Chaos theory | Fractals | Dimension theory | Dynamical systems
In mathematics, more specifically in fractal geometry, a fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern (strictly speaking, a fractal pattern) changes with the scale at which it is measured. It has also been characterized as a measure of the space-filling capacity of a pattern that tells how a fractal scales differently from the space it is embedded in; a fractal dimension does not have to be an integer. The essential idea of "fractured" dimensions has a long history in mathematics, but the term itself was brought to the fore by Benoit Mandelbrot based on his 1967 paper on self-similarity in which he discussed fractional dimensions. In that paper, Mandelbrot cited previous work by Lewis Fry Richardson describing the counter-intuitive notion that a coastline's measured length changes with the length of the measuring stick used. In terms of that notion, the fractal dimension of a coastline quantifies how the number of scaled measuring sticks required to measure the coastline changes with the scale applied to the stick. There are several formal mathematical definitions of fractal dimension that build on this basic concept of change in detail with change in scale: see the section . Ultimately, the term fractal dimension became the phrase with which Mandelbrot himself became most comfortable with respect to encapsulating the meaning of the word fractal, a term he created. After several iterations over years, Mandelbrot settled on this use of the language: "...to use fractal without a pedantic definition, to use fractal dimension as a generic term applicable to all the variants." One non-trivial example is the fractal dimension of a Koch snowflake. It has a topological dimension of 1, but it is by no means rectifiable: the length of the curve between any two points on the Koch snowflake is infinite. No small piece of it is line-like, but rather it is composed of an infinite number of segments joined at different angles. The fractal dimension of a curve can be explained intuitively thinking of a fractal line as an object too detailed to be one-dimensional, but too simple to be two-dimensional. Therefore its dimension might best be described not by its usual topological dimension of 1 but by its fractal dimension, which is often a number between one and two; in the case of the Koch snowflake, it is approximately 1.2619. (Wikipedia).
Dimensions (1 of 3: The Traditional Definition - Directions)
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From playlist Exploring Mathematics: Fractals
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
Dimensions (2 of 3: A More Flexible Definition - Scale Factor)
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From playlist Exploring Mathematics: Fractals
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
Dimensions (3 of 3: Fractal Dimensions)
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From playlist Exploring Mathematics: Fractals
Summer of math exposition submission- fractal calculus
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From playlist Summer of Math Exposition Youtube Videos
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From playlist research
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
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
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Follow Tom on his journey to Delft in the Netherlands in his quest to find a 3D Mandelbrot Set, otherwise known as a 'Mandelbulb'. We begin with a discussion of the definition of a fractal, with examples from the natural world, as well as generating our very own in the form of the Koch Sn
From playlist Director's Cut
Benjamin Schweinhart (4/3/18): Persistent homology and the upper box dimension
We prove the first results relating persistent homology to a classically defined fractal dimension. Several previous studies have demonstrated an empirical relationship between persistent homology and fractal dimension; our results are the first rigorous analogue of those comparisons. Spe
From playlist AATRN 2018
Real Analysis Ep 17: The Cantor Set
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From playlist Math 3371 (Real analysis) Fall 2020
Geometer Explains One Concept in 5 Levels of Difficulty | WIRED
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From playlist Tutorials and Lectures
Ahlfors-Bers 2014 "Conformal invariance and critical behavior within critical fractal carpets"
Wendelin Werner (ETH Zürich): Some aspects of conformal invariance can survive within fractal carpets in the plane. In the present talk, I will survey how it is possible to make sense in a rather precise way of certain of these ideas in the special case of certain random -- yet very natura
From playlist The Ahlfors-Bers Colloquium 2014 at Yale
Dynamical systems, fractals and diophantine approximations – Carlos Gustavo Moreira – ICM2018
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From playlist Plenary Lectures