# Euler's Formula Animation — Plan

## Overview
- **Topic**: Euler's Identity: e^(iπ) + 1 = 0
- **Hook**: Five fundamental constants in one equation — each hiding in plain sight
- **Aha moment**: When π appears and the identity collapses into silence
- **Target audience**: Anyone who's seen complex numbers but never felt their magic
- **Length**: ~90 seconds
- **Resolution**: 480p (draft)

## Color Palette — Classic 3B1B
- Background: `#1C1C1C`
- PRIMARY: `#58C4DD` (BLUE) — mathematical notation
- SECONDARY: `#83C167` (GREEN) — geometric/visual elements
- ACCENT: `#FFFF00` (YELLOW) — highlights, key reveals
- RED: `#FF6B6B` — e constant emphasis
- ORANGE: `#FFA07A` — i (imaginary unit)

## Arc: Discovery

---

## Scene 1: Title (~8s)
**Purpose**: Hook — pose the mystery

### Visual elements
- Title text centered: "The Most Beautiful Equation in Mathematics"
- Subtitle: "e^(iπ) + 1 = 0"

### Animation
1. FadeIn title from dark
2. Subtitle appears below after 2s

### Subtitle
"Five numbers. One line. Everything connected."

---

## Scene 2: The Five Constants (~20s)
**Purpose**: Introduce each constant visually before revealing the identity

### Visual elements
- Five constants spread across screen: e, i, π, 1, 0
- Each with a one-line description

### Layout
- Constants large (font_size 72) with colored glow
- Labels below each in smaller text

### Animation (staggered reveal, 2s each)
1. e appears — "The base of natural growth"
2. i appears — "The square root of negative one"
3. π appears — "The ratio of a circle's circumference to its diameter"
4. 1 appears — "The foundation of counting"
5. 0 appears — "The concept of nothingness"

### Subtitle
"What if I told you they were all... the same thing?"

---

## Scene 3: The Complex Plane (~20s)
**Purpose**: Build geometric intuition — the unit circle

### Visual elements
- 2D coordinate plane (Axes)
- Unit circle centered at origin
- Real axis (horizontal), Imaginary axis (vertical)
- Labels: Re, Im

### Animation
1. Axes fade in with labels
2. Unit circle drawn with Create
3. A point at (1, 0) on the circle marked with a dot
4. Arc showing rotation of angle θ traced in green
5. Cos(θ), Sin(θ) labels on axes

### Subtitle
"Every point on this circle encodes a rotation."

---

## Scene 4: From Exponential to Trigonometric (~20s)
**Purpose**: The bridge — e raised to an imaginary power

### Visual elements
- Equation evolving step by step
- Real axis highlighting cos(θ)
- Imaginary axis highlighting i·sin(θ)

### Animation
1. Show: e^(iθ) — what does this even mean?
2. Argue geometrically: rotating around the circle
3. Reveal: e^(iθ) = cos(θ) + i·sin(θ)
4. Highlight each term as it maps to the circle

### Subtitle
"The exponential function and rotation are the same thing."

---

## Scene 5: The Reveal — Plug in π (~15s)
**Purpose**: The aha moment — watch everything collapse

### Visual elements
- Final equation e^(iπ) + 1 = 0
- The unit circle with angle = π (half rotation)
- Point lands at (-1, 0)

### Animation
1. Start from e^(iθ) = cos(θ) + i·sin(θ)
2. Let θ = π
3. cos(π) = -1, sin(π) = 0 substituted
4. e^(iπ) = -1
5. e^(iπ) + 1 = 0

### Subtitle
"It was there all along."

---

## Scene 6: The Silence (~10s)
**Purpose**: Let the beauty sink in

### Visual elements
- Final equation centered, glowing
- Five constants subtly colored in the equation
- Blank dark space around

### Animation
1. Equation fades in slowly (3s)
2. Each constant briefly highlights in its color
3. Hold on stillness

### Subtitle
"e, i, π, 1, 0. The entire universe of mathematics, in one breath."