Okay, I need to create a single-page web app that shows 20 numbered balls dropping from the center of a rotating heptagon. The balls should be affected by gravity and friction, collide with each other and the walls, and their numbers should rotate when they spin. All collision detection must be implemented manually without using physics engines. Let's break this down.

First, the structure of the HTML file. I'll need a canvas element to render everything. The CSS should set the canvas to fill the screen. The JavaScript will handle the physics and rendering.

Starting with the HTML: a basic structure with a canvas tag. The style will center the canvas and make it full-screen. The script tag will contain all the code.

Next, the JavaScript. I'll need to set up the canvas context, define the heptagon vertices, create ball objects with properties like position, velocity, radius, etc. The heptagon should rotate over time, so I'll track its rotation angle and update it each frame.

For the balls, each should have position (x, y), velocity (vx, vy), radius, mass (based on radius), and rotation angle. Gravity will affect the vy each frame. Friction will reduce velocity over time.

Collision detection between balls: check pairwise distances. If the distance is less than the sum of radii, they collide. Calculate the overlap and adjust velocities using conservation of momentum and restitution (elasticity). Also, update their rotation based on collision direction.

Collision with the heptagon walls: each side of the heptagon is a line segment. For each ball, check if it's colliding with any of the heptagon's edges. This involves checking the distance from the ball to each edge and determining if it's within the ball's radius. When a collision occurs, reflect the velocity based on the edge's normal. Also, the heptagon is rotating, so the edges' positions and normals change over time.

The heptagon's rotation speed should be such that it completes a full rotation every 5 seconds. So the angular velocity is 2π radians / 5 seconds. Each frame, increment the rotation angle by angular velocity multiplied by the time delta.

For rendering, clear the canvas each frame. Draw the heptagon, then each ball. The balls' numbers should be rotated according to their own rotation angle. Use ctx.save(), ctx.translate(), ctx.rotate(), then draw the number, then ctx.restore().

Implementing all this requires careful calculations. Let's outline the steps:

1. Set up canvas and context.
2. Define the heptagon's vertices based on a radius and center point. Since it's a heptagon, 7 sides. Calculate each vertex's position using angles (2π/7 increments).
3. Create an array of ball objects with initial positions at the center, velocities, etc.
4. In the animation loop:
   a. Update the heptagon's rotation angle.
   b. Apply gravity and friction to each ball's velocity.
   c. Update each ball's position based on velocity.
   d. Check for collisions between balls and adjust velocities.
   e. Check for collisions between balls and heptagon edges, adjust velocities.
   f. Clear the canvas, draw the heptagon, then each ball with their numbers rotated.
5. Handle the time delta to make the animation frame rate independent.

Potential challenges: Collision detection with moving walls (the heptagon is rotating). Each frame, the heptagon's edges are in new positions, so the collision checks need to account for the current rotation. Also, ensuring that the collision response between balls and walls is accurate, considering the wall's normal at the point of collision.

For collision between a ball and a line segment (heptagon edge), I'll need to find the closest point on the segment to the ball's center. If that distance is less than the ball's radius, there's a collision. Then calculate the normal vector at that point and reflect the ball's velocity.

Another thing: since the heptagon is rotating, the ball's collision with the walls will depend on the current rotation angle. So during each frame, when checking collisions, the heptagon's current vertices are calculated based on the current rotation angle.

Implementing all this in JavaScript, using only basic DOM and canvas APIs, without any external libraries. Also, making sure the code is efficient enough to run smoothly in the browser.

Now, putting all this into code. Let's start writing the HTML structure, then the JavaScript code inside a script tag. The CSS will be in a style tag.

First, the HTML:

<!DOCTYPE html>
<html>
<head>
    <title>Rotating Heptagon with Balls</title>
    <style>
        body, html {
            margin: 0;
            padding: 0;
            overflow: hidden;
        }
        canvas {
            display: block;
        }
    </style>
</head>
<body>
    <canvas id="canvas"></canvas>
    <script>
        // JavaScript code here
    </script>
</html>

Then, in the script, set up the canvas and context. Define constants like canvas size, gravity, friction, etc.

Next, define the heptagon. Let's assume the heptagon is centered at (canvasWidth/2, canvasHeight/2), with a certain radius. The vertices can be calculated using trigonometry.

For the balls, create an array of 20 objects. Each with x, y, vx, vy, radius, mass (maybe mass is proportional to radius squared?), and rotation angle.

In the animation loop, use requestAnimationFrame. Calculate the delta time between frames to make the physics time-independent.

For each ball, apply gravity (vy += gravity * delta) and friction (vx *= friction, vy *= friction). Then update position (x += vx * delta, y += vy * delta).

Check collisions between all pairs of balls. For each pair, calculate the distance between centers. If distance < r1 + r2, they overlap. Compute the collision normal (dx, dy normalized), then compute velocities along that normal. Apply the collision response using conservation of momentum and coefficient of restitution (maybe 0.8 for some bounciness).

For ball-wall collisions: for each heptagon edge (line segment), check if the ball's center is within the radius distance of the edge. If so, compute the normal and reflect the velocity.

The heptagon's rotation angle increments over time. Each frame, angle += angularSpeed * delta. The angularSpeed is (2 * Math.PI) / 5 seconds. So angularSpeed = (2 * Math.PI) / 5.

When drawing the heptagon, each vertex is calculated based on the current rotation angle. Then, draw the heptagon as a polygon.

For each ball, when drawing, save the context, translate to the ball's position, rotate by the ball's rotation angle, then draw the number. Restore the context.

But how to track the ball's rotation? Each time a ball collides with another ball or a wall, it should gain some rotation. For simplicity, maybe when two balls collide, they exchange some angular momentum, causing their rotation angles to change. Alternatively, the rotation could be based on the direction of collision.

This part might be a bit tricky. For the purpose of this demo, perhaps when two balls collide, their rotation angles are updated based on the collision normal and relative velocity. But to keep it simple, maybe just add some rotation when a collision occurs. For example, when ball A collides with ball B, their rotation angles could be adjusted by some value based on the collision impulse.

Alternatively, each collision could induce a rotation based on the relative velocity perpendicular to the collision normal. But that might require more complex calculations. Since the user wants realistic bouncing, perhaps a basic approach where each collision adds a small rotation to the balls involved.

But for the scope of this problem, perhaps a simplified approach is acceptable. For example, when two balls collide, each gains a rotation proportional to their relative velocity and the collision normal.

However, to keep the code manageable, maybe just increment the rotation angle of each ball by a small amount when they collide. Or track the rotation based on their own velocity's direction.

Alternatively, the rotation of the number could be tied to the ball's own rotation, which is influenced by collisions. For example, each time a ball collides with a wall or another ball, it gains some angular velocity, which affects its rotation over time.

But since the problem requires implementing all collision detection manually, perhaps the rotation of each ball's number should reflect their own spin, which is caused by collisions. So, each ball has an angular velocity (omega), which is updated during collisions. Then, in each frame, the rotation angle is updated based on omega and delta time.

This would make the rotation more realistic. So during each frame, for each ball, rotation += omega * delta. Then, when collisions happen, omega is adjusted based on the collision.

But how to compute omega during collisions? For example, when two balls collide, the collision could induce a torque that changes their angular velocities. However, calculating this requires knowing the moment of inertia and the torque applied, which might be complex.

To simplify, perhaps during a collision between two balls, we can apply an impulse that changes their linear velocities and also adds some angular velocity. For example, the direction of the collision could influence the rotation. If the collision is off-center, it could cause the balls to spin.

But for the sake of time and code complexity, maybe we can approximate the rotation by adjusting the ball's omega when they collide. For instance, when two balls collide, each gains a small omega in the direction perpendicular to the collision normal.

This is a simplification, but it would make the numbers on the balls rotate, which is what the user wants.

Putting it all together, the code would have:

- Ball objects with x, y, vx, vy, radius, mass, rotation angle, omega.
- On collision between balls, adjust their velocities and add some omega.
- On collision with walls, adjust velocity and add some omega.
- In each frame, update rotation angle based on omega.

Now, implementing all of this in code.

Potential code steps:

1. Initialize canvas and context.
2. Set up constants: gravity, friction, heptagon radius, center, number of balls, etc.
3. Create balls with initial positions at the center, random initial velocities (maybe zero, then let gravity take over).
4. Animation loop:
   a. Update heptagon rotation angle.
   b. For each ball, apply gravity and friction.
   c. Update ball positions.
   d. Check for collisions between balls and resolve.
   e. Check for collisions between balls and heptagon walls and resolve.
   f. Update rotation angles of balls based on their omega.
   g. Clear canvas.
   h. Draw heptagon.
   i. Draw each ball with their numbers rotated.
5. Collision detection functions.

Now, implementing collision between balls:

For each pair of balls, calculate dx = x2 - x1, dy = y2 - y1. distance squared = dx^2 + dy^2. If distance < r1 + r2, collision occurs.

Calculate overlap, then adjust positions to prevent overlap. Then compute new velocities based on conservation of momentum and restitution.

For the rotation, when two balls collide, their omega can be adjusted based on the collision normal and relative velocity. For example, if the collision is head-on, maybe no rotation, but if it's glancing, add some omega.

But again, for simplicity, maybe just add a small amount to omega for each collision.

Alternatively, when balls collide, their omega could be set to some value based on the relative velocity. For example, omega1 += (relative velocity) cross product with collision normal, scaled by some factor.

But this is getting complex. For the sake of time, perhaps just add a random small rotation when they collide. Or set omega to a value based on the collision direction.

Alternatively, during each collision, compute the tangential velocity and apply it to the omega.

This is getting complicated. Maybe for the purpose of this demo, we can just have the balls' numbers rotate when they collide, without precise physics. For example, each time a ball collides with another or a wall, increment its rotation angle by a fixed amount.

But the user wants the rotation to reflect the ball's spin realistically. So perhaps when a ball collides with a wall, the direction of the collision affects the rotation. For example, if a ball hits a wall and bounces off, the point of contact could induce a spin.

But to implement this, we need to calculate the torque. The torque is the cross product of the collision point relative to the ball's center and the force applied. However, without full physics simulation, this is challenging.

Given time constraints, perhaps a simpler approach is acceptable. For example, when a ball collides with another ball, their rotation angles are updated in opposite directions based on the collision normal. Or when a ball collides with a wall, its rotation is adjusted based on the direction of the collision.

In code:

When two balls collide:

- Compute the collision normal (dx, dy normalized).
- Compute the relative velocity between the two balls.
- Calculate the impulse based on the normal and relative velocity.
- Apply the impulse to change the velocities.
- For rotation, perhaps apply an impulse that causes angular velocity. For example, if the collision is off-center, the ball will spin. But since we're not modeling the actual point of contact, maybe just add a small rotation to each ball in opposite directions.

But again, this is a simplification. However, for the purpose of the demo, it's acceptable.

Now, putting all this into code.

First, the ball objects:

Each ball has:

- x, y: position
- vx, vy: velocity
- radius
- mass (maybe mass is proportional to radius squared)
- rotation: current rotation angle
- omega: angular velocity (rotation per second)

In the animation loop:

For each ball:

- Apply gravity: vy += gravity * delta
- Apply friction: vx *= friction, vy *= friction
- Update position: x += vx * delta, y += vy * delta
- Update rotation: rotation += omega * delta

Now, collision between balls:

For each pair (i, j) where i < j:

dx = xj - xi
dy = yj - yi
distanceSquared = dx^2 + dy^2
sumRadii = r_i + r_j
overlap = sumRadii^2 - distanceSquared

If overlap > 0:

// Resolve overlap
overlap = Math.sqrt(overlapSquared) - distance (if distance is zero, set to a small value)
// Separate the balls
separationVector = normalize(dx, dy) * overlap / 2
xi -= separationVector.x * (mass_i / (mass_i + mass_j))
xj += separationVector.x * (mass_j / (mass_i + mass_j))
// same for y

// Calculate velocities
v1 = {vx: balls[i].vx, vy: balls[i].vy}
v2 = {vx: balls[j].vx, vy: balls[j].vy}

// Relative velocity
rvx = v2.vx - v1.vx
rvy = v2.vy - v1.vy

// Velocity along normal
normal = {x: dx, y: dy}
normalize normal
normal.x /= distance
normal.y /= distance

velocityAlongNormal = rvx * normal.x + rvy * normal.y

// Check if moving towards each other
if velocityAlongNormal > 0:
    continue // no collision response needed

// Calculate impulse
e = restitution (e.g., 0.8)
impulse = -(1 + e) * velocityAlongNormal / (1/mass_i + 1/mass_j)
impulse.x = impulse * normal.x
impulse.y = impulse * normal.y

// Apply impulse
balls[i].vx -= impulse.x / mass_i
balls[i].vy -= impulse.y / mass_i
balls[j].vx += impulse.x / mass_j
balls[j].vy += impulse.y / mass_j

// For rotation, add some angular velocity based on impulse direction
// For example, the direction of the impulse can influence the rotation
// Let's assume that the impulse direction is perpendicular to the normal
// So, the tangential direction is (-normal.y, normal.x)
tangent = {x: -normal.y, y: normal.x}
// Apply some torque based on the impulse and tangent direction
// For simplicity, just add a small rotation to each ball
balls[i].omega += tangent.x * impulse.magnitude * someFactor
balls[i].omega += tangent.y * impulse.magnitude * someFactor
// Similarly for ball j, but opposite direction?

This is getting quite complex, but perhaps for the demo, we can simplify. Maybe when two balls collide, each gains some omega in opposite directions based on the collision normal.

But given time constraints, perhaps it's better to just add a small rotation to each ball when they collide. For example:

balls[i].rotation += 0.1
balls[j].rotation -= 0.1

But that's arbitrary. However, for the purpose of showing the numbers rotating, it's sufficient.

Now, collision with heptagon walls:

For each ball, check collision with each edge of the heptagon. The heptagon is a rotating polygon, so each edge is a line segment between two consecutive vertices.

For each edge (defined by two points p1 and p2), find the closest point on the edge to the ball's center. If the distance from the ball's center to this closest point is less than the ball's radius, there's a collision.

The steps to check collision between a ball and a line segment:

1. Find the closest point on the line segment to the ball's center.
2. Calculate the distance from the ball's center to this point.
3. If the distance is less than the ball's radius, collision occurs.
4. Compute the normal vector at the collision point (perpendicular to the line segment).
5. Reflect the ball's velocity based on this normal.

Implementing this requires a function to find the closest point on a line segment to a point.

Once the collision is detected, adjust the ball's velocity as in the ball-ball collision, using the normal of the wall.

Additionally, the ball's rotation could be affected by the collision with the wall. For example, when the ball hits the wall, it gains some rotation based on the direction of the collision.

Again, for simplicity, when a ball collides with a wall, we can adjust its omega in some way. For example, based on the direction of the normal.

Putting all this into code.

Now, considering all these steps, the code will be quite lengthy, but it's necessary to implement all the collision detection and physics manually.

Finally, rendering the balls with their numbers rotated. For each ball, save the context, translate to the ball's position, rotate by the ball's rotation angle, then fillText the number. Restore the context.

Putting all together into the index.html file.

Okay, I need to create a single-page web app that shows 20 numbered balls dropping from the center of a rotating heptagon. The balls should be affected by gravity and friction, collide with each other and the walls, and their numbers should rotate realistically. All collision detection must be implemented manually without using physics engines.

First, I'll start by setting up the HTML structure with a canvas element. The CSS will ensure the canvas fills the screen. The JavaScript will handle the physics and rendering.

For the heptagon, I'll calculate its vertices based on a rotating angle. The heptagon should spin 360 degrees every 5 seconds, so I'll set the angular speed accordingly.

Each ball will have properties like position, velocity, radius, mass, rotation, and angular velocity (omega). The balls start at the center and are affected by gravity and friction. Their velocities are updated each frame.

Collision detection between balls involves checking the distance between their centers. If they overlap, I'll adjust their positions and velocities using conservation of momentum and restitution. Also, each collision will add some rotation to the balls.

For collisions with the heptagon walls, I'll check the distance from each ball to the edges of the heptagon. If a collision is detected, the ball's velocity is reflected, and its rotation is adjusted.

The animation loop updates the heptagon's rotation, applies physics to the balls, checks collisions, and redraws everything. The balls' numbers are rotated based on their omega value.

I'll implement all this in a single HTML file with inline CSS and JavaScript. The final code will be self-contained, using only the canvas and no external resources.