 is the internal direct product of the dual to permutations and the dual to G. Musical examples include Liszt, R. W. Venezia, S. 201 and Schoenberg, String Quartet Number 1, Opus 7. We also prove that the Fiore–Noll construction of the dual group in the finite case works, and clarify the relationship of permutations with the RICH transformation.

I saw this on the arXiv’s math.GR new preprints feed and put it in my collection before I really understood what it was about. Apparently some music theorists get a bit too theoretical and start applying group theory. Richard Green wrote an explanation of what it’s all about on Google+.

A note on paradoxical metric spaces

Via a submission to the arXiv with an intriguing title – “Invariant means of the wobbling group” – I found this paper which gives a definition of a wonderful term: a wobbling bijection. According to the paper on the wobbling group, they’re used in the solution to Tarski’s circle-squaring problem (exposited accessibly in this article from Math Horizons). Furthermore, according to the abstract of the paper “Geometrical bijections in discrete lattices”, wobbling mappings occur during earthquakes! Bonanza!

Conway’s rational tangles

I learned about this “trick” in a lecture by John Conway a number of years ago. He calls it “Rational Tangles” and there is plenty of information about it on the internet. Since then I have used it myself in classrooms of students of middle school age and older. The underlying mathematics is very interesting, but it is not necessary that the students understand the mathematics to get a lot out of the trick. In fact, some of the mathematics I do not understand.

Algebra and ropes pair up for a second time today.

Markets are efficient if and only if P = NP

I prove that if markets are efficient, meaning current prices fully reflect all information available in past prices, then P = NP, meaning every computational problem whose solution can be verified in polynomial time can also be solved in polynomial time. I also prove the converse by showing how we can “program” the market to solve NP-complete problems. Since P probably does not equal NP, markets are probably not efficient. Specifically, markets become increasingly inefficient as the time series lengthens or becomes more frequent. An illustration by way of partitioning the excess returns to momentum strategies based on data availability confirms this prediction.

Economists love assuming that markets are efficient, because it lets them apply all sorts of maths to them. Well, this chap says in this maths-light, word-heavy paper that maths might prove markets to be inefficient. What nonsense.

What are some of the most ridiculous proofs in mathematics?

Respondents seem to be conflating “ridiculous” with “short” but the top answer, which uses Fermat’s last theorem to prove $\sqrt[3]{2}$ is irrational, is sublime.

Embedding countable groups in 2-generator groups

One of the postgrads I share an office with said they’d heard something about every countable group being generated by two elements. Here’s a proof. It’s such a pleasing fact about the universe.

The Muddy Children: a logic for public announcement

Public announcement logic is pretty cool. One of the very first entries in my collection is the paper “‘Knowable’ as ‘known after an announcement’” (closed access, Cambridge, £30). These slides present the idea through a very accessible problem – three kids each may or may not have mud on their faces, and a fourth kid (being wilfully obtuse) tells them that at least one of them is muddy. Is that information to tell if you have mud on your face? By repeatedly announcing whether they know if they have mud on their faces, all three kids can eventually work out if they need to make a trip to the bathroom.

Circular reasoning: who first proved that $C/d$ is a constant?

We answer the question: who first proved that $C/d$ is a constant? We argue that Archimedes proved that the ratio of the circumference of a circle to its diameter is a constant independent of the circle and that the circumference constant equals the area constant ($C/d=A/r^{2}$). He stated neither result explicitly, but both are implied by his work. His proof required the addition of two axioms beyond those in Euclid’s Elements; this was the first step toward a rigorous theory of arc length. We also discuss how Archimedes’s work coexisted with the 2000-year belief — championed by scholars from Aristotle to Descartes — that it is impossible to find the ratio of a curved line to a straight line.

Zeroless Arithmetic: representing integers ONLY using ONE

We use recurrence equations (alias difference equations) to enumerate the number of formula-representations of positive integers using only addition and multiplication, and using addition, multiplication, and exponentiation, where all the inputs are ones. We also describe efficient algorithms for the random generation of such representations, and use Dynamical Programming to find a shortest possible formula representing any given positive integer.

Doron Zeilberger and an acolyte cement their place in my bad books by denying the utility of zero. Along the way, they neglect the entirety of Peano arithmetic by coming up with some approximate solutions to problems which almost definitely have closed form answers. He’s got a nerve.

About the author

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

One Response to “Interesting Esoterica Summation, volume 6”

  1. David Roberts

    There’s probably an explanation of the results in the Gnang-Zeilberger paper using the fragment of category theory known as combinatorial species, which DZ has admitted he likes.

    Reply

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