The following discussion may be of interest on account of the way in which it leads to the axial and focal properties.

I. Let the locus of a point P such that A(CDEP) = B(CDEP), where A, B, C, D, E are fixed points, be called the conic (AB, CDE). Then it is clear that permuting A, B or C, D, E does not affect the conic, and that the conic passes through C, D, E, and also through A, B. (Give BP the four positions BC, BD, BE, BA and then give AP the position AB.)

Figures 1–4 set the scene, with configurations of tangent circles that may perhaps be new. The labels are curvatures: the curvature or bend being the reciprocal of the radius. By convention, a circle that surrounds another has negative bend and radius. In the first part of this paper I give some easy formulae for configurations like these (see Note a). I show how we can develop fractal structures—I call them froths—having an infinity of tangent circles in each ‘triangular’ region; and give methods for finding integral froths. The second part extends these ideas to systems of tangent spheres, which may also be integral. Throughout, an integral object is one whose bend is an integer, circles and spheres are generally referred to by their bends, and examples are marked with bullets (•).

The way this problem is stated tempts one to think of it as an isolated puzzle. But there are a number of reasons why the problem may be worth investigating more closely. Perhaps you found it so mind-boggling that you couldn’t do it at all. Or you may have succeeded in finding one such list, but could not find a convincing proof that your list was the only possible solution. In either case it would have been worth having a go at a simpler version of the same problem first. But even if you solved the puzzle completely, the mathematician in you should be mildly inquisitive as to why it asks for a list of eight whole numbers, rather than say seven, or nine, or even six hundred and thirty four.

The polar equation to any curve can be written r=f(θ).

If P and Q are inverse points with respect to a circle whose centre O is at the pole and whose radius is k, then, since OP OQ = k2, it follows that the polar equation to the inverse curve is k2/r=f(θ).

It is proposed to find by inversion some properties of the limaçon and the cardioid that follow from certain well-known properties of the conies : others will suggest themselves. For convenience the same letter is used to denote both a point on the original figure and the corresponding point on the figure obtained by inversion.

BROWSING for amusement in a book that I never saw at school, Casey's Sequel to Euclid, I was introduced to something new to me. It was my first encounter with harmonic polygons, and it was interesting to trace how the painstaking geometers built up a series of propositions applicable to such figures. What annoyed me, however, was the attempt to derive some of their properties by “inversion” of a regular polygon. Again it seemed illogical to build up a proof of general properties by proving a succession of mensurations. This feeling was intensified when it came to deal with the “Brocard” properties and I started to build up the theory from a new angle.