Bouncing balls in a heptagon · ling-3.0-flash

Let me break down the requirements:

1. A 2D physics simulation of 20 numbered balls inside a rotating heptagon on a canvas
2. Balls numbered 1-20, initially placed in compact non-overlapping arrangement near center, then fall under gravity
3. Heptagon rotates at 1 revolution per 5 seconds, fully visible
4. Ball-ball and ball-wall collision detection (no physics library), including moving-wall velocity in bounce response, with damping/friction
5. Each ball has angular motion and its number rotates with the ball (spin visible)
6. Reasonable sizes/constants for stability - balls contained, no persistent overlap, no unbounded speed
7. Pause/Resume and Reset buttons; Reset restores initial arrangement and wall orientation

Let me design this step by step:

**Heptagon:**
- 7-sided regular polygon
- Center of canvas
- Radius large enough to contain balls
- Rotates at 1 rev/5 sec = 0.4π rad/s or 72°/s

**Balls:**
- 20 balls, numbered 1-20
- Radius ~12-15px each
- Initial compact arrangement near center (like a grid or circle)
- Gravity pulls them down
- Bounce off walls and each other

**Physics:**
- Gravity: ~0.2-0.5 px/frame²
- Damping: coefficient of restitution ~0.8 for ball-ball, ~0.7 for ball-wall
- Friction on walls
- Moving wall velocity consideration for ball-wall collisions

**Collision Detection:**
- Ball-ball: distance between centers < sum of radii
- Ball-wall: distance from ball center to nearest edge of heptagon < ball radius
- For ball-wall with moving walls: compute wall velocity at contact point and include in bounce response

**Ball-wall collision:**
- For each edge of the heptagon, find the closest point on the edge to the ball center
- If distance < ball radius, collision
- Wall velocity at contact point = angular velocity × cross product of (contact_point - center) with the rotation axis
- Use this to compute relative velocity for bounce

Let me think about the heptagon wall collision more carefully.

A regular heptagon centered at (cx, cy) with radius R and rotation angle θ:
- Vertices at angles: θ + 2πk/7 for k=0..6
- Edges connect consecutive vertices

For ball-wall collision with a rotating wall:
- The wall is rotating, so each point on the wall has a tangential velocity
- At a point P on the wall, the velocity is ω × (P - center), where ω is the angular velocity
- For 2D: if center is (cx, cy) and P is (px, py), then:
  - vx = -ω * (py - cy)
  - vy = ω * (px - cx)

For the bounce response with moving walls:
1. Find the contact point on the wall
2. Compute wall velocity at that point
3. Compute relative velocity of ball w.r.t. wall
4. Reflect the relative velocity along the wall normal
5. Apply restitution coefficient
6. Add back the wall velocity component

Let me code this up.

For ball-ball collision with spin (angular motion):
- Each ball has position (x, y), velocity (vx, vy), angular velocity (ω), and angle (θ)
- On collision, transfer some angular momentum
- The number drawn on the ball rotates with the ball's angle

Let me think about the initial compact arrangement. I'll place them in a grid-like pattern near the center, with slight offsets to avoid overlap.

Ball radius: ~12px
Canvas: 800×800 or so
Heptagon radius: ~300px

Let me write the code now.

```html
<!DOCTYPE html>
<html>
<head>
<style>
  body { margin: 0; display: flex; flex-direction: column; align-items: center; background: #1a1a2e; color: #eee; font-family: sans-serif; }
  canvas { border: 2px solid #333; margin-top: 10px; }
  .controls { margin: 10px; display: flex; gap: 10px; }
  button { padding: 8px 20px; font-size: 16px; cursor: pointer; border: none; border-radius: 4px; background: #4a4a8a; color: #eee; }
  button:hover { background: #5a5a9a; }
</style>
</head>
<body>
<div class="controls">
  <button id="pauseBtn">Pause</button>
  <button id="resetBtn">Reset</button>
</div>
<canvas id="canvas" width="800" height="800"></canvas>
<script>
// ... physics code
</script>
</body>
</html>
```

Let me think more carefully about the heptagon collision detection.

For a regular heptagon centered at (cx, cy) with circumradius R and rotation angle θ:
- Vertices: v_k = (cx + R*cos(θ + 2πk/7), cy + R*sin(θ + 2πk/7)) for k=0..6
- Edges: from v_k to v_{k+1}

For ball-wall collision with edge from A to B:
1. The edge normal (pointing inward) can be computed
2. Distance from ball center to the edge line segment
3. If distance < ball radius, collision

The inward normal of edge AB:
- Edge direction: D = B - A
- Normal candidates: (-D.y, D.x) and (D.y, -D.x)
- Choose the one pointing toward the center

For a point P on the edge, the wall velocity due to rotation:
- v_wall = ω × (P - center)
- In 2D: v_wall_x = -ω * (P_y - center_y)
- v_wall_y = ω * (P_x - center_x)

For the bounce response:
- Compute relative velocity: v_rel = v_ball - v_wall_at_contact
- Decompose into normal and tangential components
- Reflect normal component with restitution
- Apply friction to tangential component
- Update ball velocity

Let me also think about the ball-ball collision with spin:
- When two balls collide, they exchange momentum along the collision normal
- They also exchange angular momentum (or transfer some)
- The ball's visual angle rotates with its angular velocity

Let me code this properly now.

For the heptagon, I need to compute the closest point on each edge to the ball center, then check if the distance is less than the ball radius.

Actually, for a convex polygon, I can use the separating axis theorem or just check each edge.

Let me think about the edge collision more carefully:

For edge from vertex A to vertex B:
1. Compute edge direction: E = B - A
2. Compute inward normal N (normalized, pointing toward center)
3. Compute distance from ball center C to the edge line: d = dot(C - A, N)
4. If d < ball_radius, there's a potential collision
5. Find the closest point on the edge segment to C
6. Check if the closest point is within the segment (or handle corner cases)

Actually, for a convex polygon, checking the signed distance to each edge's inward normal is sufficient if I handle the corners properly. But for simplicity, I'll check the distance from the ball center to each edge segment.

Let me think about the closest point on a line segment:

Given segment from A to B, and point P:
- Project P onto the line AB: t = dot(P - A, B - A) / dot(B - A, B - A)
- Clamp t to [0, 1]
- Closest point = A + t * (B - A)

If distance from P to closest point < ball_radius, collision.

For the contact point, I'll use the closest point on the edge.

For the wall normal, I'll use the inward normal of the edge.

Let me now write the full code.

Actually, let me reconsider the initial placement. I want 20 balls in a compact non-overlapping arrangement near the center. A hexagonal close-packing or a grid would work. Let me use a grid pattern with slight random offsets.

Ball radius: 12px
Grid spacing: 28px (so balls don't overlap initially)

Arrangement: 5 columns × 4 rows = 20 balls, centered near the canvas center.

Let me also think about the heptagon size. Canvas is 800×800, center at (400, 400). Heptagon circumradius of about 320px should give enough room.

Ball radius 12px, heptagon radius 320px - that should work. The balls need to fit inside the heptagon. The inradius of a regular heptagon with circumradius R is R * cos(π/7) ≈ R * 0.9009. So inradius ≈ 288px. With ball radius 12px, the effective containment radius is about 276px. That's plenty of room.

Now let me write the full implementation:

```javascript
const canvas = document.getElementById('canvas');
const ctx = canvas.getContext('2d');
const W = canvas.width;
const H = canvas.height;
const CX = W / 2;
const CY = H / 2;

// Physics constants
const GRAVITY = 0.15;
const RESTITUTION = 0.75; // coefficient of restitution
const WALL_RESTITUTION = 0.7;
const FRICTION = 0.99; // velocity damping per frame
const ANGULAR_FRICTION = 0.995;
const BALL_RADIUS = 12;
const HEP_RADIUS = 320; // circumradius of heptagon
const HEP_SIDES = 7;
const ROTATION_SPEED = 2 * Math.PI / 5; // 1 rev per 5 seconds
const BALL_COUNT = 20;

// State
let heptagonAngle = 0;
let paused = false;
let balls = [];

function initBalls() {
  balls = [];
  // Place 20 balls in a grid near center
  const cols = 5;
  const rows = 4;
  const spacing = BALL_RADIUS * 2.5;
  const startX = CX - (cols - 1) * spacing / 2;
  const startY = CY - (rows - 1) * spacing / 2;
  
  let idx = 0;
  for (let r = 0; r < rows; r++) {
    for (let c = 0; c < cols; c++) {
      if (idx >= BALL_COUNT) break;
      balls.push({
        x: startX + c * spacing + (Math.random() - 0.5) * 4,
        y: startY + r * spacing + (Math.random() - 0.5) * 4,
        vx: 0,
        vy: 0,
        angle: 0,
        angVel: 0,
        number: idx + 1,
        radius: BALL_RADIUS
      });
      idx++;
    }
  }
}

function getHepVertices(angle) {
  const verts = [];
  for (let i = 0; i < HEP_SIDES; i++) {
    const a = angle + 2 * Math.PI * i / HEP_SIDES;
    verts.push({
      x: CX + HEP_RADIUS * Math.cos(a),
      y: CY + HEP_RADIUS * Math.sin(a)
    });
  }
  return verts;
}

function getEdgeNormal(A, B, inward) {
  const dx = B.x - A.x;
  const dy = B.y - A.y;
  // Two possible normals
  const nx1 = -dy;
  const ny1 = dx;
  const nx2 = dy;
  const ny2 = -dx;
  // Choose inward
  const midX = (A.x + B.x) / 2;
  const midY = (A.y + B.y) / 2;
  const toCenterX = CX - midX;
  const toCenterY = CY - midY;
  const dot1 = nx1 * toCenterX + ny1 * toCenterY;
  if (dot1 > 0 === inward) return { x: nx1, y: ny1 };
  return { x: nx2, y: ny2 };
}

function normalize(v) {
  const len = Math.sqrt(v.x * v.x + v.y * v.y);
  if (len === 0) return { x: 0, y: 0 };
  return { x: v.x / len, y: v.y / len };
}

function dot(a, b) {
  return a.x * b.x + a.y * b.y;
}

function closestPointOnSegment(P, A, B) {
  const dx = B.x - A.x;
  const dy = B.y - A.y;
  const len2 = dx * dx + dy * dy;
  if (len2 === 0) return { x: A.x, y: A.y };
  let t = ((P.x - A.x) * dx + (P.y - A.y) * dy) / len2;
  t = Math.max(0, Math.min(1, t));
  return { x: A.x + t * dx, y: A.y + t * dy };
}

function wallVelocityAtPoint(P) {
  // Velocity of a point on the rotating heptagon
  // v = omega × r (in 2D)
  const rx = P.x - CX;
  const ry = P.y - CY;
  return {
    x: -ROTATION_SPEED * ry,
    y: ROTATION_SPEED * rx
  };
}

function updatePhysics() {
  const dt = 1; // frame-based
  
  // Update heptagon rotation
  heptagonAngle += ROTATION_SPEED * dt;
  
  const verts = getHepVertices(heptagonAngle);
  
  // Update ball positions and velocities
  for (const ball of balls) {
    ball.vy += GRAVITY;
    ball.vx *= FRICTION;
    ball.vy *= FRICTION;
    ball.angVel *= ANGULAR_FRICTION;
    ball.angle += ball.angVel;
    ball.x += ball.vx;
    ball.y += ball.vy;
  }
  
  // Ball-ball collisions
  for (let i = 0; i < balls.length; i++) {
    for (let j = i + 1; j < balls.length; j++) {
      const a = balls[i];
      const b = balls[j];
      const dx = b.x - a.x;
      const dy = b.y - a.y;
      const dist = Math.sqrt(dx * dx + dy * dy);
      const minDist = a.radius + b.radius;
      if (dist < minDist && dist > 0) {
        // Collision response
        const nx = dx / dist;
        const ny = dy / dist;
        
        // Relative velocity
        const dvx = a.vx - b.vx;
        const dvy = a.vy - b.vy;
        const dvn = dvx * nx + dvy * ny;
        
        if (dvn > 0) { // Approaching
          const impulse = dvn * (1 + RESTITUTION) / 2;
          a.vx -= impulse * nx;
          a.vy -= impulse * ny;
          b.vx += impulse * nx;
          b.vy += impulse * ny;
          
          // Angular momentum transfer
          const tangentialSpeed = (a.angVel + b.angVel) * 0.5;
          a.angVel = tangentialSpeed * 1.2;
          b.angVel = -tangentialSpeed * 1.2;
          
          // Separate balls
          const overlap = minDist - dist;
          const sep = overlap / 2 + 0.5;
          a.x -= sep * nx;
          a.y -= sep * ny;
          b.x += sep * nx;
          b.y += sep * ny;
        }
      }
    }
  }
  
  // Ball-wall collisions
  for (const ball of balls) {
    for (let i = 0; i < HEP_SIDES; i++) {
      const A = verts[i];
      const B = verts[(i + 1) % HEP_SIDES];
      
      const closest = closestPointOnSegment(ball, A, B);
      const dx = ball.x - closest.x;
      const dy = ball.y - closest.y;
      const dist = Math.sqrt(dx * dx + dy * dy);
      
      if (dist < ball.radius && dist > 0) {
        // Normal pointing inward (from wall toward ball center)
        const nx = dx / dist;
        const ny = dy / dist;
        
        // Wall velocity at contact point
        const wallVel = wallVelocityAtPoint(closest);
        
        // Relative velocity of ball w.r.t. wall
        const relVx = ball.vx - wallVel.x;
        const relVy = ball.vy - wallVel.y;
        const relVn = relVx * nx + relVy * ny;
        
        if (relVn < 0) { // Ball moving toward wall
          // Reflect normal component
          ball.vx -= (1 + WALL_RESTITUTION) * relVn * nx;
          ball.vy -= (1 + WALL_RESTITUTION) * relVn * ny;
          
          // Tangential friction
          const tx = -ny;
          const ty = nx;
          const relVt = relVx * tx + relVy * ty;
          ball.vx -= 0.1 * relVt * tx;
          ball.vy -= 0.1 * relVt * ty;
          
          // Angular velocity from friction
          ball.angVel += relVt * 0.05;
          
          // Push ball out of wall
          const penetration = ball.radius - dist;
          ball.x += penetration * nx;
          ball.y += penetration * ny;
        }
      }
    }
  }
}
```

Wait, I need to think more carefully about the wall normal direction. The inward normal of the heptagon edge points toward the center. When a ball is inside the heptagon and collides with a wall, the ball is on the inward side, so the normal from the wall toward the ball center should point inward.

Let me reconsider. For a convex polygon, the inward normals point toward the center. When a ball is inside and hits a wall, the collision normal should point from the wall toward the ball (i.e., inward).

In my closestPointOnSegment approach, the vector from closest point to ball center gives the direction from wall to ball, which is inward. That's correct for the normal.

But wait - I should verify that this normal is consistent with the wall's orientation. For a convex polygon where vertices are ordered counterclockwise, the inward normal of edge A→B is to the left of the edge direction, which is (-dy, dx) normalized.

Let me just use the vector from closest point to ball center as the normal - that's always correct for the collision response regardless of polygon orientation.

Actually, there's a subtlety. The wall velocity at the contact point should be computed correctly. For a rotating polygon, the velocity of a point P is ω × r where r is the vector from center to P. In 2D with counterclockwise rotation (positive ω):
- vx = -ω * ry
- vy = ω * rx

Since ROTATION_SPEED is positive (counterclockwise), this should be correct.

Let me also think about the ball-ball collision angular momentum transfer. I want the balls to spin visibly when they collide. Let me make the angular transfer more noticeable.

Also, I need to make sure the balls don't escape the heptagon. Let me add a safety check - if a ball goes too far from center, push it back.

Let me also think about the Reset functionality. Reset should:
1. Stop the simulation
2. Reset heptagonAngle to 0
3. Reinitialize balls to their starting positions
4. Resume (or stay paused, depending on state)

Actually, the spec says "Reset restores the initial ball arrangement and wall orientation." So reset should restore the heptagon angle to 0 and balls to initial positions.

Let me also think about the Pause/Resume button. When paused, the physics shouldn't update but the canvas should still be drawn (so we can see the current state).

Let me refine my approach and write the complete code now.

One more thing - I need to handle the case where a ball is near a vertex of the heptagon. The closest point on the edge might be a vertex, and the normal might not be well-defined. But using the vector from closest point to ball center should handle this naturally.

Also, I should be careful about the ball-ball collision angular momentum. Let me make it so that the collision imparts spin proportional to the tangential component of the relative velocity.

Let me write the complete code now:

Actually, I realize I should also think about the drawing. The heptagon should be drawn as a filled or stroked polygon. The balls should be drawn as circles with their numbers, and the numbers should be rotated by the ball's angle.

For drawing the number rotated with the ball:
```javascript
ctx.save();
ctx.translate(ball.x, ball.y);
ctx.rotate(ball.angle);
ctx.fillText(ball.number.toString(), 0, 0);
ctx.restore();
```

Let me also add some visual polish - maybe a gradient background, colored balls, etc.

Let me finalize the code. I'll make the heptagon semi-transparent with a visible border, and the balls will have distinct colors based on their number.

Actually, let me keep it simple but visually clear. White or light-colored balls with numbers, a visible heptagon outline.

Let me also think about the initial placement more carefully. I want the balls to be compact and non-overlapping. With ball radius 12, the minimum center-to-center distance is 24px. A grid with spacing 28px should work.

Let me also add some random jitter to the initial positions so they're not perfectly aligned.

Here's my plan for the complete file:

```html
<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="UTF-8">
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<title>Rotating Heptagon Physics</title>
<style>
  /* styles */
</style>
</head>
<body>
  <div class="controls">
    <button id="pauseBtn">Pause</button>
    <button id="resetBtn">Reset</button>
  </div>
  <canvas id="canvas" width="800" height="800"></canvas>
  <script>
    // Full physics simulation code
  </script>
</body>
</html>
```

Let me now write the complete JavaScript. I need to be careful about:

1. The heptagon wall collision - need to check all 7 edges
2. Ball-ball collision - O(n²) for 20 balls is fine
3. Moving wall velocity in bounce response
4. Angular motion of balls
5. Damping/friction

Let me also think about stability. With gravity at 0.15 per frame and 60fps, that's about 9 px/s². That seems reasonable. The balls should settle quickly.

Actually, wait. I should think about what "frame" means. I'll use requestAnimationFrame which runs at ~60fps. So gravity of 0.15 px/frame² means about 0.15 * 60² = 540 px/s², which is about 0.055g. That's a bit low. Let me increase it to maybe 0.3 px/frame² for a more visible fall.

Actually, let me think about this differently. I want the balls to fall at a reasonable speed and bounce nicely. Let me use:
- Gravity: 0.3 px/frame²
- Ball radius: 12px
- Restitution: 0.75 for ball-ball, 0.65 for wall
- Friction: 0.995 per frame (velocity damping)

Let me also add a small random initial velocity to make things more interesting at the start.

Actually, the spec says "let them fall under gravity" so they should start from rest. The compact arrangement near center means they'll fall and spread out.

Let me write the complete code now. I'll be very careful about the physics.

One more consideration: the heptagon rotation speed. 1 rev per 5 seconds = 2π/5 rad/s. At 60fps, that's 2π/5/60 = π/150 rad per frame ≈ 0.0209 rad/frame. That's a slow, visible rotation.

The wall velocity at the edge of the heptagon (radius 320px) is:
v = ω * r = (2π/5) * 320 ≈ 402 px/s ≈ 6.7 px/frame

That's significant enough to affect ball trajectories.

Let me finalize and write the code.

I also want to make sure the heptagon is fully visible. With canvas 800×800 and heptagon radius 320, centered at (400, 400), the heptagon extends from 80 to 720 in both x and y. That's well within the canvas.

Let me write the complete code now.

Actually, I want to reconsider the ball size and spacing. With 20 balls of radius 12 in a 5×4 grid with spacing 28px:
- Total width: 4 * 28 = 112px (from first to last ball center)
- Total height: 3 * 28 = 84px

That's a compact arrangement near the center. Good.

Let me also think about what happens when balls hit the wall. The heptagon has 7 edges. The distance from center to each edge (apothem) is R * cos(π/7) ≈ 320 * 0.9009 ≈ 288px. So balls can get as close as 288 - 12 = 276px from center before hitting a wall. That's fine.

Now, for the ball-wall collision, I need to be careful about the normal direction. When a ball is inside the heptagon and collides with an edge, the normal should point from the wall toward the ball (inward). The vector from the closest point on the edge to the ball center gives this direction naturally.

Let me also add a small amount of random velocity to the balls at initialization so they don't all just sit there. Actually, the spec says "let them fall under gravity" which implies starting from rest. I'll start them at rest.

Wait, but if they start at rest at the center, they'll just fall straight down and pile up at the bottom. That might be boring. Let me add a tiny random velocity to each ball (like ±0.5 px/frame) so they spread out a bit.

Actually, let me re-read the spec: "Initially place them in a compact, non-overlapping arrangement near the enclosure center and let them fall under gravity." So yes, they should start from rest and fall under gravity. The compact arrangement means they're all near the center initially, and then gravity pulls them down and they spread out.

Let me also think about whether I need to worry about the balls being placed exactly at the center. If all balls start at the same point, they'll overlap. I need to place them in a non-overlapping arrangement.

OK, I think my grid approach works. Let me finalize the code.

One more thing: I should handle the case where a ball is already partially outside the heptagon (penetrating a wall) at the start of a frame. I should push it back inside.

Let me also add a boundary check - if a ball goes too far from center (beyond the heptagon), push it back. This is a safety measure.

Actually, the wall collision detection should handle this. But I should make sure the collision resolution is robust.

Let me also think about the angular motion of balls. When a ball bounces off a wall, the tangential friction should impart some spin. When two balls collide, the tangential component of the collision should also impart spin.

For ball-ball collisions, I'll add angular velocity transfer based on the tangential component of the relative velocity at the contact point.

For ball-wall collisions, the tangential friction already imparts some spin (I have that in my code above).

Let me finalize the code now. I'll write it all out.

Actually, I realize I should also think about the visual appearance more carefully. Let me make the heptagon have a nice stroke, maybe with a gradient fill, and the balls should be clearly numbered and colored.

For ball colors, I can use HSL with the ball number determining the hue:
```javascript
const hue = (ball.number / BALL_COUNT) * 360;
```

This gives each ball a distinct color.

OK, let me write the complete code now. I'll be thorough and make sure everything works.

Let me also think about the Reset button behavior. When reset is clicked:
1. Set heptagonAngle = 0
2. Reinitialize balls to their starting positions
3. If paused, stay paused; if running, stay running

Actually, I think Reset should just restore the initial state regardless of pause state. The user can then press Pause/Resume as needed.

Let me also make sure the canvas is properly sized and the layout looks good.

Here's my final plan for the code structure:

1. HTML structure with canvas and controls
2. CSS for layout and styling
3. JavaScript:
   a. Constants and configuration
   b. State variables
   c. Initialization function (initBalls)
   d. Heptagon vertex computation
   e. Collision detection and response
   f. Physics update
   g. Rendering
   h. Animation loop
   i. Event handlers for buttons

Let me write it all now.

Hmm, I realize I should also think about the ball-ball collision more carefully. When two balls collide, I need to:
1. Detect overlap (distance < sum of radii)
2. Compute collision normal (from ball A to ball B)
3. Compute relative velocity along normal
4. If approaching (relative velocity along normal < 0), apply impulse
5. Separate the balls
6. Transfer angular momentum

For the angular momentum transfer, I'll use a simple model:
- The tangential component of the relative velocity at the contact point imparts spin
- spin_impulse = friction_coeff * tangential_relative_velocity

Actually, for simplicity, let me just transfer some angular velocity proportional to the collision impulse. When two balls collide, each ball gets a spin proportional to the impulse and the tangential direction.

Let me think about this more carefully. The contact point between two balls is at the midpoint of the line connecting their centers (since they're equal radius). The tangential direction at the contact point is perpendicular to the collision normal.

The relative velocity at the contact point has both normal and tangential components. The normal component is handled by the restitution. The tangential component should impart spin.

For ball i and ball j colliding:
- Normal direction: n = (pos_j - pos_i) / |pos_j - pos_i|
- Tangential direction: t = (-n.y, n.x)
- Relative velocity at contact: v_rel = v_i - v_j (since contact point is same for both, and both have same radius)
- Normal component: v_rel_n = dot(v_rel, n)
- Tangential component: v_rel_t = dot(v_rel, t)

The tangential component causes spin. I'll add angular velocity to each ball based on the tangential impulse.

Actually, for simplicity and visual effect, let me just do:
- When balls collide, add a small angular velocity to each ball based on the tangential relative velocity
- ball_i.angVel += v_rel_t * 0.02
- ball_j.angVel -= v_rel_t * 0.02

This will make the balls spin when they collide, which is the desired visual effect.

OK, I think I have enough planned. Let me write the complete code.

One more thing about the wall collision. I need to make sure the wall normal is pointing inward (toward the center of the heptagon). When I compute the closest point on an edge to the ball center, the vector from closest point to ball center points from the wall toward the ball. If the ball is inside the heptagon, this vector points inward, which is the correct collision normal.

But wait - what if the ball is outside the heptagon? Then the vector would point outward. I should make sure balls stay inside the heptagon. If a ball somehow escapes, I should push it back in.

To be safe, I'll add a check: if the ball center is farther from the heptagon center than the heptagon radius minus the ball radius, and the closest edge distance check fails, I'll push the ball back toward the center.

Actually, the edge distance check should handle this. If the ball is inside the heptagon and close to an edge, the closest point on that edge will be found and the collision will be detected. If the ball somehow goes outside, the closest point on an edge might still be found, and the normal will point outward, which would push the ball back in.

Hmm, but if the ball is outside the heptagon, the closest point on the edge might not be the right edge. Let me think...

For a convex polygon, if a point is outside, the closest point on the boundary will be on one of the edges, and the vector from closest point to the point will point outward. In my collision response, I check if relVn < 0 (ball moving toward wall). If the ball is outside and moving outward, relVn > 0, so no collision response is applied. But the ball is already outside!

I need to handle the case where a ball is outside the heptagon. I should:
1. Check if the ball center is inside the heptagon
2. If not, push it back inside

To check if a point is inside a convex polygon, I can check that it's on the inward side of all edges.

For edge from A to B with inward normal N:
- Point P is inside if dot(P - A, N) >= 0 for all edges

If a ball is outside, I'll push it back to the nearest point on the boundary.

Actually, let me simplify. I'll just check the distance from the ball center to each edge. If the ball is outside (distance to edge > 0 in the outward direction), I'll push it back.

Wait, I'm overcomplicating this. Let me just use the signed distance to each edge. For a convex polygon with inward normals, the signed distance is positive inside and negative outside. If any signed distance is less than -ball_radius, the ball is outside and I need to push it back.

Let me compute the signed distance properly:

For edge from A to B with inward normal N (pointing toward center):
signed_distance = dot(ball_center - A, N)

If signed_distance < -ball_radius, the ball is penetrating outside the wall. Push it back.

But I also need the correct inward normal. Let me compute it properly.

For a regular heptagon with vertices in counterclockwise order:
- Edge from v[i] to v[i+1]
- Edge direction: D = v[i+1] - v[i]
- Left normal (inward for CCW polygon): N = (-D.y, D.x) normalized

Let me verify: for a CCW polygon, the left normal of each edge points inward. Yes, that's correct.

So:
```javascript
function getEdgeNormal(A, B) {
  const dx = B.x - A.x;
  const dy = B.y - A.y;
  const len = Math.sqrt(dx * dx + dy * dy);
  return { x: -dy / len, y: dx / len }; // left normal for CCW
}
```

And the signed distance from point P to edge AB:
```javascript
const N = getEdgeNormal(A, B);
const dist = (P.x - A.x) * N.x + (P.y - A.y) * N.y;
```

If dist < 0, P is outside the edge (on the outward side).
If dist < -ball_radius, the ball is penetrating outside.

For the collision response when the ball is inside and hitting the wall (dist < ball_radius and dist > 0):
- Normal is N (inward)
- Wall velocity at contact point
- Relative velocity along normal
- If ball is moving toward wall (relVn < 0), bounce

For the case where the ball is outside (dist < 0):
- Push it back inside
- Maybe also reflect velocity

Let me handle both cases in the wall collision code.

OK, I think I've planned enough. Let me write the complete code now.

Let me also think about performance. With 20 balls and 7 edges, the wall collision check is 20 * 7 = 140 per frame. Ball-ball is 20*19/2 = 190 per frame. That's very fast, no performance concerns.

Let me write the final code:

```javascript
(function() {
  const canvas = document.getElementById('canvas');
  const ctx = canvas.getContext('2d');
  const W = canvas.width;
  const H = canvas.height;
  const CX = W / 2;
  const CY = H / 2;
  
  // Constants
  const GRAVITY = 0.25;
  const BALL_RADIUS = 12;
  const HEP_RADIUS = 310;
  const HEP_SIDES = 7;
  const ROTATION_SPEED = 2 * Math.PI / 5; // rad/s, 1 rev per 5 sec
  const BALL_COUNT = 20;
  const RESTITUTION = 0.75;
  const WALL_RESTITUTION = 0.65;
  const FRICTION = 0.998;
  const ANGULAR_FRICTION = 0.995;
  const WALL_FRICTION = 0.15; // tangential friction for wall collisions
  
  let heptagonAngle = 0;
  let paused = false;
  let balls = [];
  let lastTime = 0;
  
  function initBalls() {
    balls = [];
    const cols = 5;
    const rows = 4;
    const spacing = BALL_RADIUS * 2.6;
    const startX = CX - (cols - 1) * spacing / 2;
    const startY = CY - (rows - 1) * spacing / 2 - 30; // slightly above center so they fall
    
    let idx = 0;
    for (let r = 0; r < rows; r++) {
      for (let c = 0; c < cols; c++) {
        if (idx >= BALL_COUNT) break;
        balls.push({
          x: startX + c * spacing + (Math.random() - 0.5) * 3,
          y: startY + r * spacing + (Math.random() - 0.5) * 3,
          vx: (Math.random() - 0.5) * 0.5,
          vy: (Math.random() - 0.5) * 0.5,
          angle: Math.random() * Math.PI * 2,
          angVel: 0,
          number: idx + 1,
          radius: BALL_RADIUS
        });
        idx++;
      }
    }
  }
  
  function getHepVertices(angle) {
    const verts = [];
    for (let i = 0; i < HEP_SIDES; i++) {
      const a = angle + 2 * Math.PI * i / HEP_SIDES;
      verts.push({
        x: CX + HEP_RADIUS * Math.cos(a),
        y: CY + HEP_RADIUS * Math.sin(a)
      });
    }
    return verts;
  }
  
  function closestPointOnSegment(P, A, B) {
    const dx = B.x - A.x;
    const dy = B.y - A.y;
    const len2 = dx * dx + dy * dy;
    if (len2 === 0) return { x: A.x, y: A.y };
    let t = ((P.x - A.x) * dx + (P.y - A.y) * dy) / len2;
    t = Math.max(0, Math.min(1, t));
    return { x: A.x + t * dx, y: A.y + t * dy };
  }
  
  function wallVelocityAtPoint(P, omega) {
    const rx = P.x - CX;
    const ry = P.y - CY;
    return {
      x: -omega * ry,
      y: omega * rx
    };
  }
  
  function updatePhysics(dt) {
    // Update heptagon rotation
    heptagonAngle += ROTATION_SPEED * dt;
    
    const verts = getHepVertices(heptagonAngle);
    const omega = ROTATION_SPEED;
    
    // Update ball positions
    for (const ball of balls) {
      ball.vy += GRAVITY * dt;
      ball.vx *= Math.pow(FRICTION, dt);
      ball.vy *= Math.pow(FRICTION, dt);
      ball.angVel *= Math.pow(ANGULAR_FRICTION, dt);
      ball.angle += ball.angVel * dt;
      ball.x += ball.vx * dt;
      ball.y += ball.vy * dt;
    }
    
    // Ball-ball collisions
    for (let i = 0; i < balls.length; i++) {
      for (let j = i + 1; j < balls.length; j++) {
        const a = balls[i];
        const b = balls[j];
        const dx = b.x - a.x;
        const dy = b.y - a.y;
        const distSq = dx * dx + dy * dy;
        const minDist = a.radius + b.radius;
        if (distSq < minDist * minDist && distSq > 0.0001) {
          const dist = Math.sqrt(distSq);
          const nx = dx / dist;
          const ny = dy / dist;
          
          const dvx = a.vx - b.vx;
          const dvy = a.vy - b.vy;
          const dvn = dvx * nx + dvy * ny;
          
          if (dvn > 0) {
            const impulse = dvn * (1 + RESTITUTION) / 2;
            a.vx -= impulse * nx;
            a.vy -= impulse * ny;
            b.vx += impulse * nx;
            b.vy += impulse * ny;
            
            // Tangential spin
            const tx = -ny;
            const ty = nx;
            const dvt = dvx * tx + dvy * ty;
            a.angVel += dvt * 0.03;
            b.angVel -= dvt * 0.03;
            
            // Separate
            const overlap = minDist - dist;
            const sep = overlap / 2 + 0.5;
            a.x -= sep * nx;
            a.y -= sep * ny;
            b.x += sep * nx;
            b.y += sep * ny;
          }
        }
      }
    }
    
    // Ball-wall collisions
    for (const ball of balls) {
      for (let i = 0; i < HEP_SIDES; i++) {
        const A = verts[i];
        const B = verts[(i + 1) % HEP_SIDES];
        
        const dx = B.x - A.x;
        const dy = B.y - A.y;
        const len = Math.sqrt(dx * dx + dy * dy);
        // Inward normal (left normal for CCW polygon)
        const nx = -dy / len;
        const ny = dx / len;
        
        // Signed distance from ball center to edge line
        const dist = (ball.x - A.x) * nx + (ball.y - A.y) * ny;
        
        if (dist < ball.radius) {
          // Ball is close to or penetrating this edge
          const closest = closestPointOnSegment(ball, A, B);
          const cdx = ball.x - closest.x;
          const cdy = ball.y - closest.y;
          const cdist = Math.sqrt(cdx * cdx + cdy * cdy);
          
          if (cdist < ball.radius && cdist > 0.0001) {
            // Collision normal from wall to ball
            const collNx = cdx / cdist;
            const collNy = cdy / cdist;
            
            // Wall velocity at contact point
            const wVel = wallVelocityAtPoint(closest, omega);
            
            // Relative velocity
            const relVx = ball.vx - wVel.x;
            const relVy = ball.vy - wVel.y;
            const relVn = relVx * collNx + relVy * collNy;
            
            if (relVn < 0) {
              // Bounce
              ball.vx -= (1 + WALL_RESTITUTION) * relVn * collNx;
              ball.vy -= (1 + WALL_RESTITUTION) * relVn * collNy;
              
              // Tangential friction
              const tx = -collNy;
              const ty = collNx;
              const relVt = relVx * tx + relVy * ty;
              ball.vx -= WALL_FRICTION * relVt * tx;
              ball.vy -= WALL_FRICTION * relVt * ty;
              
              // Spin from friction
              ball.angVel += relVt * 0.02;
              
              // Push ball out of wall
              const penetration = ball.radius - cdist;
              ball.x += penetration * collNx;
              ball.y += penetration * collNy;
            }
          } else if (cdist <= 0.0001 && dist < ball.radius) {
            // Ball center is exactly on the edge line, handle specially
            // Just push outward along the edge normal
            ball.x += (ball.radius - dist + 1) * nx;
            ball.y += (ball.radius - dist + 1) * ny;
          }
        }
      }
    }
    
    // Safety: keep balls inside heptagon
    for (const ball of balls) {
      const dx = ball.x - CX;
      const dy = ball.y - CY;
      const distFromCenter = Math.sqrt(dx * dx + dy * dy);
      const maxDist = HEP_RADIUS * Math.cos(Math.PI / HEP_SIDES) - ball.radius - 5;
      if (distFromCenter > maxDist) {
        const scale = maxDist / distFromCenter;
        ball.x = CX + dx * scale;
        ball.y = CY + dy * scale;
        ball.vx *= 0.5;
        ball.vy *= 0.5;
      }
    }
  }
  
  function draw() {
    // Clear
    ctx.fillStyle = '#1a1a2e';
    ctx.fillRect(0, 0, W, H);
    
    // Draw heptagon
    const verts = getHepVertices(heptagonAngle);
    ctx.beginPath();
    ctx.moveTo(verts[0].x, verts[0].y);
    for (let i = 1; i < verts.length; i++) {
      ctx.lineTo(verts[i].x, verts[i].y);
    }
    ctx.closePath();
    ctx.strokeStyle = '#4a9eff';
    ctx.lineWidth = 3;
    ctx.stroke();
    ctx.fillStyle = 'rgba(74, 158, 255, 0.05)';
    ctx.fill();
    
    // Draw balls
    for (const ball of balls) {
      ctx.save();
      ctx.translate(ball.x, ball.y);
      ctx.rotate(ball.angle);
      
      // Ball body
      const hue = (ball.number / BALL_COUNT) * 360;
      ctx.beginPath();
      ctx.arc(0, 0, ball.radius, 0, Math.PI * 2);
      ctx.fillStyle = `hsl(${hue}, 70%, 60%)`;
      ctx.fill();
      ctx.strokeStyle = `hsl(${hue}, 70%, 40%)`;
      ctx.lineWidth = 1.5;
      ctx.stroke();
      
      // Number
      ctx.fillStyle = '#fff';
      ctx.font = 'bold 11px sans-serif';
      ctx.textAlign = 'center';
      ctx.textBaseline = 'middle';
      ctx.fillText(ball.number.toString(), 0, 0);
      
      ctx.restore();
    }
  }
  
  function loop(timestamp) {
    if (!lastTime) lastTime = timestamp;
    const dt = Math.min((timestamp - lastTime) / 16.67, 3); // normalize to ~60fps, cap at 3
    lastTime = timestamp;
    
    if (!paused) {
      updatePhysics(dt);
    }
    
    draw();
    requestAnimationFrame(loop);
  }
  
  // Event handlers
  document.getElementById('pauseBtn').addEventListener('click', function() {
    paused = !paused;
    this.textContent = paused ? 'Resume' : 'Pause';
  });
  
  document.getElementById('resetBtn').addEventListener('click', function() {
    heptagonAngle = 0;
    initBalls();
  });
  
  // Initialize and start
  initBalls();
  requestAnimationFrame(loop);
})();
```

Wait, I need to reconsider the dt handling. Using `dt` as a multiplier for physics means I need to be careful. If I use `dt` as a frame multiplier (where 1.0 is one frame at 60fps), then:
- Gravity should be multiplied by dt
- Velocity should be multiplied by dt for position updates
- But friction should be applied as `friction^dt` or just `friction` per frame

Actually, let me simplify. I'll use a fixed timestep approach where dt is always 1 (one frame), and I'll just use requestAnimationFrame without worrying about variable dt. This is simpler and more stable for a canvas simulation.

But wait, requestAnimationFrame can have variable frame times. If the tab is in the background, dt could be very large. Let me cap dt at 2 or 3.

Actually, let me just use a simple approach: each frame, advance physics by 1 step. If the frame rate drops, the simulation slows down but remains stable. This is the simplest approach and works fine for this kind of simulation.

Let me simplify:

```javascript
function updatePhysics() {
  heptagonAngle += ROTATION_SPEED / 60; // assuming 60fps
  // ... rest of physics with dt=1
}
```

Actually, using a fixed dt of 1/60 second would be more correct. Let me just use dt = 1 for simplicity and adjust the constants accordingly. The physics will run at the display refresh rate.

Hmm, let me just keep it simple and use dt = 1 per frame. The constants I've chosen should work well at 60fps.

Let me also reconsider the FRICTION value. 0.998 per frame means after 1000 frames (about 17 seconds), velocity is reduced to 0.998^1000 ≈ 0.135 of original. That seems reasonable.

ANGULAR_FRICTION of 0.995 per frame means after 1000 frames, angular velocity is 0.995^1000 ≈ 0.0067. That's very damped. Let me increase it to 0.998 for less damping.

OK let me finalize the code. I think I have a good plan. Let me write it out completely.

One more thing - I should make sure the initial ball positions are inside the heptagon. The center of the canvas is (400, 400), and the heptagon has circumradius 310. The apothem (distance from center to edge) is 310 * cos(π/7) ≈ 279. So the center is well inside. The balls start within about 60px of center, so they're definitely inside.

Let me also add a subtle glow or shadow to the heptagon for visual appeal.

Let me also think about the Reset behavior more carefully. When Reset is clicked:
1. heptagonAngle = 0
2. Balls are reinitialized to their starting positions
3. The simulation continues (doesn't pause)

This matches the spec: "Reset restores the initial ball arrangement and wall orientation."

OK, I'm ready to write the final code. Let me also add some visual polish:
- Gradient background
- Glow effect on the heptagon
- Ball shadows
- Smooth animations

Actually, let me keep it clean and not overdo the visual effects. The physics simulation is the focus.

Let me write the complete index.html file now.

I realize I should also think about what happens when the simulation is paused and then resumed. The heptagon should continue rotating from where it left off (since the angle is still being tracked, even if physics isn't updating). Wait, no - when paused, the heptagon shouldn't rotate either. Let me re-read the spec...

"Provide Pause/Resume and Reset." - Pause should stop all motion, Resume should restart it.

So when paused:
- Balls don't move
- Heptagon doesn't rotate

When resumed:
- Everything continues from where it left off

I should not update the heptagon angle or ball positions when paused.

OK, let me finalize the code now. I'll write it to index.html.

Let me also reconsider the ball-ball collision angular momentum transfer. I want the spin to be visible. Let me increase the tangential impulse coefficient.

Also, I want to make sure the balls don't gain too much angular velocity. Let me cap the angular velocity.

Let me also think about the number rendering. The numbers should be clearly visible and rotate with the balls. Using `ctx.rotate(ball.angle)` before drawing the number should achieve this.

For the font size, with ball radius 12, a font size of 10-11px should fit inside the ball.

Let me write the final code now.

Actually, one more thought on the heptagon wall collision. I'm computing the closest point on the edge segment to the ball center, and then checking if the distance is less than the ball radius. This is correct for detecting collisions with the wall.

But I also need to make sure the collision normal points inward (toward the center of the heptagon). The vector from the closest point to the ball center gives this direction naturally when the ball is inside the heptagon.

However, there's an edge case: what if the ball is near a vertex of the heptagon? The closest point might be a vertex, and the normal from the vertex to the ball center might not align well with either adjacent edge's normal. But for a convex polygon, this should still work - the ball will be pushed away from the vertex along the correct direction.

I think my approach is sound. Let me write the code.

One more thing: I should make the heptagon rotation visible. With 1 rev per 5 seconds, the rotation is slow enough to be clearly visible. At 60fps, that's about 0.021 radians per frame, or about 1.2 degrees per frame. That's a nice, visible rotation.

OK, writing the final code now.

Let me also think about whether I need to handle the case where two balls are at the exact same position (dist = 0). I have a check `distSq > 0.0001` which handles this.

Also, I should make sure that after a ball-ball collision, the balls don't overlap in the next frame. The separation step should handle this, but if the balls are moving very fast, they might still overlap. With the constants I've chosen, this shouldn't be a problem.

Let me also add a maximum velocity cap to prevent balls from gaining too much speed:
```javascript
const maxSpeed = 15;
const speed = Math.sqrt(ball.vx * ball.vx + ball.vy * ball.vy);
if (speed > maxSpeed) {
  ball.vx *= maxSpeed / speed;
  ball.vy *= maxSpeed / speed;
}
```

This is a good safety measure.

OK, I'm going to write the complete code now. Let me be thorough and make sure everything is correct.

Let me also reconsider the initial ball placement. I want them in a "compact, non-overlapping arrangement near the enclosure center." A 5×4 grid with spacing 28px (2.33 * ball diameter) should work. The grid will be centered near the canvas center, slightly above so the balls fall down initially.

Actually, let me place them slightly below center so they have room to spread. Or maybe right at center - they'll fall down and spread out naturally.

Let me place the grid centered at the canvas center. The grid extends about ±56px horizontally and ±42px vertically from center. That's well within the heptagon.

One more consideration: I should make sure the initial positions don't have any balls overlapping. With spacing 28px and ball radius 12px, the minimum center-to-center distance is 28px, and the sum of radii is 24px. So there's a 4px gap between balls. Good, no overlap.

Wait, I said spacing of BALL_RADIUS * 2.6 = 12 * 2.6 = 31.2px. That gives a gap of 31.2 - 24 = 7.2px between balls. Even better.

OK, let me write the final code. Here goes:

Actually, I want to reconsider one thing about the physics. I'm using a simple Euler integration, which can be unstable for stiff collisions. For 20 balls with moderate speeds, it should be fine. But I should make sure the time step is small enough.

At 60fps with dt=1, the maximum velocity a ball can have per frame is limited by gravity and wall bounces. With gravity 0.25/frame², after 10 frames of falling, a ball has vy = 2.5 px/frame. After bouncing off a wall with restitution 0.65, it reverses with 0.65 * 2.5 = 1.6 px/frame. These are all small enough that Euler integration is stable.

OK, I'm confident the simulation will be stable. Let me write the code.

Let me also add a subtle trail or motion blur effect... no, that might be too complex. Let me keep it simple.

Here's my final code plan:

```html
<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="UTF-8">
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<title>Rotating Heptagon - Ball Physics</title>
<style>
  * { margin: 0; padding: 0; box-sizing: border-box; }
  body {
    display: flex;
    flex-direction: column;
    align-items: center;
    justify-content: center;
    min-height: 100vh;
    background: #0f0f23;
    font-family: 'Segoe UI', sans-serif;
    color: #e0e0e0;
  }
  h1 {
    margin: 15px 0 5px;
    font-size: 1.4em;
    color: #7ab8ff;
  }
  .controls {
    margin: 5px 0 10px;
    display: flex;
    gap: 12px;
  }
  button {
    padding: 8px 24px;
    font-size: 15px;
    border: none;
    border-radius: 6px;
    cursor: pointer;
    background: #2a3a6a;
    color: #c0d8ff;
    transition: background 0.2s;
  }
  button:hover { background: #3a4a8a; }
  canvas {
    border: 2px solid #2a3a5a;
    border-radius: 4px;
  }
  .info {
    margin-top: 8px;
    font-size: 12px;
    color: #667;
  }
</style>
</head>
<body>
<h1>20 Balls in a Rotating Heptagon</h1>
<div class="controls">
  <button id="pauseBtn">Pause</button>
  <button id="resetBtn">Reset</button>
</div>
<canvas id="canvas" width="800" height="800"></canvas>
<div class="info">Heptagon rotates at 1 rev/5s · Gravity · Ball-ball &amp; ball-wall collisions</div>

<script>
(function() {
  // ... all the code
})();
</script>
</body>
</html>
```

Now let me write the complete JavaScript. I'll be very careful about the physics.

Let me reconsider the wall collision normal. I want to make sure the inward normal is correct for a counterclockwise-ordered polygon.

For a regular heptagon with vertices at angles θ + 2πk/7 (counterclockwise):
- Vertex k: (cos(θ + 2πk/7), sin(θ + 2πk/7))
- Edge from vertex k to vertex k+1

The edge direction is from vertex k to vertex k+1, which goes counterclockwise. The left normal of this edge (perpendicular, pointing left when walking along the edge) points inward.

Left normal of (dx, dy) is (-dy, dx). So:
- Edge direction: (cos(θ + 2π(k+1)/7) - cos(θ + 2πk/7), sin(θ + 2π(k+1)/7) - sin(θ + 2πk/7))
- Left normal: (-(sin(θ + 2π(k+1)/7) - sin(θ + 2πk/7)), cos(θ + 2π(k+1)/7) - cos(θ + 2πk/7))

This should point toward the center. Let me verify with a simple case: θ=0, k=0.
- Vertex 0: (1, 0)
- Vertex 1: (cos(2π/7), sin(2π/7)) ≈ (0.623, 0.782)
- Edge direction: (-0.377, 0.782)
- Left normal: (-0.782, -0.377)

This points toward the lower-left, which is... hmm, let me check. The midpoint of this edge is at approximately (0.811, 0.391). The vector from this midpoint to the center (0,0) is (-0.811, -0.391). The left normal (-0.782, -0.377) is in roughly the same direction. Good, it points inward.

So my formula for the inward normal is correct: for edge from A to B (CCW order), the inward normal is (-dy, dx) normalized, where (dx, dy) = B - A.

Now, in my collision code, I compute the closest point on the edge to the ball center, and then the vector from closest point to ball center gives the collision normal (from wall to ball). This is always correct regardless of polygon orientation.

But I also need the wall's inward normal for the signed distance check. Let me use the left normal formula for that.

Actually, I think I can simplify. Instead of using signed distance to check if the ball is near an edge, I'll just:
1. Find the closest point on each edge to the ball center
2. If the distance from ball center to closest point < ball radius, there's a collision
3. The collision normal is the vector from closest point to ball center (normalized)
4. The wall velocity at the contact point is computed from the rotation

This approach naturally handles all cases (inside, outside, near vertices) because the closest point on the edge always gives the correct direction.

The only issue is if the ball is inside the heptagon but the closest point on an edge is far away (the ball is near the center). In that case, the distance will be large and no collision will be detected, which is correct.

Let me also add a check to make sure the ball doesn't escape the heptagon. If after all collision responses, a ball is still outside the heptagon (all edge distances are negative), I'll push it back to the center.

Actually, let me just compute the signed distance to each edge and check if the ball is inside. If not, push it back.

OK, I think I've been overthinking this. Let me just write the code and it'll work.

Let me write the complete code now. I'll be careful but not overthink every detail.

Here's my final implementation:

```javascript
(function() {
  const canvas = document.getElementById('canvas');
  const ctx = canvas.getContext('2d');
  const W = canvas.width;
  const H = canvas.height;
  const CX = W / 2;
  const CY = H / 2;
  
  // Physics constants
  const GRAVITY = 0.3;
  const BALL_RADIUS = 12;
  const HEP_RADIUS = 310;
  const HEP_SIDES = 7;
  const ROTATION_SPEED = 2 * Math.PI / 5; // rad/s
  const BALL_COUNT = 20;
  const RESTITUTION = 0.75;
  const WALL_RESTITUTION = 0.65;
  const VELOCITY_DAMPING = 0.998;
  const ANGULAR_DAMPING = 0.997;
  const WALL_FRICTION = 0.12;
  const MAX_SPEED = 20;
  
  let heptagonAngle = 0;
  let paused = false;
  let balls = [];
  
  function initBalls() {
    balls = [];
    const cols = 5;
    const rows = 4;
    const spacing = BALL_RADIUS * 2.5;
    const startX = CX - (cols - 1) * spacing / 2;
    const startY = CY - (rows - 1) * spacing / 2;
    
    let idx = 0;
    for (let r = 0; r < rows; r++) {
      for (let c = 0; c < cols; c++) {
        if (idx >= BALL_COUNT) break;
        balls.push({
          x: startX + c * spacing + (Math.random() - 0.5) * 2,
          y: startY + r * spacing + (Math.random() - 0.5) * 2,
          vx: 0,
          vy: 0,
          angle: 0,
          angVel: 0,
          number: idx + 1,
          radius: BALL_RADIUS
        });
        idx++;
      }
    }
  }
  
  function getHepVertices(angle) {
    const verts = [];
    for (let i = 0; i < HEP_SIDES; i++) {
      const a = angle + 2 * Math.PI * i / HEP_SIDES;
      verts.push({
        x: CX + HEP_RADIUS * Math.cos(a),
        y: CY + HEP_RADIUS * Math.sin(a)
      });
    }
    return verts;
  }
  
  function closestPointOnSegment(px, py, ax, ay, bx, by) {
    const dx = bx - ax;
    const dy = by - ay;
    const len2 = dx * dx + dy * dy;
    if (len2 < 0.0001) return { x: ax, y: ay };
    let t = ((px - ax) * dx + (py - ay) * dy) / len2;
    t = Math.max(0, Math.min(1, t));
    return { x: ax + t * dx, y: ay + t * dy };
  }
  
  function updatePhysics() {
    // Update heptagon rotation
    heptagonAngle += ROTATION_SPEED;
    
    const verts = getHepVertices(heptagonAngle);
    
    // Update ball positions
    for (const ball of balls) {
      ball.vy += GRAVITY;
      ball.vx *= VELOCITY_DAMPING;
      ball.vy *= VELOCITY_DAMPING;
      ball.angVel *= ANGULAR_DAMPING;
      ball.angle += ball.angVel;
      ball.x += ball.vx;
      ball.y += ball.vy;
    }
    
    // Ball-ball collisions
    for (let i = 0; i < balls.length; i++) {
      for (let j = i + 1; j < balls.length; j++) {
        const a = balls[i];
        const b = balls[j];
        const dx = b.x - a.x;
        const dy = b.y - a.y;
        const distSq = dx * dx + dy * dy;
        const minDist = a.radius + b.radius;
        if (distSq < minDist * minDist && distSq > 0.01) {
          const dist = Math.sqrt(distSq);
          const nx = dx / dist;
          const ny = dy / dist;
          
          const dvx = a.vx - b.vx;
          const dvy = a.vy - b.vy;
          const dvn = dvx * nx + dvy * ny;
          
          if (dvn > 0) {
            const impulse = dvn * (1 + RESTITUTION) / 2;
            a.vx -= impulse * nx;
            a.vy -= impulse * ny;
            b.vx += impulse * nx;
            b.vy += impulse * ny;
            
            // Tangential spin
            const tx = -ny;
            const ty = nx;
            const dvt = dvx * tx + dvy * ty;
            a.angVel += dvt * 0.04;
            b.angVel -= dvt * 0.04;
            
            // Separate
            const overlap = minDist - dist;
            const sep = overlap / 2 + 0.5;
            a.x -= sep * nx;
            a.y -= sep * ny;
            b.x += sep * nx;
            b.y += sep * ny;
          }
        }
      }
    }
    
    // Ball-wall collisions
    for (const ball of balls) {
      for (let i = 0; i < HEP_SIDES; i++) {
        const A = verts[i];
        const B = verts[(i + 1) % HEP_SIDES];
        
        const closest = closestPointOnSegment(ball.x, ball.y, A.x, A.y, B.x, B.y);
        const cdx = ball.x - closest.x;
        const cdy = ball.y - closest.y;
        const cdist = Math.sqrt(cdx * cdx + cdy * cdy);
        
        if (cdist < ball.radius && cdist > 0.001) {
          const nx = cdx / cdist;
          const ny = cdy / cdist;
          
          // Wall velocity at contact point
          const rx = closest.x - CX;
          const ry = closest.y - CY;
          const wvx = -ROTATION_SPEED * ry;
          const wvy = ROTATION_SPEED * rx;
          
          // Relative velocity
          const relVx = ball.vx - wvx;
          const relVy = ball.vy - wvy;
          const relVn = relVx * nx + relVy * ny;
          
          if (relVn < 0) {
            // Bounce
            ball.vx -= (1 + WALL_RESTITUTION) * relVn * nx;
            ball.vy -= (1 + WALL_RESTITUTION) * relVn * ny;
            
            // Tangential friction
            const tx = -ny;
            const ty = nx;
            const relVt = relVx * tx + relVy * ty;
            ball.vx -= WALL_FRICTION * relVt * tx;
            ball.vy -= WALL_FRICTION * relVt * ty;
            
            // Spin from friction
            ball.angVel += relVt * 0.015;
            
            // Push ball out of wall
            const penetration = ball.radius - cdist;
            ball.x += penetration * nx;
            ball.y += penetration * ny;
          }
        } else if (cdist <= 0.001 && cdist >= 0) {
          // Ball center is on the edge line - push outward along edge normal
          const dx = B.x - A.x;
          const dy = B.y - A.y;
          const len = Math.sqrt(dx * dx + dy * dy);
          const enx = -dy / len;
          const eny = dx / len;
          const sd = (ball.x - A.x) * enx + (ball.y - A.y) * eny;
          if (sd < ball.radius) {
            ball.x += (ball.radius - sd + 1) * enx;
            ball.y += (ball.radius - sd + 1) * eny;
          }
        }
      }
    }
    
    // Cap velocities
    for (const ball of balls) {
      const speed = Math.sqrt(ball.vx * ball.vx + ball.vy * ball.vy);
      if (speed > MAX_SPEED) {
        ball.vx *= MAX_SPEED / speed;
        ball.vy *= MAX_SPEED / speed;
      }
      // Cap angular velocity
      if (Math.abs(ball.angVel) > 0.5) {
        ball.angVel = Math.sign(ball.angVel) * 0.5;
      }
    }
  }
  
  function draw() {
    // Background
    ctx.fillStyle = '#0f0f23';
    ctx.fillRect(0, 0, W, H);
    
    // Draw heptagon
    const verts = getHepVertices(heptagonAngle);
    
    // Glow
    ctx.save();
    ctx.shadowColor = '#4a9eff';
    ctx.shadowBlur = 15;
    ctx.beginPath();
    ctx.moveTo(verts[0].x, verts[0].y);
    for (let i = 1; i < verts.length; i++) {
      ctx.lineTo(verts[i].x, verts[i].y);
    }
    ctx.closePath();
    ctx.strokeStyle = '#4a9eff';
    ctx.lineWidth = 2.5;
    ctx.stroke();
    ctx.restore();
    
    // Fill
    ctx.beginPath();
    ctx.moveTo(verts[0].x, verts[0].y);
    for (let i = 1; i < verts.length; i++) {
      ctx.lineTo(verts[i].x, verts[i].y);
    }
    ctx.closePath();
    ctx.fillStyle = 'rgba(74, 158, 255, 0.03)';
    ctx.fill();
    
    // Draw balls
    for (const ball of balls) {
      ctx.save();
      ctx.translate(ball.x, ball.y);
      ctx.rotate(ball.angle);
      
      const hue = (ball.number / BALL_COUNT) * 360;
      
      // Shadow
      ctx.beginPath();
      ctx.arc(2, 2, ball.radius, 0, Math.PI * 2);
      ctx.fillStyle = 'rgba(0,0,0,0.3)';
      ctx.fill();
      
      // Ball
      ctx.beginPath();
      ctx.arc(0, 0, ball.radius, 0, Math.PI * 2);
      const grad = ctx.createRadialGradient(-3, -3, 2, 0, 0, ball.radius);
      grad.addColorStop(0, `hsl(${hue}, 80%, 75%)`);
      grad.addColorStop(1, `hsl(${hue}, 70%, 45%)`);
      ctx.fillStyle = grad;
      ctx.fill();
      ctx.strokeStyle = `hsl(${hue}, 80%, 35%)`;
      ctx.lineWidth = 1;
      ctx.stroke();
      
      // Number
      ctx.fillStyle = '#fff';
      ctx.font = 'bold 10px sans-serif';
      ctx.textAlign = 'center';
      ctx.textBaseline = 'middle';
      ctx.fillText(ball.number.toString(), 0, 0);
      
      ctx.restore();
    }
  }
  
  function animate() {
    if (!paused) {
      updatePhysics();
    }
    draw();
    requestAnimationFrame(animate);
  }
  
  // Controls
  document.getElementById('pauseBtn').addEventListener('click', function() {
    paused = !paused;
    this.textContent = paused ? 'Resume' : 'Pause';
  });
  
  document.getElementById('resetBtn').addEventListener('click', function() {
    heptagonAngle = 0;
    initBalls();
  });
  
  // Start
  initBalls();
  animate();
})();
```

Wait, I need to double-check the wall collision normal direction. When I compute `closest = closestPointOnSegment(ball.x, ball.y, A.x, A.y, B.x, B.y)`, and then `nx = cdx / cdist` where `cdx = ball.x - closest.x`, this gives a vector from the closest point on the wall to the ball center. If the ball is inside the heptagon, this vector points inward (toward the ball from the wall). That's the correct collision normal for the bounce response.

The wall velocity at the contact point is computed correctly using the rotation formula.

The relative velocity `relVx = ball.vx - wvx` gives the ball's velocity relative to the wall at the contact point. If `relVn < 0`, the ball is moving toward the wall, and we should bounce it.

The bounce response reflects the normal component of the relative velocity:
`ball.vx -= (1 + WALL_RESTITUTION) * relVn * nx`

This is correct for a wall with restitution. The wall is assumed to be infinitely massive (doesn't move due to the collision), but its velocity is accounted for in the relative velocity calculation.

The tangential friction reduces the tangential component of the relative velocity, which also imparts spin to the ball.

I think this is all correct. Let me also add a small check to make sure the ball doesn't tunnel through the wall in a single frame. With the maximum speed of 20 px/frame and ball radius of 12px, the ball could potentially tunnel through a thin wall. But since the heptagon edges are line segments (not thin walls), and I'm checking distance to the closest point on the edge, tunneling shouldn't be an issue as long as the ball doesn't move more than its radius in a single frame. With max speed 20 and ball radius 12, there's a risk of tunneling. Let me reduce MAX_SPEED to 15 or add continuous collision detection.

Actually, for a simple simulation like this, tunneling is unlikely to be a major issue. The balls won't be moving that fast in practice because of the damping and the containment. Let me keep MAX_SPEED at 20 and see how it goes.

One more thing: I should make sure the heptagon is drawn on top of the balls, or the balls on top of the heptagon. I think drawing the heptagon first and then the balls on top looks better, so the balls are clearly visible.

Let me finalize the code. I think it's ready.

Actually, wait. I just realized there might be an issue with the ball-wall collision when the ball is very close to a vertex. The closest point on the edge might be a vertex, and the normal from the vertex to the ball center might not be perpendicular to the edge. This is actually fine - the normal from the closest point to the ball center is always the correct collision normal for a circle-vs-line-segment collision.

But there's a subtle issue: when the ball is near a vertex and colliding with two edges simultaneously, my code will process each edge separately, which might cause the ball to be pushed in different directions by each edge. This could cause jittering. To handle this properly, I'd need to find the edge with the smallest penetration and only respond to that one. But for a simple simulation, processing all edges and letting the ball settle is fine.

Actually, let me think about this more carefully. If a ball is near a vertex and penetrating two edges, my code will push it out of both edges. This could over-correct and push the ball too far. But with the small overlap amounts and the damping, this should be fine in practice.

OK, I'm going to write the final code now. Let me also make sure the canvas and layout look good.

Let me also reconsider the initial ball placement. I want them "near the enclosure center." The center of the canvas is (400, 400). My grid starts at:
- startX = 400 - (5-1) * 30 / 2 = 400 - 60 = 340
- startY = 400 - (4-1) * 30 / 2 = 400 - 45 = 355

So the grid spans from (340, 355) to (460, 445) approximately. That's centered at (400, 400). Good.

Wait, let me recalculate. spacing = BALL_RADIUS * 2.5 = 12 * 2.5 = 30.
- cols = 5, rows = 4
- startX = 400 - 4 * 30 / 2 = 400 - 60 = 340
- startY = 400 - 3 * 30 / 2 = 400 - 45 = 355

Grid spans:
- x: 340 to 340 + 4*30 = 460
- y: 355 to 355 + 3*30 = 445

Center of grid: (400, 400). Perfect.

The grid is 120px wide and 90px tall. The heptagon has an inradius of about 279px. So the balls are well within the heptagon. Good.

Now let me write the final HTML file. I'll put it all together.

Actually, I want to reconsider one more thing. The spec says "Give each ball angular motion and rotate its number with the ball so spin is visible." I need to make sure the angular motion is clearly visible. The ball's `angle` property should rotate the entire ball (including the number) as it moves.

When a ball bounces off a wall, the tangential friction imparts spin. When two balls collide, the tangential relative velocity also imparts spin. I think my current implementation does this.

But I should also consider: when a ball is rolling on a wall (sliding with friction), it should gradually gain spin until it matches the wall's surface velocity. This is more realistic but also more complex. For simplicity, I'll just use the tangential friction impulse approach, which gives a reasonable visual effect.

Let me also make the spin more visible by increasing the angular velocity transfer coefficients. Currently I have 0.04 for ball-ball and 0.015 for ball-wall. Let me increase these to 0.06 and 0.025 respectively.

OK, final code time. Let me write it all out.

Hmm, one more thing. I should also add some visual indicators for the heptagon rotation, like a small marker or arrow at one vertex, so the user can clearly see the rotation.

Let me add a small dot or triangle at one vertex of the heptagon to make the rotation visible.

Actually, the heptagon border itself is enough to show rotation. But a marker would make it more obvious. Let me add a small bright dot at one vertex.

OK, I'll add that. Let me finalize the code now.

Let me also think about the Reset behavior. When Reset is clicked:
1. heptagonAngle = 0
2. initBalls() is called, which creates new balls with random jitter
3. The simulation continues (doesn't pause)

This seems correct. The user can then observe the balls falling again from the same initial arrangement (with the same random jitter, since Math.random() is called again).

Wait, actually, the random jitter means the Reset won't produce exactly the same arrangement. Is that OK? The spec says "Reset restores the initial ball arrangement and wall orientation." The "initial ball arrangement" is the compact non-overlapping grid near the center, which is restored. The random jitter is just a small perturbation. I think this is fine.

Alternatively, I could use a seeded random number generator so the same jitter is applied each time. But that's overkill for this simulation. The small random jitter is fine.

OK, writing the final code now. For real this time.

Let me also consider: should I use `requestAnimationFrame` with delta time or a fixed time step? I'll use a fixed time step (1 frame = 1 physics step) for simplicity and stability. The simulation will run at the display refresh rate.

Actually, using a fixed time step means the simulation speed depends on the frame rate. If the frame rate drops, the simulation slows down. This is acceptable for this kind of visualization.

But I should make sure the physics is stable even if the frame rate varies. With the damping and velocity caps, it should be fine.

Let me write the code now.

One last thing: I want to make sure the canvas is centered on the page and looks good. I'll use flexbox to center it.

OK here's my final code:

Actually, I realize I should also think about what happens when the user resizes the browser window. The canvas is fixed at 800x800, so it won't resize. That's fine for this simulation.

Let me also add a subtle grid or background pattern to make the canvas look nicer... no, let me keep it simple. A dark background is fine.

OK, writing the code now. Final version.

Let me also double-check: the spec says "Account for moving-wall velocity in bounce response." I'm doing this by computing the wall velocity at the contact point and using it in the relative velocity calculation. ✓

"include damping or friction" - I have velocity damping (VELOCITY_DAMPING = 0.998), angular damping (ANGULAR_DAMPING = 0.997), and wall friction (WALL_FRICTION = 0.12). ✓

"Give each ball angular motion and rotate its number with the ball so spin is visible" - I have angVel and angle for each ball, and I rotate the canvas before drawing the number. ✓

"Provide Pause/Resume and Reset" - I have both buttons. ✓

"Reset restores the initial ball arrangement and wall orientation" - Reset sets heptagonAngle = 0 and calls initBalls(). ✓

I think I'm ready to write the final code. Let me do it now.

Let me also reconsider the physics constants one more time:

- GRAVITY = 0.3 px/frame² (at 60fps, this is about 0.3 * 3600 = 1080 px/s² ≈ 0.11g, reasonable)
- BALL_RADIUS = 12px
- HEP_RADIUS = 310px (circumradius)
- RESTITUTION = 0.75 (ball-ball)
- WALL_RESTITUTION = 0.65 (ball-wall)
- VELOCITY_DAMPING = 0.998 per frame (at 60fps, this is about 0.998^60 ≈ 0.886 per second)
- ANGULAR_DAMPING = 0.997 per frame (at 60fps, about 0.997^60 ≈ 0.824 per second)
- WALL_FRICTION = 0.12 (tangential friction coefficient for wall collisions)
- MAX_SPEED = 20 px/frame (at 60fps, about 1200 px/s)

These seem reasonable. Let me finalize.

Wait, I should also think about whether the balls will ever gain enough speed to tunnel through the heptagon wall in a single frame. The heptagon edge length is 2 * HEP_RADIUS * sin(π/7) ≈ 2 * 310 * 0.4339 ≈ 269px. The wall velocity at the edge is ω * R = (2π/5) * 310 ≈ 390 px/s ≈ 6.5 px/frame at 60fps. The maximum ball speed is 20 px/frame. So the relative speed between ball and wall could be up to about 26.5 px/frame. With ball radius 12px, the ball could potentially tunnel through if it's moving fast enough. But with the velocity cap of 20 px/frame and the ball radius of 12px, the ball moves 20px per frame, which is less than its diameter of 24px. So it shouldn't tunnel through a wall in a single frame. Good.

Actually, wait. The wall is moving too. If the ball is moving at 20 px/frame toward a wall that's moving at 6.5 px/frame in the same direction, the relative speed is 26.5 px/frame. But the wall is a line segment, not a thin barrier. The ball needs to penetrate the wall by its radius (12px) before a collision is detected. So the ball could potentially move 20px in one frame, which is less than 24px (2 * radius), so it should be detected by the distance check.

Actually, the issue is more subtle. The ball moves 20px per frame. If it's 13px away from the wall (just outside), after moving 20px, it's 7px inside the wall. The collision detection checks if the distance from ball center to closest point on wall is < ball radius (12px). Since 7px < 12px, the collision is detected. The penetration is 12 - 7 = 5px, and the ball is pushed out by 5px. This works.

But what if the ball is 11px away and moving at 20px/frame toward the wall? After one frame, it's 9px inside the wall (11 - 20 = -9, meaning 9px past the wall). The distance from ball center to closest point on wall is 9px, which is < 12px, so collision is detected. Penetration is 12 - 9 = 3px. Ball is pushed out by 3px. This works too.

What if the ball is 5px away and moving at 20px/frame? After one frame, it's 15px inside the wall. Distance is 15px > 12px, so no collision detected! The ball has tunneled through.

Hmm, this is a problem. But in practice, with the velocity cap of 20 and the wall being a line segment (not a thin barrier), the ball would need to be very close to the wall and moving very fast to tunnel through. And with the damping, balls don't accelerate to very high speeds in this simulation.

To be safe, I could reduce MAX_SPEED to 15 or add a sub-stepping mechanism. But I think for this simulation, it'll be fine. The balls won't reach MAX_SPEED in practice because of the damping and the containment.

Let me reduce MAX_SPEED to 15 to be safe. And I'll also add a check: if a ball is more than 2 * ball.radius away from the heptagon center, push it back toward the center. This is a safety net.

OK, I'm writing the final code now. No more deliberation.

Let me also think about one more edge case: what if two balls are exactly at the same position at the start? With my random jitter of ±1px, this is extremely unlikely but possible. My collision detection handles this with the `distSq > 0.01` check.

OK, final code:

I realize I should also handle the case where `cdist` is 0 (ball center is exactly on the closest point of the edge). In this case, I can't compute a normal from `cdx/cdist`. I handle this with the `cdist > 0.001` check, and have a fallback for `cdist <= 0.001`.

In the fallback case, I use the edge normal (perpendicular to the edge, pointing inward) to push the ball out. This should work.

OK, writing the code now. For real.

Let me also make the heptagon look nicer. I'll draw it with a gradient fill and a glowing border.

Actually, let me keep the drawing simple and clean. A stroke with a subtle fill is enough.

Here's the final code:

Let me also think about the number font size. With ball radius 12, the ball diameter is 24px. A font size of 10px should fit comfortably inside the ball. The number "20" is the widest, at about 10 * 0.6 * 2 = 12px wide, which fits within the 24px diameter. Good.

OK, I'm writing the final code now. Let me put it all together.

Actually, I want to reconsider one thing about the angular velocity transfer during ball-ball collisions. Currently I'm using:
```javascript
a.angVel += dvt * 0.04;
b.angVel -= dvt * 0.04;
```

This means the spin is proportional to the tangential relative velocity. The coefficient 0.04 controls how much spin is imparted. Let me increase this to 0.06 for more visible spin.

Also, for ball-wall collisions:
```javascript
ball.angVel += relVt * 0.015;
```

Let me increase this to 0.03 for more visible spin from wall collisions.

OK, final code. Writing it now.

Let me also add a small detail: when the simulation is paused, I'll draw a "PAUSED" text on the canvas so the user knows it's paused.

Actually, the button text changes to "Resume" when paused, so that's sufficient. But I could also add a subtle overlay. Let me skip that for now.

One more thing: I should make sure the canvas is properly sized on different screens. The canvas is 800x800, which might be too large for small screens. But the spec doesn't mention responsiveness, so I'll keep it fixed.

OK, writing the final code now. Here it is:

Let me reconsider the initial placement one more time. I want the balls to be "near the enclosure center" and "compact." My 5×4 grid centered at (400, 400) with spacing 30px gives a grid that's 120px × 90px. That's compact and near the center. Good.

But I also want the balls to "fall under gravity" after being placed. So they should start with zero velocity. My initBalls() sets vx = 0 and vy = 0 for all balls. Good.

However, I also add a tiny random velocity (±0.5 px/frame) to make the initial fall more interesting. Actually, the spec says "let them fall under gravity," which implies starting from rest. Let me remove the random initial velocity and just let gravity do its work.

Wait, if all balls start at rest in a compact arrangement, they'll just sit there until gravity pulls them down. The bottom balls will hit the wall and bounce, then the wave of collisions will spread the balls outward. This is actually a nice, physically interesting scenario.

But with the compact arrangement near the center, the balls are all close together and will immediately start colliding with each other as they fall. This should create an interesting cascade of collisions.

Let me remove the random initial velocity for a cleaner start.

Actually, let me add just a tiny tiny velocity (like ±0.1 px/frame) to break the perfect symmetry. Otherwise, the bottom row of balls will all hit the wall at exactly the same time and the simulation might look too uniform.

OK, I'll add ±0.2 px/frame random velocity. Very small, just enough to break symmetry.

Final code, for real this time:

Let me also think about whether I need to worry about the heptagon vertices being drawn in the correct order. For a regular heptagon with vertices at angles θ + 2πk/7, the vertices are in counterclockwise order (since the angle increases with k). This means the edges go counterclockwise, and the left normal of each edge points inward. Good.

OK, I'm going to write the complete HTML file now. Let me be very careful and thorough.

One more consideration: I should make sure the heptagon is fully visible on the canvas. With circumradius 310 and center at (400, 400), the heptagon extends from 400-310=90 to 400+310=710 in both x and y. The canvas is 800×800, so there's a 90px margin on each side. Good, the heptagon is fully visible.

Let me also add some padding/margin around the heptagon for visual clarity. I could reduce HEP_RADIUS to 300 to give more margin. Or I could keep it at 310 and accept the 90px margin. 90px is plenty.

OK, writing the final code now. No more changes.

Let me write it out completely and carefully.

Actually, I want to reconsider the wall collision one more time. I want to make sure the collision response is physically correct.

When a ball collides with a moving wall:
1. Compute the contact point on the wall
2. Compute the wall velocity at the contact point: v_wall = ω × r (where r is from center to contact point)
3. Compute the relative velocity: v_rel = v_ball - v_wall
4. Decompose v_rel into normal and tangential components
5. If the ball is moving toward the wall (v_rel · n < 0), apply the bounce:
   - v_rel_n_new = -e * v_rel_n (where e is the coefficient of restitution)
   - v_rel_n_change = v_rel_n_new - v_rel_n = -(1 + e) * v_rel_n
   - Apply this change to the ball's velocity (in the world frame, since the wall velocity is already subtracted)
   - v_ball_new = v_ball - (1 + e) * v_rel_n * n
6. Apply friction to the tangential component:
   - v_rel_t_new = (1 - μ) * v_rel_t (where μ is the friction coefficient)
   - v_ball_new = v_ball_new - (v_rel_t - v_rel_t_new) * t
   - This simplifies to: v_ball_new -= μ * v_rel_t * t (approximately, for small μ)

Wait, I need to be more careful. Let me redo this.

The ball's velocity in the world frame is v_ball. The wall's velocity at the contact point is v_wall. The relative velocity is v_rel = v_ball - v_wall.

The normal component of relative velocity: v_rel_n = v_rel · n
The tangential component: v_rel_t = v_rel · t

After collision (with restitution e and friction μ):
- v_rel_n_new = -e * v_rel_n
- v_rel_t_new = (1 - μ) * v_rel_t (or just 0 for perfectly rough)

The change in relative velocity:
- Δv_rel_n = v_rel_n_new - v_rel_n = -(1 + e) * v_rel_n
- Δv_rel_t = v_rel_t_new - v_rel_t = -μ * v_rel_t

Since the wall is assumed to be infinitely massive (doesn't move due to the collision), the change in ball velocity equals the change in relative velocity:
- Δv_ball_n = Δv_rel_n = -(1 + e) * v_rel_n
- Δv_ball_t = Δv_rel_t = -μ * v_rel_t

So:
- v_ball_new = v_ball + Δv_ball_n * n + Δv_ball_t * t
- v_ball_new = v_ball - (1 + e) * v_rel_n * n - μ * v_rel_t * t

Wait, but this isn't quite right either. The friction should only be applied if the ball is sliding, and it should be limited by the normal impulse. In a simple simulation, we can just apply a fraction of the tangential relative velocity as a friction impulse.

Let me simplify: I'll apply the normal bounce as described, and then apply a fraction of the tangential relative velocity as friction. The fraction (WALL_FRICTION) controls how much friction is applied.

In my code:
```javascript
ball.vx -= (1 + WALL_RESTITUTION) * relVn * nx;
ball.vy -= (1 + WALL_RESTITUTION) * relVn * ny;
// Tangential friction
ball.vx -= WALL_FRICTION * relVt * tx;
ball.vy -= WALL_FRICTION * relVt * ty;
```

Wait, this is subtracting WALL_FRICTION * relVt * t from the ball's velocity. But relVt is the relative tangential velocity (ball relative to wall). If we want to reduce the relative tangential velocity, we should change the ball's velocity in the tangential direction.

If relVt > 0 (ball moving tangentially in the +t direction relative to the wall), then applying a friction impulse in the -t direction to the ball reduces relVt. So:
- Δv_ball_t = -WALL_FRICTION * relVt

This is what I have: `ball.vx -= WALL_FRICTION * relVt * tx` and `ball.vy -= WALL_FRICTION * relVt * ty`.

But wait, the sign depends on the convention. Let me think again.

v_rel_t = relVx * tx + relVy * ty

If relVt > 0, the ball is moving in the +t direction relative to the wall. Friction opposes this motion, so we want to reduce the ball's velocity in the +t direction. The change in ball velocity in the t direction should be negative:
Δv_ball_t = -μ * relVt

So:
v_ball_new_t = v_ball_t - μ * relVt

In terms of x,y components:
v_ball_new_x = v_ball_x - μ * relVt * tx
v_ball_new_y = v_ball_y - μ * relVt * ty

This matches my code. Good.

But I also need to account for the fact that the wall is moving. The relative tangential velocity includes the wall's motion. When I apply the friction impulse to the ball, I'm changing the ball's velocity in the world frame, which changes the relative velocity.

Actually, I think my approach is correct for a simple simulation. The wall is assumed to be infinitely massive, so its velocity doesn't change due to the collision. The friction impulse only affects the ball.

Let me also think about the spin. When a ball slides along a wall, the friction should impart spin. The spin rate is proportional to the tangential friction force. In a simple model:

Δω = (friction_force * R) / I

where R is the ball radius and I is the moment of inertia. For a solid sphere, I = (2/5) * m * R². But since I'm using a simplified model (no mass, just velocity), I can just use a proportionality constant:

Δω = k * relVt / R

where k is a tunable coefficient. In my code, I have:
```javascript
ball.angVel += relVt * 0.015;
```

This is a simple proportionality without the R factor. I think this is fine for a visual simulation.

OK, I'm truly done deliberating. Let me write the final code now.

Hmm, one more tiny thing. I should make sure the ball number is drawn centered at (0, 0) after the translate and rotate. With `ctx.textAlign = 'center'` and `ctx.textBaseline = 'middle'`, the text should be centered. Good.

Also, I should make sure the font size is appropriate. With ball radius 12, the ball diameter is 24px. A font size of 10px should fit comfortably. The number "20" at 10px is about 12px wide, which is less than the 24px diameter. Good.

OK, writing the final code now. This is it.

Let me also add a subtle animation to the heptagon - maybe a slight glow that pulses. Actually, no, let me keep it simple. The rotating heptagon is visually interesting enough on its own.

Let me also add vertex markers (small dots at each vertex) to make the heptagon shape more visible and the rotation more obvious.

OK, final code:

```html
<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="UTF-8">
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<title>Rotating Heptagon - Ball Physics</title>
<style>
  * { margin: 0; padding: 0; box-sizing: border-box; }
  body {
    display: flex;
    flex-direction: column;
    align-items: center;
    justify-content: center;
    min-height: 100vh;
    background: #0a0a1a;
    font-family: 'Segoe UI', Tahoma, Geneva, Verdana, sans-serif;
    color: #c0d0e0;
  }
  h1 {
    margin: 12px 0 4px;
    font-size: 1.3em;
    color: #6ab0ff;
    letter-spacing: 0.5px;
  }
  .controls {
    margin: 4px 0 8px;
    display: flex;
    gap: 14px;
  }
  button {
    padding: 7px 22px;
    font-size: 14px;
    border: 1px solid #3a5a8a;
    border-radius: 5px;
    cursor: pointer;
    background: #1a2a4a;
    color: #b0c8e8;
    transition: background 0.2s, border-color 0.2s;
  }
  button:hover { background: #2a3a6a; border-color: #5a8acc; }
  canvas { display: block; }
  .info {
    margin-top: 6px;
    font-size: 11px;
    color: #445;
  }
</style>
</head>
<body>
<h1>20 Balls in a Rotating Heptagon</h1>
<div class="controls">
  <button id="pauseBtn">Pause</button>
  <button id="resetBtn">Reset</button>
</div>
<canvas id="canvas" width="800" height="800"></canvas>
<div class="info">Heptagon: 1 rev / 5s · Gravity · Ball-ball &amp; ball-wall collisions · Damping &amp; friction</div>

<script>
(function() {
  const canvas = document.getElementById('canvas');
  const ctx = canvas.getContext('2d');
  const W = canvas.width;
  const H = canvas.height;
  const CX = W / 2;
  const CY = H / 2;

  // Physics constants
  const GRAVITY = 0.28;
  const BALL_RADIUS = 12;
  const HEP_RADIUS = 310;
  const HEP_SIDES = 7;
  const ROTATION_SPEED = 2 * Math.PI / 5; // rad/s, 1 rev per 5s
  const BALL_COUNT = 20;
  const RESTITUTION = 0.72;
  const WALL_RESTITUTION = 0.62;
  const VELOCITY_DAMPING = 0.997;
  const ANGULAR_DAMPING = 0.996;
  const WALL_FRICTION = 0.10;
  const BALL_FRICTION = 0.08;
  const MAX_SPEED = 18;
  const MAX_ANG_VEL = 0.4;

  let heptagonAngle = 0;
  let paused = false;
  let balls = [];

  function initBalls() {
    balls = [];
    const cols = 5;
    const rows = 4;
    const spacing = BALL_RADIUS * 2.5;
    const startX = CX - (cols - 1) * spacing / 2;
    const startY = CY - (rows - 1) * spacing / 2;

    let idx = 0;
    for (let r = 0; r < rows; r++) {
      for (let c = 0; c < cols; c++) {
        if (idx >= BALL_COUNT) break;
        balls.push({
          x: startX + c * spacing + (Math.random() - 0.5) * 1.5,
          y: startY + r * spacing + (Math.random() - 0.5) * 1.5,
          vx: (Math.random() - 0.5) * 0.3,
          vy: (Math.random() - 0.5) * 0.3,
          angle: Math.random() * Math.PI * 2,
          angVel: 0,
          number: idx + 1,
          radius: BALL_RADIUS
        });
        idx++;
      }
    }
  }

  function getHepVertices(angle) {
    const verts = [];
    for (let i = 0; i < HEP_SIDES; i++) {
      const a = angle + 2 * Math.PI * i / HEP_SIDES;
      verts.push({
        x: CX + HEP_RADIUS * Math.cos(a),
        y: CY + HEP_RADIUS * Math.sin(a)
      });
    }
    return verts;
  }

  function closestPointOnSegment(px, py, ax, ay, bx, by) {
    const dx = bx - ax;
    const dy = by - ay;
    const len2 = dx * dx + dy * dy;
    if (len2 < 1e-6) return { x: ax, y: ay };
    let t = ((px - ax) * dx + (py - ay) * dy) / len2;
    t = Math.max(0, Math.min(1, t));
    return { x: ax + t * dx, y: ay + t * dy };
  }

  function updatePhysics() {
    heptagonAngle += ROTATION_SPEED;

    const verts = getHepVertices(heptagonAngle);

    // Update ball positions and velocities
    for (const ball of balls) {
      ball.vy += GRAVITY;
      ball.vx *= VELOCITY_DAMPING;
      ball.vy *= VELOCITY_DAMPING;
      ball.angVel *= ANGULAR_DAMPING;
      ball.angle += ball.angVel;
      ball.x += ball.vx;
      ball.y += ball.vy;
    }

    // Ball-ball collisions
    for (let i = 0; i < balls.length; i++) {
      for (let j = i + 1; j < balls.length; j++) {
        const a = balls[i];
        const b = balls[j];
        const dx = b.x - a.x;
        const dy = b.y - a.y;
        const distSq = dx * dx + dy * dy;
        const minDist = a.radius + b.radius;
        if (distSq < minDist * minDist && distSq > 0.01) {
          const dist = Math.sqrt(distSq);
          const nx = dx / dist;
          const ny = dy / dist;

          const dvx = a.vx - b.vx;
          const dvy = a.vy - b.vy;
          const dvn = dvx * nx + dvy * ny;

          if (dvn > 0) {
            const impulse = dvn * (1 + RESTITUTION) / 2;
            a.vx -= impulse * nx;
            a.vy -= impulse * ny;
            b.vx += impulse * nx;
            b.vy += impulse * ny;

            // Tangential spin transfer
            const tx = -ny;
            const ty = nx;
            const dvt = dvx * tx + dvy * ty;
            a.angVel += dvt * BALL_FRICTION * 0.5;
            b.angVel -= dvt * BALL_FRICTION * 0.5;

            // Separate overlapping balls
            const overlap = minDist - dist;
            if (overlap > 0) {
              const sep = overlap / 2 + 0.3;
              a.x -= sep * nx;
              a.y -= sep * ny;
              b.x += sep * nx;
              b.y += sep * ny;
            }
          }
        }
      }
    }

    // Ball-wall collisions
    for (const ball of balls) {
      for (let i = 0; i < HEP_SIDES; i++) {
        const A = verts[i];
        const B = verts[(i + 1) % HEP_SIDES];

        const closest = closestPointOnSegment(ball.x, ball.y, A.x, A.y, B.x, B.y);
        const cdx = ball.x - closest.x;
        const cdy = ball.y - closest.y;
        const cdist = Math.sqrt(cdx * cdx + cdy * cdy);

        if (cdist < ball.radius && cdist > 0.001) {
          const nx = cdx / cdist;
          const ny = cdy / cdist;

          // Wall velocity at contact point (due to rotation)
          const rx = closest.x - CX;
          const ry = closest.y - CY;
          const wvx = -ROTATION_SPEED * ry;
          const wvy = ROTATION_SPEED * rx;

          // Relative velocity of ball w.r.t. wall
          const relVx = ball.vx - wvx;
          const relVy = ball.vy - wvy;
          const relVn = relVx * nx + relVy * ny;

          if (relVn < 0) {
            // Normal bounce
            ball.vx -= (1 + WALL_RESTITUTION) * relVn * nx;
            ball.vy -= (1 + WALL_RESTITUTION) * relVn * ny;

            // Tangential friction
            const tx = -ny;
            const ty = nx;
            const relVt = relVx * tx + relVy * ty;
            ball.vx -= WALL_FRICTION * relVt * tx;
            ball.vy -= WALL_FRICTION * relVt * ty;

            // Spin from wall friction
            ball.angVel += relVt * 0.02;

            // Push ball out of wall
            const penetration = ball.radius - cdist;
            if (penetration > 0) {
              ball.x += penetration * nx;
              ball.y += penetration * ny;
            }
          }
        } else if (cdist <= 0.001) {
          // Ball center is on the edge line
          const dx = B.x - A.x;
          const dy = B.y - A.y;
          const len = Math.sqrt(dx * dx + dy * dy);
          const enx = -dy / len;
          const eny = dx / len;
          const sd = (ball.x - A.x) * enx + (ball.y - A.y) * eny;
          if (sd < ball.radius) {
            ball.x += (ball.radius - sd + 0.5) * enx;
            ball.y += (ball.radius - sd + 0.5) * eny;
          }
        }
      }
    }

    // Safety: cap velocities and keep balls inside heptagon
    for (const ball of balls) {
      const speed = Math.sqrt(ball.vx * ball.vx + ball.vy * ball.vy);
      if (speed > MAX_SPEED) {
        ball.vx *= MAX_SPEED / speed;
        ball.vy *= MAX_SPEED / speed;
      }
      if (Math.abs(ball.angVel) > MAX_ANG_VEL) {
        ball.angVel = Math.sign(ball.angVel) * MAX_ANG_VEL;
      }

      // Safety: keep inside heptagon
      const dx = ball.x - CX;
      const dy = ball.y - CY;
      const distFromCenter = Math.sqrt(dx * dx + dy * dy);
      const maxDist = HEP_RADIUS * Math.cos(Math.PI / HEP_SIDES) - ball.radius - 8;
      if (distFromCenter > maxDist) {
        const scale = maxDist / distFromCenter;
        ball.x = CX + dx * scale;
        ball.y = CY + dy * scale;
        ball.vx *= 0.3;
        ball.vy *= 0.3;
      }
    }
  }

  function draw() {
    // Background
    ctx.fillStyle = '#0d0d20';
    ctx.fillRect(0, 0, W, H);

    // Subtle radial gradient background
    const bgGrad = ctx.createRadialGradient(CX, CY, 0, CX, CY, HEP_RADIUS + 50);
    bgGrad.addColorStop(0, 'rgba(20, 30, 60, 0.5)');
    bgGrad.addColorStop(1, 'rgba(10, 10, 25, 0)');
    ctx.fillStyle = bgGrad;
    ctx.fillRect(0, 0, W, H);

    const verts = getHepVertices(heptagonAngle);

    // Draw heptagon fill
    ctx.beginPath();
    ctx.moveTo(verts[0].x, verts[0].y);
    for (let i = 1; i < verts.length; i++) {
      ctx.lineTo(verts[i].x, verts[i].y);
    }
    ctx.closePath();
    ctx.fillStyle = 'rgba(40, 80, 140, 0.04)';
    ctx.fill();

    // Draw heptagon edges with glow
    ctx.save();
    ctx.shadowColor = '#3a7acc';
    ctx.shadowBlur = 12;
    ctx.beginPath();
    ctx.moveTo(verts[0].x, verts[0].y);
    for (let i = 1; i < verts.length; i++) {
      ctx.lineTo(verts[i].x, verts[i].y);
    }
    ctx.closePath();
    ctx.strokeStyle = '#3a8add';
    ctx.lineWidth = 2.5;
    ctx.stroke();
    ctx.restore();

    // Draw vertex markers
    for (const v of verts) {
      ctx.beginPath();
      ctx.arc(v.x, v.y, 4, 0, Math.PI * 2);
      ctx.fillStyle = '#5aacff';
      ctx.fill();
    }

    // Draw balls
    for (const ball of balls) {
      ctx.save();
      ctx.translate(ball.x, ball.y);
      ctx.rotate(ball.angle);

      const hue = (ball.number / BALL_COUNT) * 360;

      // Ball shadow
      ctx.beginPath();
      ctx.arc(2, 3, ball.radius, 0, Math.PI * 2);
      ctx.fillStyle = 'rgba(0, 0, 0, 0.25)';
      ctx.fill();

      // Ball body
      ctx.beginPath();
      ctx.arc(0, 0, ball.radius, 0, Math.PI * 2);
      const grad = ctx.createRadialGradient(-3, -3, 1, 0, 0, ball.radius);
      grad.addColorStop(0, `hsl(${hue}, 75%, 72%)`);
      grad.addColorStop(1, `hsl(${hue}, 65%, 42%)`);
      ctx.fillStyle = grad;
      ctx.fill();
      ctx.strokeStyle = `hsl(${hue}, 75%, 30%)`;
      ctx.lineWidth = 0.8;
      ctx.stroke();

      // Number
      ctx.fillStyle = '#ffffff';
      ctx.font = 'bold 10px sans-serif';
      ctx.textAlign = 'center';
      ctx.textBaseline = 'middle';
      ctx.fillText(ball.number.toString(), 0, 0.5);

      ctx.restore();
    }
  }

  function animate() {
    if (!paused) {
      updatePhysics();
    }
    draw();
    requestAnimationFrame(animate);
  }

  // Controls
  document.getElementById('pauseBtn').addEventListener('click', function() {
    paused = !paused;
    this.textContent = paused ? 'Resume' : 'Pause';
  });

  document.getElementById('resetBtn').addEventListener('click', function() {
    heptagonAngle = 0;
    initBalls();
  });

  // Initialize and start
  initBalls();
  animate();
})();
</script>
</body>
</html>
```

Wait, I just realized I should double-check the wall normal direction for the edge case where cdist <= 0.001. In that case, I'm using the edge normal (-dy/len, dx/len) which is the left normal (inward for CCW polygon). But I should verify this is the correct direction.

For a CCW polygon, the left normal of edge A→B points inward. The edge direction is (dx, dy) = B - A. The left normal is (-dy, dx). Normalized: (-dy/len, dx/len). This points inward. Good.

But wait, I should also check the signed distance to make sure the ball is on the inward side of the edge. If sd < 0, the ball is outside the edge (on the outward side), and I should push it inward. If sd > 0, the ball is inside, and I should push it further inward if it's penetrating.

In my code, I check `if (sd < ball.radius)` and push the ball outward along the edge normal. But if the ball is already outside (sd < 0), pushing it outward along the inward normal makes it go further outside! I should push it inward instead.

Let me fix this:
```javascript
if (sd < ball.radius) {
  // Push ball inward along the edge normal
  ball.x += (ball.radius - sd) * enx;
  ball.y += (ball.radius - sd) * eny;
}
```

Wait, if sd < 0 (ball outside), then (ball.radius - sd) > ball.radius, which pushes the ball further in the +en direction (inward). That's correct - it pushes the ball back inside.

If sd > 0 but < ball.radius (ball inside but penetrating), then (ball.radius - sd) > 0, which pushes the ball further inward. That's also correct - it pushes the ball away from the wall.

So the code is correct! The edge normal enx, eny points inward, and we always push the ball in the inward direction when it's too close to the edge.

Good.

Actually wait, I need to reconsider. If the ball is outside the heptagon (sd < 0), and I push it inward, that's correct. But if the ball is inside and close to the wall (0 < sd < ball.radius), pushing it further inward is also correct - it moves the ball away from the wall.

But what if the ball is inside but very close to the wall (sd ≈ 0)? Then (ball.radius - sd) ≈ ball.radius, and we push the ball inward by about ball.radius. That might push it too far from the wall. But this is a rare edge case and the correction is small.

I think the code is fine. Let me finalize it.

Actually, I realize there's another issue. When I check `cdist <= 0.001`, this means the ball center is essentially on the edge line. In this case, I use the edge normal to push the ball. But I should also check if the ball is on the correct side (inside the heptagon). If sd < 0, the ball is outside, and I should push it inward. If sd > 0, the ball is inside but very close to the edge, and I should push it away from the edge.

In both cases, pushing in the +en direction (inward normal) is correct:
- If sd < 0 (outside), pushing inward moves the ball back inside
- If 0 < sd < ball.radius (inside, penetrating), pushing inward moves the ball away from the wall

So my code is correct for this edge case too.

OK, I'm truly done now. Let me write the final file.

Actually, one more thought: I should also handle the case where a ball is outside the heptagon and not near any edge (i.e., it escaped through a vertex). My safety check at the end of updatePhysics handles this by pushing the ball back toward the center if it's too far from the center.

OK, truly final code now. Let me write it.

I also want to make sure the Reset button works correctly. When clicked:
1. heptagonAngle = 0
2. initBalls() is called, which creates new balls with new random jitter
3. The simulation continues

This restores the wall orientation (angle = 0) and the initial ball arrangement (grid near center). The random jitter means the exact positions are slightly different each time, but the overall arrangement is the same.

If the user wants exactly the same arrangement on reset, I could use a fixed seed for the random number generator. But that's overkill for this simulation. The small random jitter is fine.

OK, writing the final code now. I'll also make sure the file is well-formatted and readable.

Let me also reconsider whether I should use `Math.pow(FRICTION, dt)` for velocity damping when using variable dt. Since I'm using a fixed dt of 1 (one frame per physics step), I can just multiply by the damping factor directly:

```javascript
ball.vx *= VELOCITY_DAMPING;
ball.vy *= VELOCITY_DAMPING;
```

This is what I have. Good.

One final thought: I should make sure the simulation looks good from the start. When the page loads, the balls are in a compact grid near the center, and gravity immediately starts pulling them down. The first few frames should show the balls starting to fall and the bottom balls hitting the heptagon wall. This should be visually interesting from the start.

OK, I'm writing the final code now. Let me also add a small detail: I'll draw a small circle at the center of the heptagon to mark the rotation center.

Actually, that might be distracting. Let me skip it.

Here's the final code. I'm writing it to index.html now.

The file was written but it seems to have been cut off. Let me check the file and rewrite it completely.

The file was truncated. Let me write the complete file in a single write call, making sure it's not too long. I need to be more concise in my approach.

The file was written successfully. Let me verify it's complete and then open it in the browser to test.

Good, the file is complete. Let me open it in the browser to test it.

Python3 isn't available. Let me try another approach to serve the file or just open it directly in the browser.

No Python or browser available directly. Let me try using the browser tool to open the file.

The page loaded but the canvas isn't showing because the browser tool doesn't support images. Let me check the console for any errors and verify the simulation is working properly.

The browser tool doesn't support images, so I can't visually verify the canvas rendering. But I can check the console for JavaScript errors and verify the DOM structure. Let me also try to take a screenshot to see the canvas.

Let me check for console errors first, then try a screenshot.

The browser tool doesn't support images/screenshots. Let me check the console for errors and verify the simulation is working by checking the DOM and any console output. Let me also try to use node.js to verify the JavaScript syntax is valid.

Good, the JavaScript syntax is valid. Now let me check the simulation more carefully by running it in a headless browser or by examining the code logic more carefully. Let me also try to use the browser tool to interact with the simulation.

Let me also verify the key behaviors:
1. 20 balls numbered 1-20
2. Heptagon rotates at 1 rev/5s
3. Ball-ball and ball-wall collisions
4. Moving wall velocity in bounce response
5. Angular motion with number rotation
6. Pause/Resume and Reset
7. Initial compact arrangement near center

Let me check the console for any runtime errors by trying to interact with the page.

The browser tool doesn't seem to support screenshots or images. Let me try a different approach - I'll use node.js to run a more thorough verification of the code logic, and also try to use the browser tool's motion or type actions to interact with the simulation.

Let me also try to verify the simulation is working by checking the canvas content or console logs more carefully. Let me also try to take a screenshot with a different approach.

Actually, let me focus on what I can verify:
1. The JavaScript syntax is valid (confirmed)
2. The DOM structure is correct (confirmed - title, buttons, canvas, info text)
3. The simulation logic looks correct

Let me try to interact with the Pause button to verify it works, and then check the Reset button too. I should also try to verify the simulation runs correctly by checking console output after some time.

Let me also try to use the browser tool's motion feature to see if I can get any visual feedback, or try a different approach to verify the canvas rendering.

Actually, let me just verify the code logic more carefully by reviewing it, and also try to run a quick node.js test to simulate some physics steps.

The simulation is working well:
- 20 balls numbered 1-20 ✓
- All balls near center initially ✓
- No initial overlaps ✓
- After 5 seconds, all balls still in bounds ✓
- All balls have spin (angular velocity) ✓
- Max speed is reasonable (1.32 px/frame) ✓
- Numbers are correct (1-20) ✓

Wait, the heptagon angle shows 0.000 rad after 300 frames. That's because I didn't update heptAngle in the test script... let me check. Actually, looking at my test code, I did update heptAngle += ROT_SPEED each frame. Let me check: ROT_SPEED = 2*PI/5, so after 300 frames, heptAngle = 300 * 2*PI/5 = 120*PI. 120*PI mod 2*PI = 0. So the angle is exactly 0 because 300 frames * (2π/5) / (2π) = 300/5 = 60 full revolutions. So the angle mod 2π is 0. That's correct!

But wait, that means after exactly 5 seconds (300 frames at 60fps), the heptagon has completed exactly 60 full revolutions and is back to its starting orientation. That's actually a good sign - it means the rotation is working correctly.

Let me also verify the heptagon rotation speed is correct: 1 rev per 5 seconds = 60 rev per 300 frames. Yes, that's correct.

Now let me also check the Reset functionality and verify the simulation is stable over a longer period. Let me also check that the wall collision with moving wall velocity works correctly.

Actually, let me also verify the simulation over a longer period (say 600 frames = 10 seconds) to make sure balls don't gain unbounded energy.

The simulation looks good! Let me check a few things:

1. Long simulation (30s): All balls stay in bounds ✓, max speed is reasonable (0.76 px/frame) ✓, heptagon completes 360 revolutions ✓
2. Reset verification: Ball count is 20 ✓, numbers 1-20 ✓, positions near center ✓
3. Wall collision: No collision detected in the test because the ball at (CX, CY + HEP_R * 0.8) = (400, 648) is inside the heptagon but not close enough to any wall. The heptagon's apothem is about 279px, so the inner edge is at about 279px from center. The ball at 248px from center (0.8 * 310) is inside the heptagon but not near the wall.

Wait, 0.8 * 310 = 248, and the apothem is 310 * cos(π/7) ≈ 279. So the ball at 248px is well inside the heptagon. That's why no wall collision was detected. That's fine - the test was just to verify the wall collision code path works.

The simulation looks stable and correct. Let me also verify the heptagon rotation is visible by checking that the angle changes over time.

Actually, I notice that after 30 seconds (1800 frames), the balls haven't settled (0 balls with speed < 0.1). That's because the damping is very gentle (0.997 per frame). After 1800 frames, the velocity damping factor is 0.997^1800 ≈ 0.005, so velocities should be very small. But the max speed is 0.76 px/frame, which is small but not zero. The balls are still slowly moving around.

Wait, 0.997^1800 = exp(1800 * ln(0.997)) = exp(1800 * -0.003004) = exp(-5.407) ≈ 0.0045. So velocities should be reduced by a factor of ~200. If a ball started with vy = 5 px/frame, after 1800 frames it would be 5 * 0.0045 ≈ 0.022 px/frame. That's very slow.

But the max speed is 0.76 px/frame, which suggests some balls are still moving at moderate speed. This could be because the gravity is continuously adding velocity, and the damping is balancing it out. In a steady state, balls that are resting on the wall will have a small terminal velocity due to gravity being balanced by the wall normal force and damping.

Actually, 0.76 px/frame is quite slow - that's about 46 px/s, which is reasonable for a settling simulation. The balls are basically settled.

Let me also check that the heptagon rotation is working correctly. 360 revolutions in 30 seconds = 12 rev/s... wait, that can't be right. 1800 frames at 60fps = 30 seconds. ROT_SPEED = 2π/5 rad/s. In 30 seconds, the angle should be 30 * 2π/5 = 12π rad = 6 revolutions. But the test says 360 revolutions!

Oh wait, I see the issue. In my test code, I'm calling simulate() with frames=1800, and each frame adds ROT_SPEED to the angle. So heptAngle = 1800 * 2π/5 = 720π. 720π / (2π) = 360 revolutions. But that's in 1800 frames, not 1800 seconds. At 60fps, 1800 frames = 30 seconds. So in 30 seconds, the heptagon rotates 360 times, which is 12 rev/s. But the spec says 1 rev per 5 seconds = 0.2 rev/s.

Wait, I think I made an error. Let me recalculate:
- ROT_SPEED = 2π/5 rad/s
- In 30 seconds, angle = 30 * 2π/5 = 12π rad
- 12π / (2π) = 6 revolutions in 30 seconds
- That's 6/30 = 0.2 rev/s = 1 rev per 5 seconds ✓

But in my test, I'm using frames, not seconds. Each frame adds ROT_SPEED to the angle, but ROT_SPEED is in rad/s, not rad/frame. So I'm treating each frame as 1 second, which is wrong.

In the actual browser code, each frame of requestAnimationFrame calls updatePhysics() once, and the angle advances by ROT_SPEED. But ROT_SPEED is 2π/5 ≈ 1.257 rad/frame, which at 60fps would be 75.4 rad/s = 12 rev/s. That's way too fast!

I need to fix this. The rotation speed should be per-frame, not per-second. Let me recalculate:

1 rev per 5 seconds = 2π/5 rad per 5 seconds = 2π/25 rad per second.
At 60fps, that's 2π/25 / 60 = 2π/1500 rad per frame ≈ 0.00419 rad per frame.

So ROT_SPEED should be 2 * Math.PI / 5 / 60 = Math.PI / 150 ≈ 0.02094 rad per frame.

Wait, but in my browser code, I'm using requestAnimationFrame which runs at ~60fps. Each call to updatePhysics() advances the simulation by one frame. So I need ROT_SPEED to be in rad per frame, not rad per second.

Let me fix this: ROT_SPEED = 2 * Math.PI / 5 / 60 ≈ 0.02094

Actually, let me think about this more carefully. In my browser code:
```javascript
function updatePhysics() {
  heptagonAngle += ROTATION_SPEED;
  ...
}
```

And ROTATION_SPEED = 2 * Math.PI / 5. This means each frame, the angle advances by 2π/5 ≈ 1.257 radians. At 60fps, that's 1.257 * 60 = 75.4 rad/s = 12 rev/s. That's way too fast!

I need to divide by 60 to get the per-frame rotation:
ROTATION_SPEED = 2 * Math.PI / 5 / 60 = Math.PI / 150 ≈ 0.02094 rad/frame

This is a significant bug. Let me fix it.

Also, in my wall velocity calculation, I'm using ROTATION_SPEED as the angular velocity:
```javascript
var wvx = -ROTATION_SPEED * ry;
var wvy = ROTATION_SPEED * rx;
```

This should be the angular velocity in rad/s if we're using real time, or rad/frame if we're using frame-based time. Since I'm using frame-based physics, I should use the per-frame angular velocity here too.

Let me fix the code.

Actually wait, let me reconsider. In my Node.js test, I was using the same ROT_SPEED value (2π/5) and treating each iteration as a frame. So in my test, 1800 frames = 30 seconds of simulated time, but the heptagon rotated 360 times in that period, which is 12 rev/s. That's 60x too fast.

The fix is to change ROT_SPEED to 2π/5/60 = π/150.

Let me also reconsider the wall velocity calculation. The wall velocity at a point on the rotating heptagon should be:
v = ω × r

where ω is the angular velocity in rad/s and r is the position vector from the center. But since I'm using frame-based physics, I should use the per-frame angular velocity:
v_frame = ω_frame × r

where ω_frame = 2π/5/60 rad/frame.

This is consistent with my approach of using per-frame values for everything else (gravity per frame, velocity damping per frame, etc.).

Let me fix the code now.

Actually, I realize there's another approach. Instead of using frame-based physics, I could use real time with dt. But that's more complex and the frame-based approach is simpler. I just need to make sure the constants are correct.

Let me fix ROT_SPEED and also check if there are other issues.

Also, I should double-check the gravity constant. GRAVITY = 0.28 px/frame². At 60fps, that's 0.28 * 3600 = 1008 px/s² ≈ 0.1g. That seems reasonable.

Let me fix the ROT_SPEED and rewrite the file.

I see the bug - `ROT_SPEED = 2 * Math.PI / 5` is per-second, but the physics updates once per frame (at ~60fps). So the heptagon rotates 60x too fast. I need to divide by 60 to get per-frame rotation.

Also, the wall velocity calculation `wvx = -ROT_SPEED * ry` uses the same per-second value, which needs to be per-frame too.

Let me fix both issues. I'll change:
- `ROT_SPEED = 2 * Math.PI / 5 / 60` (per frame at 60fps)
- The wall velocity uses the same ROT_SPEED which is now per-frame

Let me also reconsider: the `requestAnimationFrame` loop doesn't guarantee 60fps. But for simplicity, I'll assume ~60fps and use per-frame constants. The simulation will be slightly faster or slower depending on actual frame rate, but that's acceptable for this kind of visualization.

Actually, a better approach is to use delta time. Let me use a time-based approach where I track elapsed time and compute dt in seconds. This way the simulation runs at the correct speed regardless of frame rate.

Let me rewrite the updatePhysics to use delta time.

Good, the file was written. Now let me verify the JavaScript syntax is valid and then test the simulation in the browser.

Good, the JavaScript syntax is valid. Now let me open it in the browser and verify it works properly. Let me also run a more thorough physics test with the corrected delta-time approach.