Extensions

CliffordNumbers.jl provides some extensions to allow for interoperability with other packages which we anticipate to be commonly used alongside it.

[Quaternions.jl]

[Quaternions.jl] provides the Quaternion type.

This extension treats Quaternion{T} as identical to EvenCliffordNumber{VGA(3),T,4}, and each type can be constructed from the other. This also extends to AbstractCliffordNumber{VGA(3)}.

julia> q = Quaternion(0, 1, 2, 3)
Quaternion{Int64}(0, 1, 2, 3)

julia> e = EvenCliffordNumber{VGA(3)}(q)
4-element EvenCliffordNumber{VGA(3), Int64}:
1e₁e₂ + 2e₁e₃ + 3e₂e₃

julia> Quaternion(e)
Quaternion{Int64}(0, 1, 2, 3)

julia> KVector{2,VGA(3)}(q)
3-element KVector{2, VGA(3), Int64}:
1e₁e₂ + 2e₁e₃ + 3e₂e₃

However, conversion can fail if the target type cannot contain the result:

julia> convert(KVector{2,VGA(3)}, q)
ERROR: InexactError: convert(KVector{2, VGA(3)}, EvenCliffordNumber{VGA(3), Int64}(0, 1, 2, 3))
...

Promotion attempts to preserve the semantics of CliffordNumbers.jl, and therefore prefers to return an AbstractCliffordNumber{VGA(3)}.

julia> promote_type(EvenCliffordNumber{VGA(3),Int}, QuaternionF64)
EvenCliffordNumber{VGA(3), Float64, 4}

julia> promote_type(KVector{2,VGA(3),Int}, QuaternionF64)
EvenCliffordNumber{VGA(3), Float64, 4}

julia> promote_type(KVector{1,VGA(3),Int}, QuaternionF64)
CliffordNumber{VGA(3), Float64, 8}

[Unitful.jl]

[Unitful.jl] provides support for quantities with associated units.

To make a Clifford number into a unitful quantity, simply multiply it by a unit or other unitful quantity:

julia> KVector{1, VGA(3)}(1, 2, 3)u"m"
(1e₁ + 2e₂ + 3e₃) m

julia> KVector{1, VGA(3)}(1, 2, 3) * 1.5u"m"
(1.5e₁ + 3.0e₂ + 4.5e₃) m

Constructing Clifford numbers from `Quantity` coefficients fails

Attempts to construct a Clifford number from Quantity objects will fail, because Quantity does not subtype Real or Complex:

julia> KVector{1,VGA(3)}(1u"m", 2u"m", 3u"m")
ERROR: ArgumentError:
...

For this reason, Clifford numbers with mixed units cannot be constructed, and this is intentional: AbstractCliffordNumber assumes an orthonormal basis.

Supported operations include the geometric product and wedge product.

[Quaternions.jl]: https://github.com/JuliaGeometry/Quaternions.jl [Unitful.jl]: https://github.com/PainterQubits/Unitful.jl