Benoit Mandelbrot is one of the Fathers of fractal geometry and the geometry of nature. Geometry is a systematic means to meter the Geo aka measure the world especially using scalable relationships and integrals.
Here is an example using similar triangles.
I especially like the last comment that the relationship between triangles changes as the sun roars across the heaven because shadow accuracy is dependent on the position of the sun. The point being the triangle of the tree and it’s shadow contains a fixed ratio of information which can be unpacked to provide specific answers to specific problems. That’s what a fractal is. It a way to formally describe similarity, or disprove similarity. If you used a 2 yard stick at the 5ft point the triangles would be completely dissimilar but move the 2 yard stick to the 10 ft point on the shadow and similarity has returned.
Here is a fractal view of volatility. It looks like smoke from a candle. It’s subdivided into 3 parts
The first part
describes flow in a laminar way. The profile is Gaussian:
Laminar flow is controlled by a boundary that exists around the column, the outside vectors experience more friction than the inside and so a Gaussian profile develops.
Here you see the boundary envelope begin to decay. The column becomes wavy and irregular and finally circular around an axis perpendicular to the column. The column now instead of being described by Gaussian math is described by angular math.
The flow is now at the very base around 2 axis and starts to become very complex, disorganized and translated.
Financially for me this picture represents volatility regimens in the market. The first is low volatility. Here things have low volatility and excellent predictability. A volatility measure between 5 and 15 might represent assets in low highly predictable vol. Since things are predictable bets tend to pay off. You might compare this vol to the weather between dead calm and mild summer storm. Shelters are all made to provide safety in this regimen. The second is higher vol between 15 and 30. Here vol is much less predictable and much more energetic. The circular flow is like the eye of a hurricane and the damage it can cause can test the stability of a roof. Recall the equation for kinetic energy is ke=1/2mv^2 so the energy is second order exponential. The third regimen is like a tornado. It simply destroys what it touches. If your butt is hanging in the breeze soon enough you have no butt.
You make money in region 1 You can easily loose in 2 and you are playing with fire in 3
This is a fractal similar sets describing quite well variant phenomena The kinetic energy concept (which is the integral of velocity and the second integral of acceleration) pretty well describes the controlling math. It goes from point to line to plane to volume in terms of complexity.