<video><title>Fractopia: 1. A History of Fractals</title><description>Join Sebastian Schepis in a fascinating exploration of fractals - their history, their presence in our arts, architecture, bodies, and environment. We'll also explore how fractals are used in technology, examining the use of fractals in compression, machine learning and much more.</description><transcript><transcript><line from="00:00:00,000" to="00:00:06,280">Welcome to Fractopia, a podcast exploring the fascinating world of fractals, fractals</line><line from="00:00:06,280" to="00:00:13,240">or geometric shapes that exhibit self-similarity and infinite complexity.</line><line from="00:00:13,240" to="00:00:19,220">They have fractional dimensions that capture how much space they take up in between whole</line><line from="00:00:19,220" to="00:00:25,420">number dimensions, fractals are found extensively in nature, mathematics, technology, and</line><line from="00:00:25,420" to="00:00:31,620">art. In this first episode, we'll provide an introduction to the basics of fractal geometry.</line><line from="00:00:31,620" to="00:00:37,300">We'll go over a brief history of how fractals were discovered. Talk about their key properties,</line><line from="00:00:37,300" to="00:00:43,540">like scale invariance and fractional dimensions, go over some real world examples. And discuss</line><line from="00:00:43,540" to="00:00:48,420">the types of fields where fractals are now being applied. Fractals are deeply connected</line><line from="00:00:48,420" to="00:00:55,060">to patterns in nature, financial markets, artistic beauty, and even the very fabric of space</line><line from="00:00:55,060" to="00:01:01,660">in time. By journeying into the endlessly intricate world of fractals, we can gain insight into</line><line from="00:01:01,660" to="00:01:07,020">complexity, self-organization, and the fundamental behavior of systems beyond our everyday</line><line from="00:01:07,020" to="00:01:12,460">experience. This podcast will be our guide down the infinitely twisting paths and recursive</line><line from="00:01:12,460" to="00:01:17,340">patterns of the fractal realm, so get ready to think outside the whole numbers. Look beyond</line><line from="00:01:17,340" to="00:01:22,660">the squares and spheres of Euclidean geometry and embark on an expedition through the most</line><line from="00:01:22,700" to="00:01:30,580">complex yet beautiful structures in the universe. Let's fractal. The story of fractals begins</line><line from="00:01:30,580" to="00:01:37,340">in the late 1800s with a mathematician named George Cantor. He was studying mathematical</line><line from="00:01:37,340" to="00:01:43,580">sets and realized that some sets can have non-intigure dimensions. For example, a line</line><line from="00:01:43,580" to="00:01:48,580">is one dimensional, but a zigzagging line that goes back and forth, infinitely could have</line><line from="00:01:48,580" to="00:01:55,780">a dimension between one and two. This was a radical idea at the time. In 1904, Helgy,</line><line from="00:01:55,780" to="00:02:01,420">Von Coach took this further by inventing the coach Snowflake. This starts with an equal</line><line from="00:02:01,420" to="00:02:08,260">lateral triangle. Then you divide each line segment into thirds and replace the middle third</line><line from="00:02:08,260" to="00:02:14,380">with two new segments to make a star shape. When you repeat this process, infinitely,</line><line from="00:02:14,380" to="00:02:20,620">the coach Snowflake emerges. With an infinite perimeter enclosing a finite area, Gaston</line><line from="00:02:20,620" to="00:02:28,020">Julie extended the idea of non-intigure dimensions to iterative functions in the early 1900s.</line><line from="00:02:28,020" to="00:02:32,900">By recursively applying functions, he generated intriguing fractal patterns like the</line><line from="00:02:32,900" to="00:02:39,180">Julia set. This laid the groundwork for fractal geometry, but it wasn't until the 1970s</line><line from="00:02:39,180" to="00:02:45,700">that the field really took off. Thanks to a mathematician named Benoy Mendelbrot. He coined</line><line from="00:02:45,700" to="00:02:52,940">the term fractal, studied fractal shapes in nature, and created the famous Mandelbrot set.</line><line from="00:02:52,940" to="00:02:57,700">Mandelbrot brought fractals into the mainstream of mathematics and science. These pioneering</line><line from="00:02:57,700" to="00:03:03,420">thinkers showed that fractional dimensions and recursive self-similarity were not just</line><line from="00:03:03,460" to="00:03:10,460">mathematical oddities, but intrinsic to the geometry of nature. There were created the foundations</line><line from="00:03:10,460" to="00:03:15,900">for the field of fractal geometry that will be exploring in this podcast. One of the hallmarked</line><line from="00:03:15,900" to="00:03:22,900">features of fractals is self-similarity. This means the shape contains smaller copies of itself</line><line from="00:03:22,900" to="00:03:28,700">embedded recursively within in mathematical fractals like the Sierra Penske, Triangle.</line><line from="00:03:28,740" to="00:03:35,700">The self-similarity is exact. Magnify, any small region, and it looks identical to the whole</line><line from="00:03:35,700" to="00:03:42,060">shape. These perfect fractals reveal the recurrence and infinity underlying geometric forms</line><line from="00:03:42,060" to="00:03:48,940">in natural fractals. The self-similarity is statistical. For example, segments of Iraqi coastline</line><line from="00:03:48,940" to="00:03:55,380">resemble the overall pattern. The replication is approximate rather than exact, but across</line><line from="00:03:55,380" to="00:04:00,820">scales the essential structure remains. This self-referencing recursiveness is why fractals</line><line from="00:04:00,820" to="00:04:08,820">can pack infinite complexity into finite spaces. The complexity cascades through increasing orders</line><line from="00:04:08,820" to="00:04:14,820">of magnitude via the repetition. Artists leverage this cascading self-similarity to create</line><line from="00:04:14,820" to="00:04:22,020">dazzling fractal designs. Substructures like the Mandelbrot set can be drawn at any magnification,</line><line from="00:04:22,340" to="00:04:30,420">revealing more intricate worlds within world. So whether in nature, math or art, self-similarity</line><line from="00:04:30,420" to="00:04:36,260">generates the infinite depth associated with fractals by repeating patterns across scales.</line><line from="00:04:36,260" to="00:04:42,740">It captures the recurrence of form that generates endless complexity fractals obtain their name</line><line from="00:04:42,740" to="00:04:49,380">from their fractional dimensions, values that fall between the familiar integers of Euclidean</line><line from="00:04:49,460" to="00:04:55,620">geometry. I think 1.5 dimensions instead of one or two. These non-hole number dimensions arise</line><line from="00:04:55,620" to="00:05:03,620">from the fractals' infinite self-similarity for a rough line that zig-zags its topology is one</line><line from="00:05:03,620" to="00:05:09,060">like a regular line. But it fills more space than a straight line. It's fractal dimension</line><line from="00:05:09,060" to="00:05:16,500">of say 1.7 captures its complexity between a 1.2D shape. We can calculate fractal dimensions</line><line from="00:05:16,500" to="00:05:23,940">using techniques like box counting. This tiles the shape with boxes of decreasing size and analyzes</line><line from="00:05:23,940" to="00:05:30,900">how the number of boxes required changes by scale. The power law relationship exposes the fractional</line><line from="00:05:30,900" to="00:05:36,580">dimension. So if we tried boxing the coach snowflake, we'd see it takes many more boxes</line><line from="00:05:37,140" to="00:05:44,180">than a straight one line segment, but fewer than a 2D triangle. Fractal dimensions quantify this</line><line from="00:05:44,260" to="00:05:49,540">in between complexity. In nature, fractional dimensions describe the mixed smoothness and</line><line from="00:05:49,540" to="00:05:56,660">raggedness of coastlines, organic shapes, and other fractal patterns. The non-integer values</line><line from="00:05:56,660" to="00:06:02,100">reflect the fractional structured complexity. Fractal dimensions characterize the infinity</line><line from="00:06:02,100" to="00:06:08,580">between finite dimensions. They mathematically capture the endlessly recursive nature of</line><line from="00:06:08,580" to="00:06:14,260">fractal geometry. A key property of fractals is that they exhibit similar structure at all scales</line><line from="00:06:14,260" to="00:06:21,380">of magnification. Giving them scale invariance, zoom in on the Mandelbrot set, and you'll see the</line><line from="00:06:21,380" to="00:06:28,500">same swirl of boundaries and buddh repeating. There is no characteristic scale at which the pattern</line><line from="00:06:28,500" to="00:06:34,900">changes. It retains its fractal essence infinitely. This scale free self-similarity results from</line><line from="00:06:34,980" to="00:06:41,940">the recursion generating fractals, iteratively through feedback. Tiny and large features get woven</line><line from="00:06:41,940" to="00:06:48,020">together into an inseparable web, in contrast, ordinary geometric shapes like squares only look,</line><line from="00:06:48,740" to="00:06:55,060">similar within a certain scale range, before the straight sides start to curve. Fractals never</line><line from="00:06:55,060" to="00:07:01,140">smooth out, nature leverages scale invariance for structural strength, like in branching rivers and</line><line from="00:07:01,220" to="00:07:08,180">trees, distributing weight across scales. Networks exhibit scale free structure for resilience and</line><line from="00:07:08,180" to="00:07:16,980">efficient flows, scale invariance, signals, deep connections between microscopic and macroscopic levels</line><line from="00:07:16,980" to="00:07:23,860">in physics, biology, and ecology. What starts small propagates through the system, fractals teach us</line><line from="00:07:23,860" to="00:07:30,100">that the large comes from the small in an endless feedback of form, the pioneering work on fractal</line><line from="00:07:30,180" to="00:07:37,380">geometry, opened up a whole new world of applications in diverse fields. In art, fractals enabled</line><line from="00:07:37,380" to="00:07:44,420">creating intricate recursive designs using simple computing power. Fractal-based environments</line><line from="00:07:44,420" to="00:07:51,780">and organic shapes change computer graphics. In finance, analyst found markets exhibit fractals</line><line from="00:07:51,780" to="00:07:59,540">statistics indicating self-similarity across time scales. This led to new models of volatility</line><line from="00:07:59,620" to="00:08:05,780">using fractal dimension in technology, fractal networks balance connectivity with resilience.</line><line from="00:08:06,580" to="00:08:12,740">Fractal antennas improved efficiency by increasing surface area. In physics, fractals appeared</line><line from="00:08:12,740" to="00:08:20,660">in quantum systems, statistical mechanics, and theories of quantum gravity potentially linking space</line><line from="00:08:20,660" to="00:08:27,380">in time. In biology, researchers modeled organisms using fractal growth rules and found</line><line from="00:08:27,380" to="00:08:33,300">self-similar structures in bodies. Fractal geometry described complexity. Across domains,</line><line from="00:08:33,300" to="00:08:39,300">fractals provided a new mathematical lens for understanding complexity, self-organization,</line><line from="00:08:39,860" to="00:08:45,780">and scale invariance in system. The seeds planted by early pioneers blossomed into a rich field,</line><line from="00:08:46,340" to="00:08:52,580">touching every corner of science, nature, and art. This concludes our first episode introducing</line><line from="00:08:52,660" to="00:08:58,180">the wondrous world of fractal geometry. We covered the history of pioneers like Cantor,</line><line from="00:08:58,180" to="00:09:04,420">Cuck, Julia, and Mendelbrot who developed the foundations of fractals. We explored the key</line><line from="00:09:04,420" to="00:09:11,140">properties like self-similarity, fractional dimensions, and scale invariance that create fractals</line><line from="00:09:11,140" to="00:09:16,900">as endless complexity. We discussed how fractals are found extensively in the natural world.</line><line from="00:09:17,540" to="00:09:23,700">Mathematical constructions, art, and computer graphics, and technologies ranging from networks to</line><line from="00:09:23,700" to="00:09:29,940">antennas. Fractals provide a lens for analyzing and generating complexity. In the next episode,</line><line from="00:09:30,580" to="00:09:36,580">we'll dive deeper into fractal patterns, founded nature across scales from coastlines to plants</line><line from="00:09:36,580" to="00:09:42,180">to the human body. We'll see how fractals capture fundamental recurrence and connectivity</line><line from="00:09:42,260" to="00:09:48,660">in living systems, for now pondering finnity by gazing at a fractal zoom video, marvel at</line><line from="00:09:48,660" to="00:09:54,420">self-similarity in ahead of Rominesco Broccoli, and reflect on how simple rules propagate</line><line from="00:09:54,420" to="00:09:59,940">to create immense beauty and complexity. The recursive journey has just begun.</line></transcript></transcript></video>