Bouncing balls in a heptagon · deepseek-v4.1-flash

Let me build this. I'll create a self-contained index.html with a physics simulation.

Let me think about the physics carefully.

Heptagon rotating at 1 rev / 5 s. Center (cx, cy), radius R. 7 vertices at angles θ0 + i*2π/7, where θ0 = wallAngle (rotating).

Balls: radius ~10-12 px, numbered 1-20. Initial arrangement compact near center: e.g., grid 4x5 spacing 2.2*r, centered.

Gravity ~ 500 px/s². Ball-wall collision: for each edge (segment between consecutive vertices), find closest point on segment to ball center. If distance < ball radius, resolve. Account for wall velocity: wall point velocity = ω × r (perpendicular). v_wall = ω * (-(p.y-cy), (p.x-cx)). Then relative velocity, reflect normal component with restitution, apply friction on tangential.

Also spin: ball has angle, angular velocity. On collision, tangential relative velocity at contact includes spin: v_contact = v_ball + ω_ball × r_contact. Compute friction impulse to modify spin.

Let me implement a reasonably standard impulse-based resolution.

For ball-ball: elastic collision with restitution, plus friction affecting spin.

Simpler stable approach: use positional correction + velocity reflection.

Let me write it.

Constants:
- gravity g = 900 px/s²
- restitution e = 0.55 for walls, 0.5 for balls
- friction coefficient
- air damping: v *= 0.999 per frame-ish, use exponential.
- angular damping.

Time step: fixed dt = 1/120 with substeps, or use accumulator. Let's do accumulator with fixed dt=1/120, max steps.

Ball radius: choose 11. Heptagon radius R = 300 (canvas 700x700 maybe). 20 balls in grid 4 cols x 5 rows spacing 24 → width 3*24=72, height 4*24=96. Fits.

Initial arrangement: place at center, columns x rows. Center them. Non-overlapping with spacing 2*r + 2 = 24.

Actually balls numbered 1-20 in order left-to-right, top-to-bottom.

Wall collision detection: for each of 7 edges, closest point. Also need to handle corners but closest-point on segment handles it.

Let me handle moving wall velocity at contact point:
omega = 2π/5 rad/s. Wall point velocity = omega * (-(py-cy), (px-cx)) since rotating with angle increasing (CCW in math but canvas y down means clockwise visually).

Actually in canvas, angle increases clockwise visually. Wall angle θ. Vertex = center + R*(cos θ, sin θ). dθ/dt = omega. Velocity = R*omega*(-sin θ, cos θ) = omega * (-(vy), (vx)) relative to center. Yes: d/dt (cosθ, sinθ) = omega*(-sinθ, cosθ). And (-(y-cy), (x-cx)) = R*(-sinθ, cosθ). Correct.

Collision resolution:
- n = normal from wall closest point to ball center, normalized. (pointing into interior, toward ball)
- pen = r - dist. If pen > 0, push ball out by pen along n.
- contact point on wall pc.
- v_wall at pc.
- relative velocity vrel = v_ball - v_wall.
- vn = vrel·n. If vn < 0 (moving into wall), apply impulse.
- For wall (infinite mass), j = -(1+e)*vn. v_ball += j*n.
- friction: tangential t = vrel - vn*n. vt = vrel·t. Apply friction impulse opposing tangential slide, magnitude min(mu*|j|, |vt|)... plus spin coupling.

Simpler: apply tangential friction that also changes angular velocity. Let's include spin:
Contact point relative to ball center: rc = -n * r (from center to contact).
v_contact = v_ball + ω × rc. In 2D ω×rc = ω * (-rc.y, rc.x).
vrel = v_contact - v_wall.
vn = vrel·n.
tangential t direction = vrel - vn n, normalize.
vt = vrel·t.

Impulse magnitude for friction: jt = -vt / (1/m + r²/I). For a disk I = 0.5*m*r², so r²/I = 2/m. So denominator = 3/m, jt = -vt*m/3. Clamp to mu*|jn| where jn = -(1+e)*vn*m... but wall infinite: jn = -(1+e)*vn (per unit mass). Hmm let's just use per-mass impulses.

Let me use mass m=1 for all balls.

Normal impulse (ball-wall): scalar jn = -(1+e)*vn. v_ball += jn * n.
Friction: vt vector. denom = 1 + r²/I = 3. jt = -vt/3. Clamp |jt| <= mu*|jn|. Apply v_ball += jt*t. ω += (rc × (jt*t))/I ... cross in 2D: torque = rc.x*F.y - rc.y*F.x. Δω = torque/I.

I = 0.5*r². rc = -n*r.

For ball-ball: normal from A to B. relative velocity at contact including spin.
Both mass 1, I same.
jn = -(1+e)*vn / (1/mA + 1/mB) = -(1+e)*vn/2.
Apply vA -= jn*n? Wait convention. n from A to B. vrel = vB - vA (at contact). vn = vrel·n. If vn < 0 approaching. jn = -(1+e)*vn/2. vB += jn*n, vA -= jn*n.
Friction similar with denom terms.

This is getting complex but doable. Let me keep spin friction simple but present.

Actually to ensure stability and visible spin, I'll do collisions and also add spin coupling. Numbers rotate with ball angle.

Let me write the code.

Positional correction for overlaps also important to avoid persistent overlap. After velocity resolution, separate balls by moving each half of penetration along n. For walls push out fully.

Let me consider using substeps and a few iterations of collision each frame.

I'll implement fixed timestep dt = 1/240 maybe, 4 substeps per frame at 60fps. That's 240 steps/sec. Fine for 20 balls.

Actually accumulator: while(acc >= dt){ step(dt); acc-=dt; }.

In step:
- integrate velocities (gravity, damping)
- integrate positions
- enforce wall collisions
- ball-ball collisions (with positional correction)

Damping: v *= (1 - 0.02*dt*?) ... use v *= Math.pow(0.999, dt*60)? Let's just v *= 0.999 per step at dt=1/240 → too much. Use linear drag: v *= Math.exp(-0.15*dt). Angular: ω *= Math.exp(-0.4*dt).

Restitution 0.6. mu = 0.3.

Speed cap: clamp speeds to e.g. 3000 px/s.

Let me now think sizes. Canvas 760x760. Center 380,380. Heptagon circumradius R = 300. Balls radius 12. 20 balls grid 4x5 spacing 26 → 78 x 104. Compact near center. Good.

But if heptagon rotates fast (72°/s), balls will be flung. Gravity 900. Should be contained since wall pushes. Should be okay.

Let me write the HTML.

Number rendering: draw ball circle, then save, translate to ball pos, rotate(ball.angle), draw text centered.

Let me write initial arrangement function that computes positions once and stores for reset.

Reset restores positions, zero velocities, angles=0, wallAngle=0.

Let me build.

Canvas resizing: I'll use fixed logical size and CSS scale to fit. Use devicePixelRatio maybe. Keep simple: canvas width/height 760, CSS max-width 100%.

Let me write.

For wall color/styling, draw heptagon stroke with lineWidth 8, and maybe fill translucent.

Let me now write the file.

I'll write carefully.

Code:

```html
<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Rotating Heptagon — Numbered Balls</title>
<style>
  html,body{margin:0;height:100%;}
  body{background:radial-gradient(circle at 50% 30%, #1b2233, #0a0d14 70%);color:#e8eef7;font-family:system-ui,-apple-system,Segoe UI,Roboto,sans-serif;display:flex;flex-direction:column;align-items:center;justify-content:center;min-height:100vh;gap:14px;padding:16px;box-sizing:border-box;}
  h1{font-size:16px;font-weight:600;letter-spacing:.08em;text-transform:uppercase;color:#9fb3d1;margin:0;}
  #wrap{position:relative;}
  canvas{display:block;background:#0d1320;border-radius:12px;box-shadow:0 18px 50px rgba(0,0,0,.6), inset 0 0 0 1px rgba(255,255,255,.06);max-width:92vmin;max-height:78vmin;width:760px;height:760px;}
  #controls{display:flex;gap:12px;}
  button{cursor:pointer;border:1px solid rgba(140,180,255,.35);background:linear-gradient(180deg,#243352,#1a2438);color:#dbe7fb;padding:9px 20px;border-radius:9px;font-size:14px;font-weight:600;letter-spacing:.02em;transition:transform .08s, background .2s;}
  button:hover{background:linear-gradient(180deg,#2d4066,#202c46);}
  button:active{transform:translateY(1px);}
  #hud{font-size:12px;color:#7f95b5;letter-spacing:.03em;}
</style>
</head>
<body>
<h1>20 Balls in a Rotating Heptagon</h1>
<div id="wrap"><canvas id="c" width="760" height="760"></canvas></div>
<div id="controls">
  <button id="pause">Pause</button>
  <button id="reset">Reset</button>
</div>
<div id="hud">1 revolution / 5 s &middot; gravity + damping &middot; ball-ball &amp; ball-wall collisions</div>
<script> ... </script>
```

Now JS.

```js
(() => {
const canvas = document.getElementById('c');
const ctx = canvas.getContext('2d');
const W = canvas.width, H = canvas.height;
const CX = W/2, CY = H/2;
const R = 300;          // heptagon circumradius
const N = 7;            // sides
const BALL_R = 12;
const N_BALLS = 20;
const OMEGA = 2*Math.PI/5;  // rad/s
const G = 1000;
const REST_WALL = 0.6, REST_BALL = 0.55;
const MU = 0.35;
const DRAG = 0.12;       // linear
const ANG_DRAG = 0.45;
const MAX_SPEED = 3500;
const I = 0.5*BALL_R*BALL_R; // mass 1

let balls = [];
let wallAngle = 0;
let running = true;
let last = performance.now();
let acc = 0;
const FIXED = 1/240;

function initialLayout(){
  const cols = 4, rows = 5;
  const sp = BALL_R*2 + 2.5;
  const startX = -(cols-1)*sp/2;
  const startY = -(rows-1)*sp/2;
  const arr = [];
  let n = 1;
  for(let r=0;r<rows;r++){
    for(let c=0;c<cols;c++){
      arr.push({x: CX + startX + c*sp, y: CY + startY + r*sp, vx:0, vy:0, angle: (n*0.37)%Math.PI, omega:0, n:n});
      n++;
    }
  }
  return arr;
}

function reset(){
  balls = initialLayout();
  wallAngle = 0;
  running = true;
  pauseBtn.textContent = 'Pause';
}
```

Wait, reset should restore wall orientation — wallAngle=0. But canvas draw uses wallAngle. Good.

Hmm, initial arrangement "near enclosure center" with random-ish angles fine. My angle formula uses n*0.37, deterministic so reset restores it. Good.

Gravity acts downward in canvas (+y).

Wall vertices:

```js
function vertices(angle){
  const vs = [];
  for(let i=0;i<N;i++){
    const a = angle + i*2*Math.PI/N;
    vs.push({x: CX + R*Math.cos(a), y: CY + R*Math.sin(a)});
  }
  return vs;
}
```

Physics step:

```js
function step(dt){
  wallAngle += OMEGA*dt;
  // integrate
  for(const b of balls){
    b.vy += G*dt;
    const d = Math.exp(-DRAG*dt);
    b.vx *= d; b.vy *= d;
    b.omega *= Math.exp(-ANG_DRAG*dt);
    // cap
    const sp = Math.hypot(b.vx,b.vy);
    if(sp>MAX_SPEED){ const k=MAX_SPEED/sp; b.vx*=k; b.vy*=k; }
    b.x += b.vx*dt;
    b.y += b.vy*dt;
    b.angle += b.omega*dt;
  }
  const vs = vertices(wallAngle);
  collideWalls(vs);
  collideBalls();
}
```

Wait: wallAngle update should be consistent with wall velocity used in collisions. If we update wallAngle before collisions, the vertex positions correspond to new angle and velocity omega. Fine.

But the angular velocity at a point is based on current angle. Use omega consistent.

collideWalls:

```js
function collideWalls(vs){
  for(let i=0;i<N;i++){
    const a = vs[i], b = vs[(i+1)%N];
    for(const ball of balls){
      // closest point on segment
      const ex = b.x-a.x, ey = b.y-a.y;
      const len2 = ex*ex+ey*ey;
      let t = ((ball.x-a.x)*ex + (ball.y-a.y)*ey)/len2;
      t = Math.max(0, Math.min(1, t));
      const px = a.x + t*ex, py = a.y + t*ey;
      let nx = ball.x - px, ny = ball.y - py;
      let dist = Math.hypot(nx,ny);
      if(dist === 0){ nx = CX-ball.x; ny=CY-ball.y; dist=Math.hypot(nx,ny)||1; }
      nx/=dist; ny/=dist;
      const pen = BALL_R - dist;
      if(pen > 0){
        // positional correction
        ball.x += nx*pen;
        ball.y += ny*pen;
        // wall point velocity
        const wvx = -OMEGA*(py-CY);
        const wvy = OMEGA*(px-CX);
        // contact point velocity of ball
        const rcx = -nx*BALL_R, rcy = -ny*BALL_R;
        const cvx = ball.vx + ball.omega*(-rcy);
        const cvy = ball.vy + ball.omega*(rcx);
        let rvx = cvx - wvx, rvy = cvy - wvy;
        const vn = rvx*nx + rvy*ny;
        if(vn < 0){
          const jn = -(1+REST_WALL)*vn;
          ball.vx += jn*nx;
          ball.vy += jn*ny;
          // friction: recompute rel vel
          let rvx2 = (ball.vx + ball.omega*(-rcy)) - wvx;
          let rvy2 = (ball.vy + ball.omega*(rcx)) - wvy;
          const vn2 = rvx2*nx + rvy2*ny;
          let tx = rvx2 - vn2*nx, ty = rvy2 - vn2*ny;
          const tlen = Math.hypot(tx,ty);
          if(tlen > 1e-6){
            tx/=tlen; ty/=tlen;
            let jt = -tlen/3; // denominator (1/m + r^2/I)=1+2=3
            const maxj = MU*Math.abs(jn);
            if(jt < -maxj) jt = -maxj;
            ball.vx += jt*tx;
            ball.vy += jt*ty;
            const torque = rcx*(jt*ty) - rcy*(jt*tx);
            ball.omega += torque/I;
          }
        }
      }
    }
  }
}
```

Wait, careful: after applying normal impulse, I recompute relative velocity including updated ball velocity but I use old omega. That's fine.

But applying normal impulse then friction: friction impulse changes velocity and omega. Good.

Hmm the applied friction: v_ball += jt*t. torque = rc × F = rcx*Fy - rcy*Fx where F = jt*t. So torque = rcx*(jt*ty) - rcy*(jt*tx). Δω = torque/I. Good.

Note rc = -n*R, so contact point relative to ball center is toward the wall. For a ball resting on floor (wall below, n pointing up i.e. ny=-1 in canvas? Let's check: wall segment below ball, closest point below, nx,ny = ball - p, points upward so ny negative). rc = -n*R points downward toward contact. Correct.

The sign of ω: positive ω rotates clockwise in canvas (since y down and standard rotation matrix). v at contact = v + ω × rc where × gives (-ω*rcy, ω*rcx)... actually for 2D, angular velocity ω (scalar), velocity = ω * (-rcy, rcx) if positive ω counterclockwise in math coords. In canvas y-down, positive ω appears clockwise. Whatever, consistent.

Now ball-ball collisions:

```js
function collideBalls(){
  for(let i=0;i<balls.length;i++){
    for(let j=i+1;j<balls.length;j++){
      const A=balls[i], B=balls[j];
      let dx=B.x-A.x, dy=B.y-A.y;
      let d2=dx*dx+dy*dy;
      const minD = BALL_R*2;
      if(d2 < minD*minD){
        let d=Math.sqrt(d2);
        if(d<1e-6){ dx=1; dy=0; d=1e-6; }
        const nx=dx/d, ny=dy/d;
        const pen=minD-d;
        // positional correction half each
        A.x -= nx*pen/2; A.y -= ny*pen/2;
        B.x += nx*pen/2; B.y += ny*pen/2;
        // contact rel velocity including spin
        const rA_x = nx*BALL_R, rA_y = ny*BALL_R;   // A center to contact
        const rB_x = -nx*BALL_R, rB_y = -ny*BALL_R;
        const cvA_x = A.vx + A.omega*(-rA_y);
        const cvA_y = A.vy + A.omega*(rA_x);
        const cvB_x = B.vx + B.omega*(-rB_y);
        const cvB_y = B.vy + B.omega*(rB_x);
        let rvx = cvB_x - cvA_x, rvy = cvB_y - cvA_y;
        const vn = rvx*nx + rvy*ny;
        if(vn < 0){
          const jn = -(1+REST_BALL)*vn/2;
          A.vx -= jn*nx; A.vy -= jn*ny;
          B.vx += jn*nx; B.vy += jn*ny;
          // friction
          let rvx2 = (B.vx + B.omega*(-rB_y)) - (A.vx + A.omega*(-rA_y));
          let rvy2 = (B.vy + B.omega*(rB_x)) - (A.vy + A.omega*(rA_x));
          const vn2 = rvx2*nx+rvy2*ny;
          let tx = rvx2 - vn2*nx, ty = rvy2 - vn2*ny;
          const tlen=Math.hypot(tx,ty);
          if(tlen>1e-6){
            tx/=tlen; ty/=tlen;
            // denom = 1/mA+1/mB + (rA×t)^2/IA + (rB×t)^2/IB
            const rAxt = rA_x*ty - rA_y*tx;
            const rBxt = rB_x*ty - rB_y*tx;
            const denom = 2 + (rAxt*rAxt)/I + (rBxt*rBxt)/I;
            let jt = -tlen/denom;
            const maxj = MU*Math.abs(jn);
            if(jt < -maxj) jt = -maxj;
            A.vx -= jt*tx; A.vy -= jt*ty;
            B.vx += jt*tx; B.vy += jt*ty;
            A.omega -= (rA_x*(jt*ty)-rA_y*(jt*tx))/I;
            B.omega += (rB_x*(jt*ty)-rB_y*(jt*tx))/I;
          }
        }
      }
    }
  }
}
```

Wait sign of friction for A: force on A is -jt*t (since jt negative, -jt*t points opposite t = along tangential relative velocity? Let's verify: jt = -tlen/denom negative. F_A = -jt*t = positive t direction. Hmm. relative tangential velocity of B relative to A is along t (positive). Friction should oppose relative motion: force on B should be -t direction, force on A +t. So F_A = -jt*t? jt negative → -jt positive → along t. Yes F_A along +t, correct. Good.

Torque on A: rcA × F_A = rA_x*F_Ay - rA_y*F_Ax. F_Ax = -jt*tx, F_Ay=-jt*ty. So torque = rA_x*(-jt*ty) - rA_y*(-jt*tx) = -jt*(rA_x*ty - rA_y*tx) = -jt*rAxt. Δω_A = -jt*rAxt/I. My code: A.omega -= (rA_x*(jt*ty)-rA_y*(jt*tx))/I = -(jt*rAxt)/I. Correct. Good.

Now rendering. Draw background, heptagon, balls.

```js
function draw(){
  ctx.clearRect(0,0,W,H);
  // subtle floor grid? skip
  const vs = vertices(wallAngle);
  ctx.save();
  ctx.beginPath();
  ctx.moveTo(vs[0].x, vs[0].y);
  for(let i=1;i<N;i++) ctx.lineTo(vs[i].x, vs[i].y);
  ctx.closePath();
  ctx.fillStyle='rgba(60,90,150,0.10)';
  ctx.fill();
  ctx.lineJoin='round';
  ctx.lineWidth=9;
  ctx.strokeStyle='#5b8fd6';
  ctx.shadowColor='#4a7fd0'; ctx.shadowBlur=18;
  ctx.stroke();
  ctx.restore();
  // balls
  for(const b of balls){
    ctx.save();
    ctx.translate(b.x,b.y);
    // body
    const grd=ctx.createRadialGradient(-BALL_R*0.35,-BALL_R*0.35,BALL_R*0.1,0,0,BALL_R);
    grd.addColorStop(0,'#ffe9a8'); grd.addColorStop(1,'#e0a53c');
    ctx.beginPath(); ctx.arc(0,0,BALL_R,0,Math.PI*2);
    ctx.fillStyle=grd; ctx.fill();
    ctx.lineWidth=1.5; ctx.strokeStyle='rgba(60,30,0,0.5)'; ctx.stroke();
    // number
    ctx.rotate(b.angle);
    ctx.fillStyle='#2a1a00';
    ctx.font='bold 13px system-ui, sans-serif';
    ctx.textAlign='center'; ctx.textBaseline='middle';
    ctx.fillText(b.n, 0, 1);
    ctx.restore();
  }
}
```

Hmm textBaseline middle, fillText at (0,1). fine.

Also draw center marker maybe.

Main loop:

```js
function loop(now){
  const dtReal = Math.min(0.05,(now-last)/1000);
  last=now;
  if(running){
    acc += dtReal;
    let steps=0;
    while(acc>=FIXED && steps<40){ step(FIXED); acc-=FIXED; steps++; }
    if(steps===40) acc=0;
  }
  draw();
  requestAnimationFrame(loop);
}
```

Pause: running=false. Resume true.

Buttons.

Let me double check wall containment. Balls could tunnel at high speed even with dt=1/240. Max speed 3500 * 1/240 ≈ 14.6 px per step. Ball radius 12. Could tunnel through wall if moving 14px per step and wall thickness... The wall is a line, collision detection is point-to-segment distance each step. If ball passes completely to other side and distance > radius, no collision → escapes. 14.6 px vs radius 12 → could skip. To be safe, reduce MAX_SPEED to 2500 (10.4px/step) or increase substeps (dt=1/300). Let's use FIXED=1/300 and MAX_SPEED=2600 → 8.7px/step < 12. Good.

Also positional correction: pushing out by pen along n where n computed from side ball is currently. If it tunnels to outside, n points outward (from wall to ball on outside), pushing further out. But with no tunneling this is fine. Also add a safety: if ball distance from center > R (outside polygon), pull back? Could add containment fallback: if distance from center > R*1.05, clamp to inside. Let me add a safety net: after wall collisions, for each ball if outside the polygon (point-in-polygon test) or too far, move it back toward nearest edge and reflect. Simpler: compute distance from center; if > R+some, it's likely escaped; but corners... Actually a convex polygon: ball inside if for all edges, signed distance from edge line (with inward normal) >= -? Let's just rely on good constants and also cap by radial clamp: if dist from center > R + BALL_R*3, teleport to nearest vertex direction * R*0.8 and zero velocity. That's a failsafe.

Actually better containment: use inward normals of edges. For each edge, inward direction. The closest-point method already handles. Add failsafe.

Let me also ensure balls don't overlap persistently. Positional correction each substep should keep them apart. With 20 balls and gravity they'll pack at bottom. Fine.

Also gravity with damping—balls settle. Rotating polygon continuously churns them, keeps motion. Good.

Let me also consider that wall rotation may push balls strongly. Restitution 0.6, omega 1.2566 rad/s, wall speed at R=300 is 377 px/s. Bounce adds. Damping handles.

Now, initial fall: they start near center and fall to bottom of heptagon. But heptagon rotates, so bottom vertex orientation changes. Fine.

Let me write file.

I need pauseBtn reference defined before reset uses it; I'll query after creating. Order: define buttons after functions, then call reset() at end.

Let me write full script.

Also I should set canvas CSS size responsive. I set width/height fixed 760 and max-width vmin. Fine.

Let me produce.

One more: the `I` used for ball spin = 0.5 * m * r². With m=1, I = 72. Fine.

Angular damping: 0.45 may be strong but okay; spin visible. Maybe reduce to 0.25 so numbers keep rotating. Let's use 0.3.

Let me also apply a bit of spin from wall friction; friction coefficient 0.35.

Write it.

Let me run it in browser and inspect. Let me check console and initial screenshot.

Initial view looks good, 20 balls in grid, heptagon visible. Console clean.

Now let me sample a few moments to verify rotation, falling, collisions, spin. Since simulation runs in real time, I can't control time easily. Let me examine via evaluate? There's no evaluate tool. I can use motion to capture frames over time. Let me capture motion frames.

Actually browser tools only: reload, inspect, screenshot, motion, click, drag, key, type, scroll. No JS eval. Let me use motion to see animation over time.

Let me take a few screenshots separated by doing motion. Actually each tool call takes real time so screenshots after delays show progression.

Let me check console for errors after running a bit. I'll do a screenshot now after some time has passed (the thinking + tool call latency).

Balls have fallen and settled along the lower-left edges. Heptagon has rotated (vertex orientation changed). Numbers show varied rotations → spin visible. Console still clean.

Let me take another sample after some time to see the balls being churned as the heptagon rotates. Let me use motion to capture multiple frames.

Actually, let me check the balls remain contained over a full rotation. I'll wait and screenshot again. Since tool latency provides time, let me do a couple.

Let me test Pause/Resume and Reset. First click Pause then screenshot to confirm frozen. Actually let's verify rotation at two distinct times.

Let me screenshot once more to see rotation continued.

The heptagon has rotated further; balls are being churned along the wall, spread out, still contained, numbers rotated. All 20 visible presumably (some overlapping). Good.

Now test Pause. Click pause button at approx x=550, y=706 (based on screenshot coordinates: Pause button center ~(550,706)). Then screenshot, then wait and screenshot again to confirm frozen. Actually latency between calls may be small. Let me click pause and take two screenshots.

Then Reset test: click reset, screenshot compare to initial.

Let me click Pause.

Button changed to "Resume". Let me take another screenshot after a pause to confirm frozen. But tool latency could be very short. Let me use motion to see if frames change — motion captures multiple frames over a period, which should reveal whether animation is running. Actually while paused, motion should show identical frames. Let me use motion.

Actually the two screenshots after pause: let me just take screenshot immediately; if identical to paused frame, good. But I need to be sure enough time passed. Let me use motion which samples over ~1s.

All 4 motion frames are identical → pause works, no motion. Balls all contained, numbers varied rotation.

Now test Resume and Reset. Click Resume, then after a bit click Reset and compare to initial. Let me first click Resume then reset.

Actually to test Reset: click Reset while paused or running. Then screenshot should show initial grid at center with wallAngle=0. Let me click Reset (x=657, y=706) then screenshot. Since reset sets running=true, it will immediately fall. To compare arrangement to initial I'd need to catch it fast, but reset gives exact initial state before the first frame. However, screenshot latency means it may have fallen a bit. Better: pause first, then Reset. Reset sets running=true though. Hmm. Reset also sets text to Pause and running true.

I could click Reset then quickly screenshot — likely a few hundred ms, balls fall visibly. To compare wall orientation is easier: after reset wallAngle=0. Let me just verify reset restores wall orientation and grid by clicking reset and immediately screenshotting. Actually order: I want to pause, then reset, but reset unpauses. 

Alternative: click Reset, then immediately click Pause, then screenshot. But timing.

Simpler: the initial screenshot at reload already showed the correct initial arrangement and wall orientation. After reset we can check heptagon orientation returns to the same as initial (flat-ish top vertex). Let me click Resume then Reset then screenshot immediately.

Let me click Resume first (running again), verify it resumes via motion later. Then click Reset.

Actually let me do Reset now while paused, then screenshot — the arrangement should be back at center grid even if slightly fallen, and wall orientation back to initial. Let me do it.

Reset restored the compact 4x5 grid at center (numbers 1-20, non-overlapping) and the wall orientation matches the initial view (same heptagon vertex orientation as first reload screenshot). Button back to "Pause". Simulation resumed (running=true). 

Everything checks out. Let me verify no console errors and confirm running resumed. The screenshot shows balls at center just after reset, about to fall. Good.

One concern: the reset screenshot shows numbers already rotated to their initial angles (deterministic), matching the first frame. Good.

Let me do a final check that resume works (the Reset already set running=true and balls are falling from center as seen). That's effectively resume behavior verified too since it's animating. Actually the screenshot shows grid still compact because captured at frame 0. Let me take one more screenshot to confirm it's animating (falling) without clicking Resume separately — proves running state after reset.

Also I should verify the required exact-20 balls. The grid is 4x5 = 20, numbers 1-20 all present. Good.

Let me take one final screenshot to confirm ongoing fall and no errors.

Everything works. The sim resumed after reset, balls fell and settled, heptagon rotated. Console clean throughout.

Let me write final report. I'll note behaviors checked and unverified.

Unverified: exact 5-second period timing (I observed rotation across samples but didn't precisely time one revolution), long-term stability beyond ~10 seconds, precise angular velocity transfer to spin (qualitative only), no persistent overlap over very long runs. But I did observe containment and clean console.

I should be honest. Let me finalize.