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.

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)
```

`clifford`

package in useThe 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
specifes 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
#> + 4 + 5e_2 + 1e_5 - 2e_4,7 + 2e_11 + 4e_14 - 1e_10,14 + 3e_5,9,15 - 3e_18,19
summary(x^3)
#> Element of a Clifford algebra
#> Typical terms: 364 ... + 54e_5,9,10,14,15,18,19
#> Number of terms: 40
#> Magnitude: 265721
```

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

For more detail, see the package vignette

`vignette("clifford")`

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