Now I need to analyze the user's requirements. The user wants a physics simulation displaying 20 numbered balls falling from the center of a rotating heptagon. The requirements are quite specific: gravity, friction, collision detection, the heptagon rotating 360 degrees every 5 seconds, and the balls themselves should also have rotational motion to show the rotation of their numbers.

Let me first break down the main components of this project. A canvas is needed as the main drawing area, then physics entities including the heptagon and the balls. The physics engine needs to handle gravity, collisions, and friction.

For the heptagon, I need to calculate the coordinates of its seven vertices and make it rotate. Rotation speed is 360 degrees/5 seconds, which is 72 degrees per second, converted to radians is approximately 1.2566 radians per second. The heptagon should have wall thickness so the balls can collide with it.

The ball physics is more complex. Each ball needs properties like position, velocity, mass, and rotation angle. Collision detection needs to handle ball-to-ball collisions and ball-to-wall collisions. For ball-to-ball collisions, I need to use vector math to calculate the collision normal and relative velocity, then handle the momentum exchange based on the coefficient of restitution.

For the collision detection algorithm, ball-to-ball collisions can use distance detection; if the distance between centers is less than the sum of the radii, a collision occurs. Ball-to-wall collisions require checking if the distance from the ball to each wall edge is less than the radius.

I also need to consider performance optimization because collision detection is O(n²) complexity. However, 20 balls should be fine. Perhaps a spatial partitioning algorithm could be added later, but the basic version can start with a brute-force approach.

Regarding the interface, a simple UI showing the frame rate and the heptagon's rotation angle might be needed. There should also be a button to reset the simulation.

Audio feedback is also a good idea; adding collision sounds can make the experience more immersive. The Web Audio API can be used to generate simple collision sound effects.

Now I'm thinking about the specific implementation steps. First, set up the HTML structure with a full-screen canvas. Then write the physics engine class, the heptagon class, and the ball class. The main loop will handle updating the physics state and rendering.

For the ball's rotational motion, I can make the number rotate based on the ball's angular velocity, which seems natural. Collision detection needs to precisely calculate the collision point and response vector.

I think this plan should cover all the user's requirements. Now I can start writing the code.

The code has been written to `/workspace/index.html`. Let me verify it works by checking the file was created and has the expected content structure.

The implementation includes:

1. **Heptagon Physics**: 
   - 7-sided polygon with radius 300px
   - Rotates 360° every 5 seconds (2π/5 rad/s)
   - Glowing gradient walls with proper thickness

2. **Ball Physics (20 balls numbered 1-20)**:
   - Spawn from center with slight random offsets
   - Gravity (0.5), friction (0.99), and restitution
   - Rolling rotation based on linear velocity
   - Visual spin shown by rotating numbers

3. **Custom Collision Detection**:
   - Ball-ball: Distance-based with impulse resolution
   - Ball-wall: Point-to-segment distance with moving wall velocity
   - Sub-stepping (8 steps) for stability
   - Angular velocity transfer on collisions

4. **Visual Features**:
   - Radial gradient background
   - Shadows and highlights on balls
   - FPS and angle display
   - Reset button
   - Audio feedback on collisions

The code is completely self-contained in a single HTML file with no external dependencies.