From: 3blue1brown
Fractals are often admired for their blend of simplicity and complexity, frequently featuring infinitely repeating patterns [00:00:05]. Programmers, in particular, appreciate how a small amount of code can generate images far more intricate than any human could draw [00:00:11].
A common misconception is that fractals are shapes that are perfectly self-similar [00:00:29]. Examples like the Von Koch snowflake and the Sierpinski triangle exhibit perfect self-similarity, meaning when you zoom in, you see an identical copy of the original [00:00:34]. While these shapes are beautiful and serve as good introductory models, Benoit Mandelbrot, the “father of fractal geometry,” had a much broader conception in mind [00:00:24], [00:01:01].
Beyond Perfect Self-Similarity
Mandelbrot’s view was driven by a pragmatic desire to model nature, specifically to capture its inherent roughness [00:01:05]. This perspective can be seen as a rebellion against traditional calculus, which assumes objects become smooth when sufficiently zoomed in [00:01:12]. Mandelbrot found this assumption overly idealized, leading to models that neglected important fine details [00:01:20].
While self-similar shapes provide a basis for understanding regularity in some forms of roughness, the popular belief that fractals only include perfectly self-similar shapes is an over-idealization that goes against the pragmatic spirit of fractal geometry’s origins [00:01:32], [00:01:40].
The true definition of fractals is tied to the concept of fractal dimension [00:01:49]. In this sense, the Sierpinski triangle has a dimension of approximately 1.585, the Von Koch curve around 1.262, and even the coastline of Britain is about 1.21D [00:02:01]. This concept allows shapes to have any positive real number as their dimension, not just whole numbers [00:02:12].
Initially, the idea of a fractional dimension might seem nonsensical, as dimension is typically understood in terms of natural numbers (e.g., a line is 1D, a plane is 2D) [00:02:22], [00:02:30]. However, the utility of fractal dimension lies in its ability to model the world [00:02:55].
Defining Fractal Dimension
The concept of dimension can be understood by examining how a shape’s “mass” (or measure) changes when it’s scaled [00:04:40].
Self-Similarity Dimension
Consider common geometric shapes:
- Line: If a line is scaled down by a factor of 1/2, its mass (length) is also scaled down by 1/2. Two smaller copies combine to form the original [00:04:55].
- Square: Scaling a square down by 1/2 reduces its mass (area) by a factor of (1/2)², or 1/4. Four smaller copies form the original [00:05:07].
- Cube: Scaling a cube down by 1/2 reduces its mass (volume) by a factor of (1/2)³, or 1/8. Eight smaller copies form the original [00:05:19].
In these cases, the change in mass is related to the scaling factor raised to the power of the shape’s dimension [00:05:45]. For a d-dimensional shape scaled by a factor of ‘s’, its mass changes by s^d [00:06:09].
Now, consider the Sierpinski triangle:
- Sierpinski Triangle: When scaled down by a factor of 1/2, its mass (measure) reduces by 1/3, because three smaller copies are needed to form the original [00:05:31], [00:05:39]. To find its dimension (d), we solve (1/2)^d = 1/3, which is equivalent to 2^d = 3 [00:06:38], [00:06:59]. Using logarithms (log base 2 of 3), the dimension is approximately 1.585 [00:07:05], [00:07:09].
- This implies that neither length (1D) nor area (2D) are fitting notions to describe its measure, as its length would be infinite and its area zero [00:07:27]. Instead, it requires a 1.585-dimensional analog of length [00:07:38].
Other self-similar fractals include:
- Von Koch Curve: Composed of four smaller copies scaled down by 1/3 [00:07:49]. Its dimension is log base 3 of 4, approximately 1.262 [00:08:03], [00:08:15].
- Right-angled Koch Curve variant: Made of eight copies scaled down by 1/4 [00:08:36]. Its dimension is log base 4 of 8, which is exactly 1.5 [00:08:44], [00:08:56].
This “self-similarity dimension” method makes the concept of fractional dimension tangible, but it’s not a general notion because most shapes are not perfectly self-similar [00:09:17].
Box-Counting Dimension (More General Method)
For non-self-similar shapes, a more rigorous mathematical approach is needed to define “mass” or measure. One common method is the “box-counting” method [00:10:25].
- Grid Coverage: Cover the shape with a grid and count the number of grid squares that touch the shape [00:10:29].
- Scaling and Counting: Scale the shape by a factor (S) and recount the number of touching grid squares [00:10:46].
- Proportionality: For a shape of dimension ‘D’, the number of boxes touched should be proportional to S^D [00:11:42], [00:13:42].
- Log-Log Plot: To find ‘D’, take the logarithm of both sides of the relationship. Plotting the log of the scaling factor against the log of the number of boxes should yield a linear relationship, where the slope of the line is the dimension ‘D’ [00:14:08], [00:14:21].
This method allows us to measure the dimension of shapes with inherent roughness, like the coastline of Britain [00:13:00]. Its dimension of approximately 1.21 means that when scaled, the number of boxes touching it increases roughly in proportion to the scaling factor raised to the power of 1.21 [00:13:07], [00:13:16].
What Defines a Fractal?
Essentially, fractals are shapes whose dimension is not an integer, but a fractional amount [00:15:10]. This provides a quantitative way to describe shapes that are “rough” and remain rough even when magnified [00:15:17], [00:15:20].
In pure mathematics, the dimension is often defined as the limit of this value at infinitely close zoom levels [00:17:00]. This means a geometric fractal must continue to look rough no matter how much you zoom in [00:17:21].
In applied settings, such as analyzing the coastline of Britain, it’s not feasible to zoom infinitely [00:17:33]. Instead, a shape is typically considered a fractal if its measured dimension remains approximately constant across a wide range of scales [00:17:52]. For example, the coastline of Britain retains its ~1.21 dimension even when zoomed in by a factor of a thousand [00:18:00], [00:18:05]. This persistent roughness across scales is the true sense in which many natural shapes exhibit self-similarity, though not perfect self-similarity [00:18:11].
Applications in Nature
Fractal dimension offers a quantitative way to describe roughness, making it a powerful tool for analyzing natural phenomena [00:18:56], [00:18:59].
Examples:
- The coastline of Norway, with a fractal dimension of approximately 1.52, is numerically communicated as being significantly more jagged than Britain’s coastline (1.21D) [00:19:03].
- A calm ocean surface might have a fractal dimension just above 2, while a stormy ocean might approach 2.3 [00:19:12], [00:19:17].
Fractal dimension frequently appears in nature and seems to be a key differentiator between naturally occurring objects and man-made ones [00:19:21], [00:19:24].