The classic Geomag bars are approximately 27mm long and the new ones are 58mm,

the balls remain 12.7mm diameter.

Rounding slightly: that's 1" for old bars, 2-1/4" for new bars, and 1/2" balls.

For polyhedron building, what matters is the "distance between centers" which is,

therefore, 1-1/2" for classic and 2-3/4" for kids.

The ratio between these two is 11:6 (1.83) which isn't close to any

useful ratio I can think of.

Two ratios that would have obvious uses are: the cube diagonal (1.73) and the

golden ratio (1.61).

I happen to have a few 3/4" balls and see that for this size of ball the ratio is 1.71.

Here's a pentagram with 3/4" balls:

See that the edges aren't quite straight. Also, at the sharp points the balls don't quite seat correctly because the rods interfere with each other.