# Interesting Esoterica Summation, volume 3

Summing up some more interesting esoterica seems like the right thing to do at the moment, so here’s that.

A reminder: 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.

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.

### Passage to the limit in Proposition I, Book I of Newton’s Principia

The purpose of this paper is to analyze the way in which Newton uses his polygon model and passes to the limit in Proposition I, Book I of his Principia. It will be evident from his method that the limit of the polygon is indeed the orbital arc of the body and that his approximation of the actual continuous force situation by a series of impulses passes correctly in the limit into the continuous centripetal force situation. The analysis of the polygon model is done in two ways: (1) using the modern concepts of force, linear momentum, linear impulse, and velocity, and (2) using Newton’s concepts of motive force and quantity of motion. It should be clearly understood that the term “force” without the adjective “motive,” is used in the modern sense, which is that force is a vector which is the time rate of change of the linear momentum. Newton did not use the word “force” in this modern sense. The symbol $$F$$ denotes modern force. For Newton “force” was “motive force,” which is measured by the change in the quantity of motion of a body. Newton’s “quantity of motion” is proportional to the magnitude of the modern vector momentum. Motive force is a scalar and the symbol $$F_m$$ is used for motive force.

A short history of maths article about how Newton approximated arcs using polygons whose boundaries tend to the actual arc as the number of sides increases. All this infinitesimal /passage to the limit stuff is very fiddly and sort of at the core of sorting out the calculus, but this article isn’t particularly interesting.

### Orange peels and Fresnel integrals

There are two standard ways of peeling an orange: either cut the skin along meridians, or cut it along a spiral. We consider here the second method, and study the shape of the spiral strip, when unfolded on a table. We derive a formula that describes the corresponding flattened-out spiral. Cutting the peel with progressively thinner strip widths, we obtain a sequence of increasingly long spirals. We show that, after rescaling, these spirals tends to a definite shape, known as the Euler spiral. The Euler spiral has applications in many fields of science. In optics, the illumination intensity at a point behind a slit is computed from the distance between two points on the Euler spiral. The Euler spiral also provides optimal curvature for train tracks between a straight run and an upcoming bend. It is striking that it can be also obtained with an orange and a kitchen knife.

I’ve always found Euler spirals, also known as Cornu curves, interesting. They’re curves with a constant change in radius of curvature, so if you were driving around one you’d turn your steering wheel at a constant rate. They first found an application in train tracks, but have a variety of applications in optics and computer graphics.

This paper is about a cool fact that the authors noticed, which is that if you peel an orange in a spiral and lay it flat, it makes an Euler spiral.

Here’s a paper with some historical information about the Euler spiral: The Euler spiral: a mathematical history.

### Further evidence for addition and numerical competence by a Grey parrot (Psittacus erithacus)

A Grey parrot (Psittacus erithacus), able to quantify sets of eight or fewer items (including heterogeneous subsets), to sum two sequentially presented sets of 0–6 items (up to 6), and to identify and serially order Arabic numerals (1–8), all by using English labels, was tested on addition of two Arabic numerals or three sequentially presented collections (e.g., of variously sized jelly beans or nuts). He was, without explicit training and in the absence of the previously viewed addends, asked, “How many total?” and required to answer with a vocal English number label. In a few trials on the Arabic numeral addition, he was also shown variously colored Arabic numerals while the addends were hidden and asked “What color number (is the) total?” Although his death precluded testing on all possible arrays, his accuracy was statistically significant and suggested addition abilities comparable with those of nonhuman primates.

The late parrot Alex was quite famous. His owners were conducting an experiment to see, first of all, how much language he could understand and use and, later, whether he could understand quantity and number. This paper describes how Alex could add up collections of objects and say the answer out loud.

### Barcodes: the persistent topology of data

This article surveys recent work of Carlsson and collaborators on applications of computational algebraic topology to problems of feature detection and shape recognition in high-dimensional data. The primary mathematical tool considered is a homology theory for point-cloud data sets–persistent homology–and a novel representation of this algebraic characterization–barcodes. We sketch an application of these techniques to the classification of natural images.

This is really fascinating. Suppose you have a cloud of points in a space with a lot of dimensions. We can only perceive two dimensions directly, so you have to project the points down onto the plane. Can you work out the shape of the points from the projection? The author compares this to seeing what a connect-the-dots drawing is of before drawing the lines.

The barcodes mentioned in the title are representations of how connected the data are as you try to join them into simplices. From these you can make a guess at the homology of the space the points come from, which associates a group with its topology. The homology is a quick way of identifying the shape of the cloud. I’m not doing a great job of explaining all this but the paper does, and it gets bonus points for a completely gratuitous figure of a Klein bottle in front of a barcode.

### Poe, E. – Near A Raven

The world’s most heroic mnemonic for $$\pi$$. The lengths of the words represent the (base-10) digits of $$\pi$$. If I ever felt the need to memorise a lot of digits of $$\pi$$, I think I’d use this. I won’t, though.

### Quotients Homophones des Groupes Libres / Homophonic Quotients of Free Groups

This paper has two titles. It has an excellent slogan – “Ah, la recherche! Du temps perdu.” It’s about what happens if you identify homophones in the free group generated by the letters of the alphabet. It has results in both French and English. I love this paper.

### Random walks reaching against all odds the other side of the quarter plane

For a homogeneous random walk in the quarter plane with nearest-neighbor transitions, starting from some state $$(i_0,j_0)$$, we study the event that the walk reaches the vertical axis, before reaching the horizontal axis. We derive an exact expression for the probability of this event, and derive an asymptotic expression for the case when $$i_0$$ becomes large, a situation in which the event becomes highly unlikely. The exact expression follows from the solution of a boundary value problem and is in terms of an integral that involves a conformal gluing function. The asymptotic expression follows from the asymptotic evaluation of this integral. Our results find applications in a model for nucleosome shifting, the voter model and the asymmetric exclusion process.

I wish I could remember why I saved this! The title is quite fun, but I have a feeling the result has an interesting application. It might be something to do with the application to the voter model they mention, but I can’t remember what.

### Undecidable Problems: A Sampler

After discussing two senses in which the notion of undecidability is used, we present a survey of undecidable decision problems arising in various branches of mathematics.

Just a good reference of undecidable problems.

### Doc, what are my chances?

When a physician has to deliver bad news to a patient, the bottom-line question is, “Doc, what are my chances?” The doctor’s answers are then usually couched in the language of probability.

Recently, both patients and insurance companies have become more questioning, because they want to understand the doctor’s reasoning. What persists is an explanation gap, the distance between the doctor’s best assessment and the patient’s comprehension of both the probability and how it was obtained.

We seek to reveal the logic behind the medical diagnostic decision making process and arm the patient with the appropriate vocabulary. Because decisions hinge on probabilities—“we’ll proceed to this treatment if we can ascertain that there is at least a 50% chance you have the disease” — we demonstrate how the numbers are determined. This approach removes the fuzziness from the process, and the required arithmetic is not intimidating when taken one step at a time. To demonstrate the method, we work a sample problem from start to ﬁnish.

We illustrate how a graphical solution called a nomogram [Doerﬂer 2009] delivers numerical results easily. Moreover, nomograms can offer advantages over other computational methods; an appendix gives a comparison.

I’ve seen a few nomograms in my various encounters with serious doctors. They’re a nice simple way of visualising how parameters affect some measurement. The authors of this paper have created a fantastic nomogram for using Bayesian reckoning to decide whether a test is worth doing. They also sell posters, which I’ve considered buying for the department’s Bayesians.

### A new approximation to $$\pi$$ (conclusion)

The March 16, 1946 edition of Nature contained a very short announcement that William Shanks (resident of Houghton-le-Spring, just down the road from where I grew up) had got his calculation of $$\pi$$ wrong after the 528th digit.

In July 1946, Mathematical Tables and Computation published a note with the digits corrected up to the 620th.

In April1947, an article titled A new approximation to $$\pi$$ gave D.F. Ferguson and Dr. John Wrench’s independent calculations up to 808 digits. That wasn’t quite the end of the story. In the article I linked to in the headline above, both men published a few corrections to Dr Wrench’s longer calculation. And that was the end of the matter! Definitely!

Several sources note that Ferguson was the first to use a mechanical desk calculator to do his computations. I’d really like to know what it was.

### Topic-based Vector Space Model

This paper motivates and presents the Topic-based Vector Space Model (TVSM), a new vector-based approach for document comparison. The approach does not assume independence between terms and it is flexible regarding the specification of term-similarities. Stopword-list, stemming and thesaurus can be fully integrated into the model. This paper shows further how the TVSM can be fully implemented within the context of relational databases. This facilitates the use of this approach by generic applications. At the end short comparisons with other vector-based approaches namely the Vector Space Model (VSM) and the Generalized Vector Space Model (GVSM) are presented.

I found this while trying to find references for a story we linked to on The Aperiodical about BBC R&D’s efforts to automatically tag transcripts of radio programmes. The paper itself isn’t enormously interesting but it’s good to know it’s been done.

### Gaussian prime spirals

Imagine a particle in the complex plane, starting at $$c_0$$, a Gaussian integer, moving initially $$\pm$$ in the horizontal or vertical directions. When it hits a Gaussian prime, it turns left $$90^{\circ}$$. Does the spiral always form a cycle?

One of that kind of MathOverflow question that makes you forget the incomprehensible self-esteem-destroying horror of the rest of the site. A nice and simply stated question about an automaton walking between the Gaussian integers, with some clever insights and lots of nice pictures.

### The bitangent sphere problem

Here is the problem: choose a point po on a (smooth, closed, embedded) surface $$M$$ in $$\mathbb{R}_3$$. Does there always exist a sphere which is tangent to $$M$$ at $$p_o$$ and at some other point $$p \in M$$?
Here is the answer: no. But the answer is very nearly yes, for there is always a sphere or plane having the required property…

A nice bit of geometry.

### Statistical Laws Governing Fluctuations in Word Use from Word Birth to Word Death

We analyze the dynamic properties of $$10^7$$ words recorded in English, Spanish and Hebrew over the period 1800–2008 in order to gain insight into the coevolution of language and culture. We report language independent patterns useful as benchmarks for theoretical models of language evolution. A significantly decreasing (increasing) trend in the birth (death) rate of words indicates a recent shift in the selection laws governing word use. For new words, we observe a peak in the growth-rate fluctuations around 40 years after introduction, consistent with the typical entry time into standard dictionaries and the human generational timescale. Pronounced changes in the dynamics of language during periods of war shows that word correlations, occurring across time and between words, are largely influenced by coevolutionary social, technological, and political factors. We quantify cultural memory by analyzing the long-term correlations in the use of individual words using detrended fluctuation analysis.

### The mate-in-n problem of infinite chess is decidable

Infinite chess is chess played on an infinite edgeless chessboard. The familiar chess pieces move about according to their usual chess rules, and each player strives to place the opposing king into checkmate. The mate-in-n problem of infinite chess is the problem of determining whether a designated player can force a win from a given finite position in at most n moves. A naive formulation of this problem leads to assertions of high arithmetic complexity with 2n alternating quantifiers—there is a move for white, such that for every black reply, there is a counter-move for white, and so on. In such a formulation, the problem does not appear to be decidable; and one cannot expect to search an infinitely branching game tree even to finite depth. Nevertheless, the main theorem of this article, confirming a conjecture of the first author and C. D. A. Evans, establishes that the mate-in-n problem of infinite chess is computably decidable, uniformly in the position and in n. Furthermore, there is a computable strategy for optimal play from such mate-in-n positions. The proof proceeds by showing that the mate-in-n problem is expressible in what we call the first-order structure of chess, which we prove (in the relevant fragment) is an automatic structure, whose theory is therefore decidable. Indeed, it is definable in Presburger arithmetic. Unfortunately, this resolution of the mate-in-n problem does not appear to settle the decidability of the more general winning-position problem, the problem of determining whether a designated player has a winning strategy from a given position, since a position may admit a winning strategy without any bound on the number of moves required. This issue is connected with transfinite game values in infinite chess, and the exact value of the omega one of chess is not known.

“Infinite chess” and “mental poker” are just wonderful things to say you’re researching. I need to think of an unsolved problem in something like “chaotic tiddlywinks”.

### Cake Cutting Mechanisms

We examine the history of cake cutting mechanisms and discuss the efficiency of their allocations. In the case of piecewise uniform preferences, we define a game that in the presence of strategic agents has equilibria that are not dominated by the allocations of any mechanism. We identify that the equilibria of this game coincide with the allocations of an existing cake cutting mechanism.

An undergrad’s dissertation giving an exhaustive survey of ways of dividing up cake between several people, who might want different amounts or pieces. I lost interest by the time I got to the game theory (nothing is more tedious!) but I’m always forgetting cake-slicing mechanisms so this will be good to have as a reference. It’s got a lot of good quotes too. Keen undergrads are what make life bearable.

### Navigating Hyperbolic Space with Fibonacci Trees

I recently found a joyful game called HyperRogue II: a roguelike played in hyperbolic space. I’ve always wanted to do something similar, so seeing that it’s possible has prompted me to look up the maths behind tiling and navigating hyperbolic space. This post sums up, in a very handwavey way, how Maurice Margenstern does it, using Fibonacci trees. Bonus points for a justifiable application of the fibonacci sequence; deeply satisfying.

It took me a while to find an accessible copy of the paper the author refers to, Cellular automata in the hyperbolic plane: proposal for a new environment.

That’s it for now!