Octonion Multiplication & Fano Plane

Multiplication Result

Select Basis Elements

About Octonions

• 8-dimensional algebra over reals (1 + e₁...e₇)
• Non-associative: (ab)c ≠ a(bc) in general
• Alternative: (aa)b = a(ab) and (ab)b = a(bb)
• Division algebra: Every non-zero element has inverse
• The Fano plane encodes all multiplication rules

How to Read the Fano Plane

Each line (including the circle) contains 3 elements. Following the arrows: multiplying two adjacent elements gives the third with sign determined by direction.

Example: e₁ × e₂ = e₃ (following arrow), but e₂ × e₁ = -e₃ (against arrow)