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Interesting Esoterica Summation, volume 7

Do you ever collect too much fun maths stuff to keep to yourself, and then start a website just so you’ve got somewhere to put it? That happens to me sometimes.

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.

Problems to sharpen the young

An annotated translation of Propositiones ad acuendos juvenes, the oldest mathematical problem collection in Latin, attributed to Alcuin of York.

This came up when we talked to David Singmaster. Alcuin was a cool dude! Closed access on JSTOR, $14.

Das 2:3-Ei – ein praktikables Eimodell

Or, “The 2:3 egg – a practical egg model”. Lengthy discussion (in German) about what makes a good egg shape, and how to construct one.

Constructing the Tits ovoid from an elliptic quadratic

This came up while I was searching for information on egg construction.

A smaller sleeping bag for a baby snake

By a sleeping bag for a baby snake in $d$ dimensions we mean a subset of $R^d$ which can cover, by rotation and translation, every curve of unit length. We construct sleeping bags which are smaller than any previously known in dimensions 3 and higher. In particular, we construct a three-dimensional sleeping bag of volume approximately 0.075803. For large $d$ we construct $d$-dimensional sleeping bags with volume less than $\frac{(c \sqrt{ \log d})^d}{d^{3d/2}}$ for some constant $c$.

To obtain the last result, we show that every curve of unit length in $R^d$ lies between two parallel hyperplanes at distance at most $c_1 d^{-3/2} \sqrt{\log d}$, for some constant $c_1$.

David Cushing told us about this puzzle at Newcastle MathsJam. This paper is also available in added-value form from Springer for £29.95.

The Urinal Problem

A man walks into a men’s room and observes n empty urinals. Which urinal should he pick so as to maximize his chances of maintaining privacy, i.e., minimize the chance that someone will occupy a urinal beside him? In this paper, we attempt to answer this question under a variety of models for standard men’s room behavior. Our results suggest that for the most part one should probably choose the urinal furthest from the door (with some interesting exceptions). We also suggest a number of variations on the problem that lead to many open problems.

Good knowledge to have, and I’m glad someone’s thinking about it.

Circuitry in 3d chess

This is the second of a projected three-part series of articles, which will ultimately prove the Turing-completeness of three-dimensional chess. In the first article, I described the basic rules of the game. In this article, I shall show how to construct basic logic gates using rooks, kings and pawns. The final article, which demonstrates Turing completeness, requires a fourth piece, namely the knight.

Familial Sinistrals Avoid Exact Numbers

We report data from an internet questionnaire of sixty number trivia. Participants were asked for the number of cups in their house, the number of cities they know and 58 other quantities. We compare the answers of familial sinistrals – individuals who are left-handed themselves or have a left-handed close blood-relative – with those of pure familial dextrals – right-handed individuals who reported only having right-handed close blood-relatives. We show that familial sinistrals use rounder numbers than pure familial dextrals in the survey responses. Round numbers in the decimal system are those that are multiples of powers of 10 or of half or a quarter of a power of 10. Roundness is a gradient concept, e.g. 100 is rounder than 50 or 200. We show that very round number like 100 and 1000 are used with 25% greater likelihood by familial sinistrals than by pure familial dextrals, while pure familial dextrals are more likely to use less round numbers such as 25, 60, and 200. We then use Sigurd’s (1988, Language in Society) index of the roundness of a number and report that familial sinistrals’ responses are significantly rounder on average than those of pure familial dextrals. To explain the difference, we propose that the cognitive effort of using exact numbers is greater for the familial sinistral group because their language and number systems tend to be more distributed over both hemispheres of the brain. Our data support the view that exact and approximate quantities are processed by two separate cognitive systems. Specifically, our behavioral data corroborates the view that the evolutionarily older, approximate number system is present in both hemispheres of the brain, while the exact number system tends to be localized in only one hemisphere.

A Do-It-Yourself Paper Digital Computer, 1959.

I made one of these. It didn’t really work.

Division of labor in child care: A game-theoretic approach

This paper uses a game of repeated play to model parental child care in order to examine the gap between the expectations of egalitarian-minded couples before the transition to parenthood and the reality of parenthood, with its gendered roles. This is done first in a gender-free context in order to examine the mechanism by which the division of labor is established in a family – it is this same process through which gendered expectations have an impact. The analysis shows that it is difficult to achieve the equilibrium of equal sharing of child care, even when this is the preference of the parents. This leads to a discussion of alterations and meta-strategies for couples who want to share care equally. Gender differences between parents are also modeled, including the impact these have on outcomes and equilibria.

Cyclic twill-woven objects

Classical (or biaxial) twill is a textile weave in which the weft threads pass over and under two or more warp threads, with an offset between adjacent weft threads to give an appearance of diagonal lines. This paper introduces a theoretical framework for constructing twill-woven objects, i.e., cyclic twill-weavings on arbitrary surfaces, and it provides methods to convert polygonal meshes into twill-woven objects. It also develops a general technique to obtain exact triaxial-woven objects from an arbitrary polygonal mesh surface.

Richard Green posted about this paper on Google+. Closed access, Elsevier, $31.50.

Kindergarten Quantum Mechanics

These lecture notes survey some joint work with Samson Abramsky as it was presented by me at several conferences in the summer of 2005. It concerns `doing quantum mechanics using only pictures of lines, squares, triangles and diamonds’. This picture calculus can be seen as a very substantial extension of Dirac’s notation, and has a purely algebraic counterpart in terms of so-called Strongly Compact Closed Categories (introduced by Abramsky and I in quant-ph/0402130 and [4]) which subsumes my Logic of Entanglement quant-ph/0402014. For a survey on the `what’, the `why’ and the `hows’ I refer to a previous set of lecture notes quant-ph/0506132. In a last section we provide some pointers to the body of technical literature on the subject.

Using Monoidal Categories in the Transformational Study of Musical Time-Spans and Rhythms

Transformational musical theory has so far mainly focused on the study of groups acting on musical chords, one of the most famous example being the action of the dihedral group D24 on the set of major and minor chords. Comparatively less work has been devoted to the study of transformations of time-spans and rhythms. D. Lewin was the first to study group actions on time-spans by using a subgroup of the affine group in one dimension. In our previous work, the work of Lewin has been included in the more general framework of group extensions, and generalizations to time-spans on multiple timelines have been proposed. The goal of this paper is to show that such generalizations have a categorical background in free monodical categories generated by a group-as-category. In particular, symmetric monodical categories allow to deal with the possible interexchanges between timelines. We also show that more general time-spans can be considered, in which single time-spans are encapsulated in a “bracket” of time-spans, which allow for the description of complex rhythms.

I must look more closely into the combination of group theory and music theory at some point.

Six Ways to Sum a Series

A discussion of the sum of squares of the recipricals of the positive integers with a review of several proofs.

The Ubiquitous $\pi$

Some well-known and little-known appearances of $\pi$ in a variety of problems.

Closed access, JSTOR, $12.

Non-sexist solution of the ménage problem

The ménage problem asks for the number of ways of seating $n$ couples at a circular table, with men and women alternating, so that no one sits next to his or her partner. We present a straight-forward solution to this problem. What distinguishes our approach is that we do not seat the ladies first.

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