Bouncing balls in a heptagon · particle-1

Let me build this. I'll write an index.html with canvas, 20 balls in a compact grid arrangement near center, heptagon rotating one rev per 5s, ball-wall collisions with moving wall velocity, ball-ball collisions, spin, damping, pause/resume/reset.

Let me plan the physics.

Canvas size responsive, say full window. Heptagon centered, radius maybe min(w,h)*0.38. Ball radius ~ 13-14 px, scale with heptagon (e.g. radius = R*0.045).

20 balls compact arrangement: 4x5 grid with spacing 2.1*r starting near center, gravity falls.

Rotation: angle = 2π * t / 5. Vertices at angle + i*2π/7.

Ball-wall collision: for each edge, compute distance from ball center to the segment/line. Since it's a convex polygon and the ball is inside, we can use the inward normals. For each edge from A to B (in order), compute outward normal. Compute signed distance d = dot(p - A, n_out) where n_out points outward. If d > -r, penetration = d + r, push ball inward by penetration along -n_out, and reflect velocity relative to wall velocity.

Wall velocity at contact point = ω × r_vec, i.e. for rotation about center c with angular velocity ω: v_wall = ω * (-(p.y - c.y), (p.x - c.x)).

Then relative velocity vrel = v - v_wall. vn = dot(vrel, n_out). If vn > 0 (moving outward), reflect: vrel -= (1+e) * vn * n_out. Then add friction on tangential component: vt = vrel - vn*n_out; vrel_t *= (1 - friction). Then v = vrel + v_wall.

Careful with velocity-based collisions; use position correction plus velocity. Also spin: on collision, apply spin change from tangential relative velocity. Keep simple: ball has angular velocity omega_b, and on wall collision, tangential impulse changes spin. Also ball-ball collisions: apply friction and spin transfer.

To keep stable, use substeps (e.g. 4-8 per frame), gravity ~ 1400 px/s² scaled with size.

Let's normalize: define polygon circumradius R = min(w,h)*0.40. Ball radius r = R*0.042 → for R=340, r≈14.3. 20 balls in 5x4 grid spacing 2.1r ≈ 30 → width 4*30=120, height 3*30=90. Fits.

Gravity = maybe 900 px/s². Restitution 0.55, friction 0.2, air damping 0.999.

Rotation rate ω = 2π/5 ≈ 1.2566 rad/s. Wall speed at R: ~427 px/s. That's fine.

Also to prevent tunneling, use substeps with dt small (dt/8) and speed cap.

Spin rendering: rotate text with ball angle. Ball rotation driven by omega; add angular damping. On ball-ball collision, change spins.

Ball-ball collision: relative velocity at contact includes spin? Simplify: normal impulse with restitution, plus tangential friction that changes both velocities and spins. Let's do standard:

For ball-ball:
- normal n from a to b.
- relative velocity at contact points: va + ωa × ra - (vb + ωb × rb). ra = n*ra, rb = -n*rb. In 2D, ω × r = ω * (-r.y, r.x).
- vn = dot(vrel, n). if vn > 0 (separating) skip.
- jn = -(1+e)*vn / (1/ma + 1/mb)  (equal masses: /(2/m)) → jn = -(1+e)*vn * m/2.
- apply impulses ±jn*n.
- tangential: t = perpendicular of n. vt = dot(vrel, t). jt = -vt / (1/ma+1/mb + ra²/Ia + rb²/Ib). with I = 0.5*m*r². Clamp jt to μ*jn.
- apply angular: ωa -= cross(ra, jt*t)/Ia ... sign conventions.

Use mass proportional to r² (equal here). All balls same radius, so simpler. Actually equal radius → all same.

Ball-wall: treat wall as infinite mass. Similar impulse with spin.

I'll implement a reasonably careful impulse solver.

Let me define constants:
- r ball radius.
- m = 1.
- I = 0.5 * m * r * r.

Ball-wall:
n = outward normal (from polygon interior toward outside). Ball hitting wall means moving outward, vn = dot(vrel, n) > 0.
contact point = p + n * r (approximately). r_vec = contact - center_poly.
v_wall = ω_poly × r_vec.
vrel = v - v_wall.
vn = dot(vrel,n).
if vn > 0:
  jn = -(1+e)*vn / (1/m) → jn = -(1+e)*vn*m (negative). impulse = jn*n pushes inward.
  v += jn*n/m.
  tangential rc = contact - ball centre = n*r. Actually the contact point relative to ball centre is n*r (outward). r_c = n*r.
  vt = dot(vrel, t).
  jt = -vt / (1/m + (r_c × t)²/I). compute cross(r_c, t) = r_c.x*t.y - r_c.y*t.x. With r_c = r*n and t perpendicular to n, cross = ±r.
  clamp |jt| ≤ μ*|jn|.
  v += jt*t/m
  ω_ball += cross(r_c, jt*t)/I.
  Position correction: push inward by penetration.

Note the moving wall pushing ball: if ball static and wall moves toward it, vrel = -v_wall, so dot with n... Let me check. Ball at rest, wall rotating inward toward the ball. The wall's surface moves inward, so v_wall points inward, vrel = -v_wall points outward, so vn > 0 → collision response pushes ball inward with speed. Good, that gives kick. Actually it might cause energy pump, leading to "unbounded speed" — cap speed and use damping. With restitution < 1 and friction, plus gravity, it may be jittery but bounded. Cap speed at, say, 2000 px/s and angular at 30 rad/s.

Positional correction: move ball inward by penetration (d + r) along -n. But when the wall sweeps, this is fine.

Also to avoid balls escaping, after the collision loop do a containment check: if ball center distance from polygon center > R - r (using inradius/apothem), clamp back inside. Better: for each edge, ensure dot(p - A, n_out) <= -r; if violated more than some epsilon, project. I'll do the per-edge loop correction each substep; the heptagon interior is intersection of half-planes so pushing for the most violated edge handles it well. Actually doing all edges sequentially is fine for a convex polygon since corrections may conflict near vertices; do a couple iterations.

Let's compute edges with vertices in CCW order. For CCW polygon in screen coords (y down), hmm. Let's just compute outward normal by checking against center: n = normalize(perp(B-A)); ensure dot(n, A - center) > 0 else flip. That handles orientation robustly.

Initial arrangement: 5 columns × 4 rows, spacing 2.15*r, centered at polygon center, maybe slightly above? "compact, non-overlapping arrangement near the enclosure center and let them fall under gravity." Place centered with small jitter? Keep deterministic grid for reset comparison. Let's place centered, rows with slight offset maybe not needed. Balls numbered 1..20 in order.

Since heptagon rotates, the initial arrangement is axis-aligned grid at center; polygon rotates from angle 0. Reset restores angle = 0 and the same grid positions.

Timing: use requestAnimationFrame with dt clamped, simulation time accumulates only when not paused. Rotation angle = 2π * simTime / 5.

Rendering: dark background, heptagon stroke with glow, balls with radial gradient, numbers rotated by ball.angle. Add HUD: title, pause/resume button, reset button, maybe spin indicator. Use DOM buttons.

Handle resize: recompute R, r, positions? On resize, simplest: recompute geometry and rescale ball positions relative to center proportionally, or just re-init. I'll rescale positions by the ratio of new R to old R around the center. Keep it simple: rebuild geometry, scale positions to fit.

Let me be careful the ball radius scales: define baseR = min(w,h)*0.40; r = baseR/24 perhaps → R=340 → r=14.2. Grid width 5 balls * 2.15*14.2 = 152, height 4*30.5=91. Fine.

Actually with 20 balls radius 14 at R 340, total area 20*π*14² = 12300 vs heptagon area ~ 2.736*R² = 316k. Fine.

DT: fixed step 1/240 with accumulation? Simpler: physics per frame with substeps: substeps = 6, h = dt/substeps, dt clamped to 1/30.

Let me code it.

```js
const canvas = document.getElementById('c');
const ctx = canvas.getContext('2d');
let W=0,H=0,DPR=1;
const state = { t:0, paused:false, balls:[], cx:0, cy:0, R:0, r:0, angle:0 };
const OMEGA = 2*Math.PI/5;
```

Geometry setup:

```js
function setupGeometry(){
  W = canvas.clientWidth; H = canvas.clientHeight;
  const dpr = Math.min(window.devicePixelRatio||1, 2);
  canvas.width = Math.round(W*dpr); canvas.height = Math.round(H*dpr);
  ctx.setTransform(dpr,0,0,dpr,0,0);
  state.cx = W/2; state.cy = H/2;
  state.R = Math.min(W,H)*0.40;
  state.r = state.R/24;
}
```

Hmm, on resize positions need scaling. I'll store positions in normalized units relative to R? Simpler: on resize, scale ball positions and velocities by ratio and rebuild. Let's do:

```js
function resize(){
  const oldR = state.R || 1;
  const oldX = state.cx, oldY = state.cy;
  setupGeometry();
  const s = state.r / oldR;
  ...
}
```

Actually simpler: store ball positions relative to center in units of R. Then on draw/physics use cx + x*R. Hmm, but physics in pixel units. Alternative: store normalized coords and convert each frame. That's clean: ball.px = normalized x (in units of R). Then physics works in normalized units with gravity in R/s²! Let's do that: work in normalized units where polygon circumradius = 1. Ball radius = 1/24. Gravity = e.g. 6 R-units/s²? For R=340, gravity 6*340=2040 px/s². Hmm that's large but with r=14 px, ball falling... Time to fall 1 unit: sqrt(2/6)=0.58s. That seems reasonable. Let's pick gravity g = 4.5 (units/s²) → 1530 px/s². Ball falls from center to bottom ~0.6 of R = 204px in sqrt(2*0.6/4.5)=0.52s. Good.

Speed cap in normalized units: 12 units/s (≈4000px/s) — high. Wall velocity near rim = ω*1 = 1.257 units/s. Hmm, that's much slower than balls falling. That's realistic: rotation period 5s.

Actually with such a fast-falling ball vs slow wall, the wall's moving velocity contribution is small but still present. It's fine, but "account for moving-wall velocity in bounce response" is satisfied.

Maybe make gravity smaller to be more visible: g = 2.5 units/s². Fall time from center to bottom 0.7s. Let's use 2.8.

Ball radius normalized 1/24 ≈ 0.0417. Grid spacing 2.15*r = 0.0896 units. 5 cols → 0.36 wide. Good.

Restitution 0.5, friction μ = 0.25 for balls, ball-wall 0.3.

Now collisions in normalized units: m=1, I = 0.5*m*r² = 0.5*0.001736 = 8.68e-4. Fine.

Let me write the collision code carefully.

Wall data: vertices computed each substep from angle. Edges array {ax,ay,bx,by,nx,ny}.

```js
function polygonVerts(angle){
  const pts=[];
  for(let i=0;i<7;i++){
    const a = angle + i*2*Math.PI/7;
    pts.push({x:Math.cos(a), y:Math.sin(a)});
  }
  return pts;
}
```
Wait — does the heptagon shape rotate about the center properly? Vertices at angle + offset → yes it rotates.

Edges: for i, A=pts[i], B=pts[(i+1)%7]. Outward normal: e=(B-A), n=(e.y,-e.x) normalized; if dot(n, A) < 0 (center at origin) flip. Then n outward. Distance from center to edge line = dot(A,n) = apothem > 0. For a point p inside, dot(p-A, n) < 0. Penetration when dot(p-A,n) > -r.

Ball-wall resolution:

```js
for each edge:
  const d = (p.x-A.x)*n.x + (p.y-A.y)*n.y; // signed, negative inside
  if (d > -r){
     // contact
     const pen = d + r;
     p.x -= n.x*pen; p.y -= n.y*pen;
     // contact point approx: p + n*r  (after correction)
     const cxp = p.x + n.x*r, cyp = p.y + n.y*r;
     const rvx = cxp, rvy = cyp; // relative to polygon center at origin
     const vwx = -OMEGA*rvy, vwy = OMEGA*rvx;
     let relvx = b.vx - vwx, relvy = b.vy - vwy;
     const vn = relvx*n.x + relvy*n.y;
     if (vn > 0){
       const jn = -(1+e)*vn / (1/m);  // m=1
       b.vx += jn*n.x/m; b.vy += jn*n.y/m;
       relvx = b.vx-vwx; relvy = b.vy-vwy;
       // tangent
       let tx = -n.y, ty = n.x;
       const vt = relvx*tx + relvy*ty;
       const rcx = n.x*r, rcy = n.y*r;
       const crossRT = rcx*ty - rcy*tx;  // = r*(n.x*n.x + n.y*n.y)?? compute: rcx*ty - rcy*tx = r*n.x*n.x - r*n.y*(-n.y) = r. yes = r
       const denom = 1/m + crossRT*crossRT/I;
       let jt = -vt/denom;
       const maxJt = mu*Math.abs(jn);
       if (Math.abs(jt) > maxJt) jt = Math.sign(jt)*maxJt;
       b.vx += jt*tx/m; b.vy += jt*ty/m;
       b.omega += crossRT*jt/I;  // hmm sign: torque = r_c × J, J = jt*t. cross = rcx*jt*ty - rcy*jt*tx = jt*crossRT. angular += torque/I
     }
  }
```

Wait sign of the friction impulse direction in the relative-velocity equation: tangential relative velocity at contact point includes spin. Let me be more careful with the standard formulation.

Contact point on ball: q = p + r*n. Velocity of ball material point at q: v + ω × (q - p) = v + ω * perp(r*n) where perp(a) = (-a.y, a.x). So vq = v + ω*(-r*n.y, r*n.x).

Relative velocity of ball material point w.r.t. wall material point: vrel = vq - v_wall.

vn = dot(vrel, n). If vn > 0 → separating... no wait, pushing outward means penetrating (since n is outward). We resolve if vn > 0 (moving further into wall). Hmm, in the standard contact convention with n pointing from wall toward ball (i.e., inward here), vn < 0 means approaching. Let me use n_in = -n (pointing inward, from wall toward ball interior). Then approaching when dot(vrel, n_in) > 0.

Let me just define n_in = -n_out. Contact point q = p + r*n_out = p - r*n_in. Then vq = v + ω * perp(-r*n_in) = v - ω*r*perp(n_in) where perp(n_in)=(-n_in.y, n_in.x).

Impulse direction along n_in (kicking the ball inward): J = j * n_in with j > 0.

Change in v: v += j*n_in/m. Change in ω: ω += cross(rc, J)/I where rc = q - p = -r*n_in. cross(rc, J) = j*cross(-r*n_in, n_in) = 0. Good, normal impulse creates no torque (as expected since rc is along n_in).

Relative velocity after: vrel' = (v + j*n_in/m) + (ω+Δω)*... tangential unaffected by j. vn' = dot(vrel, n_in) + j/m (since n_in·n_in=1). For restitution: vn' = -e*vn. So j = -(1+e)*vn*m. With vn = dot(vrel, n_in) > 0 (approaching). So j negative? That's wrong sign. Let me recheck: vn = dot(vrel, n_in) > 0 means the ball is approaching the wall? n_in points from the wall into the polygon (inward). The ball moving outward → v·n_in < 0. Hmm. Ball moving toward the wall = moving outward = v·n_out > 0 → v·n_in < 0. So approaching is vn < 0 with n_in. Standard: vn' = -e*vn requires vn < 0. Then vn' = vn + j/m → j = (-e*vn - vn)*m = -(1+e)*vn*m > 0. Good, positive.

So with n_in:
- contact q = p - r*n_in... wait p is inside, q is on the surface toward the wall which is outward direction = n_out = -n_in. So q = p + r*n_out = p - r*n_in. Yes.

Hmm hold on: if n_out is outward and the wall is outside, the contact point is in the outward direction. Yes q = p + r*n_out.

Let me just implement with n_out and handle sign explicitly:

```js
const d = dot(p - A, n_out);  // negative inside
if (d > -r) {
  pen = d + r;
  p -= n_out * pen;   // push inward
  // contact point
  qx = p.x + n_out.x*r; qy = p.y + n_out.y*r;
  ...
}
```
After correction, q lies exactly on the wall line.

Then:
- v_wall at q: ω_poly * perp(q) = ω_poly*(-q.y, q.x).
- vq = v + ω_ball * perp(q - p) = v + ω_ball * (-r*n_out.y, r*n_out.x).
- vrel = vq - v_wall.
- vn_out = dot(vrel, n_out). Approaching if vn_out > 0.
- jn = -(1+e)*vn_out*m. Apply v += jn*n_out/m.
- friction: t = perp(n_out) = (-n_out.y, n_out.x). vt = dot(vrel_after_normal? use pre-impulse vrel, tangential unchanged), jt = -vt/denom where denom = 1/m + (crossRC_T)²/I, with rc = q - p = r*n_out, and cross(rc, t) = r*n_out.x*t.y - r*n_out.y*t.x = r*(n_out.x*n_out.x - n_out.y*(-n_out.y)) = r. So crossRC_T = r. denom = 1/m + r²/I. With I=0.5 m r², r²/I = 2/m. denom = 3/m. So jt = -vt*m/3.
- clamp |jt| ≤ μ|jn|.
- v += jt*t/m; ω_ball += cross(rc, jt*t)/I = jt*r/I.

Check spin direction: ball with pure tangential motion vt>0 along t, no spin. jt = -vt*m/3 < 0 → v += jt*t/m reduces tangential velocity, ω += jt*r/I = negative. Rolling: ω negative means... for a ball moving in +t direction on a surface, it should develop ω such that contact point velocity decreases, i.e. ω × rc contributes -t direction. rc = r*n_out, and the contact velocity from spin is ω*perp(rc) = ω*r*perp(n_out) = ω*r*t. To reduce +t velocity, ω must be negative. ✓. And rolling without slipping condition later: ω = -vt/r. OK consistent.

Ball-ball: standard with both spins.

```
n from a to b.
d = dist. if d < ra+rb:
  overlap, push apart half each.
  contact point: q = a.p + n*ra.
  va_q = a.v + a.ω*perp(q - a.p) = a.v + a.ω*perp(ra*n)
  vb_q = b.v + b.ω*perp(q - b.p) = b.v + b.ω*perp(-rb*n)
  vrel = va_q - vb_q
  vn = dot(vrel, n)  // approaching if vn < 0 (since n points from a to b)
  if vn < 0:
    jn = -(1+e)*vn / (1/ma + 1/mb)   → positive
    a.v += jn*n/ma; b.v -= jn*n/mb
    t = perp(n)
    vt = dot(vrel, t)
    jt = -vt / (1/ma + 1/mb + (cross(ra_vec,t))²/Ia + (cross(rb_vec,t))²/Ib)
    ra_vec = ra*n, cross(ra_vec,t) = ra; rb_vec = -rb*n, cross(rb_vec,t) = -rb.
    With equal radii and masses: denom = 2/m + ra²/I + rb²/I = 2/m + 2*(2/m) = 6/m.
    clamp |jt| ≤ μ|jn|
    a.v += jt*t/ma; b.v -= jt*t/mb
    a.ω += cross(ra_vec, jt*t)/Ia = jt*ra/Ia
    b.ω += cross(rb_vec, -jt*t)/Ib = (-jt)*(-rb)/Ib = jt*rb/Ib
```
Hmm wait b receives impulse -jt*t at rb_vec = -rb*n. Torque = cross(rb_vec, -jt*t) = (-rb)*(cross(n,t))*(-jt) = (-rb)*(1)*(-jt) = rb*jt. Hmm: cross(n,t) where t=perp(n)=(-n.y,n.x): cross(n,t) = n.x*t.y - n.y*t.x = n.x*n.x + n.y*n.y = 1. So cross(rb_vec, Jb) with rb_vec = -rb*n, Jb = -jt*t: = (-rb)*(-jt)*cross(n,t) = rb*jt. So b.ω += rb*jt/Ib.

And a: ra_vec = ra*n, Ja = jt*t: cross = ra*jt. a.ω += ra*jt/Ia.

OK.

Also apply angular damping and linear air drag each substep: v *= (1 - 0.02*h*?); let's do v *= Math.exp(-0.15*h) or just v *= 0.9995 per substep. And omega *= 0.999.

Speed cap: if |v| > 8 units/s, scale down. Spin cap 40 rad/s.

Also if a ball somehow escapes (distance from center > 1.2), reposition to center-ish and zero velocity. Safety.

Now numbering and spin rendering: draw ball circle with gradient, then save, translate to ball center, rotate(b.angle), draw the number with textAlign center, baseline middle. Draw something to show spin? The number rotates, that's enough. Maybe add a small notch/dot on the rim to make rotation more visible. The requirement says "rotate its number with the ball so spin is visible" — number rotation suffices.

Ball angle: b.angle += b.ω*h.

Rendering polygon: use same verts, path with rounded joins, stroke + subtle fill. Add a rim glow. Add gradient background. Add HUD DOM with buttons and info text.

Pause/Resume: toggles text. Reset: rebuild balls to initial arrangement, simTime=0, angle=0.

Note: rotation should be tied to sim time so pause freezes it — yes.

Let me also add fingers? No.

Now write the code.

For the initial arrangement, use rows: 4 rows of 5. Let me compute positions in normalized units.

```js
const COLS=5, ROWS=4;
const sp = 2.12*R_BALL; // R_BALL=1/24
for (let j=0;j<ROWS;j++) for (let i=0;i<COLS;i++){
  const x = (i-(COLS-1)/2)*sp;
  const y = (j-(ROWS-1)/2)*sp;
  ...
}
```
sp = 2.12/24 = 0.0883. width = 4*0.0883 = 0.353 → half 0.177. Fine inside apothem ~0.9.

Check no overlap: 2.12r > 2r ✓.

Numbers 1..20, index = j*5+i+1.

I'll add a tiny bit of initial downward velocity? No, they just fall.

Also ensure the grid is centered vertically but the polygon rotates — initially the polygon at angle 0 has a vertex at right and flat-ish? With 7 sides at angle 0,2π/7,..., vertices include (1,0). Bottom of polygon: minimum y of vertices = sin of angles closest to π/2 → angle values 0, 0.897, 1.795, 2.692, 3.590, 4.488, 5.386 rad. sin(1.795)=0.975, sin(2.692)=0.434. So bottom region around apothem 0.9. The balls at y up to 0.13 are near center. They fall onto the bottom facet region.

Potential issue: rotating heptagon corners sweeping through balls could push them fast. With ω=1.257 and R=1, max wall speed 1.257 units/s vs ball speeds of a few units/s — fine.

Now let's think about stability: with restitution 0.5 and gravity 2.8, in a rotating container, balls settle at the bottom, sliding. Should be stable.

Substeps: 8 per frame at dt=1/60 → h ≈ 0.002s. With max speed 8 units/s, movement per step 0.0017 units = 0.6px. Ball radius 14px. No tunneling.

Let me write the file now.

Also add a subtle trail? Not needed.

HUD: top-left title "Heptagon Physics", controls at bottom center or top-right. Buttons styled.

I'll include a small "spin" hint? Not necessary.

Let me write code.

```html
<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Heptagon Bounce</title>
<style>
  :root{ --ink:#e8eefc; }
  *{box-sizing:border-box}
  html,body{height:100%;margin:0}
  body{
    background: radial-gradient(circle at 50% 40%, #17203a 0%, #0b0f1c 55%, #05070d 100%);
    color:var(--ink);
    font-family: ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, sans-serif;
    overflow:hidden;
    -webkit-font-smoothing:antialiased;
  }
  #stage{position:fixed;inset:0;display:grid;place-items:center}
  canvas{display:block;width:100%;height:100%;touch-action:none}
  header{...}
```

Use absolute positioned HUD.

Let me write everything concretely.

Canvas sizing: I'll set canvas CSS to fill; use devicePixelRatio.

For rendering I want R based on min(W,H). With HUD overlays on top.

Let me write the JS.

```js
(() => {
  'use strict';
  const canvas = document.getElementById('sim');
  const ctx = canvas.getContext('2d');

  const SIDES = 7;
  const REV_TIME = 5;
  const OMEGA_POLY = 2*Math.PI/REV_TIME;
  const R_BALL = 1/24;
  const MASS = 1;
  const INERTIA = 0.5*MASS*R_BALL*R_BALL; // ~0.000868
  const GRAVITY = 2.6;
  const WALL_REST = 0.55;
  const BALL_REST = 0.5;
  const WALL_FRICTION = 0.35;
  const BALL_FRICTION = 0.22;
  const AIR_DRAG = 0.06;
  const SPIN_DRAG = 0.5;
  const MAX_SPEED = 7;
  const MAX_SPIN = 45;

  const sim = {
    W:0,H:0,cx:0,cy:0,R:1,
    t:0, angle:0,
    paused:false,
    balls:[],
    last:0, acc:0
  };
```

Note: In normalized space, ball radius 1/24 and inertia tiny, so angular updates jt*r/I will be big. jt ~ vt*m/3, ω += jt*r/I = (-vt/3)*r/(0.5 r²) = -2vt/(3r) = -2vt/3 * 24 = -16 vt. For vt ~ 2, ω ~ -32 rad/s. That's very fast spin, visible as crazy spinning. Hmm. Realistically, a ball with friction would spin fast, but visually it'd be a blur. Let me reduce friction coefficient (0.15) and/or scale down the spin response by treating spin as more visual: I could increase inertia artificially (I = MASS*R_BALL*R_BALL*k) to make spin slower. Use I = 0.9*MASS*R_BALL² maybe. Or multiply the applied angular impulse by a factor 0.35 to make rotation readable.

Vanishing friction on a small ball does result in big spin rates. But for visual clarity, use I = 2.0*m*r²? Then denom changes. Let me use I = MASS*R_BALL*R_BALL (2× real) and cap spin at 25 rad/s (about 4 rev/s — visible, numbers readable? at 30fps it'd be ~0.8 rad/frame, looks fast but okay). Maybe cap at 18 rad/s (~3 rev/s). Hmm, that's spinning pretty fast; the number would be a blur but visible as rotating.

Actually, the number rotates with the ball; to make spin clearly visible, moderate spin is nicer. Let me use spin damping stronger: SPIN_DRAG such that ω *= exp(-1.2*h)? That only affects over time.

Let me pick: I = 1.6 * m * r². jt = -vt/denom with denom = 1/m + r²/I = 1 + 1/1.6 = 1.625 (for wall). jt = -vt/1.625. ω += jt*r/I = (-vt/1.625)*r/(1.6r²) = -vt/(1.625*1.6*r) = -vt/(2.6*0.04167) = -9.2*vt. Still fast for vt=1 → 9 rad/s = 1.4 rev/s. Reasonable and readable. Cap 25.

Air drag on v: 0.06/s is very light. Roll resistance: apply small torque proportional to ω: ω -= ω*0.6*h? Let's keep SPIN_DRAG=0.5/s.

Fine. Let me also add a "restitution for spin"? No.

One thing: with tiny inertia and friction, the tangential velocity gets killed quickly by jt up to μ|jn|; that's fine.

Now the render function. Let's make it pretty:

- Background: canvas clear (CSS body gradient shows through if canvas transparent). I'll not fill background, or fill with subtle radial gradient. Clear with transparent.
- Heptagon: fill with slight gradient (rgba white 0.03), stroke rgba(120,190,255,0.55), lineWidth ~ 3, shadowBlur glow.
- Inner faint concentric heptagon lines? A nice touch: draw a faint inner ring at 0.92 scale.
- Balls: gradient from lighter to darker based on hue. Colors per ball index (HSL hue = (i*17) % 360 with nice saturation). Draw circle with radial gradient offset highlight. Stroke rim slightly brighter. Number in dark color for contrast? If ball colors are saturated but medium-dark, white text with dark shadow works.

Number font size ≈ r*1.0, bold.

Also draw a small arc marker on the rim so spin reads even for single-digit... number rotation is enough.

Let me write with colors: hues from 200 to 350 etc. Use hue = (i*137.5) % 360.

Balls gradient: `hsl(h, 75%, 62%)` center highlight to `hsl(h,70%,38%)`. Text white bold.

Draw order: polygon fill+stroke, then balls.

Also draw a soft shadow under each ball? Skip.

HUD elements:
```html
<div id="hud">
  <div class="brand">HEPTAGON <span>PHYSICS</span></div>
  <div class="stats">20 balls · 5.0 s / rev</div>
</div>
<div id="controls">
  <button id="pauseBtn">Pause</button>
  <button id="resetBtn">Reset</button>
</div>
```
Buttons with pointer-events, styled with borders, hover.

Maybe also a speed readout? Not needed.

Physics step:

```js
function step(h){
  sim.t += h;
  sim.angle = sim.t * OMEGA_POLY;  // wait, angle should = sim.t*omega, and t accumulates only when unpaused. Yes.
  // gravity + integrate
  for (const b of sim.balls){
    b.vy += GRAVITY*h;
    // air drag
    const drag = Math.exp(-AIR_DRAG*h);
    b.vx *= drag; b.vy *= drag;
    b.omega *= Math.exp(-SPIN_DRAG*h);
    // clamp
    ...
    b.x += b.vx*h; b.y += b.vy*h;
    b.angle += b.omega*h;
  }
  // ball-ball (a couple of iterations for stability)
  for (let it=0; it<2; it++) collideBalls();
  // walls
  collideWalls();
}
```

Wait: collideBalls twice per substep might double the impulse response? Usually okay (positional correction + velocity). I'll do 1 iteration of velocity resolution per pair per substep with 8 substeps — sufficient at these speeds.

Let me merge: ball-ball collision does positional split and impulse. Run once per substep.

collideWalls: for each ball, loop edges, resolve. Also iterate twice for corners.

Boundaries: use the edge loop. If ball is far outside (|p| > 1.05), reset toward center.

Let me code collideWalls:

```js
function collideWalls(){
  const verts = polyVerts(sim.angle);
  for (const b of sim.balls){
    for (let iter=0; iter<2; iter++){
      for (let i=0;i<SIDES;i++){
        const A=verts[i], B=verts[(i+1)%SIDES];
        let ex=B.x-A.x, ey=B.y-A.y;
        const el = Math.hypot(ex,ey) || 1;
        let nx = ey/el, ny = -ex/el;
        // outward
        if (nx*A.x + ny*A.y < 0){ nx=-nx; ny=-ny; }
        const d = (b.x-A.x)*nx + (b.y-A.y)*ny;
        if (d > -R_BALL){
          const pen = d + R_BALL;
          b.x -= nx*pen; b.y -= ny*pen;
          // contact point
          const qx = b.x + nx*R_BALL, qy = b.y + ny*R_BALL;
          const vwx = -OMEGA_POLY*qy, vwy = OMEGA_POLY*qx;
          const rcx = nx*R_BALL, rcy = ny*R_BALL;
          const rvx = b.vx + b.omega*(-rcy) - vwx;  // hmm: spin velocity at contact = ω × rc = ω*(-rcy, rcx)
          const rvy = b.vy + b.omega*(rcx) - vwy;
          const vn = rvx*nx + rvy*ny;
          if (vn > 0){
            const jn = -(1+WALL_REST)*vn*MASS;
            b.vx += jn*nx/MASS; b.vy += jn*ny/MASS;
            // tangential
            const tx = -ny, ty = nx;
            const vt = rvx*tx + rvy*ty;
            const denom = 1/MASS + (R_BALL*R_BALL)/INERTIA;
            let jt = -vt/denom;
            const lim = WALL_FRICTION*Math.abs(jn);
            if (Math.abs(jt) > lim) jt = Math.sign(jt)*lim;
            b.vx += jt*tx/MASS;
            b.vy += jt*ty/MASS;
            const crossRT = rcx*ty - rcy*tx; // = R_BALL
            b.omega += crossRT*jt/INERTIA;
          }
        }
      }
    }
  }
}
```

Hmm, one issue: the position correction is applied for every edge each iteration and the ball gets pushed based on d > -r. After resolving, the ball is at d = -r exactly. Good.

Also the velocity: after the normal impulse, the ball's velocity may still have outward component if e>0? vn' = -e*vn relative to wall. Fine.

For the spin clamp: after wall collisions, clamp omega. Do it in integration clamp each substep.

Ball-ball:

```js
function collideBalls(){
  const n = sim.balls.length;
  for (let i=0;i<n;i++){
    const a = sim.balls[i];
    for (let j=i+1;j<n;j++){
      const b = sim.balls[j];
      let dx = b.x-a.x, dy = b.y-a.y;
      let dist2 = dx*dx+dy*dy;
      const minD = R_BALL*2;
      if (dist2 >= minD*minD || dist2 === 0) continue;
      let dist = Math.sqrt(dist2);
      let nx = dx/dist, ny = dy/dist;
      const pen = minD - dist;
      // positional correction
      a.x -= nx*pen*0.5; a.y -= ny*pen*0.5;
      b.x += nx*pen*0.5; b.y += ny*pen*0.5;
      // contact velocities
      const rax = nx*R_BALL, ray = ny*R_BALL;
      const rbx = -nx*R_BALL, rby = -ny*R_BALL;
      let rvx = (a.vx + a.omega*(-ray)) - (b.vx + b.omega*(-rby));
      let rvy = (a.vy + a.omega*(rax)) - (b.vy + b.omega*(rbx));
      const vn = rvx*nx + rvy*ny;
      if (vn > 0) continue; // separating (n points a→b)
      const jn = -(1+BALL_REST)*vn/(2/MASS);
      a.vx += jn*nx/MASS; a.vy += jn*ny/MASS;
      b.vx -= jn*nx/MASS; b.vy -= jn*ny/MASS;
      // friction
      const tx=-ny, ty=nx;
      const vt = rvx*tx + rvy*ty;
      const crossA = rax*ty - ray*tx; // = R_BALL
      const crossB = rbx*ty - rby*tx; // = -R_BALL
      const denom = 1/MASS + 1/MASS + crossA*crossA/INERTIA + crossB*crossB/INERTIA;
      let jt = -vt/denom;
      const lim = BALL_FRICTION*Math.abs(jn);
      if (Math.abs(jt) > lim) jt = Math.sign(jt)*lim;
      a.vx += jt*tx/MASS; a.vy += jt*ty/MASS;
      b.vx -= jt*tx/MASS; b.vy -= jt*ty/MASS;
      a.omega += crossA*jt/INERTIA;
      b.omega += crossB*(-jt)/INERTIA;
```

Check b torque: b gets impulse -jt*t at rb_vec; torque = cross(rb_vec, -jt*t) = (-jt)*cross(rb_vec, t) = (-jt)*crossB. So b.omega += (-jt)*crossB/INERTIA. With crossB = -R_BALL: = jt*R_BALL/INERTIA. Same as before ✓.

Note a.omega update: cross(ra_vec, jt*t) = jt*crossA ✓.

Now point: rvx uses a.omega*(-ray) = ω × r for a. ✓ and for b: ωb × rb where rb = (-R*n) → (-rby, rbx) → rvx contribution b.omega*(-rby) ✓.

Now speed clamp function.

Ball-ball relative velocity sign: vn > 0 means separating since n from a to b ✓.

Then rendering.

Timing loop:

```js
let prev = performance.now();
function frame(now){
  const dt = Math.min((now - prev)/1000, 1/30);
  prev = now;
  if (!sim.paused) {
    const SUB = 8;
    const h = dt/SUB;
    for (let i=0;i<SUB;i++) step(h);
  }
  draw();
  requestAnimationFrame(frame);
}
```

Note: with dt clamped at 1/30 and SUB=8, h ≤ 0.00417. Fine.

Hmm, if a frame hiccups (tab switch), paused? We clamp. Fine.

Reset function:

```js
function reset(){
  sim.t = 0;
  sim.angle = 0;
  sim.balls = buildBalls();
  sim.paused = false;  // hmm should reset unpause? Spec: Reset restores initial ball arrangement and wall orientation. Keep paused state? I'll leave pause state as-is, but resuming is nicer? Let's keep paused unchanged to avoid surprises... Actually resetting while paused should show the initial arrangement frozen — that's a good verification. Keep paused state.
  pauseBtn.textContent = sim.paused ? 'Resume' : 'Pause';
}
```

Hmm, initial angle 0, and draw at t=0.

buildBalls: create with x,y from grid, vx=0, vy=0, omega=0, angle=0, hue.

Actually give a tiny random-ish spin? For reset determinism, use deterministic small values based on index: b.omega = 0, b.angle = 0. Maybe set initial angle = 0 so numbers upside? fine.

Add slight variation so it doesn't look perfectly rigid? The requirement says compact non-overlapping; grid is fine. But balls in a perfect grid falling onto a rotating floor can be a bit static. I'll add tiny deterministic offsets (alternating ±0.15r) while keeping non-overlap.

Let me place with offset: y rows alternate small x-shift? With spacing 2.12r, shifting by 0.15r keeps non-overlap. Sure.

Actually keep it simple and safe: grid, no offsets. 2.12r spacing leaves 0.12r gap. Fine. But hmm, they fall and stack; perfectly aligned grid might have numerical symmetry issues (all falling straight). Ball-ball collisions occur when touching. It's fine, slight asymmetry from polygon rotation.

I'll add deterministic jitter of ±0.1r to x and y for a nicer look.

Let me now consider draw():

```js
function draw(){
  const {W,H,cx,cy,R} = sim;
  ctx.clearRect(0,0,W,H);
  const verts = polyVerts(sim.angle);
  // glow
  ctx.save();
  ctx.translate(cx,cy);
  ctx.scale(R,R);
  // outer glow stroke
  ctx.beginPath();
  verts.forEach((p,i)=> i?ctx.lineTo(p.x,p.y):ctx.moveTo(p.x,p.y));
  ctx.closePath();
  // fill
  const g = ctx.createRadialGradient(0,0,0.1,0,0,1.05);
  ...
```

Careful: scaling by R then lineWidth must be in normalized units (e.g., 3/R). Text also scaled. Easier: draw scene with transform translate/scale and set lineWidth = 4/R, font sizes in normalized units. Alternatively compute pixel coords. I'll use the transform approach for polygon but balls in pixel space (their radius = R_BALL*R).

Polygon rendering with transform:
- path
- fill with radial gradient in normalized space: createRadialGradient(0,0,0.2,0,0,1.02) colors rgba(140,180,255,0.06) → rgba(90,120,220,0.02).
- stroke: shadowColor/blue, shadowBlur = 18 (shadow blur is in device px, unaffected? With transform, shadow blur is applied in device space I believe... shadowBlur is not scaled by transform in most browsers? Actually shadowBlur is in the current transform space? MDN says shadowBlur is not affected by transform matrix. Should be fine either way.)

I'll keep transform for polygon only.

Also draw faint spokes / tick marks along the rim to emphasize rotation? Nice touch: draw 7 short ticks at vertices. Already visible via vertices.

Ball drawing:

```js
for (const b of sim.balls){
  const px = cx + b.x*R, py = cy + b.y*R, rad = R_BALL*R;
  // glow
  ctx.beginPath();
  ctx.arc(px,py,rad,0,Math.PI*2);
  hex gradient...
}
```
Use two circles: shadow, base fill, then rotate and text.

Text font: `bold ${rad*0.95}px` with textAlign center, textBaseline middle. Draw with slight offset for optical centering (numbers look better with +0.02*rad? Use baseline middle and manual offset 0).

Add a small tick mark near the rim to make spin obvious? The number rotation is enough, but a rim notch helps. I'll draw a tiny darker dot at angle 0 relative to ball rotation. Actually a "seam" line: draw a small line from center to rim in rotated frame, low alpha. Good for visibility.

Now, ensure numbers stay upright legible? No — requirement explicitly says rotate the number with the ball.

Let me write the code now fully.

Also handle window resize: recompute canvas size and geometry; normalized coords mean balls stay put relative to polygon, which is good. Just update W,H,cx,cy,R.

The circle radius R_BALL*R updates automatically.

Now check the "keep it fully visible" — R = min(W,H)*0.40 with cx,cy center → diameter 0.8*min ≤ fits. Good. Maybe use 0.42.

Edge case: very wide window — fine.

Let's write out the file.

One more consideration about stability: when the polygon rotates and pushes a ball into a corner, penetration correction each substep might push balls out of the polygon. The safe-guard: after wall resolution, check if ball is outside by using all edges: compute max d; if max d > 0.02 (outside), pull back to the centroid of the polygon. Let's implement:

```js
let maxD = -Infinity;
... compute after all edges
if (maxD > 0.01) { // still outside after resolution (shouldn't happen)
   b.x = 0; b.y = 0; b.vx = 0; b.vy = 0;
}
```

Hmm, but 2 iterations of the edge loop should handle it. Also, since the polygon's rotation can sweep, a ball may end up in a spot where the rotating wall moved past. Position correction handles it.

Another subtlety: the position correction pushes the ball along -n_out, which may push it into another wall (near a vertex). Iterating twice helps.

Let's also cap penetration correction per step to avoid explosive jumps: pen could be big if a ball is deep outside; but with substeps, at most a few px.

Now the wall contact point q used for wall velocity should be the point on the wall closest to the ball center — which is p - n_out*d (i.e., before correction p + n_out*(r - pen)?). Let's use q = p + n_out*R_BALL after correction, which is exactly the projection on the wall line. Fine.

Let me now also double-check the angular velocity of the wall sign. Screen coordinates y down. Polygon vertex at angle a: (cos a, sin a). As angle increases with time, the point moves with velocity d/dt = (-sin a, cos a)*ω = (-y, x)*ω. So v_wall = ω*(-y, x) ✓ matches vwx = -ω*qy, vwy = ω*qx.

Good.

Now writing.

Also maybe add click/drag interaction to nudge balls? Not required; skip to save time. Could add a subtle "click to add impulse" but not needed.

Let me write index.html.

For the HUD numbers: display timer / angle maybe. Nice: show "rotation 123°" maybe not needed. I'll show a small stats line with elapsed time and rev count. Fine, low cost.

Let me write.

```js
function polyVerts(a){
  const pts = [];
  for (let i=0;i<SIDES;i++){
    const th = a + i*2*Math.PI/SIDES;
    pts.push({x:Math.cos(th), y:Math.sin(th)});
  }
  return pts;
}
```

Build balls with jitter:

```js
function buildBalls(){
  const balls = [];
  const sp = 2.14*R_BALL;
  const cols=5, rows=4;
  let idx=0;
  for (let r=0;r<rows;r++){
    for (let c=0;c<cols;c++){
      idx++;
      const jx = ((idx*37)%11 - 5)/5 * 0.06*R_BALL;   // tiny deterministic jitter
      const jy = ((idx*53)%13 - 6)/6 * 0.06*R_BALL;
      balls.push({
        x: (c-(cols-1)/2)*sp + jx,
        y: (r-(rows-1)/2)*sp + jy,
        vx:0, vy:0, angle:0, omega:0,
        n: idx,
        hue: (idx*137.508)%360
      });
    }
  }
  return balls;
}
```

Jitter magnitude 0.06r is tiny; keep non-overlap. Fine.

Hmm, but a perfect fall with a rotating heptagon: balls will land on the moving floor. Good.

Let me write the whole file now.

Colors: hue = (idx*137.508)%360 gives diverse nice colors. Saturation/lightness: `hsl(h,72%,58%)` bright. Text: dark? White with shadow works.

Actually for readability of numbers: use white text with a dark outline (strokeText dark, fillText white). Good.

Let me write.

Drawing detail for the ball:

```js
function drawBall(b){
  const rad = R_BALL*R;
  const px = cx + b.x*R, py = cy + b.y*R;
  ctx.save();
  ctx.translate(px,py);
  // glow
  ctx.beginPath(); ctx.arc(0,0,rad,0,TAU);
  ctx.fillStyle = 'hsla('+b.hue+',80%,60%,0.18)';
  ctx.fill();
  // body
  const g = ctx.createRadialGradient(-rad*0.35,-rad*0.4,rad*0.1, 0,0, rad*1.05);
  g.addColorStop(0, 'hsl('+b.hue+',85%,72%)');
  g.addColorStop(0.55, 'hsl('+b.hue+',72%,50%)');
  g.addColorStop(1, 'hsl('+b.hue+',70%,30%)');
  ctx.beginPath(); ctx.arc(0,0,rad,0,TAU);
  ctx.fillStyle=g; ctx.fill();
  ctx.lineWidth = Math.max(1, rad*0.09);
  ctx.strokeStyle='hsla('+b.hue+',90%,80%,0.75)';
  ctx.stroke();
  // rotate number
  ctx.rotate(b.angle);
  // seam marker
  ctx.beginPath();
  ctx.moveTo(rad*0.15,0); ctx.lineTo(rad*0.62,0);
  ctx.strokeStyle='rgba(255,255,255,0.28)';
  ctx.lineWidth=Math.max(1,rad*0.07);
  ctx.stroke();
  // number
  ctx.font = '700 ' + (rad*0.98).toFixed(1) + 'px ui-sans-serif, system-ui, sans-serif';
  ctx.textAlign='center'; ctx.textBaseline='middle';
  ctx.lineWidth = Math.max(1.2, rad*0.12);
  ctx.strokeStyle='rgba(12,16,30,0.55)';
  ctx.strokeText(String(b.n), 0, rad*0.03);
  ctx.fillStyle='#fff';
  ctx.fillText(String(b.n), 0, rad*0.03);
  ctx.restore();
}
```

Note the seam marker at 0 angle would overlap the number's left side; make it a short arc near the rim? Alternatively draw a small dot at (−0.55rad, 0)? Number occupies center. Put the marker from rad*0.72 to rad*0.86 — close to rim, might clash with rim stroke. Just skip the marker; but spin visibility for numbers is enough. Actually for single digit numbers like "1", rotation is less obvious. Hmm. A small seam near the edge helps. Let me draw a tiny circle ("pip") at (0, -rad*0.72) radius rad*0.1 in a light color, rotating with the ball. That clearly shows spin. Do that.

Number font size rad*0.98 might be big; for 2-digit numbers width ≈ 1.1*rad... font size 0.98rad means digits height 0.7rad, width ~1.1rad total for two digits. Ball diameter 2rad. Fine.

Now the polygon fill: also draw an inner dashed ring? Keep simple.

Background: I'd like a subtle vignette; body has a radial gradient. Canvas transparent so it shows.

Let me write everything out now. I'll include the DOM overlay with controls.

Also make sure the buttons don't steal keyboard? Add space to pause maybe. Nice: keydown space toggles pause, "r" resets. Prevent default scroll. Add that.

Now, potential physics issue: balls at the very center of a rotating heptagon with gravity 2.6 — they fall to the bottom and pile up. With rotation the pile gets dragged. Wall friction 0.35 and restitution 0.55 should keep things active but bounded. Speed clamp 7 units/s → 2380 px/s at R=340; that's fast but only in extreme cases.

I should also add a slight "rolling" so balls don't jitter in a pile forever: velocity threshold? Add small linear damping; fine.

Let's write the file.

Then test in browser and check console.

Let me write the code carefully with correct syntax.

Also note: `sim.t` used for angle; reset sets both to 0.

Write now.

Empty directory. Let me write index.html.

Now run in the browser. The browser tool probably serves index.html from cwd. Let me reload and inspect.

Initial view looks good. The numbers are visible, though the viewport is 1200x800 and the polygon's R = 0.4*800 = 320, so it fits.

The balls are arranged compactly. Note the balls look like they have a vertical line (the number's stroke?). Actually the number text is drawn but it looks like the numbers are behind? It looks a bit blurry at this size. Fine.

Now let's sample moments after some time. Let's take a screenshot after ~2s and ~5s. I can use motion frames perhaps. Let me wait using bash sleep? The browser tool actions happen instantly; I can use `browser motion` which captures frames over some period. Let me try motion.

It's working. Balls pile up at the bottom, polygon rotates. However, most balls are settled in a pile at the bottom — the "fall" already happened by 0.5s. The numbers appear rotated (some sideways), confirming spin. But I notice the balls seem somewhat overlapping? Looking at the pile, balls appear to touch/slightly overlap visually but that's just close packing. Let me zoom in to check the pile for overlaps and containment. Also I notice balls appear to be clustered and pressing against the wall—fine.

One concern: the pile looks a bit compressed and there might be persistent overlap. Let me take a screenshot at a later time and zoom. Perhaps better: add a debug check via inspect of internal state? The sim is inside an IIFE, not exposed. I can evaluate via console? The browser tool has no evaluate action, just inspect (which gives dom/console). Hmm. I could temporarily expose `window.__sim = sim` for debugging, check overlaps numerically, then leave it (harmless) or remove.

Actually, exposing a debug hook is fine but not necessary. Let me instead check visually with a cropped screenshot. I can't crop. Alternatively write a quick node test of the physics? That's a lot.

Better: temporarily add `window.__sim = sim;` and use... but there's no eval tool. inspect only returns DOM text and console. Could I dispatch via click? Not for reading numbers.

Alternative: create a temporary copy of index.html (e.g., test.html) that runs the physics headlessly for N seconds and prints overlap statistics into the DOM, then inspect the DOM text. That's feasible: build a test harness page including the same physics code? Duplication is bad. Better approach: make index.html expose window.__sim and then write a tiny test harness page that loads index.html in an iframe... but there's no network and file loading in iframe may work via relative path (browser tool serves directory presumably). Simpler: temporarily add a debug panel to index.html itself that computes max overlap and containment, printing into the DOM, record stats over time, then remove after verifying.

Actually simplest: add a hidden debug div updated each frame with max overlap depth (px), max speed, outside-check. Then inspect DOM. Then remove the debug hook after verification. That's a quick edit.

Alternatively I could reason: positional correction splits each pair overlap fully each substep; overlap should be minimal.

Let me do the quick debug instrumentation since it's cheap. I'll add a `stats` div with text updated every ~10 frames, check, then remove.

Also consider: the balls' numbers appear rotated, good. The pile appears stuck in bottom, with some balls climbing the walls as it rotates — that's expected.

One visual issue: the pile sometimes shows balls slightly overlapping visually because they press together — acceptable.

Another concern: the initial fall is very fast (~0.5s), so "sample several moments of initial fall" — verification requires capturing early frames. Hard with timestamps; I could use the browser screenshot immediately after reload. Let's do reload then immediate screenshot after small time.

Let me first add debug stats.

Oops I used escaped strings with \r characters for the old text? I passed "\r" sequences... The edit succeeded, meaning oldText matched? That's suspicious — the oldText contained literal `\r`? Wait I wrote the JSON with `\r` inside — those are literal carriage returns? Hmm, in JSON "\r" is a carriage return character. The tool reported success, so the files probably... no, oldText must match exactly. Weird. Let me check the file content.

Actually maybe the tool normalized. Let me check the file.

The function isn't called yet. Add call in frame after draw.

Initial arrangement is compact and non-overlapping, numbers upright (angle 0). But the initial view screenshot shows the balls already slightly falling (v=0.17). Fine.

Wait: at reload, the balls are visible but at first glimpse they were at center. Good. But notice the readout "ROTATION 5°" and polygon appears slightly rotated. Good.

Hmm, the numbers are still upright in this frame, spin=0.0 — meaning no ball-wall contact yet. Good.

Now check after some runtime for penetration/containment numbers. Let's wait a few seconds by clicking? I can use motion which takes ~2s, or use repeated inspect calls which seem to advance time. Let me call inspect a few times with sleeps in between? The browser tool calls are sequential; I can run bash sleep... but does the browser keep running between tool calls? Probably yes (it keeps frames? unclear). Let's just do motion twice and inspect.

Observations:
- Balls all contained (wall=0), max penetration 0.017 units ≈ 5.7px when ball radius is 13.3px — that's 42% overlap at worst, during collision transients. That's high. Visually in frame 1, balls look quite overlapping in the pile (e.g. 3/9, 4/5 area). Hmm.

Actually wait, the pile is resting with pressed overlaps: maxPen 0.017 units = 5.5px on radius 13.3 → centers separated by 21px vs 26.6 required. That is visible overlap. This is because the positional correction is split between two balls but the pressure from gravity + rotation re-compresses each frame; with restitution and substeps it should resolve... Let me think: the correction is applied fully each substep (pen split 50/50), so after the collision pass pairs should not overlap at all. But the subsequent wall correction or other pairs' corrections can reintroduce overlaps. With 20 balls in a pile, the sequential Gauss-Seidel approach needs iterations. Only 1 pass per substep of 8 substeps → 8 passes per frame, but the last pass leaves residual overlaps.

Also the debug measures penetration at the time of drawing, i.e. after the last substep's collideWalls. The wall correction pushes balls into each other, creating overlap up to pen.

To reduce visible overlap, add 2-3 iterations of collideBalls with a relaxation factor, or use baumgarte positional correction with slop. Since it's a pile with gravity, small overlaps are normal in impulse engines, but 40% radius is too much.

Wait, why so large? Gravity per substep adds Δv = 2.6 * h ≈ 2.6*0.00208 = 0.0054 units/s; positional drift per step is tiny (0.00001 units). The overlap of 0.017 must come from the wall push: wall correction pushes a ball inward by pen which then overlaps neighbors by that amount. Also the solver order matters: collideBalls runs before collideWalls, so wall correction at end of substep can create the overlap that persists to draw.

Solution: do the ball-ball resolution after wall resolution too, or iterate: e.g., per substep: walls, balls, walls, balls. And use multiple iterations in collideBalls with positional correction weighted.

Better approach: run collideBalls() twice per substep, and collideWalls() twice, alternating, and finish with collideBalls then a light wall pass? Ending with walls will reintroduce overlap again. Simplest is to iterate the pair (walls → balls) 2 times and accept small residual, with position corrections relaxed to 0.9.

Alternatively, do positional-only iterations after the velocity solve: iterate say 6 times over all pairs and walls with a small relaxation (0.6) each substep, no velocity changes. This gives good separation cheaply.

Let me implement a `separate()` function: repeats 4 times over ball pairs + walls with pure positional correction (push each by half the penetration, 0.8 relaxation). then final draw. Velocity solve remains as is in each substep.

Let me restructure step(h):
1. integrate
2. collideBalls (velocity + position)
3. collideWalls (velocity + position)
4. separatePositions(): 3-4 iterations of positional-only pushes for pairs and walls.

The last separation ensures no visible overlap.

Cost: 20 balls → 190 pair checks, 4 iterations = 760 per substep, ×8 = 6080 per frame, cheap.

Implement:

```js
function separatePositions(){
  var balls=sim.balls;
  for (var it=0; it<4; it++){
    for pairs: if d<minD: push each half (relax 1.0)
    // walls
    var verts=polyVerts(sim.angle);
    for each ball, for each edge: if d > -R_BALL: push inward by (d+R_BALL)
  }
}
```

Balls inside walls push, but pushing balls by walls can re-overlap pairs; after 4 iterations residual should be small.

Additionally, in the collision routine the position correction of pairs does 50/50; fine.

Also maybe reduce pile pressure by increasing substeps? Not needed.

Another visual concern: the numbers are rotated, but in the screenshots many numbers appear upside down (like "18" mirrored? no, rotated 180). That's expected and required: numbers rotate with the balls. Spin values up to 20 rad/s — quite fast, numbers blur visually. It's okay but maybe reduce MAX_SPIN to ~16 and spin from friction. Actually spin=20 rad/s = 3.2 rev/s, numbers spin quickly. The requirement just says numbers rotate with the ball so spin is visible. Fine, but let me reduce spin somewhat for aesthetics: increase INERTIA multiplier to 4? That reduces angular change by ~... denom = 1/m + r²/I; with I=4mr², crossA²/I = 0.25/m, denom = 2.25/m → jt = -vt*m/2.25 (vs m/6 before with I=1.6?). Wait let me recompute: previously I=1.6 m r², crossA²/I = r²/(1.6 m r²) = 0.625/m; denom = 2 + 1.25 = 3.25/m → jt = -vt*m/3.25. angular change = crossA*jt/I = r*jt/(1.6 m r²) = jt/(1.6 m r). With jt = -vt m/3.25 → Δω = -vt/(5.2 r) = -4.8*vt? For vt=1 → 4.8 rad/s. Combined over a pile, spins up to 20. Fine, keep.

MAX_SPIN 26 caps it.

Also should I worry that the rotation of the heptagon is hard to see? It's clearly rotating (screens show vertex at different angles).

Another issue: balls slope down to a corner and get pushed up the left wall — visible in frame 1 (balls between 280,585 and the wall). Fine.

Frame 1 showed heavy overlap which needs fixing. Let's add the separation iterations.

Also the debug shows pen measured after last collideWalls; if I add separation it should drop near zero.

Let me also double-check gravity feel: balls reach bottom quickly, then pile. All good.

Implement the fix.

Initial arrangement is clean, grid-like with tiny jitter, no overlaps, numbers upright, heptagon nearly at start orientation (8°). Good.

Now check post-collision overlaps and view. Let's use motion for ~2s and inspect the stats.

There are significant issues:

1. The heptagon looks like a *hexagon* in these screenshots! Count vertices in frame 1: top, upper-right, mid-right, bottom-right, bottom-left... Let me count visible vertices: at (661,113) top; (873,263) upper right; (873,533)? Actually I see corners at ~ (661,113), (873,263), (873,533)... Looking at frame 2: vertices at top (661,113), right-upper (873,263), right-lower (873,533)? Hmm, the shape appears to have 7 corners: top(661,113), upper-right(873,263), lower-right(873,533)? no...

Frame 2 outline: starting top vertex (661,113); edge down-right to (873,263); edge straight down to (873,533)? Actually the right side shows a vertical edge from y=263 to y=533 — that's one edge with both endpoints; then down-left to bottom (682,687); then up-left to (301,404)? and up to (412,116)? Let me count: (661,113) top, (873,263) right-top, (873,533) right-bottom, (682,687) bottom, (301,404) left-bottom, (412,116) left-top. That's 6 vertices visible... plus maybe one hidden behind? A heptagon's vertices are all on the convex hull so all 7 should be visible.

Wait — with 7 vertices, at some rotations one edge appears nearly vertical and one vertex may be hard to spot due to the near-collinear edges. Hmm, but geometrically a regular heptagon has 7 distinct corners. Let me count carefully in frame 4: I see corners at approximately:
- (661,113) top
- (873,263)
- (873,533)? Hmm in frame 4 image, right side has a corner at (838,235) and another at (838,535)? The polygon's rotation changes.

Actually the drawn shape in all frames looks like a hexagon with a somewhat flat top... It's suspicious. Let me check: maybe my detection is off due to antialiasing. I'll test by rendering at pause with angle 0 and counting corners programmatically (compute vertices). The vertices are defined as cos/sin at 7 angles, so there must be 7. Unless... SIDES=7 yes.

Hmm, wait: in frame 1, the shape looks like it has corners at: top (661,113); (873,263); (873,533)?? no, right side from (873,263) goes down to about (873,533) then to (682,687) then to (301,404) and (412,116) then back to top. So 6 corners. But 7th could coincide in the outline if two edges are almost collinear — for a heptagon, the interior angle is 128.6°, so no collinearity. Strange.

Let me look at frame 3: corners visible: top (581,143)?, (633,85)? Hmm the polygon looks rotated with vertices at (633,85), (838,235), (838,535)?, (698,724), (306,421), (385,178)? That's 6.

Something's off. Maybe I'm miscounting from a downscaled image. Let me render a debug test that logs the vertex count... Actually, easier: take a screenshot while paused at angle 0 and look. Or overlay vertex markers — I already draw vertex dots (small white dots). In the screenshots I can see small dots at the vertex positions? In the images, there are faint dots at (661,113) and (281,399) etc. — those are drawn vertex markers. Looking at frame 2 I see two faint dots: top (661,113) and left (281,399)? Those are the vertex markers.

Let me test the math: 7 vertices at angles 0..2π*6/7. Their shape is definitely a heptagon. The renderer draws them. So maybe I'm just misreading the rotated shape at low resolution. Actually wait — in frame 2, count again by the distinct direction changes: the left side from (301,404)? Hmm.

Let me just do a screenshot with the sim paused and count dots (7 vertex markers). I'll pause via click and screenshot.

2. Balls overlap the wall visibly in frames 1, 2, 4: e.g., ball 5 at bottom-left sits partly outside the rim. In frame 4, ball 16 at bottom seems to straddle the wall. Debug says wall=0 violations but pen tiny. Visually ball 5 sits half outside the edge at (373,573) in frame 1. Hmm, look at frame 2: ball "5" at (357,446) is clearly outside the left wall (the wall line passes through it). That means my separation pushes it inside but then the drawn image shows it outside... Wait debug max wall = 0 means no ball is within 1e-4 of the wall line violation. But the rendering shows a ball crossing the wall.

Hold on — is the wall being drawn at the same angle the physics uses? drawPolygon uses polyVerts(sim.angle), same as physics. So there shouldn't be a mismatch.

Unless the balls' positions are drawn with cx + b.x*R while physics normalized... same.

Hmm, but wait: in frame 2, ball 5 appears at the left wall intersection. Let me consider that ball 5 is inside the polygon but the wall drawn passes through it because the polygon's edge passes... If the ball is inside, the edge cannot pass through it. Unless the drawn polygon is different from physics.

OH WAIT. I see the issue! The screenshot shows the polygon rendered at a *later* angle than the physics during that frame? No, same frame.

Hmm, maybe the issue: the rotation of the polygon is fast (1.257 rad/s) and the ball is contained but the *rendered* polygon is drawn with sim.angle after the last substep, while balls were resolved... both after the same step. Consistent.

Let me consider the "5" ball in frame 2 more carefully: the left edge runs from about (301,404) upward to (412,116)? and the ball 5 is at (357,446), which is below the left-bottom vertex (301,404)? Wait if (301,404) is a vertex, the ball at (357,446) might be inside the polygon below that edge... Hmm, the polygon's left-bottom edge runs from (301,404) down-right to (682,687)? Then a ball at (357,446) would be near that edge, potentially inside. Actually looking at the image, the ball 5's center appears above the lower-left edge line. Let me look again: there's a line from (301,404) to (682,687)? No...

I think I'm over-analyzing low-res images. Let me do a programmatic check instead: expose sim on window and... I can't evaluate JS. But I can render diagnostics into the DOM: I already do: wall=0 (no violation). But the check uses d > -R_BALL + 1e-4, i.e., violation or contact. wall=0 means all balls are strictly inside by more than 1e-4. So the balls ARE inside. Then what I perceive as balls crossing the wall must be a vertex-marker/edge alignment illusion, or the balls are inside but adjacent to the wall.

Hmm, in frame 1, ball 5 at (373,573): the left-bottom edge seems to pass through the ball's upper-left. But per debug it's inside. Let me zoom by taking a screenshot at higher... couldn't crop.

Alternative approach: draw the polygon with a thicker stroke; if balls overlap, we'd see. I could also add a check for the *distance to each edge line* min over balls — debug already reports 0 violations.

Wait, maybe there's a subtle bug: the debug check uses the same verts computed at draw time, fine.

OK, let me look at the top area: in frame 2, the polygon's top-left vertex at (412,116)? and the top vertex at (661,113)? There seem to be two vertices near the top (one at 412,116 and one at 661,113) — that means 7 vertices: (412,116), (661,113), (873,263), (873,533)... hmm no. Let me count left side: (301,404) and (412,116) — both are distinct. So vertices: (412,116), (661,113)? Those two are very close in y — plausible for a heptagon rotated such that two vertices are near the top. Then (873,263), (873,533)? no wait if (873,263) and (873,533) both, the right side would be vertical and quite long — but a regular heptagon's edge length should be equal for all edges. Edge from (873,263) to (873,533) is 270px, while edge (661,113) to (873,263) is 256px, and (412,116) to (661,113) is 249px. Those are roughly equal, plausible! And then bottom (682,687): from (873,533) to (682,687) is 243px ✓. Then (301,404) to (682,687)? that's 467px — too long! So maybe there's a vertex between them, hidden around (450,580)? In frame 2 I see the bottom-left edge from (301,404) going down to about (470,590)? and then a shallow edge to (682,687)? Looking at the image, the lower-left boundary does appear to have a bend near (500,620)? Hard to tell.

OK, likely 7 vertices fine. Let me verify differently: pause at reset and take a screenshot; count the small white vertex dots. That's the easiest.

3. Another problem: rotation readout in frame 1 says 128° while frame 2 says 179°, frames are 600ms apart, meaning ~52°/0.6s = 87°/s... but expected 72°/s. Close enough given frame timing.

4. Balls scatter: in frame 4, ball 5 is falling from mid-canvas while the rest are piled — some balls bounce up. Fine.

Big issue: Balls in the pile look like they overlap each other a lot (e.g., balls 20, 5, 9 at left). Debug says maxPen=0.002 (0.6px) so visually they're just touching. The perception of overlap comes from the glows and colors. OK.

So ball-wall containment seems fine numerically; but visually the pile at the bottom is squeezed such that some ball centers appear outside the polygon. Let me trust the numeric check (wall=0).

Hmm, but wait: the numeric check `d > -R_BALL + 1e-4` counts *contacts* as violations. wall=0 means even contact tolerance has 0 count — that means no ball is in contact with the wall at the sample instant, which seems odd for a resting pile at the bottom where balls should touch the floor within 1e-4. Unless the check has an error! Let me examine:

```js
var d = (b.x - A.x) * nx + (b.y - A.y) * ny;
if (d > -R_BALL + 1e-4) { outside++; }
```
With R_BALL ≈ 0.0417. A resting ball touching the wall has d = -0.0417, so d > -0.0416 is false. Contact counts as d = -R_BALL exactly → not counted. So wall=0 is correct and indicates all balls are at or inside the wall. Good. That's just poorly named.

So containment is verified numerically. But visually ball 5 sits partly outside the rim line in frame 2... Since the check is exact and geometry identical, I'll trust it. Maybe the appearance is because the ball's glow circle (radius 1.9r) extends beyond the rim, and the body itself is at the rim. Looking at frame 2 ball 5 at (357,446): the rim line seems to pass through its center-right. Hmm.

Let me measure using pixel coordinates of the polygon: In frame 2, R = 320, center (600,400). The ball 5 center (357,446) → normalized (-0.759, 0.144), |p| = 0.772. The heptagon's apothem is 0.9009, so any point with |p| < 0.772 could still be outside if it's beyond a face... the maximum distance from center for an interior point is 1.0 (vertex). For direction of the ball, the boundary distance = apothem / cos(angle offset from face normal) which ranges 0.9–1.0. So a point at 0.772 is definitely inside. So the ball IS inside, and the line passing "through" it is probably the *inner ghost outline* at 0.93 scale? No, that's inside too, closer to the rim... 0.93*? The ghost outline is scaled 0.93 of vertex positions, so its "radius" ~0.837-0.93. Still outside 0.772.

Hmm, maybe what I see as a line through the ball is the reflection/glow. Enough; numeric checks say contained.

Wait, actually there IS one thing I notice in frame 2: the left wall line passes right through ball 5 and the ball appears on the inside. So it's just adjacent, overlapping the rim stroke visually by a pixel or two. Fine.

Now, real issues to fix:

A. The pile's balls spin very fast (spin up to 20 rad/s) making numbers unreadable — acceptable but maybe tone down spin to max 12? Requirement wants spin visible; a fast blur isn't as nice. I'll raise INERTIA to 3.2*m*r² and cap MAX_SPIN to 14. Then angular response halves. Let's check: Δω = crossA*jt/I. With I=3.2mr², denom = 2/m + 2*(r²/(3.2mr²)) = 2/m + 0.625/m = 2.625/m; jt=-vt*m/2.625 = -0.381 vt m; Δω = jt*r/I = -0.381 vt m r/(3.2 m r²) = -0.119 vt/r = -2.86*vt. So vt=2 → 5.7 rad/s. Wall: denom = 1/m + r²/I = 1 + 0.3125 = 1.3125/m; jt = -0.762 vt m; Δω = jt r/I = -0.762 vt r/(3.2r²)= -0.238vt/r = -5.7vt. Still fast. Cap 14 rad/s. OK.

Numbers rotating at 14 rad/s = 2.2 rev/s — will appear as a blur but rotation is evident. Fine.

B. The heptagon rotation appears faster than 5s/rev? Readout says REV 0.36 at frame 1 then 0.76 at frame 4 — frames approximately 2s apart (motion total ~2s + between calls). Frame1 REV 0.36, frame4 REV 0.76: Δ=0.4 rev in ~2.6s ≈ 0.15 rev/s → 6.5 s/rev. Hmm, that suggests dt accumulation is slower than real time. That's because rAF may be throttled during screenshots or the dt clamp (1/30) drops time. Not critical, but the rotation rate constancy relative to wall-clock could be affected by the clamp. The requirement is "constant rate of one revolution every five seconds" — sim-time based. Fine.

C. Balls settle at the bottom-left and pile up in a narrow wedge, leaving most of the polygon empty. It might look nicer if the pile were at the bottom. The pile sits at the lowest point which rotates around — at angle 128°, the lowest point is bottom-left-ish. Fine, that's physical.

D. Balls appear to sometimes escape upward? No.

Also I realize the balls' numbers are hard to read at radius 13px with font 13px and two digits — okay.

Next: verify reset restores arrangement and orientation. Click Pause then Reset, screenshot, compare with initial. But when paused, does the sim keep paused after reset? Yes, paused stays. Then the view shows balls at initial grid angle 0.

Let me test: click pause, click reset, screenshot.

Also check console errors — none.

Let me apply the spin tweak, remove debug after final verification. Actually keep debug until the end, then remove.

Let me also consider adding a subtle motion trail? No.

One more improvement: at rest, balls pile with jitter due to the rotating walls. Fine.

Apply spin tuning.

Wait, larger inertia makes spin-up slower but also makes the friction impulse *more* effective at stopping tangential slip? jt clamps at μ|jn|. Fine.

Now verify a paused reset view and the initial fall. Reload, immediately screenshot? rAF runs while we take screenshots. Let's do: reload, then click Pause quickly? The click coordinates of the pause button are (545,755). After reload, balls start falling. Hard to pause within 200ms. But I can verify reset restores arrangement: click pause at any time, then reset → paused at angle 0 with initial grid. Screenshot that and compare to initial.

Then click Resume to continue and observe.

Let me do that.

The initial view shows a heptagon with 7 vertices (dots): I count dots at top (573,81), right-upper (830,177), right-lower (908,436), bottom (766,672), bottom-left (487,671), left (327,464), left-top (298,229). Wait — 7 dots visible in this image? I see dots: (573,81), (830,177), (908,436), (766,672), (487,671), (327,464), (298,229). That's 7! But wait, the outline drawn — the top-left edge goes from (298,229) to (573,81); the right side... Hmm, but the outline seems to show only 6 corners? Because the dots at (766,672) and (487,671) both are on the bottom and the bottom edge between them looks flat-ish. Let me count edges: top-left edge (298,229)-(573,81); top-right (573,81)-(830,177); right upper (830,177)-(908,436); right lower (908,436)-(766,672); bottom (766,672)-(487,671); bottom-left (487,671)-(327,464)?? Hmm and (327,464)-(298,229). That's 7 edges! Yes, it's a heptagon. Great — earlier images were just hard to read.

But hold on: the vertex dots are 28px further out than the drawn outline? Look at the dot at (573,81) while the outline's top corner is at (573,110)? The dots appear offset outward from the outline! Hmm, in this screenshot the dot at top (573,81) is above the polygon's top corner which looks like it's at y≈110. Also dot at (327,464) vs outline's left corner near (338,470)? Slight offset.

Why? The vertex dots are drawn with ctx.arc(verts[v].x, verts[v].y, 2.4/sim.R) — 2.4/R = 0.0075 → 2.4px radius. But there's `ctx.shadowBlur = 0` reset before. Also `ctx.lineWidth` etc. Hmm, the dots appear ~30px outside at the top. Odd.

Actually, could it be `ctx.shadowColor`/shadowBlur leftover on the dots? No.

Wait, maybe the dots I see are the *previous* polygon angle's vertices? No, same frame.

Let me look again at the image description: dots at various positions. The top dot at (573,81) — the polygon's top vertex in the image looks like (573,110). Difference 29px. The right-upper dot at (830,177) vs outline corner (830,200)? Also ~23px outside. The right-lower dot (908,436) vs corner (890,436)? Hmm.

Maybe the shadowBlur=22 applied to the rim stroke makes the stroke appear shifted? No.

OH. I bet it's `ctx.lineJoin='round'` + shadow... no.

Hmm, maybe the discrepancy is that I perceive vertices at the rounded corners' outer glow edges. Let me test precisely by computing where vertices should be: angle = 7°=0.122 rad, R=320, center (600,400). V0 = (cos.122, sin.122)*320 + center = (917.6, 438.9) ✓ that's near the dot at (908,436) — matches within a few px. V1 at angle 0.122+0.897=1.019: (0.524,0.851)*320+(600,400) = (767.7, 672.3) ✓ dot at (766,672). V2 angle 1.917: (-0.340,0.940)*320 → (491,700)? Hmm dot at (487,671)?? Let me recompute: 1.917 rad: cos=-0.339, sin=0.941 → (491.5, 701.1). The dot in the image is at (487,671) — 30px off. Hmm, my pixel estimates from the image are approximate (image is 1200x800 exactly). Since the outline bottom in the image is at y≈675 and the dot at (487,671), they align. My computed y=701 doesn't match. So maybe the drawn polygon is smaller than 320?

R = 0.4*min(1200,800) = 320. Center (600,400). Bottom vertex y = 400+0.941*320 = 701. But the image's bottom edge is at y≈675. Hmm, so maybe R is 292? 0.941*292+400 = 675 ✓. And right vertex x = 600+0.972*292 = 884 ✓ (image ~890). So R ≈ 292, not 320. Why? Because window.innerHeight might be 730 (excluding some chrome)? The screenshot is 1200x800 but the page viewport might be 1200x730. Indeed 0.4*730 = 292. OK that resolves it. So the dots ARE at the vertices. 

So the heptagon is fine, and the ball pile appearance is fine.

The initial arrangement is a neat 5x4 grid, numbers upright. Good.

Now test pause/reset behavior: click Pause, click Reset, screenshot, then Resume and check.

Paused. Now the balls' numbers are rotated, spin visible. The pile is at bottom-left. Ball 9 is in the air (bouncing).

Now click Reset while paused to verify restoration.

Reset while paused restored the initial grid and wall orientation (rotation 0°, same as initial view — compare vertices: at reset, top vertex (528,110)... in the first initial screenshot at 7° the top vertex was (573,110). Close; angle 0 now).

Interesting: the polygon at "rotation 0°" is drawn slightly different from the very first screenshot (which was at 7°). Good.

Note the balls look slightly overlapping in the grid — no, they're adjacent with small gaps (spacing 2.14r). The glow makes them look overlapping. Fine.

Hmm, but the reset grid differs from the initial screenshot? In the initial screenshot rows appear with jitter (e.g. 2 higher, 7 higher). Here similarly rows staggered? Looks consistent (same deterministic jitter).

Now resume and check falling + collision bounce with a bounce. Let's resume and take a couple of screenshots. Also verify the ball-ball bounce response visible.

I should also verify the numbers rotate: seen (numbers at various angles in earlier screenshots, spin=13.4). ✓

Now check the physics stability over a longer run and that balls don't gain unbounded speed. I'll resume and let it run for ~10s while doing other things, then check v/spin/pen. The debug line gives max speed.

Let me click Resume and use motion twice.

Resumed; the balls just started falling (v=0.13). Note that clicking Resume then a screenshot at +0ms is very early.

Let me capture the falling sequence using motion, which gives frames at ~600ms intervals. But the simulation appears to run slower than real time in the browser tool? Actually the motion tool captures frames as fast as the page runs.

Let's call motion and see the fall and collisions. I want to confirm a visible bounce off a moving wall. Let me run motion twice.

Great, this looks good. The heptagon rotates, balls fall to the bottom-left wall and pile, balls are contained, numbers rotate (various orientations), one ball (19, 6) bounces up off the moving wall.

Observations:
1. The pile is riding up along the lower-left wall as the polygon rotates — realistic and shows moving-wall interaction.
2. Max penetration 0.0016 (0.5px) — excellent.
3. Max velocity 5.1 units/s — bounded (cap 7).
4. Spin 10-15 rad/s.

Potential issue: The balls pile up in a wedge and don't spread along the floor. Fine.

Visual: numbers are readable. Spin visible.

One thing: ball-ball bounces aren't super visible because the pile moves together. Since balls are all at the same spot with gravity, that's expected. Bouncing balls like 19 demonstrate contact.

Also the pile appears to be outside the wall? Ball 5 at (365,541) in the first motion frame — the wall passes near it. Numeric checks confirm containment.

Concern: balls seem to be outside the drawn outline in frame 1 (left column) — e.g. ball 7 at (370,543) seems to sit over the line. It's at the wall, so partially covering the rim stroke by glow. Fine.

Now long-run stability: let it run for ~15+ seconds and check pen/speed. Let me use several motion calls (each ~2s). Also, verify no ball trapped outside.

One additional thing to verify: Reset while running restores arrangement and orientation, and the pause button works. Already did.

Let me now remove the debug panel and do a final reload + visual check (initial view, a mid-fall, and reset). Actually keep debug for the long run test, then remove.

Let's do 4-5 motion calls to accumulate ~10s of sim time. But motion images consume context. Instead use inspect calls with sleeps? inspect returns a screenshot too. Both attach images. Cheap enough: one inspect after some seconds. But does the sim advance while no tool is running? Probably yes if the browser keeps rAF running. Let me run bash sleep 10 then inspect.

After ~4 revolutions: still contained, max speed 4.56, max spin 13.5, penetration 0.0083 (2.5px max, transient). Stable.

However, I notice the pile is stuck in the lower-left corner and doesn't distribute much — with the rotating walls it should slide around the bottom as the polygon rotates. The pile appears to remain at the bottom-left over multiple revs while the polygon rotates. Actually the bottom-left vertex is where balls collect because gravity + rotation. Hmm, in a rotating container with friction, balls generally get carried a bit then slide back. The image suggests they're pinned at the lower-left. Is the polygon rotation actually carrying them? Compare rev 3.99 and rev 0.84: the pile was at the bottom center-left, now at lower-left, so it does move with the walls somewhat. Fine.

But wait — a physically odd thing: the pile's position relative to the polygon changed (it's now near the lower-left vertex). It's fine.

One more improvement for visual appeal: correct spin direction. Looks fine.

Now potential issue: the ball 19 at (706,594) is floating/falling in open space in the middle-right — that's a ball that was bouncing. Fine.

Now remove the debug panel from the final file, then final check with reload.

Also I should double check the "20 balls" count visible: yes.

Another check: after Reset while running, the sim restarts. Already verified while paused. Good enough.

Let me remove debug code cleanly.

Remove lines 82, 495-525ish, and 551. Let me view exact boundaries.

Remove the debug panel div, the debugStats function, and the call. I'll use sed with line ranges carefully, or use the edit tool with exact text. Since the function text is long, use sed to delete lines 495-524 and the div line 82, and replace call line.

Line numbers: 495 = "var dbgPrev = 0;" through 524 = "}" (the line before blank and function draw). Let's delete 495-525 (including blank). Check line 525 is blank, 526 "function draw()".

Also remove the div at line 82. Do it: delete 495-525 then 82, and remove line "    debugStats(now);".

Use sed to delete ranges in reverse.

The initial arrangement is nice, but the digits look a bit cut/blurred — font size rad*0.98 = ~13px with font "ui-sans-serif" fallback. It reads "1 2 3 4 5..." though. Two-digit numbers seem cramped (10-20) but readable. Fine.

Wait — the ball numbering layout: row 1 has 1..5, row 2 has 6..10, etc. Yes.

The numbering text appears slightly clipped by the seam pip? No.

One issue: the digits look small relative to the ball. Fine.

Now the last thing: verify pause/resume click still works after removing debug and the reset. And confirm the final file has no debug leftovers, no console errors. Also confirm the fall again and bounce. I've checked plenty. One more quick check: pause, then reset, then resume.

Actually I already verified these behaviors before removal; the only change was removing debug code. The reload shows correct initial view and no console errors.

Let me do one quick pause/resume test to ensure the button handlers still work (they weren't touched).

Also confirm the file is self-contained (no external refs). grep for http/src.

Pause works, balls frozen against the rotating wall (some bouncing, e.g. 9 in the air, 16 falling). Numbers clearly rotated — spin visible.

One notable thing: ball 16 at (424,527) appears to be *outside* the wall (left of the wall line at x≈390 near y=520?). Actually the wall goes from (301,356) down to about (383,545)? The ball at (424,527) looks inside, just next to the wall. It's in the middle of a fall, having bounced.

Hmm, ball 9 at (387,470) also appears close to the wall line — inside.

Let me be sure about containment over a long run. I removed the debug, but I can re-add temporarily? I already verified over 4 revolutions with wall=0. Given the deterministic nature and pause tests, I'm confident.

However, the pile appears to be partly outside the wall on the left: ball 1 at (385,617) crosses the wall line? The wall near y=617 is around x≈420. Hmm. Looking at the image: the lower-left wall runs from ~(301,356) to ~(421,447)? then a vertical-ish? Let me trace: the polygon at rotation 239° has vertices at (303,356), (444,115), (707,113), (882,297), (877,533)?? and the bottom (620,687), (421,636)? Actually I see dots at (286,356), (438,125), (707,113), (901,297), ... and near (747,737), (647,700)?

I see faint dots in the image at (286,356), (438,125), (707,113), (901,297) and maybe (748,737),(650,700) and (421,636)? The bottom-left dot near (421,636) is right where ball 1 sits. So the ball is at the vertex, inside. OK good.

So containment holds.

Now I'm fairly confident in the deliverable. The file is 554 lines. Final check on the file content for leftover duplication (e.g., "function draw" blank line). Let me quickly view a few sections for sanity and confirm no syntax problems (console is clean).

One last consideration: the ball count is exactly 20 ✓; numbering 1-20 ✓.

Also requirement: "Initially place them in a compact, non-overlapping arrangement near the enclosure center and let them fall under gravity." ✓.

"Rotate the heptagon at a constant rate of one revolution every five seconds." ✓ (readout shows rev based on sim time).

"Keep it fully visible." R = 0.4*min(w,h) ✓.

"Account for moving-wall velocity in bounce response, and include damping or friction." ✓.

"Give each ball angular motion and rotate its number with the ball so spin is visible." ✓.

"Pause/Resume and Reset. Reset restores initial ball arrangement and wall orientation." ✓.

Let me finally screenshot the reset-at-paused state once more to compare with the initial view (already did, matched). Good.

I'll finish with a summary. Also clean up: no leftover temp files. Only index.html.

Wait, I should double check the reset restores the wall orientation to exactly 0 and the initial grid — verified earlier: rotation 0°, grid centered and upright numbers.

One more: after reset, if paused, resuming continues. Verified via resume click.

Final report.

Everything is in order. Final answer summary.

Let me include what I verified and what remains unverified:
Verified: initial render, 20 balls 1-20 in compact grid, falling, ball-ball/ball-wall collisions with spin (numbers rotated), containment (no wall violations over ~4 revolutions), bounded speeds/spins, penetration tiny, pause/resume, reset (while paused) restores grid + 0° orientation, no console errors.
Unverified: precise wall-clock 5s/rev timing (sim-time based; readout showed ~4 revs of sim time), long-run (>1 min) extremes, high-DPI/resize behavior, touch/mobile layout.