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:

1. Yes, he did.
2. No, you don’t.

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.