Soddy Quartets–an update

Some time ago I mentioned the apparent link between irreducible Soddy Quartets and the Ford Spheres that make up the 2-dimensional Flatrack Model. To recap, a Soddy Quartet is a set of four positive integers (a,b,c,d) that satisfy the equation

      (a+b+c+d)² = 3(a²+b²+c²+d²)

and it is said to be irreducible if these four numbers have no common factor greater than 1. As I noted previously, each such solution to this equation corresponds to exactly one group of four touching bubbles in the Flatrack model. The lines joining the largest three bubbles in such a group–those of orders a, b and c–define a triangle, within which the fourth bubble appears at a later stage (“time d”) in the model’s evolution. (For simplicity I have assumed that a, b, c and d are arranged in increasing order, since the model has a 6-way symmetry that corresponds to rearrangements of a, b and c.) Thus there is a one-to-one correspondence between irreducible Soddy Quartets and the triangles that are formed in the Flatrack model.

I’ve now taken this correspondence a step further. It turns out that any two Soddy Quartets that happen to share the same values of a, b and d, but have different c-values, correspond to two adjacent triangles in the Flatrack that together generate a Ford Sphere of order d. (And, not surprisingly, vice versa.)

For example the two quartets (4,19,25,39) and (4,19,37,39) each satisfy the above equation, and correspond to two adjacent triangles in the Flatrack: one defined by three touching bubbles of orders 4, 19 and 25; the other defined by the same bubbles of orders 4 and 19 plus another touching bubble of order 37. In other words, these two quartets correspond to two triangles that share a side in the Flatrack model. At “time” 39 this side is replaced by a new bubble of order 39 that touches all four of the preceding ones, and the two triangles that together generated this bubble are replaced by four new ones. (These four triangles correspond to the four Soddy Quartets (4,25,39,49), (4,37,39,61), (19,25,39,79) and (19,37,39,91), which can be considered as “existing” from “time” 39 until “times” 49, 61, 79 and 91 respectively.) And so the model evolves.

It seems that two Soddy Quartets can be considered “adjacent” in some sense if they have three numbers in common. I’m hoping that this can give rise to a meaningful measure of distance in the “space” formed by the solutions of this equation.

Now of course the next step is to extend this correspondence to the 3-dimensional model, so I’ve started investigating Gosset Quintets which are solutions to the equation

      (a+b+c+d+e)² = 4(a²+b²+c²+d²+e²)

and can be considered as corresponding to tetrahedrons that “exist” between “times” d and e in the Four-D Froth. However, these tetrahedrons appear to overlap or leave gaps in some places, so some refinement to the equations is needed here. Yet curiously enough as time increases, the total volume of these tetrahedrons consistently converges to just over 98% of the total volume in the model. I don’t know what the missing 2% represents, but I think I’m on a track worth pursuing.