Monday, February 20, 2006

Does your ball roll at normal speed?

Flying in the face of recent efforts to redefine the scientist stereotype, described at cosmic variance, entropy bound and inkycircus*, comes the latest rebuttle from no less thanJose Mourinho, who manages to make his feelings known during an interview about Chelsea's pitch condition prior to their Champion's League fixture with Barcelona:
'Sometimes you see beautiful people with no brains. Sometimes you have ugly people who are intelligent, like scientists,' he said.

'Our pitch is a bit like that. From the top it's a disgrace but the ball rolls at normal speed.'
*By the way this is a great science news blog that I only just cottoned onto, which I heartily recommended to you all, if you haven't been there already.

Monday, February 06, 2006

Poncelet's Porism

A week ago last Friday John Silvester from KCL's very own maths department gave us a very entertaining geometry colloquium under the esoteric title "Pendulums, Pencils, and the Poristic Polygons of Poncelet".

John began with a couple of anecdotes. Having thanked the audience for his invitation to speak, he told us a story about an unnamed mathematician who was invited to talk on a BBC radio show and was told that the fee would be £100. The mathematician thought about this and then asked if they would prefer a cheque or cash. A second anecdote concerned a London Mathematical Society president who offered a cash prize to the speaker who gave the first talk in which he did not fall asleep, and then duly claimed the prize himself when he next gave a talk.

John's talk was on the subject of Poncelet's Porism so he showed us a wonderfully stern picture of Jean-Victor Poncelet who fought in the 1812 Napoleonic attack on Moscow, was imprisoned and there turned his mind to projective geometry (I wonder how many of those presently incarcerated at her majesty's pleasure on these shores are also making strident breakthroughs in mathematics). He also has a unit of power named after him in France. John then turned his attention to the word "porism" - what does it mean? Well according to the the free dictionary it has two meanings:
n. 1. (Geom.) A proposition affirming the possibility of finding such conditions as will render a certain determinate problem indeterminate or capable of innumerable solutions.
2. (Gr. Geom.) A corollary.
Poncelet's porism refers to the first case. So what is Poncelet's porism? Take two conic sections, now if you can draw one n-sided polygon (n>2) such that its sides all touch tangentially one conic section and its vertices all lie on the other one, then you can draw infinitely many. The infinite comes about because you are able to rotate the polygon (not held fixed though) so that its vertices all rotate around the outer curve. Phew. Let's look at some pictures and see if this is understandable:
  • a triangle (sort of) with vertices on a parabola and circumscribing a circle
  • a quadrilateral with vertices on a circle and sides tangent to a parabola
  • the classic triangle and two circles
  • There are plenty more of these animations to be found here.

    John restricted our attention to the case of the triangle and the two circles. If you want to play with this set up yourself and convince yourself it really does work then there are some very nice interactive animations here (move the inner circles until the eyes open wide and then rotate the vertex on the outer circle) and here (move the pink line back and forth to change the inner circle's radius). This last link will be useful for describing John's talk since it includes a button for showing diagonals. The diagonals (for the triangle) are the lines that connect a vertex on the outer circle to the point where the polygon touches the inner circle opposite it. Push the button and see this. John was wondering about a line in the rather detailed page from mathworld concerning the porism. In particular, John was not convinced by the following line:
    "For an even-sided polygon, the diagonals are concurrent at the limiting point of the two circles, whereas for an odd-sided polygon, the lines connecting the vertices to the opposite points of tangency are concurrent at the limiting point."
    If you click on the aforementioned link showing the diagonals and move the pink line about I think you can see even there that it is not clear that the meeting point of the diagonals stays fixed. John demonstrated to us his expertise with both matlab and an excellent program called the Geometer's Sketchpad. Using matlab John took us through several pages of enormous calculations working out the locus of the meeting points of the diagonals and finally reduced the locus down to a sixth-order polynomial! Using the sketchpad John was able to convince us that the meeting point actually travels around a circle looped on top of itself three times.

    John showed us much more, including the relation of three swinging pendula to Poncelet's porism (stagger the starts of three identical pendual and then draw straight lines between their bobs, these lines are tangent to a circle...) as well as the Encyclopedia of Triangle Centres (ETC but not et cetera) where one can look up famous triangles! The talk concluded with another example of gentle humour that had pervaded, with John borrowing the phrase of the late radio four presenter John Ebdon, "if you have been, thanks for listening."

    Dark Matters

    A quick pointer to the dark matter article that's on the BBC site at the moment as well as to the articles in The Guardian, The Telegraph, The Independent, Nature and New Scientist. The story concerns the findings of the team lead by Professor Gerry Gilmore at Cambridge who, by making use of the very large telescope array, constructed 3D maps of distant "dwarf galaxies" and have inferred from their motions certain properties of dark matter. Some rather exciting things too, via the BBC:
    "The distribution of dark matter bears no relationship to anything you will have read in the literature up to now," explained Professor Gilmore.

    "It comes in a 'magic volume' which happens to correspond to an amount which is 30 million times the mass of the Sun.

    "It looks like you cannot ever pack it smaller than about 300 parsecs - 1,000 light-years; this stuff will not let you. That tells you a speed actually - about 9km/s - at which the dark matter particles are moving because they are moving too fast to be compressed into a smaller scale.

    "These are the first properties other than existence that we've been able determine."
    The BBC article notes that the research findings have yet to be submitted to a journal so hold your horses...a little.

    Updates: Courtesy of Andrew Jaffe who has a link to the preprint The internal kinematics of dwarf spheroidal galaxies and to some discussion at Dynamics of Cats.

    Thursday, February 02, 2006


    A quick couple of links to two new articles about maths/physics blogging that both, coincidentally, came out this month. First Craig Laughton, of Gooseania fame, wrote an article about mathematics blogs for Mathematics Today; it's available online, so you can read it here. Second, physicsweb have a blogging editorial and a blogging article from Physics World available online.

    I guess blog is the word of the month. Maybe the word "blog" is the blogosphere's equivalent of the word "smurf" for the smurfs. Perhaps in a few years all blog articles will be written using variations of "blog" as verbs and will be entirely about blogging. Blogging hell! I'm going to blog off now, and so on...