A fractal is a mathematical shape that is infinitely complex and repeats forever. Fractals are self-similar across different scales, meaning that every part of the fractal looks similar to the whole image, regardless of how zoomed in or out you are. This property is called self-similarity.
Let's start by looking at common Fractals. One of these is the Mandelbrot set, named after mathematician Benoit Mandelbrot who introduced the term Fractal.
The word "Fractal" originates from the Latin word frāctus, meaning "broken" or "fractured," fitting due to Fractals' fractional elements.
To be a Fractal, a shape doesn't need to be identical. It just needs to have recurring similarities. Below is the Mandelbrot Set Fractal, a well-known mathematical fractal:
Fractals demonstrate self-similarity as they are constructed from simple, consistent rules that remain unchanged regardless of the system's size. For instance, a tree-like fractal might follow this rule: grow a branch for a certain distance, then create a new branch from it. This rule is reapplied to each new branch, giving birth to the fractal. Different fractals employ different rules to form their structures.
Fractals display self-similarity because they are built from simple rules which do not change as the system grows larger.
Fractals are prevalent in nature because the universe operates under uniform rules across all scales over time. Galaxies, with their star clusters, solar systems, planetary systems, and ring systems, are built from fractals. Similarly, the human body, with its branching appendages and organs; and the Earth's watersheds, with their springs, streams, and rivers, are also constructed from fractals. All these systems are fractal in nature because their smaller components follow the same rules as their larger counterparts.
The universe is governed by a few simple, scale-invariant rules. These are the rules of thermodynamics, which dictate how changes in energy and entropy influence the behavior of systems. Without delving too deeply into the physics (explained in the linked article), it's enough to say that due to these universal rules, there are ”four primary fractal patterns” from which most natural systems are built.
There are four classes of fractals which are the most common in nature. They are found everywhere, from the planet’s surface itself to the microscopic structures within your bones. Next, we will learn about each of their structures, the rules they follow, and the functions they fulfill.
Fractal One: The Hierarchy
The hierarchy is designed to efficiently transport energy and materials in two directions — up and down.
The hierarchical fractal, such as a tree or leaf, is designed for efficient two-way transport of energy and materials. It achieves this by maximizing its surface area on an energy-collecting boundary and minimizing the distance from the boundary to the interior. This design enables reliable transportation of value (like solar energy) from the system's edges to its top or center. A tree is a notable example of a double-hierarchy, collecting energy from two boundaries: the roots and the leaves. Other natural systems based on the hierarchical fractal include watersheds, respiratory systems and corporations.
Fractal Two: The Web
A web fractal is a system configuration whereby any element can be connected to any other element. Elements in webs can therefore achieve greater equality in their access to energy.
The democratic, distributed nature of the web is characteristic of systems such as the internet. It is no accident that the internet was once commonly referred to as “the worldwide web.”
Web fractals create new nodes where there's an energy source. All nodes have the potential to generate new ones and establish connections with any other node. Energy moves across the web, diffusing towards areas of lower concentration.
The human brain is also a prominent example of a system primarily structured as a web fractal.
Growth of the web is unconstrained and there are many redundant pathways within the web. This makes the web more resilient to disturbance than the hierarchy.
Fractal Three: The Crystal
A simple crystal is a set of elements arranged at equal distance and angle from one another.
The crystal, among the four fractals studied, has the least energy and entropy. This results in highly stable and predictable structures. They are used in materials for optical, informational, and structural applications. Crystals are fundamental to the modern world, evident in fiberoptic cables, glass, and metal - examples of crystals found where people live.
Unlike our first two fractals, the hierarchy and the web, which process energy, crystals are generally formed from elements releasing energy to the surroundings.
Fractal Four: The Spiral
Spirals are lines that follow a curved path around a center point. They are found when examining the growth and movement patterns of systems.
The spiral is our final fractal. Spirals are curved lines revolving around a central point, seen in growth and movement systems. They have two ends, one spiraling out from the center, another spiraling inward.
Like hierarchies, spirals efficiently transport energy inward or outward. This is evident in whirlpools. Spirals are also low-entropy configurations, allowing growth only at two points.
A snail's shell is a common spiral in nature. The spiral shell's effectiveness lies in its ability to grow with the resident snail. Spirals can grow consistently without compromising the structure, unlike hierarchies and webs that become more vulnerable as they expand.
Spirals teach us to grow systems sustainably, leveraging what already exists. Each spiral turn represents proportional growth from its previous size, allowing further expansion with each turn.
The human nervous system is a prime example of the combination of three of our fractal patterns: the hierarchy, the web and the crystal.
The crystal element is the unchanging, consistent, and reliable character of the nervous system. It originates in our DNA, the blueprint from which the entire system is constructed.
The hierarchy of our nervous system is consists of the brain and brain stem as command centers with our peripheral nervous system branching out from our spinal column, splitting repeatedly until reaching the nerve endings on our skin. This confers the great benefit of central processing — the many sensory inputs from our body’s nerves are collected at the top of the hierarchy, which then can make an informed decision about how to react. On the other hand, damage to the top of the hierarchy in the form of mental illness, head injury, or spinal column severance creates dire consequences for all systems downstream.
To offset this weakness, the top of the hierarchy, the brain, is itself a web fractal. The greatest strength of the web, its democracy, offsets the greatest weakness of the hierarchy, its bottleneck of control.
The structure of Greek Key-inspired Fractal Architected Materials is based on the fourth fractal: The Spiral. The square Greek Key pattern is similar to a kind of square spiral, comparable to the circular shell Spiral.
Yes, spirals with different numbers of turns can be considered fractals. Fractals are geometric shapes or patterns that exhibit self-similarity at different scales, and spirals with varying numbers of turns can demonstrate this self-similarity.
When you observe spirals with different numbers of turns, you may notice that the overall shape remains similar or exhibits a recurring pattern, even though the scale or size of the spiral changes. This property aligns with the concept of self-similarity in fractals, where parts of the structure resemble the whole on different scales.
It's important to note that while spirals with different numbers of turns share some characteristics with fractals, they may not possess the infinite complexity and perfect self-similarity found in some classical fractal structures like the Mandelbrot set. Nonetheless, the self-repeating nature of spirals makes them interesting and relevant to the broader idea of fractals, particularly when considering patterns in nature or art that exhibit similar qualities.
Fractal and hierarchical structures relate to how elements are organized and arranged, yet they have unique features. The Hierarchy is a kind of Fractal.
Fractal:
- Self-Similarity: Fractals display self-similarity across various scales, meaning the overall structure of a fractal resembles its parts when viewed at different magnification levels.
- Infinite Complexity: Fractals can possess infinite complexity, indicating that similar patterns keep repeating at smaller or larger scales as you zoom in or out.
- Iterative Construction: Fractals are typically generated through an iterative or recursive process where a simple geometric shape is repeatedly modified using a set of rules or equations to form the overall fractal structure.
- Natural and Mathematical Examples: Fractals are found in natural phenomena (like coastlines, clouds, or fern leaves) and mathematical constructs (such as the Mandelbrot set).
Hierarchical:
- Nested Levels: Hierarchical structures have a system with multiple levels or layers, with each level subordinate to the one above it and possibly having sub-levels beneath.
- Organization by Rank or Order: Hierarchy suggests an order or ranking of elements, with higher levels generally having more control or influence, and lower levels handling more specific and detailed functions.
- Clear Top-Down Structure: Hierarchies usually have a clear top-down structure, with each level managing or organizing the levels below it.
- Common in Organizational Systems: Hierarchical structures are frequently used in organizational systems, where there is a clear chain of command, with higher management supervising subordinates.
Differences:
- Self-Similarity vs. Nested Levels: The primary difference is in the way repetition occurs. Fractals display self-similarity at all levels, while hierarchies have nested levels, each with unique functions.
- Infinite Complexity vs. Structured Order: Fractals tend to have infinite complexity, giving a sense of randomness or unpredictability. Hierarchies, however, show structured order with a clear organizational hierarchy.
- Natural Patterns vs. Organizational Systems: Fractals are often linked with natural patterns and phenomena, while hierarchical structures are typically used to organize systems and relationships, especially in human-made constructs.
In conclusion, fractal and hierarchical structures both involve element organization, but fractals focus on self-similarity and infinite complexity, while hierarchies concentrate on nested levels and structured order within an organizational setting.
References
https://iternal.us/what-is-a-fractal/#:~:text=A Fractal is a type,many different aspects of life
