Platonic Toys
1st August, 2009
…in which I showcase the talents of my wife.
A mathematician friend of ours recently produced offspring. Mrs Elwes created a set of welcome-to-the-world presents:
Next time, it’s the Catalan solids!
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Dürer, rhinos, and snowflakes
10th April, 2009
I’ve been a fan of the 16th century German artist Albrecht Dürer ever since I saw his print of a rhinoceros. It was put together from descriptions and someone else’s sketch: he never saw a real one. I love the scaly reptilian legs poking out beneath the sheet-metal armour, the shoulder horn, and the serrated hind-quarters. 
Dürer was also a master of religious art to rival any of his Italian contemporaries. But it’s only recently that I’ve learned of his interest in mathematics[1].
In fact it was over at the Walking Randomly blog, in a post about pentaflakes: snowflake-like fractal constructions built from pentagons (check out the link for pictures). It was Dürer who first discovered them, in the second volume of his work Underweysung der Messung (’Instruction in measurement’) in 1525 (almost 400 years before the discovery of the Koch snowflake).
(On the subject of beautiful snowflakes, please have a look at Kenneth Libbrecht’s stunning photographs of some real ones [via The Filter].)
In the spirit of my recent post, Dürer’s 1538 revision of the Underweysung is also significant for his thoughts on polyhedra. This is the first known use of nets to analyse these shapes. Here, he can also claim discovery of two of the Archimedean solids: the truncated cuboctahedron and the snub cube.
Another solid associated to Dürer is the so-called Melancholy Octahedron from his allegorical engraving Melancholia I:
Schreiber (1999) identifies it as a cube, first distorted to give rhombus faces with angles of 108° and then truncated so that its vertices lie on a sphere.
Also depicted in that picture is Europe’s first magic square:
As well as the rows, columns, diagonals, quadrants, corners and other significant 4-tuples all summing to 34, the bottom row also serves as a signature: the date 1514 is positioned inside the numbers 4 and 1: alphanumeric code for D and A.
It’s a delightful trick. But Dürer’s influence on mathematics goes deeper. He contributed to the theory of ruler and compass constructions, and studied a variety of algebraic curves in some depth, including an account of logarithmic spirals a hundred years before Descartes or Bernoulli.
The ultimate fusion of his artistic and mathematical interests came in his work on perspective, or more generally the problems of accurately representing 3-dimensional objects on a 2-dimensional space: so-called descriptive geometry. This is a fundamental question for artists, architects, and astronomers, as well as mathematicians, and its first systematic study is generally attributed to Gaspard Monge, almost 300 years later.
In short, Dürer was not an artist toying with mathematics, but a genuine polymath, whose broad interests and talents led to inspiring achievements in both art, and science.
Albrecht Dürer, self-portrait
[1] He even has his own MacTutor biography, which is where I got most of the information for this post.
Reference:
Schreiber, P. A New Hypothesis on Dürer’s Enigmatic Polyhedron in His Copper Engraving ‘Melancholia I.’, Historia Math. 26, 369-377, 1999.
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Talking knot-sense
10th April, 2009
A couple of weeks ago Matt Daws and I gave a masterclass on knot theory as part of this year’s Leeds Festival of Science [pdf].
Many thanks go to Ruth Holland and Hazel Kendrick for organising us, and Dave Pauksztello for turning up and helping out.
First we spent some time playing with knotted electric wires, investigating when two knots are equivalent (i.e one can be pulled into the shape of the other, or more technically they are ambient isotopic), and when they’re not.
Here’s a motivating example, which begins to suggest that this is a tougher question than you might first imagine.
Then we cranked up the science, working through the writhe, Kauffman’s bracket polynomial, and ultimately the Jones polynomial for a selection of knots and links.
Matt created some hand-outs which he’s put online:
- Some knots: PDF file and LaTeX source.
- The writhe: PDF file and LaTeX source.
- Kauffman’s bracket polynomial: PDF file and LaTeX source.
- The Jones polynomial: PDF file and LaTeX source.
(As he says, we did tweak the definition of the Jones polynomial, omitting the final change of variable.)
Finally, here’s my lo-tech contribution which works through calculations of the bracket polynomial for some knots and links.
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Passing Platonic Solids
20th February, 2009
I hope everyone knows that the only regular convex solids are the tetrahedron, cube, octahedron, dodecahedron, and icosahedron: the Platonic solids.
These are the only 3-dimensional solid shapes built from straight lines and flat faces which are are face-transitive (all the faces look the same), edge-transitive (all the edges look the same), and vertex-transitive (all the corners look the same), as well as being convex: if you pick any two points in the shape and join them with a straight line, then that line segment is lies totally inside the shape. (So a sphere is convex, but a curvy banana isn’t, because a straight line between its two ends passes outside the fruit.)
To generalise, we can loosen some of these requirements. 
Firstly, we can simply drop the requirement of convexity. In this case we add in four new Kepler-Poinsot polyhedra (we also have to tolerate self-intersection here - some of the faces crash through each other at ‘false edges’, and we are in the realm of polyhedra rather than solids).
If we demand convexity, edge-transitivity and vertex-transitivity, but we allow the faces to be different, then we additionally win the cuboctohedron and the icosidodecahedron.
Now dropping convexity, we obtain the great icosidodecahedron, the dodecadodecahedron, and three ditirigonal variants.
If we only insist on convexity and vertex-transitivity and allow edges and faces to be different, then we get the thirteen[1] Archimedean solids (of course including the cuboctohedron and the icosidodecahedron), plus two infinite families: the prisms and antiprisms ….that is at least if we restrict to convex vertex-transitive polyhedra with regular polygonal faces. Otherwise there are millions more, not yet fully classified.
If we additionally sacrifice convexity, so we’re looking for vertex-transitive polyhedra with regular polygonal faces, then we arrive at the 76[2] uniform polyhedra, plus infinite families of prisms and (crossed) antiprisms
What else could we do? Well, we could let vertex-transitivity slide, and instead demand face-transitivity, edge-transitivity, and convexity. At this stage we’ll have to allow faces which are not regular polygons. But it’s worth it to win the rhombic dodecahedron and the rhombic triacontahedron.
If we only care about face-transitivity and convexity (for example if we’re looking for the shapes that will make fair dice), we arrive at the thirteen[3] sumptuous Catalan solids (including the deltoidal hexecontahedron pictured above, with its sixty kite (or deltoid) shaped faces), as well as two infinite families: the bipyramids and trapezohedra.
What if we demanded only edge-transitivity but allowed faces and vertices to differ? Then we’d get nothing new! Any edge-transitive solid is also either vertex-transitive or face-transitive.[4]
However there are polyhedra which are vertex-transitive and face-transitive, but not edge-transitive. They are known as noble and are a little mysterious [pdf]. The only convex examples apart from the Platonic solids, are disphenoids: tetrahedra whose triangular faces are not equilateral. However there are non-convex examples too, such as the toroidal crown polyhedra. Again these shapes are not yet fully classified.
Another way to generalise from the Platonic solids is to forget about transitivity, and get back to regular polygons. What other convex polyhedra are there whose faces are regular polygons? The answer comes out as the the ninety-two Johnson solids (with a few near misses).
…and if we forget about convexity now? There are infinitely many possibilities, obtained by gluing Johnson solids together. For example, if you start with an octahedron and glue a tetrahedron on each face, you get a stella octangula. This process can be repeated indefinitely. What’s more, we lose simple-connectedness again, and discover more solids with holes in them: the Stewart toroids appear, along with higher genus[5] variants.
Still not satisfied? Let’s generalise to higher dimensions. Meet the pentachoron (pictured[6]), tesseract, hexadecachoron, icositetrachoron, hecatonicosachoron, and the hexacosichoron: the 4-dimensional analogues of the Platonic solids.
Now we can start all over again! Did someone say ‘non-convex’? The 4-dimensional analogues of the Kepler-Poinsot polyhedra are the ten Schläfli-Hess polychora. And here are the sixty-four convex uniform polychora, plus two infinite families of prismatic forms.
But if we go up to 5 dimensions and more, we’re in for a shock. We only ever find three regular polytopes: an n-simplex, an n-hypercube, and an n-orthoplex, the analogues of the tetrahedron, cube, and octahedron respectively. The others vanish. Similarly, there are no non-convex regular polytopes at all. This is one example of life in 3 and 4 dimensions being more complicated than in higher dimensions. (This is why low dimensional topology gets its own subject.)
What’s the point of all this? Firstly it’s to demonstrate that Wikipedia truly is an excellent resource for mathematics these days (although it has its limits). Secondly it’s to illustrate the interdependence of funky maths (in this case mad shapes) with seemingly dry maths (notions such as vertex-transitivity). Without all the hard thinking, we’d never attain full funkiness. More precisely, it’s to give an example of what mathematicians mean when they say ‘generalise’: when you look at a Platonic solid, there are many valid observations you could make. You can pull any of these criteria out, consider them on their own, and then see what else you can find that fits the bill: and that could be the first step on a very long, and very rich journey.
[1] Or fifteen if you count the left- and right-handed versions of the snub cube and snub dodecahedron separately.
[2] 75 if you discount Skilling’s figure where some pairs of edges coincide.
[3] Fifteen if you count the left- and right-handed versions of the pentagonal icositetrahedron and pentagonal hexecontahedron separately.
[4] Imagine two edges meeting at a vertex and bounding a face. There are two different ways those edges could be matched up. What happens to the vertices and faces in those two cases?
[5] More holes.
[6] If you click on it it rotates. Animation created by Jason Hise, taken from Wikipedia.
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Sketches of topology
8th February, 2009
This post, for example, perfectly illustrates the notion of blowing up a point on a manifold.
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The sine of the beast
18th December, 2008
Two things (one interesting, one very silly) that I have recently learnt about the sine function:
1. The sine rule, as most school-students know, says that in any triangle, 
is constant, whichever side A and opposite angle a you pick.
Less well-known is that this quantity actually has a geometric meaning: it gives the diameter of the triangle’s circumscribing circle. The Math Less Travelled has pictures.
2.
This may seem an innocuous enough fact. But prepare to be amazed! If you convert 
into degrees, you get 666o, which is of course the the Number of the Beast. And you might spot the ubiquitous Golden Ratio 
lurking on the other side of the equation. So now our formula becomes:
The standard numerological-satanic interpretation of this fact, I am reliably informed, is that the Devil is the opposite of God (hence the minus sign), and only half as powerful. Where exactly the trigonometry fits in, I’m not sure…
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Save the LMS?
10th December, 2008
The London Mathematical Society is a small but important institution which operates out of De Morgan House in London. It publishes a few (very high quality) books and journals, organises and supports conferences and symposia, and has small grants to give out for mathematical activities. The LMS also bestows highly regarded prizes and medals for mathematical research. Its focus is research into pure maths.
There is a firm plan to merge the LMS with the Institute of Mathematics and its Applications (IMA), no doubt a fine institution, but one about which I must plead ignorance. Its focus is applied - or applicable - areas of maths.
There have been meetings around the country to discuss the proposed New Unified Mathematical Society, with the Presidents of both societies present (but me absent).
I can’t say that I have weighed the arguments carefully myself. But certainly several mathematicians are deeply concerned about this plan. If you have a view, you can follow the debate at their Save the LMS blog.
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Knots and algorithms
28th November, 2008
The best-known techniques for telling knots apart are with knot invariants. These are algebraic objects (e.g polynomials) associated to knots. If two knots have different invariants, then you know they really are different knots rather than different configurations of the same knot.
But there is an alternative algorithmic approach to this question.
In 1970, the German-born mathematician Wolfgang Haken, at the University of Illinois, tackled the question of telling when two knots are the same: his tactic was to turn the whole problem inside out. Instead of comparing two knots floating in space, he looked at the knots’ complements: the 3-dimensional shapes that are left when you remove the knots from the surrounding matter, leaving knot-shaped holes. (Imagine setting the loosely knotted loops in blocks of glass, and then removing the strings.) If he could tell when these two objects could be deformed one into the other, then the same would go for the knots.
He set to work on a method which would take the two knot-complements, dissect them in stages, and eventually decide whether or knot they were the same. He made a great deal of progress, but Haken’s algorithm was left with holes in it when he moved onto other concerns (most notably in 1976 with Kenneth Appel he proved the famous Four Colour Theorem). However other people picked up the algorithm, notably Sergei Matveev at Chelyabinsk State University in Russia who filled in the final gap in 2003.
So in theory, mathematicians now have the foolproof[1] method for distinguishing knots that they longed for. A tremendous achievement though this is, it may be too cumbersome ever to be fully implemented on a computer in the real-world. So other, less powerful but more practical algorithms are used in current knot tabulation efforts.
Another reason for mathematicians’ reticence is that the Haken algorithm doesn’t leave any fingerprints. In theory, it can provide a yes/no answer to the question of whether two specific knots are the same. But it can’t identify or describe individual knots: algebraic invariants are needed for that.
For the specific problem of recognising configurations of the unkot, more manageable algorithms have been found. But with these too, it is an open question whether they can be made to run fast enough (i.e in Polynomial Time) to be of widespread practical use in the real world: this is the so-called unkotting problem.
[1] perhaps “watertight” would be more accurate here - not a lot of mathematics is foolproof
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Gorgeous Möbius
14th November, 2008
A short film about Möbius transformations, by Douglas Arnold and Jonathan Rogness. The music is Schumann’s “Of Foreign Lands and Peoples”, played by Donald Betts.
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Noting knots
30th October, 2008
The very first catalogue of different types of knots dates from 1876, and was the brainchild of the Scottish physicist Peter Guthrie Tait. In fact, Tait believed he was constructing the periodic table of the elements: together with his friend and fellow physicist William Thomson (later Lord Kelvin), he had developed a theory of physics in which atoms were knotted vortices in an all-pervading aether. According to that hypothesis, to tell chemical elements apart, you had to tell knots apart.
The theory of vortex-atoms was short-lived. But Tait’s labour of love, classifying different knots according to their number of crossings, continues to this day.
By 1877, Tait had single-handedly got as far as knots with 7 crossings. He was joined in his project by the Reverend Thomas Kirkman, a mathematical vicar from Lancashire in the UK, and Charles Little of the University of Nebraska. Working individually and together, they largely managed to classify knots with 8, 9, and 10 crossings, and made inroads into those with 11. You can read some of their original papers here.
Efforts continued throughout the 20th century, aided by mathematical and technological progress. The development of tangle theory by John Conway was one important ingredient.
In 1998, Hoste, Thistlethwaite, and Weeks announced a classification up to 16 crossings, amounting to 1,701,936 distinct knots.
Although no complete classification is yet known for knots or links (i.e ‘knots’ which involve more than one piece of string) with more than 16 crossings, important subfamilies have been analysed: namely prime, alternating links.
The work of Flint and Rankin has focused on these families, and their recent results show that…
“In total, there are 98,517,495,461 prime alternating links of crossing size at most 23, and there are 417,377,448,058 prime alternating links of 24 crossings.”
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