We are creating a single-page web app in one file: index.html.
 We'll use HTML, CSS, and JavaScript (with Canvas for graphics and physics).
 The requirements:
   - A heptagon (7-sided polygon) that spins 360 degrees every 5 seconds.
   - 20 numbered balls (1-20) dropping from the center.
   - Physics: gravity, friction, collisions (ball-to-ball and ball-to-walls), and bouncing.
   - Each ball shows its spin by rotating its number.

 Since this is complex, we'll break it down:

 1. Set up the HTML structure: just a canvas element.
 2. CSS: make the canvas fullscreen? or fixed size? Let's use a fixed size for simplicity, say 800x600.
 3. JavaScript:
    a. Define the canvas and context.
    b. Define the physics constants (gravity, friction, elasticity, etc.).
    c. Create a Ball class:
        - Properties: position, velocity, radius, number, rotation (angle), angular velocity.
        - Method: draw() to draw the ball and the rotated number.
    d. Create a Heptagon class (for the walls) that can rotate.
    e. Implement a function to detect and resolve collisions between balls (using momentum conservation and impulse).
    f. Implement a function to detect collisions between a ball and a rotating heptagon wall and bounce it appropriately.
    g. The walls (heptagon) are rotating at a constant angular speed (360 deg every 5s => 72 deg/s).

 4. Animation loop: update positions, handle collisions, redraw.

 However, note: We must implement collision detection ourselves.

 Steps for collision detection:

   Ball-to-ball:
     - Check distance between centers <= sum of radii.
     - If so, compute normal vector and relative velocity, then apply impulse.

   Ball-to-wall:
     - The heptagon has 7 walls (each as a line segment). Since the heptagon is rotating, we need to transform wall segments to world coordinates each frame.
     - For each wall, we check if the ball is colliding (by projecting the ball's center onto the wall segment and checking the distance to the line and the endpoints).
     - The bounce should reflect the velocity relative to the moving wall (because the wall is rotating). We need to compute the wall's velocity at the point of collision (using angular velocity and radial vector).

 5. The balls start at the center and are given random initial velocities to spread them.

 Due to complexity, we will:

   - Approximate the heptagon as a regular polygon centered at the center of the canvas.
   - The heptagon's rotation is done by keeping an angle that increases by 72 deg/s (72 * (deltaTime/1000)) per frame.

 Challenges:

   - Performance: 20 balls and 7 walls => up to 200 ball-to-ball checks and 140 ball-to-wall checks per frame. This is acceptable.

   - Realism: We'll use a simple physics model with conservation of momentum and energy (with a restitution coefficient). Friction will be applied to the tangential component.

 6. Physics formulas:

    Ball-to-ball collision:

        normal = (b2.position - b1.position).normalized();
        rel_velocity = b2.velocity - b1.velocity;
        vel_along_normal = dot(rel_velocity, normal);

        // Don't collide if moving apart
        if (vel_along_normal > 0) break;

        // Impulse calculation
        e = min(b1.restitution, b2.restitution); // coefficient of restitution
        j = -(1 + e) * vel_along_normal;
        j /= 1/b1.mass + 1/b2.mass;

        impulse = j * normal;
        b1.velocity -= (impulse / b1.mass);
        b2.velocity += (impulse / b2.mass);

        // Additionally, friction: we'll apply tangential impulse as well?

    Ball-to-wall: We treat the wall as having infinite mass.

        normal = ... // pointing outwards
        // The wall's velocity at the collision point: v_wall = angular_velocity * perpendicular(r) (rotational velocity)
        r = collision_point - center_of_heptagon; // vector from center to collision point.
        wall_velocity = new Vector(-angular_velocity * r.y, angular_velocity * r.x);  [Note: if angular_velocity is in radians, then linear velocity = angular_velocity * |r|, but direction perpendicular to r. Actually, (omega x r)].

        rel_velocity = ball.velocity - wall_velocity;
        vel_along_normal = dot(rel_velocity, normal);

        if (vel_along_normal < 0) { // moving towards the wall
            e = ball.restitution * wall_restitution; // wall_restitution = 1? 
            j = -(1 + e) * vel_along_normal;
            impulse = j * normal;
            ball.velocity += impulse;

            // Also, apply friction in the tangential direction?
        }

 7. Implementation:

    We'll set up:

        const canvas = document.getElementById('canvas');
        const ctx = canvas.getContext('2d');

        const balls = [];
        const heptagon = {
            centerX: canvas.width/2,
            centerY: canvas.height/2,
            radius: 200, // distance from center to vertex
            sides: 7,
            angle: 0, // current rotation angle in radians
            angularVelocity: 2 * Math.PI / 5  // 360 deg per 5s = 2*pi rad / 5s = (2*pi/5) rad/s
        };

    Create 20 balls at the center, but with random initial velocity.

    In animation loop:

        update(dt) {
          // Update heptagon angle
          heptagon.angle += heptagon.angularVelocity * dt;

          // Update balls: apply gravity, velocity, friction (damping)
          for (let ball of balls) {
              // Gravity
              ball.vy += gravity * dt;

              // Update position
              ball.x += ball.vx * dt;
              ball.y += ball.vy * dt;

              // Damping (friction with the ground/air)
              ball.vx *= (1 - friction * dt);
              ball.vy *= (1 - friction * dt);

              // Update rotation of the ball based on angular velocity
              ball.rotation += ball.angularVelocity * dt;
          }

          // Collision detection and resolution: ball-ball and ball-heptagon
          // We'll do multiple iterations for stability? Maybe 2-3? Or just one.

          // First, ball-ball
          for (let i = 0; i < balls.length; i++) {
              for (let j = i+1; j < balls.length; j++) {
                  if (colliding(balls[i], balls[j])) {
                      resolveBallCollision(balls[i], balls[j]);
                  }
              }
          }

          // Then, ball-heptagon
          // We need to get the current wall segments (rotated heptagon)
          let walls = getHeptagonWalls(heptagon);
          for (let ball of balls) {
              for (let wall of walls) {
                  let collision = checkBallWallCollision(ball, wall);
                  if (collision) {
                      resolveWallCollision(ball, wall, collision.normal, collision.wallVelocity);
                  }
              }
          }
        }

    Drawing:

        clear canvas
        draw heptagon: for each vertex, connect in a loop
        draw each ball: circle and then draw the number at the ball's center, rotated by ball.rotation.

 8. Heptagon walls:

      To get the vertices: 
        angleStep = (2 * Math.PI) / 7;
        for (let i=0; i<7; i++) {
            let a = heptagon.angle + i * angleStep;
            let x = heptagon.centerX + heptagon.radius * Math.cos(a);
            let y = heptagon.centerY + heptagon.radius * Math.sin(a);
        }

      Walls: array of {x1,y1, x2,y2} for each pair of consecutive vertices.

 9. Ball rotation:

        The ball's angular velocity can be initialized randomly? It might also change due to friction with the wall (tangential force), but that is complex. 
        We'll simulate spin when they bounce off walls: if the ball hits a wall off-center, it gains spin.

        For simplicity, we update the angular velocity during wall collisions:

          // At the point of contact, the relative velocity has a tangential component. We can change the linear velocity and assign some to angular?
          // This is advanced. So alternatively, we can do: 
          //   We assume that the ball is rolling? Not exactly.

        Since the requirement is to show spin via rotation of the number, and we don't have to simulate the exact physics of spin, we can:

          - After resolving the wall collision, set the angular velocity to be proportional to the tangential component of the collision?

        For now, we'll just set a random initial spin and then let it be updated by collisions? Or we skip the effect of collision on spin and just set it to zero? 

        Actually, the problem states: "show each ball's spin via the rotation of its number". So we must have rotation. We can:

          - When the ball collides with the wall, we set the ball's angular velocity to a value based on the tangential component of the impact.

        But for simplicity, we can ignore the torque and just assign the ball an initial angular velocity and then let it be constant? That might be acceptable.

        However, note that without friction, the spin wouldn't change. But we have friction? Actually, if the ball is sliding, the friction could change spin.

        We are going to skip the simulation of spin physics and just set an initial angular velocity and then let it increase arbitrarily? That might not be realistic.

        Alternatively, we can set the angular velocity of the ball to a function of its speed? But that's not physically accurate.

        Given the complexity, we'll do:

          - Initialize each ball's angular velocity to zero.
          - When the ball collides with a wall, we set the angular velocity to a fixed fraction of the inverse of its radius times the tangential component of the relative velocity? This is complex.

        Instead, we'll set the angular velocity to a random value and then update it by a fixed amount on collision? Or just make the angular velocity constant? 

        The requirement does not specify how the spin changes, only that we show it. So we can have:

          - Each ball has an angular velocity that starts at 0.
          - During a wall collision, we add (for example) a scaled tangential component to the angular velocity.

        However, we are not required to simulate the torque. So let's do:

          On wall collision, we do:

            // After resolving the linear collision, we update the angular velocity:
            let tangent = {x: normal.y, y: -normal.x}; // or normal.perp()
            let rel_tangent = dot(rel_velocity, tangent);
            // We'll set angularVelocity = rel_tangent / (ball.radius) * some_factor? 
            // The factor is arbitrary.

        Or we can skip for now? Let's just show the spin by having the ball's angular velocity change randomly at collision? 

        Since the time is short, we'll set an initial angular velocity for each ball (random) and then update it by the magnitude of the collision impulse? 

        Actually, we'll make it simple: the angular velocity of the ball remains constant unless we detect a collision. Then, on collision with a wall, we set:

            ball.angularVelocity = (Math.random() - 0.5) * 10; // arbitrary

        Or if we want to relate to the collision: we can set:

            ball.angularVelocity += 0.1 * rel_tangent / ball.radius; // in radians per second

        But note the unit: dt is in seconds, so we are updating per second. The value we set is arbitrary.

        We'll do:

            let tangent = {x: normal.y, y: -normal.x};
            let tangent_speed = dot(ball.velocity, tangent);
            ball.angularVelocity = tangent_speed / ball.radius;  // This would be for no slipping, but note: the ball might be sliding.

        This would make the ball appear to be rolling without slipping? That might be a good approximation.

        Actually, we are not constrained by strict realism. So we do:

            ball.angularVelocity = tangent_speed / ball.radius;

        Then when we draw the ball, we rotate the number by the ball.rotation angle.

 10. Implementation of ball-wall collision:

        Steps:

          a. For each wall segment (from vertex[i] to vertex[i+1]), we check the distance from the ball center to the segment.

          b. The distance to the segment: 

             let wallVec = {x: x2-x1, y: y2-y1};
             let ballToWallStart = {x: ball.x - x1, y: ball.y - y1};
             let wallLengthSquared = wallVec.x*wallVec.x + wallVec.y*wallVec.y;
             let t = dot(ballToWallStart, wallVec) / wallLengthSquared;
             t = Math.max(0, Math.min(1, t));
             let projection = {x: x1 + t*wallVec.x, y: y1 + t*wallVec.y};

             let dist = Math.sqrt((ball.x - projection.x)**2 + (ball.y - projection.y)**2);

             if (dist <= ball.radius) -> collision.

          c. The normal at the collision: pointing from the projection point to the ball center? Actually, we want the normal pointing outward from the heptagon.

             We can compute the normal to the wall: 
                normal = {x: -wallVec.y, y: wallVec.x} or {x: wallVec.y, y: -wallVec.x}; depending on the winding order.

             Our heptagon is drawn from the center? Actually, we are defining the vertices in counterclockwise? So the outward normal would be normalize( {x: wallVec.y, y: -wallVec.x} )? 

             But to get the normal that points from the wall to the ball: 
                dir = {x: ball.x - projection.x, y: ball.y - projection.y}
                if the dot product of dir and wallNormal is positive, then the normal is wallNormal, otherwise -wallNormal.

             Actually, we want the normal pointing outward from the polygon. Since we defined the heptagon in counterclockwise and the wall vector from vertex0 to vertex1, then the outward normal would be:

                normal = normalize( {x: wallVec.y, y: -wallVec.x} )

             But since the ball is inside the heptagon (the walls form the boundary of the arena), the collision normal should point inward? Wait, the heptagon is the container, so the walls are the boundaries and the inside is the playing area. Therefore, the normal for collision is from the wall into the heptagon? No, actually the ball is inside the heptagon and the wall is preventing it from going outside. So we want the normal pointing outward from the heptagon? Actually, we want the normal pointing inward to the heptagon? 

             Consider: the ball is approaching the wall from inside. We want to reflect the ball off the wall by pushing it away in the normal direction that is from the wall inward? That's not correct. The wall is the boundary. The wall blocks the ball from going through to the outside. So the normal should point from the wall toward the center? Actually, no: the normal of the wall (outward) is pointing away from the center. If the ball is inside and hits the wall, the normal we want for collision is the vector perpendicular to the wall pointing outward? Then we reflect the ball relative to that outward normal. But that would push the ball back inside? Yes.

             So: the collision normal is the vector perpendicular to the wall pointing outward from the heptagon. Therefore, for a counterclockwise winding, the outward normal is: normalize( (wallVec.y, -wallVec.x) )? No, let me check:

                Wall vector: from A to B. 
                The vector to the right (outward in a CCW polygon) is (dy, -dx). So normal = normalize( {x: wallVec.y, y: -wallVec.x} )

          d. The velocity of the wall at the collision point: the wall is moving because the entire heptagon is rotating. The center is fixed? The center is heptagon.center. The collision point (projection) is a point in space that is rotating.

                let r = {x: projection.x - heptagon.centerX, y: projection.y - heptagon.centerY};
                wall_velocity = {x: -heptagon.angularVelocity * r.y, y: heptagon.angularVelocity * r.x}; 

          e. Then we compute the relative velocity and resolve the collision.

 11. Constraints and constants:

        We'll set:
          gravity = 500; // pixels per second squared
          friction = 0.5; // coefficient for air resistance? that we apply to velocity
          ball_radius = 15; 
          elasticity = 0.8; // for ball-ball and ball-wall

 12. Initialization:

        for (let i=0; i<20; i++) {
            let ball = new Ball();
            ball.x = canvas.width/2;
            ball.y = canvas.height/2;
            ball.vx = (Math.random() - 0.5) * 200; // -100 to 100
            ball.vy = (Math.random() - 0.5) * 200;
            ball.radius = 15;
            ball.number = i+1;
            ball.rotation = 0;
            ball.angularVelocity = 0;
            ball.restitution = elasticity;
            ball.mass = Math.PI * ball.radius * ball.radius; // area as mass? or constant? Let's use 1 for all? Then in collision we don't need mass? But we do: impulses are proportional to mass. We can use 1 for simplicity? Then all balls have same mass? Or let mass = radius^2? Since 20 balls with same radius, we can set mass=1 and then we can ignore it? Actually, the mass cancels in the impulse equation if they are the same? So set to 1 for all.

            balls.push(ball);
        }

 13. Ball-ball collision resolution (for same mass):

        let dx = b2.x - b1.x;
        let dy = b2.y - b1.y;
        let dist = Math.sqrt(dx*dx + dy*dy);
        if (dist > b1.radius + b2.radius) return;

        // Normal vector
        let nx = dx / dist;
        let ny = dy / dist;

        // Relative velocity
        let dvx = b2.vx - b1.vx;
        let dvy = b2.vy - b1.vy;
        let vel_along_normal = dvx * nx + dvy * ny;

        // Do not resolve if separating
        if (vel_along_normal > 0) return;

        let e = Math.min(b1.restitution, b2.restitution);
        let j = -(1 + e) * vel_along_normal;
        j /= (1/b1.mass + 1/b2.mass); // if mass=1, j = -(1+e)*vel_along_normal / 2

        let impulse_x = j * nx;
        let impulse_y = j * ny;

        b1.vx -= impulse_x / b1.mass;
        b1.vy -= impulse_y / b1.mass;
        b2.vx += impulse_x / b2.mass;
        b2.vy += impulse_y / b2.mass;

        // Also separate the balls to avoid sticking (positional correction)

        let total_radius = b1.radius + b2.radius;
        let overlap = total_radius - dist;

        // Push each ball away by half the overlap (if masses are equal, otherwise proportional to mass)
        let percent = 0.8; // usually 80%
        let correction = percent * overlap / (b1.mass + b2.mass) * 0.5; // if masses=1 then 0.8 * overlap * 0.5
        let correction_x = nx * correction;
        let correction_y = ny * correction;

        b1.x -= correction_x * b1.mass;
        b1.y -= correction_y * b1.mass;
        b2.x += correction_x * b2.mass;
        b2.y += correction_y * b2.mass;

 14. Due to complexity, we will use the same elasticity for all.

 15. We'll write the entire code in one script tag.

 16. Let's write the index.html.

 17. Note: The simulation might be unstable if dt is large. We'll use requestAnimationFrame and compute dt.

 18. We'll limit the maximum dt to avoid large steps.

 19. We are going to assume the balls start at the center, and then we spread them a bit randomly to avoid initial big collision.

 20. We'll write everything in one file.

 Due to the complexity, we start with the structure and then fill in the functions.

 Let me outline the code:

   index.html:

   <!DOCTYPE html>
   <html lang="en">
   <head>
      <meta charset="UTF-8">
      <meta name="viewport" content="width=device-width, initial-scale=1.0">
      <title>Bouncing Balls in Rotating Heptagon</title>
      <style>
         body { margin: 0; }
         canvas { display: block; background: #eee; }
      </style>
   </head>
   <body>
      <canvas id="canvas" width="800" height="600"></canvas>
      <script>
         // Define constants and variables

         // Classes: Ball

         // Heptagon

         // Helper functions: dot product, normalize, etc.

         // Collision functions: ball-ball, ball-wall

         // Main loop

         // Initialization

         // Start animation
      </script>
   </body>
   </html>

 21. Implementation details:

      We'll use:

        let lastTime = 0;

        function loop(timestamp) {
            let dt = (timestamp - lastTime) / 1000; // in seconds
            if (dt > 0.1) dt = 0.1; // cap at 100 ms
            lastTime = timestamp;

            update(dt);
            draw();

            requestAnimationFrame(loop);
        }

        Then call requestAnimationFrame(loop) after initialization.

 22. Let's write the code.

 23. We must be careful: the heptagon wall collision detection might be the trickiest part.

 Given the time constraints, we might not get everything perfect, but we'll do our best.

 Let's code with comments.