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
where
and
are the quaternion units linked by the equations
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
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:
# install.packages("onion") # uncomment this to install the package
library("onion")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 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+00- 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
For more detail, see the package vignette
vignette("onionpaper")