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Create, coerce, and test for clifford objects

clifford.Rd

An object of class clifford is a member of a Clifford algebra. These objects may be added and multiplied, and have various applications in physics and mathematics.

Usage

clifford(terms, coeffs=1)
is_ok_clifford(terms, coeffs)
as.clifford(x)
is.clifford(x)
nbits(x)
nterms(x)
# S3 method for class 'clifford'
dim(x)

Arguments

terms

A list of integer vectors with strictly increasing entries corresponding to the basis vectors of the underlying vector space

coeffs

Numeric vector of coefficients

x

Object of class clifford

Details

  • Function clifford() is the formal creation mechanism for clifford objects. If coeffs is of length 1, it will be recycled (even if terms is empty, in which case the zero Clifford object is returned). Argument terms is passed through list_modifier(), so a zero entry is interpreted as numeric(0)

  • Function as.clifford() is much more user-friendly and attempts to coerce a range of input arguments to clifford form

  • Function nbits() returns the number of bits required in the low-level C routines to store the terms (this is the largest entry in the list of terms). For a scalar, this is zero and for the zero clifford object it (currently) returns zero as well although a case could be made for NULL

  • Function nterms() returns the number of terms in the expression

  • Function is_ok_clifford() is a helper function that checks for consistency of its arguments

References

Snygg 2012. “A new approach to differential geometry using Clifford's geometric algebra”. Birkhauser; Springer Science+Business.

Author

Robin K. S. Hankin

See also

Ops.clifford

Examples


(x <- clifford(list(1,2,1:4),1:3))   # Formal creation method
#> Element of a Clifford algebra, equal to
#> + 1e_1 + 2e_2 + 3e_1234
(y <- as.1vector(4:2))
#> Element of a Clifford algebra, equal to
#> + 4e_1 + 3e_2 + 2e_3
(z <- rcliff(include.fewer=TRUE))
#> Element of a Clifford algebra, equal to
#> + 6 - 6e_23 - 9e_25 - 2e_125 + 9e_1245 + 6e_2345 + 5e_1246 + 1e_2346 - 7e_56 -
#> 3e_3456

terms(x+100)
#> A disord object with hash 44041cf8cc66f11ef3ad34a2b8596d11d2abe80a and elements
#> [[1]]
#> integer(0)
#> 
#> [[2]]
#> [1] 1
#> 
#> [[3]]
#> [1] 2
#> 
#> [[4]]
#> [1] 1 2 3 4
#> 
#> (in some order)
coeffs(z)
#> A disord object with hash 145748b538f82204ea9b48ed4b4973253b73b962 and elements
#>  [1]  6 -6 -9 -2  9  6  5  1 -7 -3
#> (in some order)

## Clifford objects may be added and multiplied:

x + y
#> Element of a Clifford algebra, equal to
#> + 5e_1 + 5e_2 + 2e_3 + 3e_1234
x*y
#> Element of a Clifford algebra, equal to
#> + 10 - 5e_12 + 2e_13 + 4e_23 - 6e_124 + 9e_134 - 12e_234