3Blue1Brown — Euler's formula with introductory group theory

Summary: Introduces basic group theory (symmetries, composition, examples) and reframes Euler’s formula as the statement that $\exp$ is a homomorphism from the additive group of complex numbers to the multiplicative group — mapping vertical slides to pure rotations, with $e$ being the base that converts 1 unit of slide into 1 radian of rotation.

Core argument

1. A group is a set of symmetries plus a composition rule. A symmetry is any action on an object that leaves it looking the same. Two examples are emphasised:

   • Dihedral group $D_4$ — 8 symmetries of a square (identity + 3 rotations + 4 reflections). Finite.
   • Circle group — all rotations of a circle, parameterised continuously by angle in $[0,2\pi )$. Infinite. Each action can be labelled by where it sends a chosen reference point.

2. Numbers can be viewed as groups in two distinct ways.

   • Additive group of reals — every real $r$ is the sliding action on the number line that drags 0 to $r$. Composing two slides = adding the numbers.
   • Multiplicative group of positive reals — every positive real $r$ is the stretching action (fixing 0) that drags 1 to $r$. Composing two stretches = multiplying the numbers. Zero is excluded (squishing to 0 is not invertible).

3. Extension to the complex plane.

   • Additive group $(\mathbb{C},+)$ — 2D sliding actions. A slide by $a+bi$ = horizontal slide by $a$ followed by vertical slide by $b$.
   • Multiplicative group $({\mathbb{C}}^{\ast },\times )$ — stretch+rotate actions fixing 0. The action associated with $i$ is a 90° rotation (apply twice → 180° flip, hence $i\cdot i=-1$). Every action decomposes into a positive-real stretch followed by a pure rotation parameterised by the unit circle.

4. The exponential is a homomorphism. The defining identity $a^{x+y}=a^x\cdot a^y$ says “adding inputs corresponds to multiplying outputs.” In group terms, $\exp$ is a structure-preserving map from the additive group to the multiplicative group:

   • Real horizontal slides → real positive stretches
   • Imaginary vertical slides → pure rotations on the unit circle

5. The role of $e$. Different bases differ only in how fast a unit of vertical slide converts to rotation:

   • $2^{ix}$: 1 unit slide → $\ln 2\approx 0.693$ rad rotation
   • $5^{ix}$: 1 unit slide → $\ln 5\approx 1.609$ rad rotation
   • $e^{ix}$: 1 unit slide → exactly 1 radian rotation

   So $\pi i$ → half-turn, giving $e^{i\pi }=-1$.

6. Visualising $z\mapsto e^z$ on the whole plane. Vertical lines wrap into concentric circles (rotations); horizontal lines map to rays from the origin (stretches). Imagined as: roll the plane into a cylinder, then smoosh that cylinder onto the $xy$-plane around 0.

Key takeaways

• An exponential’s purpose is to bridge additive and multiplicative structures — repeated multiplication is just one specialisation of that.
• The “real ↔ stretch, imaginary ↔ rotation” split on the multiplicative side mirrors the “horizontal ↔ real, vertical ↔ imaginary” split on the additive side. Euler’s formula matches the two decompositions.
• This is intuition, not proof. The 1-radian property of $e$ is justified properly via calculus (derivative = self), not group theory alone.

Sources

• Euler’s formula with introductory group theory