Robert de Marrais says in his paper math.GM/0011260: "... We know that surprises will keep on coming at least up to the 2^8-ions, due to the 8-cyclical structure of all Clifford Algebras. And we know at least a little about what such surprises will entail: as can already be seen in low-dimensioned Clifford Algebras including non-real units which are square roots of positive one, numbers whose squares and even higher-order powers are 0 will appear. The 8-cycle implies an iterable, hence ever-compoundable pattern, implying in its turn cranking out of numbers which are 2^Nth roots of 0, approaching as a limit-case an analog of the "Argand diagram" whose infinitude of roots form a "loop" of some sort. If we can work with this it all, it could only be by having as backdrop some sort of geometrical environment with an infinite number of symmetries . . . suggesting the "loop" resides on some sort of negative-curvature surface. For the incomparably stable soliton waves which are deployed within such negatively curved arenas also are just about the only concrete wave-forms which meet the "infinite symmetries" (usually interpreted as "infinite number of conservation laws") requirement. ...".
I agree with Robert de Marrais's view of periodicity of the 2^N-ions, and it seems likely to me that the 2^8-ions might be regarded as a basic building block of number theory and group theory, just as I see the 2^8-dim Clifford algebra Cl(8) as a basic building block of physics in the D4-D5-E6-E7-E8 VoDou Physics model.
I conjecture further that these links might be usable to establish a relationship between the Riemann zeta function and quantum theory.
...since for 16-ions and larger you have interesting zero-divisor "sleeper-cell" substructures, could they be useful with respect to computational systems, perhaps doing things like forming loops that might let the computational system "adjust itself" and/or "teach itself"?
Robert de Marrais commented: "what is truly interesting is this: zero-divisor systems are, ironically, PRESERVERS of associative order! Specifically, each of the four "sails" on a box-kite can be represented (on an isomorphic box-kite diagram, in fact!) as a system of four interconnected Quaternion copies: write each vertex as a pairing of one uppercase and one lowercase letter (with the 'generator' of the given 2^n-ions being the divider of the two: e.g., with the Sedenions, g = the index-8 imaginary, and the pure Sedenions of index > 8 are "uppercase," with the Octonions thereby being written with "lowercase" letters)... each sail can be seen as an ensemble of 5 Quaternion copies (the 4 associative triplets each are completed by the real unit, and the "sterile" zero-divisor-free triplet of generator, strut constant, and their XOR makes 5). Viewing things in closest-packing-pattern style, we have 5 interacting "unit quaternion" algebras -- with the interactions entailing (1, u), where 'u' is the shared non-real unit. Interestingly, this gives a nice way to think about the Tibetan Book of the Dead's "58 angry demons and 42 happy Buddhas," 100 in all = 5 * 16 + 2*10 = 100 distinct units in the interlinked 5-fold "unit quaternion" ensemble. So one first sees the "42 Assessors," then zooms in one one of the 7 isomorphic box-kites (which, as with all isomorphies, can be seen as identical at some higher level); then, one zooms in further on the "second box-kite" which has its struts defined by upper vs. lower case letters, and the triple zigzag analog being the "all lowercase" sail..."