# 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.

### 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.