2011 will be a Löschian year

Friday, December 31, 2010 5:56 pm PST

I’ve been working on other things for a while, but as the year draws to a close I’ve finally found the time to do more work on my PQR Theory.  Specifically, I’ve been looking at the two-dimensional (Ford Spheres) model, which I call the Flatrack because it vaguely resembles a rack of pool balls, but with the bottom surface flattened off against the table.

The “sizes” of the spheres (by which I mean the reciprocal of their diameters) in the Flatrack are the Löschian (or Loeschian) numbers, which are numbers of the form x²+xy+y², where x and y are integers.  The first few such numbers are 1, 3, 4, 7, 9, 12 and 13.   2011 is a Löschian year, as 2011=10² + 10×39 + 39².  (The last Löschian year was 2007 and the next will be 2017.)

When one looks at a group of four Ford Spheres that touch each other in the Flatrack, their sizes are all Löschian numbers that satisfy the Soddy Equation: (a+b+c+d)² = 3(a²+b²+c²+d²).  (Note that this is a special case of the Soddy 5-sphere equation, where one of the spheres is the plane of the pool table, which is geometrically equivalent to a sphere of infinite radius, or of “size” zero.)  I call a set of four positive integers {a,b,c,d} that satisfy this equation a Soddy Quartet.

Interestingly, it seems that all Soddy Quartets consist of 4 Löschian numbers (or of 4 Löschian numbers times a common multiplier).  And every Soddy Quartet that does not have a common factor appears to be represented in the Flatrack by exactly one group of four mutually touching Ford Spheres (together with its symmetrical reflections).  Interesting!  Whatever next?

In the meantime, I would like to wish my reader(s) a Happy New Löschian Year.