involutions : rev(u)  invol(u)  cj(u)
 
                   reverse
 
rev(a1, a2, ... ar) = ar ... a2, a1
rev(A) = <A>0 + <A>1 - <A>2 - <A>3         ( Cl(3) )
reverse basis element order e0 -> e0, e1 -> e1, e2 -> e2, ..., e12 -> e21, e23 -> e32, ..., e123 -> e321
rev(A) =   A   -->  A is said real
rev(A) = - A    --> A is said imaginary
rev(e12) = -e12
rev(3e1+2e23+e123) = 3e1 -2e23 - e123
rev(AB) = rev(B) rev(A)               (anti automorphism)
R v rev(R)   a 3D rotation, with rotor R, preserve the reality of the transformed vector.
 
                   grade involution
invol(a1, a2, ... ar) = (−1)**r  (a1, a2, ... ar)
invol(A) = <A>0 - <A>1 + <A>2 - <A>3       ( Cl(3) )
change basis element direction e0 -> e0, e1 -> -e1, e2 -> -e2, ..., e12 -> e12, e23 -> e23, ..., e123 -> -e123
invol(e12) = e12
invol(e1+2e2) = -e1 - 2e2
invol(2e0+e3+2e12-e23) = 2e0 - e3 +2e12 - e23
invol(AB) = invol(A) invol(B)      (automorphism)
if invol(A) =  A                A is even
if invol(A) = -A               A is odd
 
                   Clifford conjugation
 cj(A) = <A>0 - <A>1 - <A>2 + <A>3         ( Cl(3) )
conjugation = reversion(involution) = involution(reversion)
 
cj(e12) = -e12
cj(e1+2e2) = -e1 - 2e2
cj(2e0+e3+2e12-e23) = 2e0 - e3 - 2e12 + e23
cj(AB) = cj(A) cj(B)                  (anti automorphism)
FUNCTIONS
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