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:

Po´rism
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."