<video><title>Fractopia: 3. Mathematical Constructions</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="00:00:02,800">Welcome back fellow fractal explorers.</line><line from="00:00:02,800" to="00:00:06,400">In our journey so far, we've encountered stunning examples</line><line from="00:00:06,400" to="00:00:08,800">of fractals throughout the natural world</line><line from="00:00:08,800" to="00:00:10,960">from coastlines to broccoli.</line><line from="00:00:10,960" to="00:00:13,840">We saw how life leverages self-similar scaling</line><line from="00:00:13,840" to="00:00:15,240">for vital functions.</line><line from="00:00:15,240" to="00:00:19,080">Today, we're going to dive deeper into the mathematical techniques</line><line from="00:00:19,080" to="00:00:22,760">used to intentionally construct fractal geometry</line><line from="00:00:22,760" to="00:00:25,720">by recursively applying simple rules.</line><line from="00:00:25,720" to="00:00:28,800">We can build fractal shapes with infinite complexity</line><line from="00:00:28,800" to="00:00:29,800">from the ground up.</line><line from="00:00:29,800" to="00:00:32,680">We'll look at classic fractals like the cantor set.</line><line from="00:00:32,680" to="00:00:34,080">Cokes, no flake.</line><line from="00:00:34,080" to="00:00:37,760">See your pensky triangle and mandal brought Julia sets.</line><line from="00:00:37,760" to="00:00:40,760">We'll see how recursion and feedback are key</line><line from="00:00:40,760" to="00:00:44,080">to generating fractals through repeating transformations.</line><line from="00:00:44,080" to="00:00:46,480">These mathematical fractals provide</line><line from="00:00:46,480" to="00:00:48,920">pure realizations of the repetitive symmetry</line><line from="00:00:48,920" to="00:00:52,960">that makes fractals so endlessly captivating.</line><line from="00:00:52,960" to="00:00:55,240">They show how fractal dimensions describe</line><line from="00:00:55,240" to="00:00:58,240">the infinite detail between whole number dimensions.</line><line from="00:00:58,240" to="00:01:00,280">Understanding these construction methods</line><line from="00:01:00,280" to="00:01:04,880">gives us tools to engineer and analyze fractal systems.</line><line from="00:01:04,880" to="00:01:08,320">So let's delve into the abstract fractal factory</line><line from="00:01:08,320" to="00:01:11,760">and see how mathematics manifests infinity.</line><line from="00:01:11,760" to="00:01:15,320">The cantor set is a classic one-dimensional fractal</line><line from="00:01:15,320" to="00:01:18,760">mathematical construction invented by George Cantor</line><line from="00:01:18,760" to="00:01:20,200">in the late 1800s.</line><line from="00:01:20,200" to="00:01:22,520">It starts with a line segment.</line><line from="00:01:22,520" to="00:01:25,880">Then you remove the middle third, leaving two segments.</line><line from="00:01:25,880" to="00:01:27,840">Next, you remove the middle third of those,</line><line from="00:01:27,840" to="00:01:29,760">leaving four segments.</line><line from="00:01:29,760" to="00:01:31,640">By repeating this process, infinitely,</line><line from="00:01:31,640" to="00:01:32,760">you get the cantor set.</line><line from="00:01:32,760" to="00:01:36,640">This fractal dust of points is several special properties.</line><line from="00:01:36,640" to="00:01:38,880">It is uncountable, meaning the number of points</line><line from="00:01:38,880" to="00:01:41,680">cannot be mapped to the natural numbers.</line><line from="00:01:41,680" to="00:01:44,040">It is also a space filling curve.</line><line from="00:01:44,040" to="00:01:46,720">The points perfectly fill the one space</line><line from="00:01:46,720" to="00:01:49,800">despite the total length shrinking to zero.</line><line from="00:01:49,800" to="00:01:53,600">The cantor set maintains a topological dimension of one.</line><line from="00:01:53,600" to="00:01:55,520">But it's housed with dimension describing</line><line from="00:01:55,520" to="00:01:59,360">its fractional complexity as approximately 0.63.</line><line from="00:01:59,360" to="00:02:02,600">The cantor set illustrates how fractals can remove area</line><line from="00:02:02,600" to="00:02:06,880">yet maintained topological features, like connectivity.</line><line from="00:02:06,880" to="00:02:09,720">This mind-bending math, laterally groundwork</line><line from="00:02:09,720" to="00:02:12,920">for fractal geometry, another pioneered mathematical</line><line from="00:02:12,920" to="00:02:16,480">fractal is the coaxnophlake, invented</line><line from="00:02:16,480" to="00:02:20,360">by Swedish mathematician Helgivoccoge in 1904.</line><line from="00:02:20,360" to="00:02:23,400">It starts with an equilateral triangle.</line><line from="00:02:23,400" to="00:02:26,520">Then you divide each line segment into thirds,</line><line from="00:02:26,520" to="00:02:29,000">and replace the middle third with two new segments</line><line from="00:02:29,000" to="00:02:30,360">forming a bump.</line><line from="00:02:30,360" to="00:02:32,600">This creates a 12-sided star.</line><line from="00:02:32,600" to="00:02:36,040">When you repeat this subdivision and replacement process,</line><line from="00:02:36,040" to="00:02:40,920">infinitely, you get the coaxnophlake with its iconic jagged,</line><line from="00:02:40,920" to="00:02:42,360">snowflake like shape.</line><line from="00:02:42,360" to="00:02:46,040">The coaxnophlake is several interesting properties.</line><line from="00:02:46,040" to="00:02:50,480">As you iterate further, the perimeter length approaches infinity.</line><line from="00:02:50,480" to="00:02:53,360">While the area remains finite, it exhibits</line><line from="00:02:53,480" to="00:02:55,720">exact self-similarity.</line><line from="00:02:55,720" to="00:02:59,680">Any subsection is an exact scale down replica,</line><line from="00:02:59,680" to="00:03:03,280">and it has a fractional dimension of approximately 1.6.</line><line from="00:03:03,280" to="00:03:06,320">The coaxnophlake demonstrates how simple recursion</line><line from="00:03:06,320" to="00:03:09,760">can quickly build geometric complexity</line><line from="00:03:09,760" to="00:03:12,440">for shadowing the beauty of fractal structures.</line><line from="00:03:12,440" to="00:03:15,360">The serpensky triangle is another famous example</line><line from="00:03:15,360" to="00:03:18,320">of a recursive mathematical fractal.</line><line from="00:03:18,320" to="00:03:22,560">It was described by Walkla Sierpensky in 1915,</line><line from="00:03:22,560" to="00:03:24,920">though similar designs appeared earlier.</line><line from="00:03:24,920" to="00:03:28,160">It starts with a solid, equal lateral triangle.</line><line from="00:03:28,160" to="00:03:32,320">Then you remove the central upside-down triangle,</line><line from="00:03:32,320" to="00:03:35,000">leaving three smaller triangles.</line><line from="00:03:35,000" to="00:03:36,800">Next, you remove the central triangles</line><line from="00:03:36,800" to="00:03:38,840">from each of those and repeat infinitely.</line><line from="00:03:38,840" to="00:03:40,920">The resulting serpensky triangle</line><line from="00:03:40,920" to="00:03:44,200">has a distinctive pattern of fractal triangular voids</line><line from="00:03:44,200" to="00:03:47,880">embedded within each other in a self-similar cascade.</line><line from="00:03:47,880" to="00:03:52,240">Logically, the construction relates to Pascal's triangle.</line><line from="00:03:52,600" to="00:03:57,120">With its fractal patterns of binomial coefficients.</line><line from="00:03:57,120" to="00:03:59,240">And geometricly, the serpensky triangle</line><line from="00:03:59,240" to="00:04:02,120">has a fractional dimension around 1.5A.</line><line from="00:04:02,120" to="00:04:04,760">The serpensky triangle elegantly demonstrates</line><line from="00:04:04,760" to="00:04:07,800">the fractal concept of infinite recursion.</line><line from="00:04:07,800" to="00:04:09,840">Carving voids with invoids,</line><line from="00:04:09,840" to="00:04:12,920">Ed Infanitum to create a complex, perforated hole,</line><line from="00:04:12,920" to="00:04:17,520">two of the most famous and dazzlingly complex fractals,</line><line from="00:04:17,520" to="00:04:20,200">are the Mandelbrot and Juliusets,</line><line from="00:04:20,200" to="00:04:23,040">developed by mathematicians, Benoid Mandelbrot,</line><line from="00:04:23,040" to="00:04:24,280">and Gaston Julius.</line><line from="00:04:24,280" to="00:04:27,200">They are constructed by recursively iterating</line><line from="00:04:27,200" to="00:04:31,440">simple quadratic polynomial functions in the complex plane.</line><line from="00:04:31,440" to="00:04:36,120">Such as EC2, CE visualizing how the outputs diverge</line><line from="00:04:36,120" to="00:04:38,960">based on the parameter C, tiny changes in C lead</line><line from="00:04:38,960" to="00:04:42,960">to radically different shapes, including swirling spirals,</line><line from="00:04:42,960" to="00:04:47,160">dendritic lightning bolts, and infinitely interlocking blobs,</line><line from="00:04:47,160" to="00:04:50,360">the hallmark of a fractal, coloring each point,</line><line from="00:04:50,360" to="00:04:53,680">based on the speed of divergence reveals the Mandelbrot</line><line from="00:04:53,680" to="00:04:57,240">and Juliusets and all their infinite psychedelic glory.</line><line from="00:04:57,240" to="00:05:00,240">Their endless boundaries embody mathematical chaos</line><line from="00:05:00,240" to="00:05:02,600">and strangeness, advanced fractal properties</line><line from="00:05:02,600" to="00:05:05,680">like periodic bulbs, buds, and antennae appear.</line><line from="00:05:05,680" to="00:05:08,280">No point can decode the entire pattern.</line><line from="00:05:08,280" to="00:05:11,080">These fractals exemplify how basic recursion</line><line from="00:05:11,080" to="00:05:13,800">creates astronomical complexity,</line><line from="00:05:13,800" to="00:05:16,800">containing infinitely regressing patterns</line><line from="00:05:16,800" to="00:05:19,280">that reveal the scope of mathematical infinite</line><line from="00:05:19,280" to="00:05:22,760">at the core of mathematically generating fractals</line><line from="00:05:22,760" to="00:05:25,200">is the use of recursion and feedback.</line><line from="00:05:25,200" to="00:05:27,240">Recursion means a function calls itself,</line><line from="00:05:27,240" to="00:05:31,680">so the output gets fed back as input indefinitely.</line><line from="00:05:31,680" to="00:05:34,080">This creates an infinite repetitive process</line><line from="00:05:34,080" to="00:05:36,320">where each iteration builds on the last.</line><line from="00:05:36,320" to="00:05:39,000">In fractals, the recursion happens by applying</line><line from="00:05:39,000" to="00:05:42,840">a transforming rule or function over and over.</line><line from="00:05:42,840" to="00:05:44,640">For example, in the coach snowflake,</line><line from="00:05:44,640" to="00:05:47,320">the segment dividing rule gets recursed infinite.</line><line from="00:05:47,320" to="00:05:50,440">This repetition builds exponential complexity rapidly</line><line from="00:05:50,440" to="00:05:52,080">through the feedback.</line><line from="00:05:52,080" to="00:05:54,680">Tiny changes get amplified over iterations</line><line from="00:05:54,680" to="00:05:56,200">leading to mass change.</line><line from="00:05:56,200" to="00:06:00,600">Recursion allows creating infinite detail from finite rules.</line><line from="00:06:00,600" to="00:06:03,040">The complexity comes not from the rule itself,</line><line from="00:06:03,040" to="00:06:05,920">but from recursively applying it within a feedback rule.</line><line from="00:06:05,920" to="00:06:08,160">This gives fractals their self-referential</line><line from="00:06:08,160" to="00:06:10,040">in scale and variant nature.</line><line from="00:06:10,040" to="00:06:12,200">Simple programs can build fractal worlds</line><line from="00:06:12,200" to="00:06:14,960">of endless depth by leveraging the power</line><line from="00:06:14,960" to="00:06:16,880">of infinite recursion and feedback.</line><line from="00:06:16,880" to="00:06:19,480">A key feature mathematical fractals</line><line from="00:06:19,480" to="00:06:22,600">is that their fractional dimensions accurately</line><line from="00:06:22,600" to="00:06:24,760">quantify their infinite complexity.</line><line from="00:06:24,760" to="00:06:26,480">Recall that topological dimensions</line><line from="00:06:26,480" to="00:06:29,120">describe the shape of misnuances.</line><line from="00:06:29,120" to="00:06:32,160">A fractal line is one with fractually more complex.</line><line from="00:06:32,160" to="00:06:36,120">Fractal dimensions capture this in-between complexity.</line><line from="00:06:36,120" to="00:06:39,320">We can calculate dimensions using the box counting method.</line><line from="00:06:39,320" to="00:06:41,400">First overlay the fractal with a grid of boxes</line><line from="00:06:41,400" to="00:06:44,720">of sizes count the number of boxes needed to cover the shape.</line><line from="00:06:44,720" to="00:06:48,120">Now, then do this for smaller box sizes,</line><line from="00:06:48,120" to="00:06:51,040">finding this for each as value.</line><line from="00:06:51,040" to="00:06:53,200">Plot this on a log log scale.</line><line from="00:06:53,200" to="00:06:55,320">The slope gives the fractal dimension.</line><line from="00:06:55,320" to="00:06:57,720">Since fractals have infinite detail,</line><line from="00:06:57,720" to="00:07:00,920">now it becomes huge even for small S.</line><line from="00:07:00,920" to="00:07:03,720">This reveals the fractional dimension value.</line><line from="00:07:03,720" to="00:07:07,560">For a line is scales linearly giving a dimension of one.</line><line from="00:07:07,560" to="00:07:09,040">But for coach snowflake,</line><line from="00:07:09,040" to="00:07:12,600">in a scales faster revealing its fractional dimension</line><line from="00:07:12,600" to="00:07:14,560">of approximately 1.6.</line><line from="00:07:14,560" to="00:07:17,360">Fractal dimensions mathematically characterize</line><line from="00:07:17,360" to="00:07:21,120">the limitless complexity of fractal shapes.</line><line from="00:07:21,120" to="00:07:24,960">Box counting is one method to estimate these fractional values.</line><line from="00:07:24,960" to="00:07:28,440">Our exploration today of mathematically constructed fractals</line><line from="00:07:28,440" to="00:07:31,600">reveals the underlying recursion and infinity</line><line from="00:07:31,600" to="00:07:33,760">at the heart of fractal geometry.</line><line from="00:07:33,760" to="00:07:36,680">By recursively applying simple replacement rules</line><line from="00:07:36,680" to="00:07:40,760">and feedback loops, mathematicians generate fractal shapes</line><line from="00:07:40,760" to="00:07:43,160">with unending nuance complexity,</line><line from="00:07:43,160" to="00:07:45,240">captured by fractional dimensions.</line><line from="00:07:45,240" to="00:07:48,160">Seeing these fractal sets curves and iterative functions</line><line from="00:07:48,160" to="00:07:52,480">helps us understand how self-similarity, scale invariance,</line><line from="00:07:52,480" to="00:07:55,920">and complexity arise from basic algorithmic processes</line><line from="00:07:55,920" to="00:07:57,160">endlessly repeated.</line><line from="00:07:57,160" to="00:07:59,240">In our next episode, we'll explore</line><line from="00:07:59,240" to="00:08:03,080">how computer graphics taps into mathematical fractals</line><line from="00:08:03,080" to="00:08:06,080">to create stunning landscapes, art,</line><line from="00:08:06,080" to="00:08:09,120">and special effects leveraging their beauty and realism.</line><line from="00:08:09,120" to="00:08:12,840">Until then, ponder infinity by zooming into the Mandelbrot set,</line><line from="00:08:12,840" to="00:08:14,800">contemplate the emergence of complexity</line><line from="00:08:14,800" to="00:08:18,760">from simplicity by gazing at a seropensky triangle</line><line from="00:08:18,760" to="00:08:21,360">and appreciate the creativity within constraints</line><line from="00:08:21,360" to="00:08:23,320">by imagining new fractal shapes.</line></transcript></transcript></video>