Quaternionic arithmetic with Clifford algebra

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.

Alternative mapping

Alternatively we might consider the even subalgebra of \(\operatorname{Cl}(0,3)\) with general element \(q_0 + q_1e_{23} -q_2e_{13} + q_3e_{12}\) (note change of sign for \(q_2\)). Thus

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

A quick-and-dirty R function might be

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)
}

Then verification is straightforward:

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

References

Hankin, R. K. S. 2006. “Normed Division Algebras with R: Introducing the ‘Onion‘ Package.” R News 6 (2): 49–52.

———. 2022. “Clifford Algebra in R.” arXiv. https://doi.org/10.48550/ARXIV.2209.13659.

Robin K. S. Hankin

Alternative mapping

References