Visualising Euler's Totient Function $\phi(n)$

Overview

A brief animation that introduces the lesser‑known arithmetic function $\phi(n)$, shows how it counts integers coprime to $n$, and visualises the values for the first twelve positive integers. The key takeaway is how $\phi$ behaves irregularly yet reveals hidden structure in the integers.

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Phases

#	Phase Name	Duration	Description
1	Intro	~3s	Title fades in, then a simple number line appears with markers 1‑12.
2	Definition	~4s	A brief visual of the formula \phi(n)=n\prod_{p\mid n}\left(1-\frac{1}{p}\r\right) appears beside the number line; the formula fades out as the next step begins.
3	Coprime Highlight	~6s	For each $n$ from 1 to 12, the animation highlights in green the integers $k$ ($1\le k\le n$) that are coprime to $n$. The count of green dots is displayed as a small bar above the corresponding $n$.
4	Pattern Reveal	~4s	Bars for all $n$ are collected into a bar‑chart style histogram on the right side, showing the sequence $\phi(1),\dots,\phi(12)$. A quick sweep emphasizes the irregular jumps.
5	Outro	~3s	The title re‑appears with the statement “$\phi$ measures hidden coprime structure”. All elements fade out.

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Layout

┌───────────────────────────────────────┐
│               TOP AREA                 │
├───────────────────────┬───────────────┤
│        LEFT AREA       │   RIGHT AREA  │
│  (Number line +       │ (Histogram of │
│   coprime highlights) │   \phi values)│
├───────────────────────┴───────────────┤
│               BOTTOM AREA              │
│   Small formula / caption (optional)   │
└───────────────────────────────────────┘

Area Descriptions

Area	Content	Notes
Top	Title "Euler's Totient Function"	Fades in at start of Intro, fades out at Outro
Left	Number line 1‑12 with dynamic green highlights for coprime integers	Primary visual; each $n$ is highlighted sequentially
Right	Bar chart of $\phi(n)$ values for $n=1\dots12$	Appears in Phase 4, stays until Outro
Bottom	Optional formula $\phi(n)=n\prod_{p\mid n}(1-1/p)$ displayed during Definition phase	Small font, fades with the formula

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Notes

• Total runtime ≈ 20 seconds, well under the 30‑second guideline.
• All visual elements are generated with a single Manim Scene class.
• No explanatory text is used beyond the brief formula; the visual highlighting conveys the concept.
• Colors: green for coprime numbers, muted gray for non‑coprime, blue for histogram bars.
• Transitions are simple fades and linear motions to keep the animation concise.