title

Fractals are typically not self-similar

description

An explanation of fractal dimension.
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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 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
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{'title': 'Fractals are typically not self-similar', 'heatmap': [], 'summary': "Explores fractal geometry and dimension, challenging the concept of self-similarity and advocating for roughness in modeling nature, emphasizing the pragmatic approach of fractal dimension. it discusses examples like the sierpinski triangle (1.585), von koch curve (1.262), and britain's coastline (1.21) to challenge traditional perceptions and discuss the potential utility of fractal dimension in modeling the world.", 'chapters': [{'end': 115.412, 'segs': [{'end': 90.445, 'src': 'embed', 'start': 61.057, 'weight': 0, 'content': [{'end': 65.581, 'text': 'But Mandelbrot had a much broader conception in mind, one motivated not by beauty,', 'start': 61.057, 'duration': 4.524}, {'end': 70.065, 'text': 'but more by a pragmatic desire to model nature in a way that actually captures roughness.', 'start': 65.581, 'duration': 4.484}, {'end': 79.577, 'text': 'In some ways, fractal geometry is a rebellion against calculus, whose central assumption is that things tend to look smooth if you zoom in far enough.', 'start': 72.372, 'duration': 7.205}, {'end': 85.381, 'text': 'But Mandelbrot saw this as overly idealized, or at least needlessly idealized,', 'start': 80.278, 'duration': 5.103}, {'end': 90.445, 'text': "resulting in models that neglect the finer details of the thing that they're actually modeling, which can matter.", 'start': 85.381, 'duration': 5.064}], 'summary': "Mandelbrot's fractal geometry captures roughness, rebelling against idealized models in calculus.", 'duration': 29.388, 'max_score': 61.057, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/gB9n2gHsHN4/pics/gB9n2gHsHN461057.jpg'}, {'end': 125.139, 'src': 'embed', 'start': 98.277, 'weight': 1, 'content': [{'end': 104.703, 'text': 'But the popular perception that fractals only include perfectly self-similar shapes is another over-idealization,', 'start': 98.277, 'duration': 6.426}, {'end': 108.927, 'text': "one that ironically goes against the pragmatic spirit of fractal geometry's origins.", 'start': 104.703, 'duration': 4.224}, {'end': 115.412, 'text': 'The real definition of fractals has to do with this idea of fractal dimension, the main topic of this video.', 'start': 109.688, 'duration': 5.724}, {'end': 125.139, 'text': 'You see, there is a sense, a certain way to define the word dimension, in which the Sierpinski triangle is approximately 1.585 dimensional,', 'start': 116.233, 'duration': 8.906}], 'summary': 'Fractals are not just self-similar shapes, but defined by fractal dimension, such as the sierpinski triangle at approximately 1.585 dimensional.', 'duration': 26.862, 'max_score': 98.277, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/gB9n2gHsHN4/pics/gB9n2gHsHN498277.jpg'}], 'start': 4.403, 'title': 'Fractal geometry and fractal dimension', 'summary': 'Delves into the concept of fractals by benoit mandelbrot, challenging the notion of self-similarity and advocating for roughness in modeling nature, emphasizing fractal dimension as a pragmatic approach.', 'chapters': [{'end': 115.412, 'start': 4.403, 'title': 'Fractal geometry and fractal dimension', 'summary': 'Discusses the broader conception of fractals by benoit mandelbrot, highlighting the misconception of self-similarity and the rebellion against the idealization of smoothness, emphasizing the pragmatic desire to model nature with roughness captured by fractal dimension.', 'duration': 111.009, 'highlights': ['The real definition of fractals involves fractal dimension, emphasizing a broader conception motivated by a pragmatic desire to model nature with roughness captured by fractal dimension.', "Mandelbrot's conception of fractals extends beyond self-similarity, rebelling against the idealization of smoothness, aiming to capture the finer details of nature's roughness.", 'Fractal geometry serves as a rebellion against the idealized smoothness assumed by calculus, providing a basis for modeling the regularity in some forms of roughness.', "Self-similar shapes are a common misconception of fractals, as Mandelbrot's conception encompasses a broader definition involving fractal dimension."]}], 'duration': 111.009, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/gB9n2gHsHN4/pics/gB9n2gHsHN44403.jpg', 'highlights': ["Mandelbrot's conception of fractals extends beyond self-similarity, aiming to capture the finer details of nature's roughness.", 'The real definition of fractals involves fractal dimension, emphasizing a broader conception motivated by a pragmatic desire to model nature.', 'Fractal geometry serves as a rebellion against the idealized smoothness assumed by calculus, providing a basis for modeling the regularity in some forms of roughness.', "Self-similar shapes are a common misconception of fractals, as Mandelbrot's conception encompasses a broader definition involving fractal dimension."]}, {'end': 433.34, 'segs': [{'end': 159.739, 'src': 'embed', 'start': 116.233, 'weight': 0, 'content': [{'end': 125.139, 'text': 'You see, there is a sense, a certain way to define the word dimension, in which the Sierpinski triangle is approximately 1.585 dimensional,', 'start': 116.233, 'duration': 8.906}, {'end': 129.681, 'text': 'that the von Koch curve is approximately 1.262 dimensional.', 'start': 125.139, 'duration': 4.542}, {'end': 137.907, 'text': "the coastline of Britain turns out to be around 1.21 dimensional and in general it's possible to have shapes whose dimension is any positive real number,", 'start': 129.681, 'duration': 8.226}, {'end': 138.748, 'text': 'not just whole numbers.', 'start': 137.907, 'duration': 0.841}, {'end': 147.514, 'text': 'I think when I first heard someone reference fractional dimension like this, I just thought it was nonsense, right?', 'start': 142.352, 'duration': 5.162}, {'end': 149.895, 'text': 'I mean mathematicians are clearly just making stuff up.', 'start': 147.654, 'duration': 2.241}, {'end': 153.756, 'text': 'Dimension is something that usually only makes sense for natural numbers, right?', 'start': 150.395, 'duration': 3.361}, {'end': 159.739, 'text': "A line is one-dimensional, a plane that's two-dimensional, the space that we live in that's three-dimensional, and so on.", 'start': 154.136, 'duration': 5.603}], 'summary': 'Shapes like sierpinski triangle have fractional dimensions, challenging traditional whole number dimensions.', 'duration': 43.506, 'max_score': 116.233, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/gB9n2gHsHN4/pics/gB9n2gHsHN4116233.jpg'}], 'start': 116.233, 'title': 'Fractal dimensions', 'summary': "Delves into the concept of fractional dimensions, using examples such as the sierpinski triangle (1.585), von koch curve (1.262), and britain's coastline (1.21). it challenges the traditional perception of dimension and discusses the potential utility of fractal dimension in modeling the world.", 'chapters': [{'end': 180.878, 'start': 116.233, 'title': 'Fractal dimensions and their significance', 'summary': "Explores the concept of fractional dimensions, citing examples such as the sierpinski triangle (1.585), von koch curve (1.262), and britain's coastline (1.21). it challenges the traditional perception of dimension as applicable only to natural numbers and discusses the potential utility of fractal dimension in modeling the world.", 'duration': 64.645, 'highlights': ['The Sierpinski triangle has an approximate dimension of 1.585, showcasing the existence of shapes with non-integer dimensions.', 'The von Koch curve demonstrates an approximate dimension of 1.262, further challenging the traditional understanding of dimension as solely applicable to whole numbers.', "Britain's coastline is estimated to be around 1.21 dimensional, indicating the possibility of non-integer dimensions in real-world phenomena.", 'The chapter challenges the conventional perception of dimension as exclusively pertaining to natural numbers, highlighting the potential utility of fractal dimension in modeling the world.']}, {'end': 433.34, 'start': 181.338, 'title': 'Fractal dimension: measuring self-similarity', 'summary': 'Discusses the concept of fractal dimension, using self-similar shapes like lines, squares, cubes, and sierpinski triangles to illustrate how mass changes as shapes are scaled, leading to the conclusion that the dimensionality of a sierpinski triangle is approximately 1.585.', 'duration': 252.002, 'highlights': ['The concept of fractal dimension is explained using self-similar shapes like lines, squares, cubes, and Sierpinski triangles, demonstrating how their mass changes as they are scaled down.', 'The dimensionality of a Sierpinski triangle is determined to be approximately 1.585 by equating the change in its mass when scaled down by a factor of one half to the power of its dimension.', 'The discussion leads to the conclusion that a Sierpinski triangle has a dimensionality of approximately 1.585, derived using the concept of logarithms to solve the equation 2^d = 3.']}], 'duration': 317.107, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/gB9n2gHsHN4/pics/gB9n2gHsHN4116233.jpg', 'highlights': ['The Sierpinski triangle has an approximate dimension of 1.585, showcasing non-integer dimensions.', 'The von Koch curve demonstrates an approximate dimension of 1.262, challenging traditional understanding.', "Britain's coastline is estimated to be around 1.21 dimensional, indicating non-integer dimensions in real-world phenomena.", 'The chapter challenges the conventional perception of dimension as exclusively pertaining to natural numbers.']}, {'end': 1170.343, 'segs': [{'end': 456.569, 'src': 'embed', 'start': 433.34, 'weight': 2, 'content': [{'end': 440.563, 'text': 'So, in this way, the Sierpinski triangle is not one-dimensional, even though you could define a curve that passes through all its points,', 'start': 433.34, 'duration': 7.223}, {'end': 443.424, 'text': 'and nor is it two-dimensional even though it lives in the plane.', 'start': 440.563, 'duration': 2.861}, {'end': 446.525, 'text': "Instead, it's 1.585-dimensional.", 'start': 444.184, 'duration': 2.341}, {'end': 451.727, 'text': 'And if you want to describe its mass, neither length nor area seem like the fitting notions.', 'start': 447.225, 'duration': 4.502}, {'end': 456.569, 'text': 'If you tried, its length would turn out to be infinite, and its area would turn out to be zero.', 'start': 452.387, 'duration': 4.182}], 'summary': 'The sierpinski triangle is 1.585-dimensional, with infinite length and zero area.', 'duration': 23.229, 'max_score': 433.34, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/gB9n2gHsHN4/pics/gB9n2gHsHN4433340.jpg'}, {'end': 531.8, 'src': 'embed', 'start': 500.598, 'weight': 0, 'content': [{'end': 507.121, 'text': 'So, in a sense, the Von Koch curve is a 1.262 dimensional shape.', 'start': 500.598, 'duration': 6.523}, {'end': 510.182, 'text': "Here's another fun one.", 'start': 509.362, 'duration': 0.82}, {'end': 513.384, 'text': 'This is kind of the right-angled version of the Koch curve.', 'start': 510.583, 'duration': 2.801}, {'end': 521.712, 'text': "It's built up of eight scaled-down copies of itself, where the scaling factor here is 1 fourth.", 'start': 516.849, 'duration': 4.863}, {'end': 531.8, 'text': 'So, if you want to know its dimension, it should be some number d, such that 1, fourth to the power of d, equals 1, eighth,', 'start': 525.015, 'duration': 6.785}], 'summary': 'The von koch curve is a 1.262 dimensional shape, and the right-angled version is built of eight scaled-down copies with a scaling factor of 1 fourth.', 'duration': 31.202, 'max_score': 500.598, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/gB9n2gHsHN4/pics/gB9n2gHsHN4500598.jpg'}, {'end': 1098.612, 'src': 'embed', 'start': 1069.952, 'weight': 3, 'content': [{'end': 1078.138, 'text': 'And in this more applied setting, a shape is typically considered to be a fractal only when the measured dimension stays approximately constant,', 'start': 1069.952, 'duration': 8.186}, {'end': 1079.879, 'text': 'even across multiple different scales.', 'start': 1078.138, 'duration': 1.741}, {'end': 1084.902, 'text': "For example, the coastline of Britain doesn't just look 1.21 dimensional at a distance.", 'start': 1080.619, 'duration': 4.283}, {'end': 1091.627, 'text': 'Even if you zoom in by a factor of a thousand, the level of roughness is still around 1.21.', 'start': 1085.403, 'duration': 6.224}, {'end': 1098.612, 'text': 'That right there is the sense in which many shapes from nature actually are self-similar, albeit not perfect self-similarity.', 'start': 1091.627, 'duration': 6.985}], 'summary': 'Fractal shapes in nature exhibit self-similarity with a measured dimension staying constant across different scales.', 'duration': 28.66, 'max_score': 1069.952, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/gB9n2gHsHN4/pics/gB9n2gHsHN41069952.jpg'}, {'end': 1161.334, 'src': 'embed', 'start': 1136.887, 'weight': 4, 'content': [{'end': 1142.608, 'text': 'What this number is, what this fractional dimension gives us, is a quantitative way to describe roughness.', 'start': 1136.887, 'duration': 5.721}, {'end': 1147.49, 'text': 'For example, the coastline of Norway is about 1.52 dimensional,', 'start': 1143.328, 'duration': 4.162}, {'end': 1152.051, 'text': "which is a numerical way to communicate the fact that it's way more jaggedy than Britain's coastline.", 'start': 1147.49, 'duration': 4.561}, {'end': 1161.334, 'text': 'The surface of a calm ocean might have a fractal dimension only barely above 2, while a stormy one might have a dimension closer to 2.3.', 'start': 1152.871, 'duration': 8.463}], 'summary': "Fractal dimension quantifies roughness, e.g. norway's coastline is 1.52 dimensional.", 'duration': 24.447, 'max_score': 1136.887, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/gB9n2gHsHN4/pics/gB9n2gHsHN41136887.jpg'}], 'start': 433.34, 'title': 'Fractal dimensions and mass', 'summary': 'Discusses the concept of fractal dimensions and their application in describing the mass of self-similar fractals, citing examples like the sierpinski triangle (dimension of 1.585), the von koch curve (dimension of 1.262), and the right-angled version (dimension of 1.5). it also explores the quantitative nature of roughness and its applicability in natural phenomena, with examples such as the coastline of britain having a dimension of 1.21.', 'chapters': [{'end': 553.736, 'start': 433.34, 'title': 'Fractal dimensions and mass', 'summary': 'Discusses the concept of fractal dimensions and how they can be used to describe the mass of self-similar fractals, with examples like the sierpinski triangle having a dimension of 1.585, the von koch curve having a dimension of 1.262, and the right-angled version having a dimension of 1.5.', 'duration': 120.396, 'highlights': ['The Sierpinski triangle is 1.585-dimensional, and its mass cannot be described by length or area.', 'The Von Koch curve has a dimension of 1.262, determined by raising 1/3 to a power, which equals 1/4.', 'The right-angled version of the Koch curve is precisely 1.5 dimensional, determined by the scaling factor and the decreased mass.']}, {'end': 1170.343, 'start': 555.126, 'title': 'Understanding fractal dimensions', 'summary': 'Explores the concept of fractal dimensions, demonstrating how shapes with fractional dimensions exhibit roughness at different scales, with examples such as a circle with a dimension of 2, sierpinski triangle with a dimension of 1.585, and the coastline of britain with a dimension of 1.21, and emphasizes the quantitative nature of roughness and its applicability in natural phenomena.', 'duration': 615.217, 'highlights': ['Shapes with fractional dimensions exhibit roughness at different scales, with examples such as a circle with a dimension of 2, Sierpinski triangle with a dimension of 1.585, and the coastline of Britain with a dimension of 1.21.', 'The coastline of Britain is found to have a dimension of 1.21, signifying its roughness and jagged nature.', 'Fractal dimension serves as a quantitative way to describe roughness, with examples such as the coastline of Norway having a dimension of about 1.52, and the distinction between natural and man-made objects based on their fractal dimensions.']}], 'duration': 737.003, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/gB9n2gHsHN4/pics/gB9n2gHsHN4433340.jpg', 'highlights': ['The right-angled version of the Koch curve is precisely 1.5 dimensional, determined by the scaling factor and the decreased mass.', 'The Von Koch curve has a dimension of 1.262, determined by raising 1/3 to a power, which equals 1/4.', 'The Sierpinski triangle is 1.585-dimensional, and its mass cannot be described by length or area.', 'Shapes with fractional dimensions exhibit roughness at different scales, with examples such as a circle with a dimension of 2, Sierpinski triangle with a dimension of 1.585, and the coastline of Britain with a dimension of 1.21.', 'The coastline of Britain is found to have a dimension of 1.21, signifying its roughness and jagged nature.', 'Fractal dimension serves as a quantitative way to describe roughness, with examples such as the coastline of Norway having a dimension of about 1.52, and the distinction between natural and man-made objects based on their fractal dimensions.']}], 'highlights': ["Mandelbrot's conception of fractals extends beyond self-similarity, aiming to capture the finer details of nature's roughness.", 'The real definition of fractals involves fractal dimension, emphasizing a broader conception motivated by a pragmatic desire to model nature.', 'Fractal geometry serves as a rebellion against the idealized smoothness assumed by calculus, providing a basis for modeling the regularity in some forms of roughness.', "Self-similar shapes are a common misconception of fractals, as Mandelbrot's conception encompasses a broader definition involving fractal dimension.", 'Shapes with fractional dimensions exhibit roughness at different scales, with examples such as a circle with a dimension of 2, Sierpinski triangle with a dimension of 1.585, and the coastline of Britain with a dimension of 1.21.', 'The coastline of Britain is found to have a dimension of 1.21, signifying its roughness and jagged nature.', 'The Sierpinski triangle has an approximate dimension of 1.585, showcasing non-integer dimensions.', 'The Von Koch curve has a dimension of 1.262, determined by raising 1/3 to a power, which equals 1/4.', 'The right-angled version of the Koch curve is precisely 1.5 dimensional, determined by the scaling factor and the decreased mass.', 'The chapter challenges the conventional perception of dimension as exclusively pertaining to natural numbers.']}