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 , are a generalization of the complex numbers represented as a four-dimensional vector space over the reals. An arbitrary quaternion represented as
[ q=a + b\mathbf{i} + c\mathbf{j}+ d\mathbf{k}](https://latex.codecogs.com/png.latex?%0Aq%3Da%20%2B%20b%5Cmathbf%7Bi%7D%20%2B%20c%5Cmathbf%7Bj%7D%2B%20d%5Cmathbf%7Bk%7D%0A " q=a + b + c+ d ")
where and are the quaternion units linked by the equations
[ \mathbf{i}2= \mathbf{j}2= \mathbf{k}^2= \mathbf{i}\mathbf{j}\mathbf{k}=-1.](https://latex.codecogs.com/png.latex?%0A%5Cmathbf%7Bi%7D%5E2%3D%0A%5Cmathbf%7Bj%7D%5E2%3D%0A%5Cmathbf%7Bk%7D%5E2%3D%0A%5Cmathbf%7Bi%7D%5Cmathbf%7Bj%7D%5Cmathbf%7Bk%7D%3D-1. " 2= 2= ^2= =-1.")
which, together with distributivity, define quaternion multiplication. We can see that the quaternions are not commutative, for while , it is easy to show that . 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}{a2 + b^2 + c^2 + d^2} \left(a-b\mathbf{i} - c\mathbf{j}- d\mathbf{k}\right)](https://latex.codecogs.com/png.latex?%0A%5Cleft%28%20a%2Bb%5Cmathbf%7Bi%7D%20%2B%20c%5Cmathbf%7Bj%7D%2B%20d%5Cmathbf%7Bk%7D%5Cright%29%5E%7B-1%7D%3D%0A%5Cfrac%7B1%7D%7Ba%5E2%20%2B%20b%5E2%20%2B%20c%5E2%20%2B%20d%5E2%7D%0A%5Cleft%28a-b%5Cmathbf%7Bi%7D%20-%20c%5Cmathbf%7Bj%7D-%20d%5Cmathbf%7Bk%7D%5Cright%29%0A " ( a+b + c+ d)^{-1}= (a-b - c- d) ")
shows that the quaternions are a division algebra: division works as expected (although one has to be careful about ordering terms).
The octonions are essentially a pair of quaternions, with a general octonion written
(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 in general.
You can install the released version of onion from CRAN with:
onion package in useThe 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 9Function 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 -1.5522800Octonions 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+00For more detail, see the package vignette
vignette("onionpaper")