Visualizing the Binomial Expansion $(a+b)^3$ with a 3‑D Cube

Overview

A three‑dimensional illustration of the identity $(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$. A large cube of side $a+b$ is first shown, then it is split into eight coloured blocks that correspond to the four terms of the expansion. The binomial formula appears at the bottom, first in compact form and then expanded.

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Phases

#	Phase Name	Duration	Description
1	Cube Outline	~2 s	The outer wireframe of the cube with side $a+b$ is drawn, establishing the visual reference frame.
2	Decompose into Pieces	~4 s	The eight coloured prisms ($a^3$, three $a^2b$ pieces, three $ab^2$ pieces, $b^3$) fade in from the back, revealing the geometric decomposition.
3	Compact Formula	~2 s	The compact binomial $(a+b)^3$ slides in at the bottom edge and is written.
4	Expanded Formula	~2 s	The compact formula morphs into the full expansion $a^3 + 3a^2b + 3ab^2 + b^3$.
5	Hold / Close‑out	~2 s	A brief pause allowing the viewer to absorb the full picture before the scene ends.

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Layout

┌─────────────────────────────────────────────┐
│                                             │
│              MAIN (3‑D cube)                │
│                                             │
├─────────────────────────────────────────────┤
│  Caption (formula) – persistent at bottom    │
└─────────────────────────────────────────────┘

Area Descriptions

Area	Content	Notes
Main	Wireframe cube, coloured prisms, camera rotation (static orientation)	Primary visual focus; occupies most of the frame.
Caption	LaTeX formula (first compact, then expanded) positioned at the bottom edge, fixed‑in‑frame.	Appears in Phase 3 and transforms in Phase 4.

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Notes

• Assumed parameters: $a = 2.4$, $b = 1.2$; colours BLUE, PURPLE, TEAL, PINK as in the original code. Camera orientation is set to $\phi=65°$, $\theta=-45°$, zoom = 1.15.
• Animation style: Fade‑in with a slight upward shift for each piece (mirroring FadeIn(..., shift=IN)) and a lagged start (lag_ratio=0.12).
• Timing: Total run time ≈ 12 seconds, well within the 30 second limit.
• No extra text is displayed other than the formula; all mathematical meaning is conveyed visually.
• The scene contains a single Scene class (BinomialCube) as required.
• All objects are added to the scene in the order that matches the phases, ensuring smooth transitions.