3D Proof of Difference and Sum of Cubes

Animation Specification (Single Manim Scene)

Purpose: Visually demonstrate the algebraic identities

$$a^3 - b^3 = (a-b)(a^2+ab+b^2)$$
$$a^3 + b^3 = (a+b)(a^2-ab+b^2)$$

by constructing 3‑D geometric models that decompose the volumes of cubes of side‑length $a$ and $b$ into rectangular prisms whose dimensions correspond to the factors on the right‑hand side. The animation will also show 2‑D area analogues for intuition.

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Overall Timing (≈ 28 s total)

Segment	Duration	Description
Intro & Setup	3 s	Camera pans to a coordinate grid; axes labeled $x, y, z$. Introduce two colored cubes: blue cube of side $a$ and red cube of side $b$.
Identity 1 – Difference of Cubes	12 s	Decompose $a^3$ and $b^3$ to illustrate $a^3-b^3 = (a-b)(a^2+ab+b^2)$.
Transition	1 s	Fade to next identity while rotating camera 180° around the scene.
Identity 2 – Sum of Cubes	12 s	Decompose $a^3$ and $b^3$ to illustrate $a^3+b^3 = (a+b)(a^2-ab+b^2)$.
Closing	0 s	End with a static view of both final decompositions.

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Detailed Breakdown

1. Intro & Setup (0‑3 s)

• Camera: Starts at a slight elevation (≈30°) looking down the $z$-axis, then slowly zooms out to reveal the full 3‑D grid.
• Objects:
  • Blue Cube: side length $a$, semi‑transparent blue (opacity 0.6), labeled with a subtle $a$ marker on one edge.
  • Red Cube: side length $b$, semi‑transparent red (opacity 0.6), placed adjacent to the blue cube sharing a corner at the origin.
• Axes: Light gray axes with arrowheads; numbers omitted for clarity.
• Transition: Fade‑in of cubes and axes over 0.5 s.

2. Identity 1 – Difference of Cubes (3‑15 s)

Goal: Show that the volume of the larger cube minus the smaller equals the product $(a-b)(a^2+ab+b^2)$.

• Step 2.1 (3‑5 s): Highlight the volume of the blue cube. A translucent overlay displays the volume label $a^3$ (with opaque background for readability).
• Step 2.2 (5‑7 s): Highlight the red cube and display $b^3$.
• Step 2.3 (7‑9 s): Subtract: the red cube fades out, leaving a "hole" inside the blue cube. The remaining shape is a L‑shaped solid representing $a^3-b^3$.
• Step 2.4 (9‑12 s): Decompose the L‑shape into three rectangular prisms:
  1. Prism P1: dimensions $(a-b)\times a\times a$ – colored teal.
  2. Prism P2: dimensions $(a-b)\times a\times b$ – colored orange.
  3. Prism P3: dimensions $(a-b)\times b\times b$ – colored purple.
  • These correspond to the terms $a^2, ab, b^2$ multiplied by $(a-b)$.
• Step 2.5 (12‑15 s): Animate the three prisms sliding apart to form a rectangular block of size $(a-b)\times (a^2+ab+b^2)\times 1$ (depth 1 unit for visual clarity). The block is colored gradient to emphasize the product. An opaque label shows the factorization $(a-b)(a^2+ab+b^2)$.
• Camera: Slowly rotates around the L‑shape during decomposition (≈30° arc) to give depth perception.

3. Transition (15‑16 s)

• Fade out the decomposition block while simultaneously rotating the camera 180° around the vertical axis, preparing for the next identity.

4. Identity 2 – Sum of Cubes (16‑28 s)

Goal: Show that the combined volume of the two cubes equals $(a+b)(a^2-ab+b^2)$.

• Step 4.1 (16‑18 s): Re‑introduce both cubes (blue and red) side‑by‑side, now positioned such that their bases share the same plane but are separated by a small gap.
• Step 4.2 (18‑20 s): Merge: a translucent green prism of dimensions $(a+b)\times a\times a$ appears, representing the term $a^2$ multiplied by $(a+b)$. Simultaneously, the blue cube fades into this prism.
• Step 4.3 (20‑22 s): Add the mixed term: a yellow prism of dimensions $(a+b)\times a\times (-b)$ (visualized as a prism extending in the opposite direction, shown with a dashed outline to indicate subtraction). This encodes the $-ab$ term.
• Step 4.4 (22‑24 s): Add the $b^2$ term: a pink prism of dimensions $(a+b)\times b\times b$ appears, merging with the red cube.
• Step 4.5 (24‑27 s): The three prisms slide together to form a single rectangular block of size $(a+b)\times (a^2-ab+b^2)\times 1$. The block is given a subtle metallic sheen.
• Step 4.6 (27‑28 s): An opaque label displays the factorization $(a+b)(a^2-ab+b^2)$ alongside the total volume label $a^3+b^3$.
• Camera: Performs a gentle orbit (≈45°) around the assembled block to showcase all faces.

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Visual Elements Summary

• Colors: Blue ($a$ cube), Red ($b$ cube), Teal, Orange, Purple (difference decomposition), Green, Yellow (dashed for subtraction), Pink (sum decomposition).
• Transparency: Cubes semi‑transparent; prisms opaque for clarity.
• Labels: All mathematical expressions appear in white text with a dark semi‑transparent rectangular background for readability.
• Lighting: Soft ambient light with a directional light from above‑right to give depth.

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Camera & Perspective

• Single camera throughout the scene.
• Initial view: isometric angle (≈30° elevation, 45° azimuth).
• Rotations: smooth arcs during each decomposition (30‑45°) to maintain viewer orientation.
• No abrupt jumps; all transitions are eased (quadratic in/out).

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Duration & Transitions

• Total runtime ≈ 28 seconds, well within the 30‑second limit.
• All fades and movements use 0.5‑second easing to keep the flow smooth.
• No text beyond essential formula labels; each label has an opaque contrasting background.

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Result: The specification describes a single Manim Scene that visually proves both the difference‑of‑cubes and sum‑of‑cubes identities using 3‑D geometric constructions, suitable for a concise educational video.