What are Fractals? – Fractal Foundation (2024)

FAQs

What are fractals in simple terms? ›

A fractal is a never-ending pattern. Fractals are infinitely complex patterns that are self-similar across different scales. They are created by repeating a simple process over and over in an ongoing feedback loop. Driven by recursion, fractals are images of dynamic systems – the pictures of Chaos.

A fractal is a recursively created never-ending pattern that is usually self-similar in nature. Separate from Euclidean geometry, fractal geometry addresses the more non-uniform shapes found in nature, such as mountains, clouds and trees. Fractals provide a systematic method to capture the “roughness” of some objects.

These patterns are called fractals. A fractal is a kind of pattern that we observe often in nature and in art. As Ben Weiss explains, “whenever you observe a series of patterns repeating over and over again, at many different scales, and where any small part resembles the whole, that's a fractal.”

Fractal dimensions

Fractals also help us make sense of weather, climate and other “chaotic” systems – those that develop along wildly different paths through a map of all their possible states in response to the tiniest change in initial conditions.

Fractal shapes exist throughout the human body, in lungs, blood vessels, and neurons. Fractals can also be used to aid diagnosis of abnormal heart rhythms and tumours. In cosmology, fractal distributions of galaxies have been detected over relatively small scales.

Three examples of fractals are the Koch snowflake, the box fractal, and the Sierpinski triangle. Some famous fractal images include the Julia set, the Mandelbrot set, and Cantor dusts.

These recent theories of quantum gravity describe a fractal structure for spacetime itself, and suggest that the dimensionality of space evolves with time. Specifically, they suggest that reality is 2D at the Planck scale, and that spacetime gradually becomes 4D at larger scales.

The heart is filled with fractal networks: the coronary arteries and veins, the fibers binding the valves to the heart wall, the cardiac muscles themselves, and the His-Purkinje system.

Nature's snowflakes have fractal-like self similarity.

The Koch snowflake is among the earliest fractal geometry work. Not surprisingly, nature's snowflakes seem to share that self similarity the Swedish mathematician Helge von Koch described.

The Most Famous Fractal by John Briggs. Largely because of its haunting beauty, the Mandelbrot set has become the most famous object in modern mathematics.

Why do humans like fractals? ›

Your brain responds to fractals positively — as in, less stress and mental fatigue. Taylor and his team found out that these negative impacts could be reduced up to 60 percent just by gazing out at nature.

The Fibonacci Spiral, which is my key aesthetic focus of this project, is a simple logarithmic spiral based upon Fibonacci numbers, and the golden ratio, Φ. Because this spiral is logarithmic, the curve appears the same at every scale, and can thus be considered fractal.

A fractal is a never-ending pattern. The patterns used in fractals can be different sizes and directions, but the pattern is used over and over to create an ongoing pattern.

The Koch Curve is one of the simplest fractal shapes, and so its dimension is easy to work out. Its similarity dimension and Hausdorff dimension are both the same. This is not true for more complex fractals.

Fractal dimension is a measure of how "complicated" a self-similar figure is. In a rough sense, it measures "how many points" lie in a given set. A plane is "larger" than a line, while S sits somewhere in between these two sets.

Our lungs, our circulatory system, our brains are like trees. They are fractal structures. Fractal geometry allows bounded curves of infinite length, and closed surfaces with infinite area. It even allows curves with positive volume, and arbitrarily large groups of shapes with exactly the same boundary.