Mathematical structures | Fractals | Topology
In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illustrated in successive magnifications of the Mandelbrot set. This exhibition of similar patterns at increasingly smaller scales is called self-similarity, also known as expanding symmetry or unfolding symmetry; if this replication is exactly the same at every scale, as in the Menger sponge, the shape is called affine self-similar. Fractal geometry lies within the mathematical branch of measure theory. One way that fractals are different from finite geometric figures is how they scale. Doubling the edge lengths of a filled polygon multiplies its area by four, which is two (the ratio of the new to the old side length) raised to the power of two (the conventional dimension of the filled polygon). Likewise, if the radius of a filled sphere is doubled, its volume scales by eight, which is two (the ratio of the new to the old radius) to the power of three (the conventional dimension of the filled sphere). However, if a fractal's one-dimensional lengths are all doubled, the spatial content of the fractal scales by a power that is not necessarily an integer and is in general greater than its conventional dimension. This power is called the fractal dimension of the geometric object, to distinguish it from the conventional dimension (which is formally called the topological dimension). Analytically, many fractals are nowhere differentiable. An infinite fractal curve can be conceived of as winding through space differently from an ordinary line – although it is still topologically 1-dimensional, its fractal dimension indicates that it locally fills space more efficiently than an ordinary line. Starting in the 17th century with notions of recursion, fractals have moved through increasingly rigorous mathematical treatment to the study of continuous but not differentiable functions in the 19th century by the seminal work of Bernard Bolzano, Bernhard Riemann, and Karl Weierstrass, and on to the coining of the word fractal in the 20th century with a subsequent burgeoning of interest in fractals and computer-based modelling in the 20th century. There is some disagreement among mathematicians about how the concept of a fractal should be formally defined. Mandelbrot himself summarized it as "beautiful, damn hard, increasingly useful. That's fractals." More formally, in 1982 Mandelbrot defined fractal as follows: "A fractal is by definition a set for which the Hausdorff–Besicovitch dimension strictly exceeds the topological dimension." Later, seeing this as too restrictive, he simplified and expanded the definition to this: "A fractal is a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole." Still later, Mandelbrot proposed "to use fractal without a pedantic definition, to use fractal dimension as a generic term applicable to all the variants". The consensus among mathematicians is that theoretical fractals are infinitely self-similar iterated and detailed mathematical constructs, of which many examples have been formulated and studied. Fractals are not limited to geometric patterns, but can also describe processes in time. Fractal patterns with various degrees of self-similarity have been rendered or studied in visual, physical, and aural media and found in , , , architecture and . Fractals are of particular relevance in the field of chaos theory because they show up in the geometric depictions of most chaotic processes (typically either as attractors or as boundaries between basins of attraction). (Wikipedia).
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From playlist research
Summer of math exposition submission- fractal calculus
Fractal Calculus
From playlist Summer of Math Exposition Youtube Videos
In this video, I define a neat concept called the fractal derivative (which shouldn't be confused with fractional derivatives). Then I provide a couple of examples, and finally I present an application of this concept to the study of anomalous diffusion in physics. Enjoy!
From playlist Calculus
mandelbrot fractal animation 5
another mandelbrot/julia fractal animation/morph.
From playlist Fractal
What Is A Fractal (and what are they good for)?
Fractals are complex, never-ending patterns created by repeating mathematical equations. Yuliya, a undergrad in Math at MIT, delves into their mysterious properties and how they can be found in technology and nature. Learn more about all the stuff that MIT is doing and researching with fr
From playlist Science Out Loud
Fractals are typically not self-similar
An explanation of fractal dimension. Help fund future projects: https://www.patreon.com/3blue1brown An equally valuable form of support is to simply share some of the videos. Special thanks to these supporters: https://3b1b.co/fractals-thanks And by Affirm: https://www.affirm.com/careers H
From playlist Explainers
Delicia Kamins - Philosophy of Fractals - CoM Oct 2020
We know that fractals are nature’s pattern makers. Fractals are in fact everywhere we look: tree bark, snowflakes, mountain ranges, cloud, rivers, seashells, all the way up to the shape of galaxies. Beyond nature, however, human beings are fractal thinkers. We depend on fractal algorithms
From playlist Celebration of Mind
The Newton Fractal Explained | Deep Dive Maths
A Newton fractal is obtained by iterating Newton's method to find the roots of a complex function. The iconic picture of this fractal is what I call The Newton Fractal, and is generated from the function f(z)=z^3-1, whose roots are the three cube roots of unity. What is the history of th
From playlist Deep Dive Maths
Geometer Explains One Concept in 5 Levels of Difficulty | WIRED
Computer scientist Keenan Crane, PhD, is asked to explain fractals to 5 different people; a child, a teen, a college student, a grad student, and an expert. Still haven’t subscribed to WIRED on YouTube? ►► http://wrd.cm/15fP7B7 Listen to the Get WIRED podcast ►► https://link.chtbl.com
From playlist Tutorials and Lectures
!!Con West 2019 - Michael Malis: Generating fractals … with SQL queries!!!
Presented at !!Con West 2019: http://bangbangcon.com/west SQL databases can do a lot. They are fantastic at making it easy to work with large amounts of data. One of the lesser-known capabilities of SQL databases is that they can be used to generate fractals! In this talk, we’ll take a l
From playlist !!Con West 2019
Marco Cavaglia - Think out of the (counting) box - IPAM at UCLA
Recorded 30 November 2021. Marco Cavaglia of the Missouri University of Science and Technology presents "Think out of the (counting) box" at IPAM's Workshop IV: Big Data in Multi-Messenger Astrophysics. Learn more online at: http://www.ipam.ucla.edu/programs/workshops/workshop-iv-big-data-
From playlist Workshop: Big Data in Multi-Messenger Astrophysics
!!Con West 2019 - Michael Malis: Generating fractals … with SQL queries!!!
Presented at !!Con West 2019: http://bangbangcon.com/west SQL databases can do a lot. They are fantastic at making it easy to work with large amounts of data. One of the lesser-known capabilities of SQL databases is that they can be used to generate fractals! In this talk, we’ll take a l
From playlist !!Con West 2019
Additive Energy of Regular Measures in One and Higher Dimensions, and the Fractal... - Laura Cladek
Analysis & Mathematical Physics Topic: Additive Energy of Regular Measures in One and Higher Dimensions, and the Fractal Uncertainty Principle Speaker: Laura Cladek Affiliation: von Neumann Fellow, School Of Mathematics Date: December 14, 2022 We obtain new bounds on the additive energy
From playlist Mathematics
Scientists Trapped Electrons In a Quantum Fractal (And It's Wild!)
Fractals aren’t just crazy cool mathematically infinite shapes. They might just have the capacity to revolutionize modern electronics as we know it. Thumbnail image courtesy of Sander Kempkes. There’s a Subterranean Biosphere Hiding in the Earth’s Crust and It’s MASSIVE - https://youtu.b
From playlist Elements | Season 4 | Seeker
In this video, I examine a sequence of strategies to try and improve the rendering speed of a Mandelbrot Fractal. Starting off with naive assumptions, I explore optimising the algorithm, then the use of vector co-processing, and finally threads and thread-pools in order to squeeze as much
From playlist Interesting Programming
You’ve heard of fracking, and you’re pretty sure lots of people don’t like it, but do you know how it actually works? Learn more at HowStuffWorks.com: http://science.howstuffworks.com/environmental/energy/hydraulic-fracking.htm Share on Facebook: http://goo.gl/M5kx1i Share on Twitter: ht
From playlist Visually-Striking Episodes From the 2010s
8.1: Fractals - The Nature of Code
This video introduces Fractals. (If I reference a link or project and it's not included in this description, please let me know!) Read along: http://natureofcode.com/book/chapter-8-fractals/ PBS Nova - Fractals - Hunting the Hidden Dimension: http://www.youtube.com/watch?v=LemPnZn54Kw
From playlist The Nature of Code: Simulating Natural Systems