**Planning physics simulation**

I need to create a single index.html file in the current directory. My focus is on a physics simulation featuring rotating heptagon walls that rotate at a constant angular velocity of 360 degrees every 5 seconds. There will also be 20 balls originating from the center, with their numbers displayed in a way that reflects their spin. I'll handle all the collision detection for both ball-ball and ball-wall interactions, which includes managing friction and restitution. It seems like a fun challenge!**Implementing ball dynamics**

I want to show the rotation of numbers on each ball, representing spin through angular velocity (ω). I’ll update their angular position and draw the text accordingly. For ball-ball collisions, I’ll consider spin exchange via tangential impulse and apply simple rotational dynamics. Each ball will have a moment of inertia (I = 0.5 * m * r^2). When two balls collide, I’ll compute normal impulses and use Coulomb friction to calculate the tangential impulse. For wall collisions, I’ll treat the walls as infinitely massive rotating segments and update the ball’s angular velocity as needed. Sounds like a solid plan!**Improving collision resolution**

I need to implement multiple iterations per frame to handle collisions more effectively. This includes position correction for any overlaps by using a percentage for correction. I’ll use iterative pair-wise resolution—integrating for each frame and repeating N times to stay thorough. For circle-line collision detection, I’ll compute the closest point on a segment and check for penetration. If there's a penetration, I’ll determine the contact normal pointing to the circle center. I also need to calculate the wall velocity considering the rotation angle, ensuring the correct vectors are used. Sounds challenging but manageable!**Setting up simulation parameters**

I’ll use restitution values of around 0.5 for ball-ball collisions and 0.4 for ball-wall interactions, with a friction coefficient of about 0.2. The balls will start at the polygon center but with small random dispersions to prevent overlaps. I could spawn them sequentially with slight offsets and small random velocities to avoid infinite overlap. Each ball will be pushed outward by random offsets. I'll also establish a gravity constant of around 1500 px/s² downward and configure the canvas size, the heptagon's center, and rotation speed accordingly. Time stepping will utilize fixed substeps for stability!**Implementing collision mechanics**

I’m focusing on implementing collision detection and resolution for ball-ball interactions. First, I’ll check if the distance between two balls is less than the sum of their radii. If so, I’ll compute the contact normal and relative velocity. For resolving collisions, I’ll calculate the impulse magnitude based on restitution and apply it to adjust the balls’ velocities, both linearly and angularly. I'll also prepare for wall collisions by calculating the closest point on the segment. Lastly, I'll ensure stability while carefully coding the physics steps in index.html!**Optimizing ball spawning and UI**

The user mentioned dropping balls from the center, so I'll spawn them right at the center but apply slight offsets to avoid overlaps. Each ball will show a number that rotates with its angle, ensuring the number texture reflects the spin. I’ll draw the filled circle and then overlay the text rotated by the ball's angle. I also want to implement a simple UI using just a canvas. For performance, I’ll set substeps to 3 and iterations to about 8 per substep. Now, I’ll prepare to write the code in index.html!