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https://github.com/RobinHankin/onion
R functionality to deal with quaternions and octonions
https://github.com/RobinHankin/onion
Last synced: 8 days ago
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R functionality to deal with quaternions and octonions
- Host: GitHub
- URL: https://github.com/RobinHankin/onion
- Owner: RobinHankin
- Created: 2018-05-17T03:27:01.000Z (over 6 years ago)
- Default Branch: master
- Last Pushed: 2024-08-27T16:13:44.000Z (4 months ago)
- Last Synced: 2024-11-28T18:40:29.132Z (14 days ago)
- Language: R
- Size: 1.77 MB
- Stars: 6
- Watchers: 2
- Forks: 1
- Open Issues: 4
-
Metadata Files:
- Readme: README.Rmd
- Citation: CITATION.cff
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- jimsghstars - RobinHankin/onion - R functionality to deal with quaternions and octonions (R)
README
---
title: "Quaternions and octonions in R"
output:
github_document:
pandoc_args: --webtex
---
```{r setup, include = FALSE}
knitr::opts_chunk$set(
collapse = TRUE,
comment = "#>",
fig.path = "man/figures/README-",
out.width = "100%"
)
```
[](https://cran.r-project.org/package=onion)
# 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](https://CRAN.R-project.org) with:
```{r, message=FALSE}
# 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:
```{r}
a <- 1:9 + Hi -2*Hj
a
a*Hk
Hk*a
```
Function `rquat()` generates random quaternions:
```{r, echo=FALSE}
set.seed(0)
```
```{r}
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:
```{r}
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
# Further information
For more detail, see the package vignette
`vignette("onionpaper")`