Quaternions and octonions in R

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
#>    [1] [2] [3] [4] [5] [6] [7] [8] [9]
#> Re   1   2   3   4   5   6   7   8   9
#> i    1   1   1   1   1   1   1   1   1
#> j   -2  -2  -2  -2  -2  -2  -2  -2  -2
#> k    0   0   0   0   0   0   0   0   0
a*Hk
#>    [1] [2] [3] [4] [5] [6] [7] [8] [9]
#> Re   0   0   0   0   0   0   0   0   0
#> i   -2  -2  -2  -2  -2  -2  -2  -2  -2
#> j   -1  -1  -1  -1  -1  -1  -1  -1  -1
#> k    1   2   3   4   5   6   7   8   9
Hk*a
#>    [1] [2] [3] [4] [5] [6] [7] [8] [9]
#> Re   0   0   0   0   0   0   0   0   0
#> i    2   2   2   2   2   2   2   2   2
#> j    1   1   1   1   1   1   1   1   1
#> k    1   2   3   4   5   6   7   8   9

Function rquat() generates random quaternions:

a <- rquat(9)
names(a) <- letters[1:9]
a
#>             a          b            c          d          e          f
#> Re  1.2629543  0.4146414 -0.005767173 -1.1476570  0.2522234 -0.2242679
#> i  -0.3262334 -1.5399500  2.404653389 -0.2894616 -0.8919211  0.3773956
#> j   1.3297993 -0.9285670  0.763593461 -0.2992151  0.4356833  0.1333364
#> k   1.2724293 -0.2947204 -0.799009249 -0.4115108 -1.2375384  0.8041895
#>              g           h          i
#> Re -0.05710677 -1.28459935 -0.4333103
#> i   0.50360797  0.04672617 -0.6494716
#> j   1.08576936 -0.23570656  0.7267507
#> k  -0.69095384 -0.54288826  1.1519118
a[6] <- 33
a
#>             a          b            c          d          e  f           g
#> Re  1.2629543  0.4146414 -0.005767173 -1.1476570  0.2522234 33 -0.05710677
#> i  -0.3262334 -1.5399500  2.404653389 -0.2894616 -0.8919211  0  0.50360797
#> j   1.3297993 -0.9285670  0.763593461 -0.2992151  0.4356833  0  1.08576936
#> k   1.2724293 -0.2947204 -0.799009249 -0.4115108 -1.2375384  0 -0.69095384
#>              h          i
#> Re -1.28459935 -0.4333103
#> i   0.04672617 -0.6494716
#> j  -0.23570656  0.7267507
#> k  -0.54288826  1.1519118
cumsum(a)
#>             a          b         c          d          e          f          g
#> Re  1.2629543  1.6775957 1.6718285  0.5241715  0.7763950 33.7763950 33.7192882
#> i  -0.3262334 -1.8661834 0.5384700  0.2490084 -0.6429127 -0.6429127 -0.1393047
#> j   1.3297993  0.4012322 1.1648257  0.8656106  1.3012939  1.3012939  2.3870632
#> k   1.2724293  0.9777089 0.1786996 -0.2328112 -1.4703496 -1.4703496 -2.1613035
#>              h          i
#> Re 32.43468886 32.0013785
#> i  -0.09257857 -0.7420502
#> j   2.15135668  2.8781074
#> k  -2.70419172 -1.5522800

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
#>          [1]           [2]           [3]           [4]           [5]
#> Re  0.000000 -5.329071e-15 -1.776357e-15 -8.881784e-16  8.881784e-16
#> i   7.201225  1.045435e+00 -3.015861e+00 -4.261327e+00  8.612680e+00
#> j   6.177845 -5.797569e+00 -5.642415e+00 -6.342342e+00  1.118819e+01
#> k  -4.917863 -4.484153e+00 -1.591524e+01 -1.119394e+00  1.571936e+01
#> l  -1.403122  1.827970e-01  7.268523e+00 -6.298392e-01 -3.564195e+00
#> il -4.950594  4.440918e+00  9.922722e+00 -7.116999e-01  7.448039e+00
#> jl  5.253879  9.239258e+00  7.195855e+00  4.224830e+00 -4.883673e+00
#> kl -2.031907  1.159402e+01 -1.147093e+01 -1.264476e+00 -2.728531e+00

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

Further information

For more detail, see the package vignette

vignette("onionpaper")

RobinHankin / onion

readme

Quaternions and octonions in R

Overview

Installation

The `onion` package in use

Octonions

References

Further information

RobinHankin / onion

readme

Quaternions and octonions in R

Overview

Installation

The onion package in use

Octonions

References

Further information

The `onion` package in use