In RobinHankin/onion: Octonions and Quaternions

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Overview

The onion package provides functionality for working with quaternions and octonions in R. A detailed vignette is provided in the package.

Informally, the quaternions, usually denoted $\mathbb{H}$, are a generalization of the complex numbers represented as a four-dimensional vector space over the reals. An arbitrary quaternion $q$ represented as

$$ q=a + b\mathbf{i} + c\mathbf{j}+ d\mathbf{k} $$

where $a,b,c,d\in\mathbb{R}$ and $\mathbf{i},\mathbf{j},\mathbf{k}$ are the quaternion units linked by the equations

$$ \mathbf{i}^2= \mathbf{j}^2= \mathbf{k}^2= \mathbf{i}\mathbf{j}\mathbf{k}=-1.$$

which, together with distributivity, define quaternion multiplication. We can see that the quaternions are not commutative, for while $\mathbf{i}\mathbf{j}=\mathbf{k}$, it is easy to show that $\mathbf{j}\mathbf{i}=-\mathbf{k}$. Quaternion multiplication is, however, associative (the proof is messy and long).

Defining

$$ \left( a+b\mathbf{i} + c\mathbf{j}+ d\mathbf{k}\right)^{-1}= \frac{1}{a^2 + b^2 + c^2 + d^2} \left(a-b\mathbf{i} - c\mathbf{j}- d\mathbf{k}\right) $$

shows that the quaternions are a division algebra: division works as expected (although one has to be careful about ordering terms).

The octonions $\mathbb{O}$ are essentially a pair of quaternions, with a general octonion written

$$a+b\mathbf{i}+c\mathbf{j}+d\mathbf{k}+e\mathbf{l}+f\mathbf{il}+g\mathbf{jl}+h\mathbf{kl}$$

(other notations are sometimes used); Baez gives a multiplication table for the unit octonions and together with distributivity we have a well-defined division algebra. However, octonion multiplication is not associative and we have $x(yz)\neq (xy)z$ in general.

Installation

You can install the released version of onion from CRAN with:

# install.packages("onion")  # uncomment this to install the package
library("onion")

The `onion` package in use

The basic quaternions are denoted H1, Hi, Hj and Hk and these should behave as expected in R idiom:

a <- 1:9 + Hi -2*Hj
a
a*Hk
Hk*a

Function rquat() generates random quaternions:

set.seed(0)

a <- rquat(9)
names(a) <- letters[1:9]
a
a[6] <- 33
a
cumsum(a)

Octonions

Octonions follow the same general pattern and we may show nonassociativity numerically:

x <- roct(5)
y <- roct(5)
z <- roct(5)
x*(y*z) - (x*y)*z

References

RKS Hankin (2006). "Normed division algebras with R: introducing the onion package". R News, 6(2):49-52
JC Baez (2001). "The octonions". Bulletin of the American Mathematical Society, 39(5), 145--205