Fractal Dimension Visualized

3 hours ago

There are plenty of times where fractals are mentioned across TA, from indicators like FRAMA, Williams Fractals, concepts like Elliott waves - all the way to my own way of breaking cycles through Fibonacci Channels. Pretty much most of them are about self-similar behavior of the market which is often invisible to unweponized eye.
In this piece I’m going to the core - fractal dimension. Don’t fixate on numbers! Instead, visualize the scaling process: how structures fills space as you zoom in and out.

Regular Dimensions
The most fundamental question is: How many copies (N) do we get with each magnification (R).

1. Line (1D): A line has only one length. If I magnify length by a factor R, the number of smaller copies that fit is N=R. (Double the length → 2 copies; in general N = R^1.)
2. Square (2D): Magnify side length by R: the big square splits into a grid of R x R old squares, so N=R^2.
3. Cube (3D): Magnify edge length by R: the big cube contains R x R x R small cubes, so N=R^3.

So in D dimensions, when you scale length by R, the count of self-similar copies is N = R^D

Hence, we can extract dimension: D = log N / log R
This is the similarity dimension formula when the object breaks into N exact copies, each scaled by 1/R in length.

Application to Fractals

• Sierpinski Triangle
  We actually start with a solid 2D equilateral triangle. Then we remove the central upside-down triangle to leave three smaller solid triangles. Now we repeat that step inside every remaining triangle, forever. As this process continues, any patch of solid area that survives will eventually be removed, so the total area shrinks toward zero while the number of pieces explodes. The limit is the Sierpiński triangle: not “just lines,” but a fractal set with no area and a non-integer dimension between 1 and 2
  
  At each step you get N=3 copies, each scaled by 1/2 (so R=2).
  D = log 3 / log 2 = 1.5850
  
  
  How to work out D in practice
  1. Identify the scaling length: by what factor R must you magnify so the large figure looks like a collage of smaller identical copies?
     
  2. Count those copies N.
     
  3. Plug into D = log N / log R.
     
  
  
  This is highly important for perceiving scaling laws not just for self-similar shapes but also other patterns.

2 hours ago

Note

• Let's also consider the Koch Curve
  The classic fractal that forms one side of the snowflake. Start with a line segment, divide it into three equal parts; remove the middle third and replace it with the two equal sides of an outward equilateral triangle. After one iteration you have 4 segments, each 1/3 the original length; repeat this process indefinitely.
  By self-similarity, magnifying the curve by a factor R=3 reveals N=4 copies of the whole, so the similarity - fractal dimension is D = log 4 / log 3 = 1.262.
  This lies between 1 and 2: the curve is “more than a line but less than a surface” - infinitely crinkly with unbounded length, while the closed snowflake made from three such sides encloses finite area.