Dark Matter Madness

17th December, 2009

There have been wild rumours flying around the web about an exciting finding at the Cryogenic Dark Matter Search.

Dark matter is a (still!) hypothetical invisible substance, which accounts for the universe behaving as if it is 5 times heavier than can be accounted for by the total visible matter. There are various searches underway for dark particles, but by definition they are hard to see, barely interacting with other particles. However they must have mass, and therefore cannot be completely imperceptible. They have been dubbed WIMPS: weakly interacting massive particles.

CDMS works by detecting heat from collisions between particles passing through discs of germanium and silicon, frozen to near absolute zero. Then they have to discount the collisions by particles we already know about.

So have they found a WIMP? Or have they just narrowed down the size that any such particle may have? This would be scientifically valuable, but less spectacular.

All should become clear at 10pm GMT today, in Jodi Cooley’s webcast. (If you’re watching it and the answer doesn’t become clear, then the answer is probably the latter.)

UPDATE: there is 77% chance that they have found WIMPs!

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The Frivolous Theorem of Arithmetic

7th December, 2009

It’s a new one on me, but I like it:


Almost all natural numbers are very, very, very large.

We could extend it by saying that for any n, almost all natural numbers are veryn large. But that might not be so frivolous.

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Platonic Toys

1st August, 2009

…in which I showcase the talents of my wife.

[Post updated: see here for a photo of some Platonic solid toys made by my wife (along with some babies, also made by her).

Next time, it’s the Catalan solids!

Categories: Art, Maths | Comments (2) | Permalink

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. durer_-_rhinoceros

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:
durer_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

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.

Categories: Art, Maths | Comments (8) | Permalink

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:

(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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A Carnival Atmosphere

1st March, 2009

carnival-of-mathematics

The Carnival of Mathematics is a fortnightly round-up of maths blogging, which has just reached its 50th incarnation. It’s a travelling show, and is currently docked at The Endeavour. Have a look, there’s some great stuff there (and a bit by me).

I will try to link to updates even when I’m not featured. But you can always find details here.

There’s also a brand new maths teachers at play carnival, focused more on school level mathematics and teaching ideas.

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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. deltoidal hexecontahedron

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.

5-cell 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.

Categories: Maths, Topology | Comments (3) | Permalink

Happy Birthday Charlie

12th February, 2009

It’s 200 years since the origin of Charles Darwin!

Without his (and admittedly others’) imagination we would still have no idea why narwhal have horns, or what happens when dung beetles lose their taste for faeces.

In fact, we would be wholly ignorant about our planet, and how the hell we got to be on it. We would die younger too [programme 5].

So here’s to you, Mr D.

Congratulations also go out today to Abraham Lincoln for making it to 200.

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Sketches of topology

8th February, 2009

Whenever I’ve met low-dimensional topologists I’ve been dazzled by the effortless way these experts can mentally manipulate the subtlest of geometric configurations. Twisting and pulling manifolds about, and sewing in projective spaces all over the place is simply the air that they breathe.
Anyone who aspires to this world, or would simply like to gaze at it in wonder, can’t do better than to browse the archives of Sketches of topology where various topological constructions have been beautifully rendered by the former conservative cabinet minister Kenneth Baker.

This post, for example, perfectly illustrates the notion of blowing up a point on a manifold.

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