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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")
These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.