https://orion.math.iastate.edu/jdhsmith/math/JS26jan4.htm
http://oas.uz.zgora.pl:7777/bib/bibwww.pdf?nIdA=14291&nIdSesji=-1
http://english.stackexchange.com/questions/234607/what-comes-after-the-ducentiquinquagintasexions
Quaternions (4-ions) Octonions (8-ions) Sedenions (16-ions) Tricenibinions / trigintabinions (32-ions) Sexageniquaternions / sexagintaquaternions (64-ions) Centeniduodetricenions / centumduodetricenions (128-ions) Duceniquinquagenisenions / ducentiquinquagintasenions (256-ions) Quingeniduodenions / quingentiduodenions (512-ions) Miliaviceniquaternions / millevigintiquaternions (1024-ions) Binamiliaduodequinquagenions / duomiliaduodequinquagenions (2048-ions) Quaternamilianonagenisenions / quattuormilianonagintasenions (4096-ions)
https://ece.uwaterloo.ca/~dwharder/C++/CQOST/
Cayley-Dickson Construction First, we define the reals R where a ∈ R implies that a* = a. Given an algebra A of diminsion n, we create, using the Cayley-Dickson construction, an algebra of dimension 2n by taking pairs (a, b) ∈ A × A and thus define the standard operations: 1 = (1, 0) −(a, b) = (−a, −b) (a, b)* = (a*, −b) (a, b) + (c, d) = (a + c, b + d) (a, b)(c, d) = (ac − d*b, da + bc*)
In order to maintain a standard behaviour, a variation of the Cayley-Dickson construction shown in wikipedia is used for calculating products in higher dimensions. The justification is that it would be nice if the original quaternion identity ijk = -1 holds — the variation on wikipedia defines ijk = 1.
https://en.wikipedia.org/wiki/Octonion
https://upload.wikimedia.org/wikipedia/commons/2/2d/FanoPlane.svg
