13. PORISMS

Introduction  

The word ‘porism’ has an old archaic meaning of ‘corollary’ which does not concern us here. It also has a much more interesting geometrical meaning which is easy to understand, but rather difficult to define. An example of such a definition is:

Porism: A proposition that uncovers the possibility of finding such conditions as to make a specific problem capable of innumerable solutions.

I defy anyone to make sense of that definition! An alternative that I grew up with is:

Porism: A situation where there are either no possible solutions or an infinite number.

This probably makes about as much sense as the first! The idea is best seen by means of example. We shall give several examples. The first of these porisms is due to the geometer Jakob Steiner – we have come across him before. It makes an ideal link with the last chapter, as it is easily proved using inversion.

Jakob Steiner

“He is a middle-aged man, of pretty stout proportions, has a long intellectual face, with beard and moustache and a fine prominent forehead, hair dark rather inclining to turn grey. The first thing that strikes you on his face is a dash of care and anxiety, almost pain, as if arising from physical suffering – he has rheumatism. He never prepares his lectures beforehand. He thus often stumbles or fails to prove what he wishes at the moment, and at every such failure he is sure to make some characteristic remark.” – Thomas Hirst

                                                                                                                                                                                                                                                                                                            


































Steiner’s Porism

We are given two fixed circles S and T, one contained within the other.

Steiner’s Porism: Either there exists a closed chain of circles tangent to circles S, T as well as to their immediate neighbors in the chain, or such a chain does not exist. In the former case, the chain can be started at an arbitrary position between the two circles S and T.

Case 1: No closed chain exists 
Case 2: Closed chain exists (I)
Case 3: Closed chain exists (II)

Proof of Steiner’s Porism

Circles S and T determine a coaxial system with a limit point X say. Inverting the figure using an inversion with centre X, S and T map to concentric circles , and the circles in the chain map to a chain of circles touching these. It is now immediately clear that for these fixed concentric circles the closure or non-closure of the chain of circles is independent of the choice of position of the initial circle in the chain.

                                                                                                                                                                         

If S and T belong to a system of coaxial circles with limiting points X and Y, then there is an orthogonal system of circles passing through X and Y. Under inversion in a circle with centre X, the orthogonal system maps to a pencil of straight lines through X', the inverse of X. Now S, T, and other circles of the original coaxial system map to a system of circles orthogonal to the lines of this pencil: that is, the system of concentric circles with centre X'.

































Poncelet’s Porism

Let S and T be two fixed circles, one contained within the other. We draw a polygonal sequence of lines with vertices on the larger circle and lines tangent to the inner circle. The sequence may close as shown to form a polygon (triangle here), or not close. Poncelet’s Porism states:

Poncelet’s Porism: If the polygonal sequence closes (alternatively, does not close) for one starting point on the outer circle, it closes (does not close) for all such points.

The illustration for closing shows a triangle, but the porism remains true for any n-sided polygon.

We notice that the previous pretty proof no longer works here. We can certainly invert circles S and T into concentric circles with common centre X, but the edges of the polygonal path map to circles through X which is not helpful. The actual proof of the Poncelet Porism is beyond us here.
Jean-Victor Poncelet

The French mathematician and engineer Jean-Victor Poncelet (pronounced Pon' sell ay) lived from 1788 to 1867. He did much to popularize projective geometry, and took a particular interest in the design of turbines and waterwheels. A French unit of power, the poncelet, was named after him.

                                                                                                                                                                     
































P

Midpoint Porism

The porisms of Steiner and Poncelet are the best known and most widely documented, but there are others.

Suppose we have a circle S and P a point on it. If starting with P as a vertex, we inscribe a triangle in S, then the three midpoints of the edges of this triangle determine a second inner circle T. We now have:

Midpoint Porism: From any point P on circle S we can inscribe in S a triangle which has the midpoints of the edges lying on the circle T.

It seems likely that this result can be extended to general n-gons.

A letter from Steve Gray which led to this porism can be found at:

http://mathforum.org/kb/message.jspa?messageID=1093830&tstart=0

A Java applet can be found at: http://www.angelfire.com/mn3/anisohedral/porism.html

It is interesting that this letter is dated 2002. This strongly suggests that there are many more mathematical discoveries to be made, including new porisms!
                                                                                                                                                                                                                                      































P

Analysis

We commented at the beginning of this chapter about the difficulty of actually defining a porism. A porism seems to involve either a moving point or set, or an infinite set of objects with some defining property.

Let’s play a little game. The figure at right illustrates the well known geometric fact that points P on the perpendicular bisector of segment AB are equidistant from the two endpoints. Suppose we rephrase this:

If point P determines a line p perpendicular to segment AB and is such that PA = PB, then for all points P on p, PA = PB. Further, if P does not lie on p, then PA PB.

Now, is this (rather trivial) statement a porism or not? In structure it does not appear very different from some of our previously stated porisms. If it does count aa a porism, we can probably rewrite just about every theorem in geometry as a porism!