Arbitrary Dimensional Clifford Algebras.

The clifford package: Clifford algebra in R

The clifford package provides R-centric functionality for working with Clifford algebras of arbitrary dimension and signature. A detailed vignette is provided in the package.

Installation

You can install the released version of the clifford package from CRAN with:

# install.packages("clifford")  # uncomment this to install the package
library("clifford")
set.seed(0)

The clifford package in use

The basic creation function is clifford(), which takes a list of basis blades and a vector of coefficients:

(a <- clifford(list(1,2,1:4,2:3),1:4))
#> Element of a Clifford algebra, equal to
#> + 1e_1 + 2e_2 + 4e_23 + 3e_1234
(b <- clifford(list(2,2:3,1:2),c(-2,3,-3)))
#> Element of a Clifford algebra, equal to
#> - 2e_2 - 3e_12 + 3e_23

So a and b are multivectors. Clifford objects are a vector space and we can add them using +:

a+b
#> Element of a Clifford algebra, equal to
#> + 1e_1 - 3e_12 + 7e_23 + 3e_1234

See how the e2 term vanishes and the e_23 term is summed. The package includes a large number of products:

a*b        # geometric product (also "a % % b")
#> Element of a Clifford algebra, equal to
#> - 16 + 6e_1 - 3e_2 - 2e_12 + 14e_3 + 12e_13 + 3e_123 - 9e_14 + 9e_34 - 6e_134
a %^% b    # outer product
#> Element of a Clifford algebra, equal to
#> - 2e_12 + 3e_123
a %.% b    # inner product
#> Element of a Clifford algebra, equal to
#> - 16 + 6e_1 - 3e_2 + 14e_3 - 9e_14 + 9e_34 - 6e_134
a %star% b # scalar product
#> [1] -16
a %euc% b  # Euclidean product
#> [1] 8

The package can deal with non positive-definite inner products. Suppose we wish to deal with an inner product of

where the diagonal is a number of terms followed by a number of terms. The package idiom for this would be to use signature():

signature(3)

Function signature() is based on lorentz::sol() and its argument specifies the number of basis blades that square to , the others squaring to . Thus and :

basis(1)
#> Element of a Clifford algebra, equal to
#> + 1e_1
basis(1)^2
#> Element of a Clifford algebra, equal to
#> scalar ( 1 )
basis(4)
#> Element of a Clifford algebra, equal to
#> + 1e_4
basis(4)^2
#> Element of a Clifford algebra, equal to
#> the zero clifford element (0)

The package uses the STL map class with dynamic bitset keys for efficiency and speed and can deal with objects of arbitrary dimensions. Thus:

options("basissep" = ",")
(x <- rcliff(d=20))
#> Element of a Clifford algebra, equal to
#> + 5 - 1e_2 + 6e_5 + 2e_4,7 - 3e_10 - 5e_11 + 3e_14 + 1e_10,14 + 4e_5,9,15 +
#> 9e_18,19
summary(x^3)
#> Element of a Clifford algebra 
#> Typical terms:  140  ...  + 216e_5,9,10,14,15,18,19 
#> Number of terms: 44 
#> Magnitude: 25376

References

• D. Hestenes 1987. Clifford algebra to geometric calculus, Kluwer.
• J. Snygg 2010. A new approach to differential geometry using Clifford’s geometric algebra. Berghauser.
• C. Perwass 2009. Geometric algebra with applications in engineering. Springer.

Further information

For more detail, see the package vignette

vignette("clifford")