e₁₂ is orthogonal to Z e₃₄
e₂₃ is orthogonal to X e₄₁
e₃₁ is orthogonal to Y e₂₄all have ring 1-2-3-4-1...
e₁ ∧ e₂₃₄ = +e₁₂₃₄ 0 swaps
e₂ ∧ e₁₃₄ = -e₁₂₃₄ 1 swap
e₃ ∧ e₁₂₄ = +e₁₂₃₄ 2 swaps
e₄ ∧ e₁₂₃ = -e₁₂₃₄ 3 swapsall good hypervolumes
Doing vR then R̃(vR) sounds tedious? It is!
v' = ~R * v * R = v - 2 * s * (B · v) + 2*B · (B · v)
v = v₁e₁ v₂e₂ v₃e₃ v₄e₄
B = R[1]e₁e₂ + R[2]e₂e₃ + R[3]e₃e₁ + R[4]e₃₄ + R[5]e₄₁ + R[6]e₂₄
s = R[0]
// only terms that share a common base vector - because
e₁₂ · e₃ = 0 overlap
B · v =