Prime Numbers and Complex Multiplication

Overview

This animation first visualizes the identification of prime numbers up to 50 using a dynamic number grid, then transitions to demonstrate multiplication of two complex numbers on the complex plane, illustrating the geometric effects of rotation and scaling. The core concepts are the Sieve of Eratosthenes for prime finding and the polar form interpretation of complex multiplication.

Phases

#	Phase Name	Duration	Description
1	Number Grid Setup	~5s	A grid of numbers from 1 to 50 appears. The number 1 is crossed out as not prime.
2	Sieve of Eratosthenes	~8s	Starting with 2, each prime is highlighted, and its multiples are sequentially crossed out in a distinct color. This continues up to 7 (since $\sqrt{50} \approx 7$).
3	Prime Highlight & Transition	~4s	All remaining uncrossed numbers (primes) are highlighted with a distinct color. The grid fades out as a complex plane fades in.
4	Complex Vectors Introduction	~5s	Two complex numbers (e.g., $z_1 = 1+2i$ and $z_2 = 2+1i$) are plotted as vectors from the origin. Their magnitudes and angles from the positive real axis are labeled.
5	Multiplication Visualization	~8s	The vectors are multiplied. The animation shows: the angle of $z_2$ being added to $z_1$ (rotation), and the length of $z_1$ being scaled by the magnitude of $z_2$. The resulting vector $z_1 \cdot z_2$ is drawn.
6	Outro	~2s	The final complex product vector pulses, and the polar multiplication formula $r_1 r_2 e^{i(\theta_1+\theta_2)}$ appears briefly.

Layout

Phases 1-3 use a full-screen layout for the number grid.
Phases 4-6 use the following split layout:

┌─────────────────────────────────────────────┐
│                   TITLE                     │
├──────────────────────┬──────────────────────┤
│                      │                      │
│   COMPLEX PLANE      │   INFO PANEL         │
│   (Main Visual)      │   (Text & Labels)    │
│                      │                      │
└──────────────────────┴──────────────────────┘

Area Descriptions

Area	Content	Notes
Title	Animation phase title (e.g., "Complex Multiplication")	Appears only in Phases 4-6, fades with phase changes.
Complex Plane	Axes, unit circle (faint), vectors for $z_1$, $z_2$, and their product.	Primary visual area. Vectors are color-coded.
Info Panel	Equations: $z_1 = r_1 e^{i\theta_1}$, $z_2 = r_2 e^{i\theta_2}$, $z_1 z_2 = r_1 r_2 e^{i(\theta_1+\theta_2)}$. Also displays current magnitudes and angles.	Text updates dynamically during Phase 5.

Notes

• Total Duration: ~32 seconds. The two topics are presented sequentially but linked by a transition (grid fade to plane fade-in).
• Prime Finding: The Sieve is demonstrated up to 7. The numbers 2, 3, 5, 7 are highlighted as the active primes during the process. Multiples are crossed out with a color matching the prime that generated them.
• Complex Numbers Example: Use $z_1 = 1+2i$ and $z_2 = 2+1i$ for clear, non-trivial rotation and scaling. Angles should be approximate (e.g., $\sim63^\circ$ and $\sim27^\circ$) for visual clarity.
• Visual Cues: During multiplication, animate $z_1$ rotating by the angle of $z_2$ and then stretching by the magnitude of $z_2$. Use an arc to show angle addition and a temporary circle to show scaling.
• Single Scene Constraint: All phases are implemented within one Scene class, using self.play() transformations and self.wait() calls for timing.