Fractals are fascinating mathematical objects that exhibit many curious properties. Here are some of them:
Self-Similarity: Fractals exhibit self-similarity, which means that they look the same at different scales. If you zoom in on a fractal, you will see smaller versions of the same pattern repeated over and over again.

Infinite Complexity: Fractals are infinitely complex, meaning that they have an infinite amount of detail that can be revealed as you zoom in closer and closer.

Non-integer Dimension: Fractals have a non-integer dimension, which means that they occupy a space between two whole numbers. For example, the Koch curve has a dimension of approximately 1.26, which is between the dimensions of a one-dimensional line and a two-dimensional shape.

Fractal Geometry in Nature: Fractals can be found in many natural phenomena, such as the branching patterns of trees, the shapes of clouds, and the coastline of a continent.
Fractal Music: Fractals have been used to generate music by mapping the patterns of a fractal onto musical notes and rhythms.

Fractal Compression: Fractals can be used for data compression by encoding a large image as a smaller set of instructions that can be used to recreate the image at different scales.

Fractal Universe: Some physicists and cosmologists speculate that the universe itself may be a fractal, with structures at different scales exhibiting self-similarity and complexity.

Fractal Art: Fractals have inspired a new form of digital art, where artists use fractal software to create intricate and beautiful patterns.

Chaos Theory: Fractals are intimately connected to chaos theory, which studies the behavior of complex and unpredictable systems.

Fractal Applications: Fractals have practical applications in many fields, such as computer graphics, engineering, finance, and medicine. They can be used to model complex systems, generate realistic terrain maps, simulate fluid dynamics, and more.