The "most beautiful equation" in one picture: eiθ is just a point rotating around a circle. Wind θ up to π and you've spun exactly half a turn — to −1. eiπ + 1 = 0.
Multiplying by i rotates by 90°. Euler's key insight: eiθ = cosθ + i·sinθ — a point one unit from the origin at angle θ. As θ sweeps 0 → π the point traces the top half of the circle and lands on (−1, 0). So eiπ = −1, and eiπ + 1 = 0 ties the five most important numbers in maths — 0, 1, e, i and π — into one line.