r =
(a, b,c, d)
r = ( a, b,c, d)
r = ( a, b,c, d)
OverBar(r) = ( d, -c,b, a)
Determinant is conserved
r*OverBar(r) =
(0, -2*c*a+2*d*b,2*a*b+2*c*d, d^2+a^2+c^2+b^2)
expect zero for s.q
r = ( a, b,c, d)
UnderBar(r) = ( d, c,-b, a)
OverBar(UnderBar(r)) = ( a, b,c, d)
UnderBar(OverBar(r)) = ( a, b,c, d)
Determinant is conserved
r*UnderBar(r) =
(0, 0,0, d^2+a^2-c^2-b^2)
expect zeroes in s.q, s.x, s.y
r = ( a, b,c, d)
Reverse(r) = ( a, b,c, -d)
Reverse(Reverse(r)) = ( a, b,c, d)
Determinant is conserved
r*Reverse(r) =
(d^2+a^2+c^2+b^2, 2*a*b+2*c*d,2*c*a-2*d*b, 0)
r = ( a, b,c, d)
Involution(r) = ( a, -b,-c, d)
Involution(Involution(r)) = ( a, b,c, d)
Determinant is conserved
r*Involution(r) =
(-d^2+a^2-c^2-b^2, -2*c*d,2*d*b, 2*d*a)
r = ( a, b,c, d)
Transpose(r) = ( a, b,c, -d)
Transpose(Transpose(r)) = ( a, b,c, d)
Determinant is conserved
r*Transpose(r) =
(d^2+a^2+c^2+b^2, 2*a*b+2*c*d,2*c*a-2*d*b, 0)
r = ( a, b,c, d)
Conjugation(r) = ( a, -b,-c, -d)
Conjugation(Conjugation(r)) = ( a, b,c, d)
Determinant is conserved
r*Conjugation(r) =
(d^2+a^2-c^2-b^2, 0,0, 0)
r = ( a, b,c, d)
CliffordConjugation(r) = ( a, -b,-c, -d)
CliffordConjugation(CliffordConjugation(r)) = ( a, b,c, d)
Determinant is conserved
r*CliffordConjugation(r) =
(d^2+a^2-c^2-b^2, 0,0, 0)
r = ( a, b,c, d)
Dual(r) = ( d, c,-b, -a)
Dual(Dual(r)) = ( -a, -b,-c, -d)
Determinant is conserved
r*Dual(r) =
(2*d*a, 2*c*a,-2*a*b, d^2-a^2-c^2-b^2)
r = ( a, b,c, d)
DorstDual(r) = ( d, -c,b, -a)
Determinant is conserved
r*DorstDual(r) =
(2*d*a, 2*d*b,2*c*d, d^2-a^2+c^2+b^2)
r = ( a, b,c, d)
DorstUnDual(r) = ( -d, c,-b, a)
DorstUnDual(DorstDual(r)) = ( a, b,c, d)
DorstDual(DorstUnDual(r)) = ( a, b,c, d)
Determinant is conserved
r*DorstUnDual(r) =
(-2*d*a, -2*d*b,-2*c*d, -d^2+a^2-c^2-b^2)
r = ( a, b,c, d)
s = ( A, B,C, D)
r + s = ( a+A, B+b,c+C, d+D)
r = ( a, b,c, d)
s = ( A, B,C, D)
r - s = ( a-A, -B+b,c-C, d-D)
r = ( a, b,c, d)
s = ( A, B,C, D)
t = ( A, B,C, D)
r != s (as expected)
s == t (as expected)
r = ( a, b,c, d)
s = ( A, B,C, D)
t = ( A, B,C, D)
r != s (as expected)
s == t (as expected)
r = ( a, b,c, d)
s = ( A, B,C, D)
u = r*s =
(c*C+B*b-d*D+a*A, -c*D+B*a+A*b+C*d,c*A-B*d+C*a+D*b, C*b+d*A-B*c+a*D)
Product is non-commutative (as expected)
r = ( a, b,c, d)
s = ( A, B,C, D)
t = ( c.a, c.b,c.c, c.d)
Product is associative (as expected)
s = Z*r = ( Z*a, Z*b,c*Z, d*Z)
s = r*Z = ( a*Z, Z*b,c*Z, d*Z)
r = ( a, b,c, d)
s = ( A, B,C, D)
u = Product(r,s) =
(c*C+B*b-d*D+a*A, -c*D+B*a+A*b+C*d,c*A-B*d+C*a+D*b, C*b+d*A-B*c+a*D)
Product(r,s) is non-commutative (as expected)
r = ( a, b,c, d)
s = ( A, B,C, D)
t = ( c.a, c.b,c.c, c.d)
Product is associative (as expected)
r = ( a, b,c, d)
u = r/2 = ( 1/2*a, 1/2*b,1/2*c, 1/2*d)
r = ( a, b,c, d)
s = ( A, B,C, D)
u = r^s =
(a*A, B*a+A*b,c*A+C*a, C*b+d*A-B*c+a*D)
r = ( a, b,c, d)
s = ( A, B,C, D)
u = Wedge(r,s) =
(a*A, B*a+A*b,c*A+C*a, C*b+d*A-B*c+a*D)
Wedge(r,s) is non-commutative (as expected)
r = ( a, b,c, d)
s = ( A, B,C, D)
t = ( c.a, c.b,c.c, c.d)
Wedge is associative (as expected)
r = ( a, b,c, d)
s = ( A, B,C, D)
u = AntiWedge(r,s) =
(C*b+d*A-B*c+a*D, B*d+D*b,c*D+C*d, d*D)
AntiWedge(r,s) is non-commutative (as expected)
r = ( a, b,c, d)
s = ( A, B,C, D)
t = ( c.a, c.b,c.c, c.d)
AntiWedge is associative (as expected)
r = ( a, b,c, d)
s = ( A, B,C, D)
u = Regressive(r,s) =
(-C*b+d*A+B*c+a*D, B*d+D*b,c*D+C*d, d*D)
Regressive(r,s) is non-commutative (as expected)
r = ( a, b,c, d)
s = ( A, B,C, D)
t = ( c.a, c.b,c.c, c.d)
Regressive is associative (as expected)
r = ( a, b,c, d)
s = ( A, B,C, D)
u = RegressiveViaFormula(r,s) =
(-C*b+d*A+B*c+a*D, B*d+D*b,c*D+C*d, d*D)
r = ( a, b,c, d)
s = ( A, B,C, D)
u = LowerRightViaFormula(r,s) =
(-d*D, -c*D+C*d,-B*d+D*b, C*b+d*A-B*c+a*D)
r = ( a, b,c, d)
s = ( A, B,C, D)
u = Expander(r,s) =
(0, 0,0, C*b-B*c)
r = ( a, b,c, d)
s = ( A, B,C, D)
u = Conserver(r,s) =
(a*A, B*a+A*b,c*A+C*a, d*A+a*D)
r = ( a, b,c, d)
s = ( A, B,C, D)
u = Shrinker(r,s) =
(c*C+B*b-d*D, -c*D+C*d,-B*d+D*b, 0)
r = ( a, b,c, d)
s = ( A, B,C, D)
u = Symmetric(r,s) =
(c*C+B*b-d*D+a*A, B*a+A*b,c*A+C*a, d*A+a*D)
r = ( a, b,c, d)
s = ( A, B,C, D)
u = AntiSymmetric(r,s) =
(0, -c*D+C*d,-B*d+D*b, C*b-B*c)
r = ( a, b,c, d)
s = ( A, B,C, D)
u = Inner(r,s) =
(c*C+B*b-d*D, -c*D+C*d,-B*d+D*b, 0)
r = ( a, b,c, d)
s = ( A, B,C, D)
u = LeftContraction(r,s) =
(c*C+B*b-d*D+a*A, -c*D+B*a,C*a+D*b, a*D)
r = ( a, b,c, d)
s = ( A, B,C, D)
u = RightContraction(r,s) =
(c*C+B*b-d*D+a*A, A*b+C*d,c*A-B*d, d*A)
X = ( 3, 5,7, 11)
Determinant(X) = 3136
X = ( 3, 5,7, 11)
u = Adjugate(X) =
(168, -280,-392, -616)
X = ( 3, 5,7, 11)
u = Reciprocal(X) =
(3/56, -5/56,-1/8, -11/56)
X*(1/X) =
(1, 0,0, 0)