Quadratic Number Fields August 4th, 2025
Patrick Stein

I want to explore arrangements of balls in n-dimensions. In particular, I am interested in exploring kissing numbers in five or more dimensions. I want to ensure that if I come up with a configuration which improves upon the known lower-bounds that I have a precise specification of where to place the kissing balls.

If I place a unit ball at the origin, all of the other unit balls that touch it have centers two units away from the origin. So, I need a set of length two vectors such that for any pair of those vectors, the dot product is at most two (the dot product is a \cdot b = |a||b| \cos\theta and with \cos\theta less than 1/2 then unit spheres at those locations would overlap each other). If I keep (n-1) of the coordinates rational, the n-th coordinate will be the square root of a rational number.

So, I want to do math involving rational numbers and square roots of rational numbers without subjecting myself to floating-point errors.

I created a library using GENERIC-CL to do this math. The library is called RATIONAL-EXTENSIONS (though I am wondering if I should call it QUADRATIC-NUMBER-FIELDS instead). You can find it here: https://github.com/nklein/rational-extensions

This library allows one do things like:

(/ (1+ (sqrt 5)) 2)  => 1/2 + 1/2·√5  ;; instead of 1.618034
(* (+ 1 (sqrt 2/3)) (- 1 (sqrt 2/3)))  => 1/3   ;; instead of 0.33333328<br>

With this, I have started working on application to allow one to explore kissing spheres in n-dimensional space. Here is an early screenshot with 13 5-dimensional balls kissing a central ball. Five of those, I calculated by hand as corners of a regular 5-simplex of side-length 2 with one vertex at the origin. The other 8 balls, I placed by rotated around in the (x,y,z) view on the left and/or the (z,w,v) view on the right and dropping a ball.

More when that app is more full-featured and less clunky.

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