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Cheat Sheet

A quick reference for common tasks, operators, and patterns in CliffordAlgebras.jl.

julia> using CliffordAlgebras; using LinearAlgebra; true
true

Algebra creation

julia> Cl = CliffordAlgebra;  # alias

julia> cl2 = Cl(2); cl3 = Cl(3); sta = Cl(:Spacetime); pga = Cl(:PGA3D); cga = Cl(:CGA3D); true
true

Unicode operators (with ASCII fallbacks)

  • Geometric product: a * b
  • Exterior (wedge): a ∧ b — ASCII: CliffordAlgebras.exteriorprod(a,b)
  • Regressive: a ∨ b — ASCII: CliffordAlgebras.regressiveprod(a,b)
  • Fat dot: a ⋅ b — ASCII: CliffordAlgebras.fatdotprod(a,b)
  • Scalar: a ⋆ b — ASCII: CliffordAlgebras.scalarprod(a,b)
  • Left contraction: a ⨼ b — ASCII: CliffordAlgebras.leftcontractionprod(a,b)
  • Right contraction: a ⨽ b — ASCII: CliffordAlgebras.rightcontractionprod(a,b)
  • Commutator: a ×₋ b — ASCII: CliffordAlgebras.commutatorprod(a,b)
  • Anti-commutator: a ×₊ b — ASCII: CliffordAlgebras.anticommutatorprod(a,b)
  • Sandwich (conjugation): A ≀ X — ASCII: CliffordAlgebras.sandwichproduct(A,X)

Working with multivectors

julia> cl3 = CliffordAlgebra(3);

julia> e1, e2, e3 = cl3.e1, cl3.e2, cl3.e3;

julia> mv = 1 + 2e1 + 3e2 + (e1 ∧ e2);  # 0-,1-,2- grades

julia> scalar(mv); grade(mv,2); even(mv); odd(mv); ~mv; dual(mv); true
true

Rotors and motors (quick)

julia> cl3 = CliffordAlgebra(3);

julia> B = (π/4) * (cl3.e1 ∧ cl3.e2); R = exp(B);

julia> v = cl3.e1; v_rot = R ≀ v; true
true

PGA motor (rotation+translation):

julia> pga = CliffordAlgebra(:PGA3D);

julia> e1, e2, e3, e0 = basevector(pga,1), basevector(pga,2), basevector(pga,3), basevector(pga,:e0);

julia> B = (π/12)*(e1 ∧ e2); T = 0.05*(e0 ∧ e1); M = exp(B+T);

julia> v = e1 + 2e2; v2 = M ≀ v; true
true

Outermorphism (linear transform)

julia> cl2 = CliffordAlgebra(2); v = cl2.e1 + cl2.e2;

julia> θ = π/6; R = [cos(θ) -sin(θ); sin(θ) cos(θ)];

julia> v2 = outermorphism(R, v); true
true

Tables and signatures

julia> cl2 = CliffordAlgebra(2);

julia> io = IOBuffer(); signaturetable(io, cl2); cayleytable(io, cl2); true
true

Quick facts

  • Algebra dimension for signature (p,q,r) with n=p+q+r is 2^n.
  • Number of grade-k elements is C(n,k).
  • Pseudoscalar I has grade n; I^2 = character(algebra).
  • Null basis vectors (r>0) square to 0 and represent infinity/conformal components depending on algebra.