In mathematics, a self-similar object is exactly or approximately similar to a part of itself (i.e. the whole has the same shape as one or more of the parts). Many objects in the real world, such as coastlines, are statistically self-similar: parts of them show the same statistical properties at many scales. Self-similarity is a typical property of fractals. Scale invariance is an exact form of self-similarity where at any magnification there is a smaller piece of the object that is similar to the whole. For instance, a side of the Koch snowflake is both symmetrical and scale-invariant; it can be continually magnified 3x without changing shape. The non-trivial similarity evident in fractals is distinguished by their fine structure, or detail on arbitrarily small scales. As a counterexample, whereas any portion of a straight line may resemble the whole, further detail is not revealed.
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
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Home page: https://www.3blue1brown.com/
One technical note: It's possible to have fractals with an integer dimension. The example to have in mind is some *very* rough curve, which just so happens to achieve roughness level exactly 2. Slightly rough might be around 1.1-dimension; quite rough could be 1.5; but a very rough curve could get up to 2.0 (or more). A classic example of this is the boundary of the Mandelbrot set. The Sierpinski pyramid also has dimension 2 (try computing it!).
The proper defi...
published: 27 Jan 2017
Self similarity
Sierpinski gasket and Koch curve
published: 07 Nov 2012
Fractals and Scaling: Self-similar and scale-free
These videos are from the Fractals and Scaling course on Complexity Explorer (complexityexplorer.org) taught by Prof. Dave Feldman. This course is intended for anyone who is interested in an overview of how ideas from fractals and scaling are used to study complex systems.
The playlist begins by viewing fractals as self-similar geometric objects such as trees, ferns, clouds, mountain ranges, and river basins. Fractals are scale-free, in the sense that there is not a typical length or time scale that captures their features. A tree, for example, is made up of branches, off of which are smaller branches, off of which are smaller branches, and so on. Fractals thus look similar, regardless of the scale at which they are viewed. Fractals are often characterized by their dimension.
In a...
Buy on iTunes: https://itunes.apple.com/album/id1243668097
Taken from Software « Chip-Meditation, Pt. 1 (Bonus Track Version) »
Get the full SOFTWARE discography @ http://software.100percentelectronica.com
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100% Electronica
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 ...
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
Home page: https://www.3blue1brown.com/
One technical note: It's possible to have fractals with an integer dimension. The example to have in mind is some *very* rough curve, which just so happens to achieve roughness level exactly 2. Slightly rough might be around 1.1-dimension; quite rough could be 1.5; but a very rough curve could get up to 2.0 (or more). A classic example of this is the boundary of the Mandelbrot set. The Sierpinski pyramid also has dimension 2 (try computing it!).
The proper definition of a fractal, at least as Mandelbrot wrote it, is a shape whose "Hausdorff dimension" is greater than its "topological dimension". Hausdorff dimension is similar to the box-counting one I showed in this video, in some sense counting using balls instead of boxes, and it coincides with box-counting dimension in many cases. But it's more general, at the cost of being a bit harder to describe.
Topological dimension is something that's always an integer, wherein (loosely speaking) curve-ish things are 1-dimensional, surface-ish things are two-dimensional, etc. For example, a Koch Curve has topological dimension 1, and Hausdorff dimension 1.262. A rough surface might have topological dimension 2, but fractal dimension 2.3. And if a curve with topological dimension 1 has a Hausdorff dimension that *happens* to be exactly 2, or 3, or 4, etc., it would be considered a fractal, even though it's fractal dimension is an integer.
See Mandelbrot's book "The Fractal Geometry of Nature" for the full details and more examples.
Music by Vince Rubinetti: https://soundcloud.com/vincerubinetti/riemann-zeta-function
Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld
------------------
3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted about new videos, subscribe, and click the bell to receive notifications (if you're into that).
If you are new to this channel and want to see more, a good place to start is this playlist: https://www.youtube.com/playlist?list=PLZHQObOWTQDPHP40bzkb0TKLRPwQGAoC-
Various social media stuffs:
Twitter: https://twitter.com/3Blue1Brown
Facebook: https://www.facebook.com/3blue1brown/
Reddit: https://www.reddit.com/r/3Blue1Brown
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
Home page: https://www.3blue1brown.com/
One technical note: It's possible to have fractals with an integer dimension. The example to have in mind is some *very* rough curve, which just so happens to achieve roughness level exactly 2. Slightly rough might be around 1.1-dimension; quite rough could be 1.5; but a very rough curve could get up to 2.0 (or more). A classic example of this is the boundary of the Mandelbrot set. The Sierpinski pyramid also has dimension 2 (try computing it!).
The proper definition of a fractal, at least as Mandelbrot wrote it, is a shape whose "Hausdorff dimension" is greater than its "topological dimension". Hausdorff dimension is similar to the box-counting one I showed in this video, in some sense counting using balls instead of boxes, and it coincides with box-counting dimension in many cases. But it's more general, at the cost of being a bit harder to describe.
Topological dimension is something that's always an integer, wherein (loosely speaking) curve-ish things are 1-dimensional, surface-ish things are two-dimensional, etc. For example, a Koch Curve has topological dimension 1, and Hausdorff dimension 1.262. A rough surface might have topological dimension 2, but fractal dimension 2.3. And if a curve with topological dimension 1 has a Hausdorff dimension that *happens* to be exactly 2, or 3, or 4, etc., it would be considered a fractal, even though it's fractal dimension is an integer.
See Mandelbrot's book "The Fractal Geometry of Nature" for the full details and more examples.
Music by Vince Rubinetti: https://soundcloud.com/vincerubinetti/riemann-zeta-function
Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld
------------------
3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted about new videos, subscribe, and click the bell to receive notifications (if you're into that).
If you are new to this channel and want to see more, a good place to start is this playlist: https://www.youtube.com/playlist?list=PLZHQObOWTQDPHP40bzkb0TKLRPwQGAoC-
Various social media stuffs:
Twitter: https://twitter.com/3Blue1Brown
Facebook: https://www.facebook.com/3blue1brown/
Reddit: https://www.reddit.com/r/3Blue1Brown
These videos are from the Fractals and Scaling course on Complexity Explorer (complexityexplorer.org) taught by Prof. Dave Feldman. This course is intended for ...
These videos are from the Fractals and Scaling course on Complexity Explorer (complexityexplorer.org) taught by Prof. Dave Feldman. This course is intended for anyone who is interested in an overview of how ideas from fractals and scaling are used to study complex systems.
The playlist begins by viewing fractals as self-similar geometric objects such as trees, ferns, clouds, mountain ranges, and river basins. Fractals are scale-free, in the sense that there is not a typical length or time scale that captures their features. A tree, for example, is made up of branches, off of which are smaller branches, off of which are smaller branches, and so on. Fractals thus look similar, regardless of the scale at which they are viewed. Fractals are often characterized by their dimension.
In addition to physical objects, fractals are used to describe distributions resulting from processes that unfold in space and/or time. Earthquake severity, the frequency of words in texts, the sizes of cities, and the number of links to websites are all examples of quantities described by fractal distributions of this sort, known as power laws. Phenomena described by such distributions are said to scale or exhibit scaling, because there is a statistical relationship that is constant across scales.
The course looks at power laws in some detail and will give an overview of modern statistical techniques for calculating power law exponents. We look more generally at fat-tailed distributions, a class of distributions of which power laws are a subset. Next we will turn our attention to learning about some of the many processes that can generate fractals. Finally, we will critically examine some recent applications of fractals and scaling in natural and social systems, including metabolic scaling and urban scaling. These are, arguably, among the most successful and surprising areas of application of fractals and scaling. They are also areas of current scientific activity and debate.
These videos are from the Fractals and Scaling course on Complexity Explorer (complexityexplorer.org) taught by Prof. Dave Feldman. This course is intended for anyone who is interested in an overview of how ideas from fractals and scaling are used to study complex systems.
The playlist begins by viewing fractals as self-similar geometric objects such as trees, ferns, clouds, mountain ranges, and river basins. Fractals are scale-free, in the sense that there is not a typical length or time scale that captures their features. A tree, for example, is made up of branches, off of which are smaller branches, off of which are smaller branches, and so on. Fractals thus look similar, regardless of the scale at which they are viewed. Fractals are often characterized by their dimension.
In addition to physical objects, fractals are used to describe distributions resulting from processes that unfold in space and/or time. Earthquake severity, the frequency of words in texts, the sizes of cities, and the number of links to websites are all examples of quantities described by fractal distributions of this sort, known as power laws. Phenomena described by such distributions are said to scale or exhibit scaling, because there is a statistical relationship that is constant across scales.
The course looks at power laws in some detail and will give an overview of modern statistical techniques for calculating power law exponents. We look more generally at fat-tailed distributions, a class of distributions of which power laws are a subset. Next we will turn our attention to learning about some of the many processes that can generate fractals. Finally, we will critically examine some recent applications of fractals and scaling in natural and social systems, including metabolic scaling and urban scaling. These are, arguably, among the most successful and surprising areas of application of fractals and scaling. They are also areas of current scientific activity and debate.
Buy on iTunes: https://itunes.apple.com/album/id1243668097
Taken from Software « Chip-Meditation, Pt. 1 (Bonus Track Version) »
Get the full SOFTWARE discog...
Buy on iTunes: https://itunes.apple.com/album/id1243668097
Taken from Software « Chip-Meditation, Pt. 1 (Bonus Track Version) »
Get the full SOFTWARE discography @ http://software.100percentelectronica.com
Production: |
100% Electronica
Buy on iTunes: https://itunes.apple.com/album/id1243668097
Taken from Software « Chip-Meditation, Pt. 1 (Bonus Track Version) »
Get the full SOFTWARE discography @ http://software.100percentelectronica.com
Production: |
100% Electronica
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
Home page: https://www.3blue1brown.com/
One technical note: It's possible to have fractals with an integer dimension. The example to have in mind is some *very* rough curve, which just so happens to achieve roughness level exactly 2. Slightly rough might be around 1.1-dimension; quite rough could be 1.5; but a very rough curve could get up to 2.0 (or more). A classic example of this is the boundary of the Mandelbrot set. The Sierpinski pyramid also has dimension 2 (try computing it!).
The proper definition of a fractal, at least as Mandelbrot wrote it, is a shape whose "Hausdorff dimension" is greater than its "topological dimension". Hausdorff dimension is similar to the box-counting one I showed in this video, in some sense counting using balls instead of boxes, and it coincides with box-counting dimension in many cases. But it's more general, at the cost of being a bit harder to describe.
Topological dimension is something that's always an integer, wherein (loosely speaking) curve-ish things are 1-dimensional, surface-ish things are two-dimensional, etc. For example, a Koch Curve has topological dimension 1, and Hausdorff dimension 1.262. A rough surface might have topological dimension 2, but fractal dimension 2.3. And if a curve with topological dimension 1 has a Hausdorff dimension that *happens* to be exactly 2, or 3, or 4, etc., it would be considered a fractal, even though it's fractal dimension is an integer.
See Mandelbrot's book "The Fractal Geometry of Nature" for the full details and more examples.
Music by Vince Rubinetti: https://soundcloud.com/vincerubinetti/riemann-zeta-function
Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld
------------------
3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted about new videos, subscribe, and click the bell to receive notifications (if you're into that).
If you are new to this channel and want to see more, a good place to start is this playlist: https://www.youtube.com/playlist?list=PLZHQObOWTQDPHP40bzkb0TKLRPwQGAoC-
Various social media stuffs:
Twitter: https://twitter.com/3Blue1Brown
Facebook: https://www.facebook.com/3blue1brown/
Reddit: https://www.reddit.com/r/3Blue1Brown
These videos are from the Fractals and Scaling course on Complexity Explorer (complexityexplorer.org) taught by Prof. Dave Feldman. This course is intended for anyone who is interested in an overview of how ideas from fractals and scaling are used to study complex systems.
The playlist begins by viewing fractals as self-similar geometric objects such as trees, ferns, clouds, mountain ranges, and river basins. Fractals are scale-free, in the sense that there is not a typical length or time scale that captures their features. A tree, for example, is made up of branches, off of which are smaller branches, off of which are smaller branches, and so on. Fractals thus look similar, regardless of the scale at which they are viewed. Fractals are often characterized by their dimension.
In addition to physical objects, fractals are used to describe distributions resulting from processes that unfold in space and/or time. Earthquake severity, the frequency of words in texts, the sizes of cities, and the number of links to websites are all examples of quantities described by fractal distributions of this sort, known as power laws. Phenomena described by such distributions are said to scale or exhibit scaling, because there is a statistical relationship that is constant across scales.
The course looks at power laws in some detail and will give an overview of modern statistical techniques for calculating power law exponents. We look more generally at fat-tailed distributions, a class of distributions of which power laws are a subset. Next we will turn our attention to learning about some of the many processes that can generate fractals. Finally, we will critically examine some recent applications of fractals and scaling in natural and social systems, including metabolic scaling and urban scaling. These are, arguably, among the most successful and surprising areas of application of fractals and scaling. They are also areas of current scientific activity and debate.
Buy on iTunes: https://itunes.apple.com/album/id1243668097
Taken from Software « Chip-Meditation, Pt. 1 (Bonus Track Version) »
Get the full SOFTWARE discography @ http://software.100percentelectronica.com
Production: |
100% Electronica
In mathematics, a self-similar object is exactly or approximately similar to a part of itself (i.e. the whole has the same shape as one or more of the parts). Many objects in the real world, such as coastlines, are statistically self-similar: parts of them show the same statistical properties at many scales. Self-similarity is a typical property of fractals. Scale invariance is an exact form of self-similarity where at any magnification there is a smaller piece of the object that is similar to the whole. For instance, a side of the Koch snowflake is both symmetrical and scale-invariant; it can be continually magnified 3x without changing shape. The non-trivial similarity evident in fractals is distinguished by their fine structure, or detail on arbitrarily small scales. As a counterexample, whereas any portion of a straight line may resemble the whole, further detail is not revealed.
Reinhart told SelfMagazine that many suffering from similar conditions lack her resources to address them, and she is motivated to continue supporting others to advocate for their own health ... To read the full interview with Self Magazine, click here.
The deepseek logo, a keyboard, and robot hands are seen in this illustration taken January 27, 2025 ... export restrictions and achieving self-sufficiency in key sectors like AI. A similar symposium in the previous year was attended by Baidu CEORobin Li.
Liang’s presence at the gathering is potentially a sign that DeepSeek’s success could be important to Beijing’s policy goal of overcoming Washington’s export controls and achieving self-sufficiency in strategic industries like AI.
However, not every case of self-diagnosis ends this well ... Similarly, a doctor interviewed by “The Independent” noted that even when patients’ self-diagnoses are "completely off base," it opens the door for meaningful conversations.
Learning to self-regulate is a valuable skill that a mental health specialist can help develop. Similarly, a dependable financial adviser can provide the guidance to recognise emotion-driven financial ...
... a knife for self-defense purposes, this may be interpreted as an unlawful intent ... If charged with carrying a knife for unlawful purposes (such as self-defense), individuals may face similar penalties.
There are undercurrents of both expressing constitutional rights, and an ability to self-protect should the need arise ... There are similar dualities in photos such as “Aaron Banks Jr and Sr, Cedar ...
The plot was always similar ... In the past few years we have seen a series of similar diplomatic stagings with the United States and Germany (standing in for Europe as a whole) ... Canada faces a similar balance-of-payments squeeze.
PM Narendra Modi on Jan ... "HappyRepublic Day ... The Param Vir Chakra is awarded for the most conspicuous act of bravery and self-sacrifice in the face of the enemy, while the Ashok Chakra is awarded for similar acts of valour and self-sacrifice ... ....
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READ MORE. Nick Cave interviewed ...Dave Benett/WireImage ... He said ... “This just collapsed completely and I just saw the folly of that, the kind of disgraceful self-indulgence of the whole thing.”.Cave made a similar remarks in his latest interview ... .
I have sensed their reduction, their growing habit of self-deprecation and apology, while men of a similar certain age may well flourish in the opposite direction, becoming distinguished and silver-foxy, weather-beaten and “experienced”.
The former star of Made In Chelsea, famous then for being his often-insufferable self, has reinvented himself in recent years to the point where he now must qualify as a professional derring-doer, if there is such a thing.
Similarly, deduction available for home loan interest on a self-occupied house property was last increased to ₹2 lakh in 2014 and has not been increased since then ... Similarly, ₹2-lakh deduction ...
</p><p>This annual protest is a reiteration of the Kashmiris' strong indignation and resistance against Indian rule, which has denied them their right to self-determination for over 77 years.
for which
