More Hermeneutics

Originally drafted on Synergeo, an eGroup, and therefore populated with references to other discussants -- Editor.

4D

Cliff is in the vicinity and that always reminds me of our shared interest in outfitting Synergetics with some Mathematica style claptrap (caltrop? -- look it up, Braley-sans-w knows), except in my case it's Pythonic harkening back to APLish (given my personal trajectory), more than Wolframic in its computerized expression [*], but that's not a problem for me if it ain't for Cliffie here.

Other names for the Quadrays namespace might be: Chakovians (after David Chako of Synergetics-L fame); Tetrays (Chako's preferred term); Caltrop Coordinates (vs. Cartesian); Simplicial Coordinates (but minus J.H. Conway's focus on the hypercross and Coxeter.4D) -- and maybe Klingon Coordinates in honor of the other big Conway (Damian).

What we do is shoot four "basis vectors" (pirating from linear algebra) to the corners (1,0,0,0) (0,1,0,0) (0,0,1,0) and (0,0,0,1). To reverse a basis vector is to flip its bits i.e. -(1,0,0,0) = (0,1,1,1) and so on. So we end up not needing negative signs in this canonical representation, though by another convention we might alternatively insist that the sum of the 4-tuples always be zero.

The Cartesians claim having 4 basis vectors is redundant, as they get by with just 3, but then they permit vector reversal to play a role in netting them 7/8ths of their space. Thanks to vector reversal and the three negative "not really" basis vectors, the Cartesians have 3 additional "ghost vectors" doing most of their dirty work i.e. pointing in that 7/8ths negatively tainted space, with only one octant (+++) remaining "positively pure" and therefore directly accessible without all the "bad neighborhood" connotations of a "non-basic" (e.g. +-+) address.

So in point of fact, the Cartesians are using essentially six reference vectors to the corners of a regular octahedron, not just three, whereas in Chakovians we're using just four, and not relying on vector reversal for back-handed access to anything.

3D

The four positive basis rays positively scale and vector-add to address any point in Positive Universe. Negative Universe is through-the-origin inside-outing of the reference tetrahedron. Your application may have no need for this other space, conjoined through (0,0,0,0), but it's there if you need it. Nor does it matter which you call Positive initially, though once you've invested, conversion may be time-consuming (just like both left and right XYZ coordinate systems have their followings).

My Synergetics on the Web @ grunch.net contains a thorough overview of Chakovians (./synergetics/quadrays.html), plus transforms for converting to-from XYZ, 4x4 rotation matrices courtesy of Tom Ace.

We don't insist on using these in public schools, e.g. won't be targeting any school boards for not, but we do bring kids through on field trips from time to time, just to remind 'em that XYZ thinking ain't all its cracked up to be in some circles.

Alternatives exist, some of them smart cookie.

Kirby

[*] e.g. in some versions of coords.py, imported by rbf.py for doing StrangeAttractors, rendering other concentric hierarchy graphics in the 4D++ IVM.FM (lots at my grunch.net and 4dsolutions.net websites).