Bee on Fractal Flower

Fractals represent a step from Plato’s ideal world towards the natural world. Even so fractals still live in Plato’s ideal world, but they do start to let us talk about the real world.

All the images below are in the realm of art; a mixture of nature and abstraction. They are delightful and fancy. They are easily recognized. They look real and surreal. AI handles these well.

Limits of Plato’s ideal World

Fun fact. In the fractal world the length of any coast line is infinite as the measurement stick gets smaller and smaller. And since in Plato’s ideal world all infinities are equal it cannot differentiate between infinities so the shorelines of all islands are equal – a meaningless and an absurd statement.

This is called the Coastline Paradox. A Paradox is something that cannot be true, yet it is true.

Limits of the Real World

Concepts like length do not exist. Things are just unique. They exist. Even the lowly yard sticks that we make are unique. None are truly precise. None are truly straight. In a sense they too are alive. They respond to their environment.

Me: Is there a paradox in fractals when we measure the length of a coastline and get infinity regardless of the size of the island?

AI: Yes, there is a paradox in fractals when we measure the length of a coastline and get infinity regardless of the size of the island. This is known as the coastline paradox¹. It arises from the difficulty of measuring shapes with complicated edges such as those of countries like Britain. As we try and be ever more accurate in our measurement of the British coastline, we get an ever larger answer!² The coastline paradox results from the fractal curve-like properties of coastlines; i.e., the fact that a coastline typically has a fractal dimension¹.
Source: Conversation with Bing, 5/27/2023
(1) Coastline paradox – Wikipedia.
(2) The Coastline Paradox and Fractional Dimensions.
(3) Fractal dimension – Wikipedia.

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 Chaos¹. Fractals are distinct from the simple figures of classical, or Euclidean, geometry—the square, the circle, the sphere, and so forth. They are capable of describing many irregularly shaped objects or spatially nonuniform phenomena in nature such as coastlines and mountain ranges².

Fractals can be found in various forms in nature, such as flowers, ferns, leaves, river channels, lightning, snowflakes, and broccoli. Trees and plants also exhibit fractal patterns at every level of the forest ecosystem. Fractal shapes are self-similar and often look alike when zooming in and out. Other examples of fractals in nature include river deltas, mountains, clouds, crystals, and bodily systems like the circulatory and respiratory systems of animals¹⁴.

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