{"id":463400,"date":"2025-06-14T21:00:46","date_gmt":"2025-06-14T21:00:46","guid":{"rendered":"http:\/\/savepearlharbor.com\/?p=463400"},"modified":"-0001-11-30T00:00:00","modified_gmt":"-0001-11-29T21:00:00","slug":"","status":"publish","type":"post","link":"https:\/\/savepearlharbor.com\/?p=463400","title":{"rendered":"The Billiard Fractals<\/span>"},"content":{"rendered":"
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Complex systems often appear chaotic or incomprehensible, yet closer examination reveals that such complexity can frequently be reduced to a simple underlying mechanism. By systematically removing layers of emergent behavior, one can uncover a fundamental rule or equation from which the entire system originates.<\/p>\n

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The work presented here is a translation and adaptation of my articles (Part 1<\/a>, Part 2<\/a>, Part 3<\/a>, Part 4<\/a>)
GitHub repository: billiard-fractals<\/a><\/p>\n

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The Billiard Fractals<\/h1>\n

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While the system described in this article may appear trivial at first glance, the resulting patterns exhibit quasi-fractal behavior that can be analyzed, encoded, and even predicted through symbolic methods. The work presented here was developed independently through direct observation, rather than derived from prior literature.<\/p>\n

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A useful way to motivate this exploration is by analogy with a common physical phenomenon \u2014 wave interference. Consider waves on the surface of a river: a wavefront moves toward the shore, reflects, and overlaps with itself. Do these reflections contain an underlying order? Is it possible to extract structure from the interference?<\/p>\n

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To investigate this, we simplify the system. Rather than modeling the full wave, we consider only the motion vector \u2014 essentially, a ray. We also smooth the \u201cshoreline\u201d and discretize the environment into a rectangular grid. From this setup emerges the core construction of this article.<\/p>\n

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simplest \u201cchaotic\u201d system<\/b> <\/p>\n
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The example of waves on the surface of a river serves as a real, intuitive starting point \u2014 an accessible physical system that demonstrates how simple rules, such as reflection and interference, can produce complex behavior. It illustrates the central idea: that what appears chaotic often emerges from deterministic structure.<\/p>\n

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The initial motivation was driven by the conviction that apparent disorder is not randomness, but the result of unresolved or hidden structure. Any system that seems chaotic is governed by rules \u2014 its complexity a consequence of perspective, not unpredictability.<\/p>\n

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To explore this further, attention turned to constructing the simplest possible system that could look chaotic yet remain fully deterministic.<\/p>\n

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One such system involved a sine wave originating from the corner of a rectangle and reflecting off its boundaries. The nonlinearity of the sine function causes it to intersect itself in complex and unintuitive ways. However, due to limited tools available at the time, the model was simplified even further.<\/p>\n

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Instead of a sine wave, a straight line was used. The line was made periodic (dashed), and the system was designed to be reproducible using only a pencil and a sheet of graph paper. Despite its simplicity, this construction revealed intricate and structured patterns-forming the foundation of what would later be described as the \u201cbilliard fractals.\u201d<\/p>\n

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Visualizing the Billiard Algorithm<\/h2>\n

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The following sequence illustrates the core mechanism of the discrete billiard system:<\/p>\n

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An animated version:<\/p>\n

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Output pattern:<\/p>\n

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A selection of patterns generated from rectangles with various dimensions:<\/p>\n

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<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n

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JavaScript implementation of this algorithm<\/a>
pattern.js<\/code><\/a> \u2014 source code<\/p>\n

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Fibonacci Numbers and Pattern Refinement<\/h2>\n

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The patterns generated by this system exhibit fractal structure: they are self-similar across scales, recursive in construction, and can be compressed symbolically. As the rectangle dimensions increase following the Fibonacci sequence, the patterns reveal increasingly detailed versions of the same underlying structure.<\/p>\n

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This refinement process is simple: given rectangle dimensions (m, n)<\/code>, new dimensions are generated by Fibonacci summation. For example, starting with 8\u00d713:<\/p>\n

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