Euler's Identity

Overview

A brief visual proof of Euler's identity $e^{i\pi}+1=0$, showing the unit circle, the exponential mapping, and the convergence of the complex exponential to the point $-1$ on the real axis.

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Phases

#	Phase Name	Duration	Description
1	Intro	~3s	Fade in a clean white background with a title "Euler's Identity" at the top. The unit circle appears centered.
2	Exponential Mapping	~8s	An arrow representing the angle $\theta$ sweeps from 0 to $\pi$ on the circle while a point moves along the circle, tracing the path of $e^{i\theta}$. A trailing line shows the radius vector.
3	Reveal Formula	~5s	As the arrow reaches $\pi$, the point lands at $-1$. The expression $e^{i\pi}$ fades in beside the point, then "+1" appears, and finally "=0" slides in, completing the identity.
4	Outro	~4s	The whole formula $e^{i\pi}+1=0$ scales up slightly, holds for a moment, then fades out together with the circle, leaving a brief caption "The most beautiful equation" at the bottom.

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Layout

┌─────────────────────────────────────┐
│               TOP AREA               │
├───────────────────────┬─────────────┤
│                       │             │
│        LEFT AREA      │  RIGHT AREA │
│   (Unit circle &      │ (Optional   │
│    moving point)      │  small note │
│                       │  if needed) │
├───────────────────────┴─────────────┤
│               BOTTOM AREA            │
└─────────────────────────────────────┘

Area Descriptions

Area	Content	Notes
Top	Title "Euler's Identity"	Fades in at phase 1, stays throughout
Left	Unit circle with moving point representing $e^{i\theta}$	Primary visual, centered vertically
Right	Small label showing current angle $\theta$ (optional)	Appears during phase 2, fades out after
Bottom	Caption "The most beautiful equation"	Appears in phase 4, fades out at end

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Notes

• Keep total runtime under 20 seconds to maintain brevity.
• Use smooth easing for the sweeping arrow and point motion.
• No additional explanatory text; the visual progression should convey the identity.
• All elements should be white on a dark background for contrast.
• The scene must be implemented as a single Manim Scene class.