✨ A First Look: Follow the Light

When you first open Reflective Bounce, you’re greeted by a circular mirror boundary with a single light ray placed at its bottom edge. The beam is already aimed upward at a gentle 30° angle, inviting you to explore. Click Start and watch as the beam springs forward, touches the curved mirror, and rebounds with perfect precision. After a few moments, a surprising picture emerges: the ray is tracing a perfect hexagon, looping endlessly as though the circle is hiding a secret geometric engine.

Try adjusting the beam direction to a few other angles — 20°, 45°, 60°, 72°, 90° are especially intriguing. Each produces its own special pattern: a 9-gon at 20°, a tidy square at 45°, a triangle at 60°, a pointed star at 72°, and a simple vertical line at 90°. These shapes appear because 360° is divisible by these angles, allowing the path to close on itself perfectly. A circle, when paired with the right angle, becomes a factory of repeating orbits.

But what about an angle that doesn’t divide the circle so neatly? Set the beam to 31° and observe again. This time the path never closes into a tidy shape. Instead, the beam traces something that looks like a drifting, slightly twisted hexagon — a pattern that keeps shifting, building a delicate net structure over time. This is your first glimpse into the difference between periodic motion (closed loops) and quasi-periodic motion (infinite drifting traces).

🛠 Exploring the Tools: Shape, Aim, and Fire

Once you’ve seen your first patterns, it’s time to design your own experiments. Start by choosing another mirror boundary — a triangle, square, rectangle, circle, or oval all behave uniquely. When a new shape appears, notice how some sides are perfectly straight while others curve smoothly. These differences determine how the tracer will bounce, because the direction of reflection depends entirely on the angle of the surface where the beam lands.

Use the corner handles to gently reshape your boundary. Stretch a square into a bigger one or pull an oval into a tall, thin corridor. Even small adjustments change the behavior of the reflections, and dramatic stretches can transform orderly paths into wildly unexpected ones. Each configuration becomes a new laboratory for exploring how shape influences motion.

Now place a tracer light source anywhere inside the boundary. A tiny glowing point marks the starting position of your beam. Aim it using the angle control, and try beginning with a diagonal direction that feels natural. This is your prediction moment: before pressing Start, try visualizing how the beam will travel and where it will meet the mirror.

Press Start to set the beam in motion. Flat surfaces produce sharp, clean bounces reminiscent of a billiard ball striking a cushion, while curved boundaries redirect the beam in smoother, often surprising arcs. Use Pause to freeze the motion mid-path, study the geometry, and adjust the aim carefully. The Reset button clears every trace without altering your mirror, allowing you to repeat the experiment just as a scientist would.

Adjust the speed slider to slow the beam for close study or speed it up for dramatic patterns. The trail fade control lets you decide how long the glowing path remains visible, helping you uncover more subtle shapes or enjoy big sweeping patterns that fill the whole canvas.

🔍 Deep Dive: Patterns, Predictions, and Experiments

After experimenting with a few straightforward angles, begin refining your control. Change the beam angle by 10°, 5°, or even 1° and watch how the entire pattern reorganizes. Although the change may seem tiny, it shifts the beam’s first collision point, which changes the mirror’s surface normal at that spot, which alters the reflection angle — and every subsequent bounce. This cascading effect demonstrates a core idea in geometric systems: small differences in starting conditions can lead to big changes in results.

As you explore different mirror shapes, you’ll encounter paths that close into stable loops. Squares, regular polygons, and circles often create repeating orbits when the angle aligns with one of their symmetries. You may find triangles that bounce in perfect three-step cycles or octagons that form elegant eight-beat rhythms. Hunting for these repeating paths is one of the most rewarding challenges in the game.

To unlock even richer behavior, add more tracer balls — up to five at once. Place them in different spots, aim them in different directions, and launch them together. Some will trace related shapes; others will diverge quickly. As the patterns overlap, you’ll witness families of trajectories emerging, revealing which starting positions behave in similar ways and which produce contrasting outcomes.

Once you’re confident with the basics, try reshaping the mirror into extreme forms: a sharply stretched oval, a flattened triangle, or an elongated polygon with uneven sides. Fire beams at unusual angles and watch as reflections become unpredictable yet still perfectly lawful. Even the most chaotic-looking motion obeys the unbreakable rule of optics: the angle of incidence always equals the angle of reflection. Understanding this rule is the key to making sense of every swirl, loop, drift, and ricochet you observe.

As you continue to explore, you’ll develop an intuition for how shapes “guide” the beam, when symmetry helps create order, and why curved surfaces can amplify small directional differences into dramatically different paths. Whether you’re seeking perfect loops or mesmerizing tangled nets, Reflective Bounce turns light itself into a playful tool for discovering the hidden geometry of motion.

🔎 Understanding How Light Reflection Works

Reflective Bounce is built on one elegant physical rule: the angle of incidence equals the angle of reflection. This principle, called specular reflection, governs how real light behaves — from laser beams and billiard balls to shimmering water highlights and everyday mirrors. In the game, each tracer ray behaves exactly like a light beam: every bounce is a precise response to the shape it meets. There is no randomness; the path is determined entirely by geometry.

📐 Flat vs. Curved Boundaries: Two Worlds of Motion

Flat-edged shapes — such as triangles, squares, and pentagons — act like collections of straight mirror segments. Because each side has a constant surface normal, reflections are predictable and angular. A beam hitting a flat edge at 30° will always bounce away at 30° on the other side of the normal. This consistency produces crisp, polygon-like paths and often allows the beam to fall into closed repeating loops. A triangle may generate a perfect three-bounce cycle, while a square can support neat four-step orbits. Flat shapes highlight the stability that comes from fixed surface orientation.

Curved boundaries — like circles and ovals — behave very differently. Here, the normal direction changes smoothly along the curve, so the outcome depends sensitively on the exact impact point. Two beams launched only a degree apart may follow completely different paths: one might loop gracefully, while another veers away after a single bounce. Curved mirrors can form perfect polygons at special “magic angles” that divide 360°, but at non-dividing angles they produce spirals, drifting nets, and quasi-periodic patterns. This contrast reveals how curvature introduces both richness and sensitivity into reflected motion.

🔶 Symmetry: The Secret Behind Repeating Shapes

Symmetry plays a crucial role in determining whether a beam’s path repeats or drifts. Shapes like squares, hexagons, and circles have strong rotational or reflective symmetry. When a ray’s starting angle aligns with one of these symmetries, the reflections reinforce the pattern, creating stable orbits such as triangles, squares, hexagons, or stars traced by the bouncing ray.

Break the symmetry — for example, by stretching a circle into a wide oval — and the behavior becomes more intricate. The beam may wander unpredictably, weave across the space, or produce paths that never exactly repeat. Yet every step still follows the same reflection rule: even complex or drifting motion remains rooted in pure geometry.

🌌 What You’ll Notice as You Explore

As patterns accumulate on the screen, you’ll begin to recognize several characteristic behaviors:

• Stable orbits: cycles that repeat exactly, forming triangles, squares, hexagons, or stars.
• Chaotic scattering: paths where tiny angle changes produce wildly different long-term behavior.
• Grazing glances: near-tangent hits that barely change the ray’s direction.

These behaviors emerge from the interplay between shape, symmetry, and curvature. With practice, you’ll start predicting where the first bounce will land, how the second will unfold, and whether the path will close cleanly or drift. Reflective Bounce transforms these ideas into visible patterns — a living illustration of how mathematics shapes motion.

📘 For Advanced Learners

The patterns in Reflective Bounce come from a beautiful area of mathematics called dynamical billiards. In a billiard system, a point particle moves in a straight line and reflects off a boundary using the same mirror rule as light. Even with this simple rule, billiards can show everything from perfectly repeating cycles to complex, chaotic motion. They are popular in mathematical physics because they reveal the full range of dynamical behavior without requiring complicated equations. Mathematician George Birkhoff even showed that billiards inside an ellipse are fully predictable, while many other shapes are far less orderly.

In the game, the behavior of a ray depends entirely on the boundary shape. Polygons give crisp, predictable reflections because each straight side acts like a flat mirror. With the right launch angle, you can form clean repeating loops. Curved shapes like circles and ovals behave differently: the surface direction changes continuously, so even tiny changes in where the beam hits can lead to dramatically different paths. This sensitivity is why curved billiards often produce spirals, drifting lattices, or sudden escapes.

Symmetry also plays a large part. In a circle, angles that neatly divide 360° produce closed polygons, while other angles create never-repeating traces. This mirrors a classic divide in dynamical systems: rational relationships lead to order, while irrational ones lead to endless variation.

Launch several rays at once and you’ll notice groups of paths that behave similarly. Others diverge quickly, revealing how the boundary guides the flow of motion. All of this emerges from the same simple rule — move straight, bounce cleanly — yet together it forms a small laboratory of geometric optics and dynamical motion.