Quaternionic arithmetic with Clifford algebra

Robin K. S. Hankin

To cite the clifford package in publications please use Hankin (2022). This short document shows how quaternionic arithmetic may be implemented as a special case of Clifford algebras. This is done for illustrative purposes only; to manipulate quaternions in R the onion package (Hankin 2006) is much more efficient and includes more transparent idiom.

Hamilton’s Broome Bridge insight:

\[ \mathbf{i}^2= \mathbf{j}^2= \mathbf{k}^2= \mathbf{i}\mathbf{j}\mathbf{k}=-1 \]

The BBI and associativity together imply

\[ \mathbf{j}\mathbf{k}=-\mathbf{k}\mathbf{j}=\mathbf{i}\qquad \mathbf{k}\mathbf{i}=-\mathbf{i}\mathbf{k}=\mathbf{j}\qquad \mathbf{i}\mathbf{j}=-\mathbf{j}\mathbf{i}=\mathbf{k}\qquad \]

and if we require a distributive algebra we get the quaternions. A general quaternion is of the form \(a+\mathbf{i}b+\mathbf{j}c+\mathbf{k}d\); addition is componentwise and multiplication follows from the above.

To express quaternionic multiplication using Clifford algebra we make the following identifications:

\[ \mathbf{i}\leftrightarrow -e_{12}\\ \mathbf{j}\leftrightarrow -e_{13}\\ \mathbf{k}\leftrightarrow -e_{23}\\ \]

Thus, for example, \(\mathbf{ii}=(-e_{12})(-e_{12})=+e_{1212}=-e_{1122}=-1\) and \(\mathbf{ij}=(-e_{12})(-e_{13})=+e_{1213}=-e_{1123}=-e_{23}=\mathbf{k}\). The default signature [in which \(e_i^2=+1\)] is fine here, but as a safety measure we can set maxdim to 3:

options(maxdim=3)  # paranoid safety measure

We might wish to multiply \(q_1=1+2\mathbf{i}+3\mathbf{j}+4\mathbf{k}\) by \(q_2=-2+\mathbf{i}-2\mathbf{j}+\mathbf{k}\):

q1 <- +1 + 2* -e(c(1,2)) + 3*-e(c(1,3)) + 4*-e(c(2,3))
q1

## Element of a Clifford algebra, equal to
## + 1 - 2e_12 - 3e_13 - 4e_23

q2 <- -2 + 1* -e(c(1,2)) - 2*-e(c(1,3)) + 1*-e(c(2,3))
q2

## Element of a Clifford algebra, equal to
## - 2 - 1e_12 + 2e_13 - 1e_23

q1*q2

## Element of a Clifford algebra, equal to
## - 2 - 8e_12 + 6e_13 + 14e_23

The product would correspond to \(-2+8\mathbf{i} -6\mathbf{j} -14\mathbf{k}\). Note that the “*” in “q1*q2” is a clifford product. It is possible to leverage the onion package and coerce between clifford objects and quaternions (but don’t actually do it, you crazy fool):

`clifford_to_quaternion` <- function(C){
    C <- as.clifford(C)
    tC <- disordR::elements(terms(C))
    stopifnot(all(c(tC,recursive=TRUE) <= 3))
    jj <- unlist(lapply(tC,length))
    stopifnot(all(jj <= 2))    # safety check
    stopifnot(all(jj%%2 == 0)) # safety check
    out <- matrix(c(const(C), -getcoeffs(C,list(c(1,2),c(1,3),c(2,3)) )))
    as.quaternion(out)
}

`quaternion_to_clifford` <- function(Q){
  Q <- as.numeric(Q)
  stopifnot(length(Q)==4)
  clifford(list(numeric(0),c(1,2),c(1,3),c(2,3)),c(Q[1],-Q[2:4]))
}

We may verify that these maps behave properly by defining some random-ish quaternions and Clifford objects:

q1 <- +1 + 2* -e(c(1,2)) + 3*-e(c(1,3)) + 4*-e(c(2,3))
q2 <- -2 + 1* -e(c(1,2)) - 2*-e(c(1,3)) + 1*-e(c(2,3))
H1 <- as.quaternion(c(3,-5,2,1),single=TRUE)
H2 <- as.quaternion(c(1,2,-2,2),single=TRUE)

First, check that they are inverses of one another:

c(  # check they are inverses of one another
q1 == quaternion_to_clifford(clifford_to_quaternion(q1)),
q2 == quaternion_to_clifford(clifford_to_quaternion(q2)),
H1 == clifford_to_quaternion(quaternion_to_clifford(H1)),
H2 == clifford_to_quaternion(quaternion_to_clifford(H2))
)

## [1] TRUE TRUE TRUE TRUE

Next, verify that they are homomorphisms:

c(
q1*q2 == quaternion_to_clifford(clifford_to_quaternion(q1)*clifford_to_quaternion(q2)),
H1*H2 == clifford_to_quaternion(quaternion_to_clifford(H1)*quaternion_to_clifford(H2))
)

## [1] TRUE TRUE

Note that in package idiom the asterisk, “*” represents either Clifford geometric product or Hamilton’s quaternionic multiplication depending on its arguments.

signature(0,3)
cliff2quat <- function(C){
  out <- getcoeffs(C,list(numeric(0),c(2,3),c(1,3),c(1,2)))
  out[2] <- -out[2]
  as.quaternion(out,single=TRUE)
}

quat2cliff <- function(H){
  jj <- c(as.matrix(H))
  jj[2] <- -jj[2]
  clifford(list(numeric(0),c(2,3),c(1,3),c(1,2)),jj)
}

c(
  cliff2quat(quat2cliff(H1)) == H1,
  cliff2quat(quat2cliff(H2)) == H2,
  quat2cliff(cliff2quat(q1)) == q1,
  quat2cliff(cliff2quat(q2)) == q2,
  cliff2quat(q1*q2) == cliff2quat(q1) * cliff2quat(q2),
  quat2cliff(H1*H2) == quat2cliff(H1) * quat2cliff(H2)
)

## [1] TRUE TRUE TRUE TRUE TRUE TRUE