勾股定理数形结合动画

Overview

A concise visual proof of the Pythagorean theorem that intertwines geometric construction with numeric area counting. The animation builds a right‑angled triangle, attaches squares on each side, decomposes the two smaller squares into pieces that perfectly fill the largest square, and culminates with the classic equation $a^{2}+b^{2}=c^{2}$.

---

Phases

#	Phase Name	Duration	Description
1	Intro	~3s	Title fades in at the top, then the scene clears to a blank canvas.
2	Construct Triangle	~4s	A right‑angled triangle with legs of lengths $a$ and $b$ and hypotenuse $c$ is drawn in the centre; the right angle is highlighted.
3	Attach Squares	~5s	Squares of side $a$ and $b$ are erected outward from the two legs, while a larger square of side $c$ is erected outward from the hypotenuse.
4	Numeric Area Highlight	~4s	The area of each square is shown by a rapid “fill‑in” of tiny unit squares (grid cells) that count up to $a^{2}$, $b^{2}$ and $c^{2}$ respectively.
5	Decomposition & Rearrangement	~6s	The unit‑square grids from the two smaller squares are animated to slide, rotate, and interlock, forming the large $c$‑square without gaps, visually demonstrating $a^{2}+b^{2}=c^{2}$.
6	Outro	~3s	The equation $$a^{2}+b^{2}=c^{2}$$ appears at the bottom, the whole figure fades out, leaving only the equation for a moment before the scene ends.

---

Layout

┌───────────────────────────────────────┐
│               TOP AREA                 │
├───────────────────────┬───────────────┤
│                       │               │
│        MAIN AREA      │   (optional)  │
│   (triangle + squares)│   RIGHT AREA  │
│                       │   (unused)    │
├───────────────────────┴───────────────┤
│               BOTTOM AREA              │
└───────────────────────────────────────┘

Area Descriptions

Area	Content	Notes
Top	Title "勾股定理数形结合" (appears only in Intro)	Fades in, then fades out before Phase 2
Main	Right‑angled triangle with attached squares; all animated transformations occur here	Central focus; occupies most of the screen
Right	Unused – left empty for possible future annotations
Bottom	Final equation $$a^{2}+b^{2}=c^{2}$$ displayed in large, clear font	Appears in Outro, stays for ~2 seconds

---

Notes

• The animation must be contained within a single Manim Scene class.
• Keep total runtime under 25 seconds to maintain brevity while fully conveying the proof.
• Use only visual elements (shapes, unit‑square grids, highlights); avoid explanatory text except for the title and final equation.
• Transitions between phases should be smooth fades or slide‑in motions to maintain visual continuity.
• The unit‑square grid should be fine enough to suggest counting without overwhelming the viewer.
• Ensure the final rearranged shape aligns perfectly with the large $c$-square to leave no visible gaps.