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Interesting Esoterica Summation, volume 5

Whoah there, traveller! Time for a break. Unhitch your wagon from the locomotive of Progress and roll into the railway siding of Idle Curiosity. I’ve got some more interesting esoterica for you.

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.

The usefulness of useless knowledge

A rather long essay from a 1939 issue of Harper’s Bazaar extolling the virtues of apparently pointless scientific inquiry. People still feel the need to make this kind of argument today; they needn’t bother – this one’s decent and most of the examples are the same ones you’d pick today.

A New Rose : The First Simple Symmetric 11-Venn Diagram

A symmetric Venn diagram is one that is invariant under rotation, up to a relabeling of curves. A simple Venn diagram is one in which at most two curves intersect at any point. In this paper we introduce a new property of Venn diagrams called crosscut symmetry, which is related to dihedral symmetry. Utilizing a computer search restricted to crosscut symmetry we found many simple symmetric Venn diagrams with 11 curves. This answers an existence question that has been open since the 1960’s. The first such diagram that was discovered is shown here.

I wrote about this in the News section a while ago, but it belongs here in the Interesting Esoterica collection.

The Fastest and Shortest Algorithm for All Well-Defined Problems

An algorithm $M$ is described that solves any well-defined problem $p$ as quickly as the fastest algorithm computing a solution to $p$, save for a factor of 5 and low-order additive terms. $M$ optimally distributes resources between the execution of provably correct $p$-solving programs and an enumeration of all proofs, including relevant proofs of program correctness and of time bounds on program runtimes. $M$ avoids Blum’s speed-up theorem by ignoring programs without correctness proof. $M$ has broader applicability and can be faster than Levin’s universal search, the fastest method for inverting functions save for a large multiplicative constant. An extension of Kolmogorov complexity and two novel natural measures of function complexity are used to show that the most efficient program computing some function $f$ is also among the shortest programs provably computing $f$.

Top titling here: some wonderfully sweep-of-the-arm terms, with a barely noticeable get-out adjective hidden in the middle. The author tackles the oldest problem in algorithm design — can I write an algorithm to do write all my other algorithms for me? — but amazingly answers in the positive. The trick is that he only looks at “well-defined” problems, “well-defined” meaning, of course, the kind of thing tractable by his approach. It’s still a pretty fantastic result, anyway, because the world of computable, well-defined problems is still pretty enormous. Just don’t expect it to have an application.

The Canonical Basis of $\dot{\mathbf{U}}$ for Type $A_{2}$

The modified quantized enveloping algebra has a remarkable basis, called the canonical basis, which was introduced by Lusztig. In this paper, all these monomial elements of the canonical basis for type $A_{2}$ are determined and we also give a conjecture about all polynomial elements of the canonical basis.

I’m not interested in the content of this paper as much as I am in its presentation. Richard Green pointed out that Theorem 2 starts on page 4 and ends on page 8, and consists solely of 66 lines of mind-numbingly similar-looking algebraic expressions which might as well be gibberish. Some times these kinds of papers have to be written, though I’d rather they weren’t.

Mastermind is NP-Complete

In this paper we show that the Mastermind Satisfiability Problem (MSP) is NP-complete. The Mastermind is a popular game which can be turned into a logical puzzle called Mastermind Satisfiability Problem in a similar spirit to the Minesweeper puzzle. By proving that MSP is NP-complete, we reveal its intrinsic computational property that makes it challenging and interesting. This serves as an addition to our knowledge about a host of other puzzles, such as Minesweeper, Mah-Jongg, and the 15-puzzle.

Yeah yeah, another problem is NP-Complete. Add it to the pile and move on.

VIP-club phenomenon: emergence of elites and masterminds in social networks

Hubs, or vertices with large degrees, play massive roles in, for example, epidemic dynamics, innovation diffusion, and synchronization on networks. However, costs of owning edges can motivate agents to decrease their degrees and avoid becoming hubs, whereas they would somehow like to keep access to a major part of the network. By analyzing a model and tennis players’ partnership networks, we show that combination of vertex fitness and homophily yields a VIP club made of elite vertices that are influential but not easily accessed from the majority. Intentionally formed VIP members can even serve as masterminds, which manipulate hubs to control the entire network without exposing themselves to a large mass. From conventional viewpoints based on network topology and edge direction, elites are not distinguished from many other vertices. Understanding network data is far from sufficient; individualistic factors greatly affect network structure and functions per se.

How far can Tarzan jump?

The tree-based rope swing is a popular recreation facility, often installed in outdoor areas, giving pleasure to thrill-seekers. In the setting, one drops down from a high platform, hanging from a rope, then swings at a great speed like “Tarzan”, and finally jumps ahead to land on the ground. The question now arises: How far can Tarzan jump by the swing? In this article, I present an introductory analysis of the Tarzan swing mechanics, a big pendulum-like swing with Tarzan himself attached as weight. The analysis enables determination of how farther forward Tarzan can jump using a given swing apparatus. The discussion is based on elementary mechanics and, therefore, expected to provide rich opportunities for investigations using analytic and numerical methods.

This was covered quite widely in the popular press. It’s in the arXiv’s Popular Physics section. A nice one to show to a keen child or under-educated adult.

A Hamiltonian circuit for Rubik’s Cube

At last, the Hamiltonian circuit problem for Rubik’s Cube has a solution! To be a little more mathematically precise, a Hamiltonian circuit of the quarter-turn metric Cayley graph for the Rubik’s Cube group has been found.

Basically it is a sequence of quarter-turn moves that would (in theory) put a Rubik’s cube through all of its 43,252,003,274,489,856,000 positions without repeating any of them, and then one more move restores the cube to the starting position. Note that if we have any legally scrambled Rubik’s Cube position as the starting point, then applying the sequence would result in the cube being solved at some point within the sequence.

I was playing with my cube one day, and I wondered if a Hamiltonian path exists. Turns out it does! If I was a well-regarded artist I would set up a robot programmed to follow this sequence of moves and install it amongst some ironic placards about the Rat Race and The Futility of Being and so on.

How Java’s Floating-Point Hurts Everyone Everywhere

Java’s floating-point arithmetic is blighted by five gratuitous mistakes:
1. Linguistically legislated exact reproducibility is at best mere wishful thinking.
2. Of two traditional policies for mixed precision evaluation, Java chose the worse.
3. Infinities and NaNs unleashed without the protection of floating-point traps and flags
mandated by IEEE Standards 754/854 belie Java’s claim to robustness.
4. Every programmer’s prospects for success are diminished by Java’s refusal to grant access
to capabilities built into over 95% of today’s floating-point hardware.
5. Java has rejected even mildly disciplined infix operator overloading, without which extensions
to arithmetic with everyday mathematical types like complex numbers, intervals, matrices,
geometrical objects and arbitrarily high precision become extremely inconvenient.
To leave these mistakes uncorrected would be a tragic sixth mistake.

The dude responsible for the international standard on floating point numbers explains at length, and with barely contained fury, why Java’s implementation is wrong even though it doesn’t need to be. In summary: dude shakes fist at world still using cheese and a spring even after he invented the perfect mousetrap. This is definitely computer science and not maths, but most people will encounter floating point weirdness if they ever do any numerical computing.

Similarly, another of Kahan’s papers jumped out at me because of its title: Beastly Numbers.

Modified Pascal Triangle and Pascal Surfaces

 We present a way of modifying the known Pascal Triangle and some regularities of the suggested model. We extend this model continuously to what we define as Pascal Surfaces and discuss some properties of the new object.

The thought occurred to me that, much like the $\Gamma$ function generalises factorials to a continuous domain, it should be possible to make a continuous version of Pascal’s triangle that fills in the gaps between the integer points. This paper was the only one I could find to do that. I’m a bit worried that I can’t find a peer-reviewed version, and that it was written by an engineering graduate student. I’d be very interested to hear from anyone who has a better reference.

Magic: The Gathering is Turing-complete

Yeah yeah, another system is Turing-complete. Add it to the pile and move on.

I’m only collecting these for completeness’s sake now. It’ll take a really mad application to rekindle my interest.

Online Dating Recommender Systems: The Split-complex Number Approach

A typical recommender setting is based on two kinds of relations: similarity between users (or between objects) and the taste of users towards certain objects. In environments such as online dating websites, these two relations are difficult to separate, as the users can be similar to each other, but also have preferences towards other users, i.e., rate other users. In this paper, we present a novel and unified way to model this duality of the relations by using split-complex numbers, a number system related to the complex numbers that is used in mathematics, physics and other fields. We show that this unified representation is capable of modeling both notions of relations between users in a joint expression and apply it for recommending potential partners. In experiments with the Czech dating website we show that our modeling approach leads to an improvement over baseline recommendation methods in this scenario.

Application of an esoteric bit of maths to an unexpected subject. Excellent!

Algebraic theory of Penrose’s non-periodic tilings of the plane

I had some fun making a bit of a Wieringa roof from bits of paper at the last Newcastle MathsJam. I tried to find some more information on the Wieringa roof but didn’t have much luck. Finally, I found this superb exposition by de Bruijn, which says on page 49 pretty much everything that needs to be said to convince you it works. I’m not sure if this paper is the origin of the phrase “Wieringa roof”. It looks like it might be.

Twin towers of Hanoi

In the Twin Towers of Hanoi version of the well known Towers of Hanoi Problem there are two coupled sets of pegs. In each move, one chooses a pair of pegs in one of the sets and performs the only possible legal transfer of a disk between the chosen pegs (the smallest disk from one of the pegs is moved to the other peg), but also, simultaneously, between the corresponding pair of pegs in the coupled set (thus the same sequence of moves is always used in both sets). We provide upper and lower bounds on the length of the optimal solutions to problems of the following type. Given an initial and a final position of N disks in each of the coupled sets, what is the smallest number of moves needed to simultaneously obtain the final position from the initial one in each set? Our analysis is based on the use of a group, called Hanoi Towers group, of rooted ternary tree automorphisms, which models the original problem in such a way that the configurations on N disks are the vertices at level N of the tree and the action of the generators of the group represents the three possible moves between the three pegs. The twin version of the problem is analyzed by considering the action of Hanoi Towers group on pairs of vertices.

I love the towers of Hanoi. It sits right at my IQ grade. This paper applies some group theory to a doubled-up version of the game. I want to have a proper read of this and try it out myself later.

The topology of the minimal regular cover of the Archimedean tessellations

In this article we determine, for an infinite family of maps on the plane, the topology of the surface on which the minimal regular covering occurs. This infinite family includes all Archimedean maps.

I think this is a pretty important result in geometric topology, but this paper was brought to my attention because on page 2 it has a picture of the Loch Ness monster.

The Loch Ness monster was named by honorary Aperiodicalist Étienne Ghys. Dude is too cool!

Earliest Uses of Symbols of Calculus

In the midst of a discussion/argument about the best notation to use for derivatives, I found this page. I’m not sure how scholarly rigorous it is, but he cites Cajori so it can’t be all lies.

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