The author could instead have used only abstract mathematical language (like "a configuration space with a basis of cardinality 4" and "parameter s defined by s

The author could instead have used only abstract mathematical language (like "a configuration space with a basis of cardinality 4" and "parameter s defined by s = ..."), but this would be accessible to fewer readers. If you prefer dryer language, read the second half of the article, or the other resources it links to. Using a configuration space that is a good fit for a specific system is not unusual in math or physics. It's the same thing that astrophysicists do when they work in orbital elements, or that engineers and programmers do when they make calculations in Fourier space. It's worth understanding these concepts even if you're not a physicist. Configuration spaces are so fundamental and useful that they are introduced in the very first lecture on classical mechanics in Leornard Susskind's excellent series of Stanford physics courses: http://theoreticalminimum.com/courses http://theoreticalminimum.com/courses (Neal Stephenson's novel Anathem also has a simple introduction to configuration spaces in an appendix, because that's the sort of thing that ends up in a Neal Stephenson novel.)