Fractal Geometry: Uses, Math & Fascinating Patterns

[Bogrune]

The very first time I ever heard about fractals was in my junior year in high school in my Algebra II class when we were studying complex numbers. I was fascinated by these wonderous objects and I've had many questions about them ever since.
Though two of my main questions have always been: how are they used in our world, and how does Fractal Geometry describe them mathematically?

 

[Anonymous217]

I would give an answer, but I think Wikipedia does it best: http://en.wikipedia.org/wiki/Fractal_geometry

 

[Bogrune]

Well, thanks for the link. I know that many fractals can be expained mathematically by an exponential expression, but can anyone tell me what Fractal Geometry is like? Also, what does it take to comprehend it (Algebra, Trigonometry, Calculus)?

 

[TGlad]

"can anyone tell me what Fractal Geometry is like?"
Generally it is geometry that is rough, like a http://en.wikipedia.org/wiki/Koch_snowflake" .
It doesn't have to be regular, for example a coastline is fractal (over a certain range).

"How are they used in the world?"
Well fractals often are optimal in some regard, for example maximum strength to weight ratio gives fractal-like structures in bird bones and in the Eiffel tower. Maximum area coverage per length gives the fractal tree-like shape of rivers... and similarly for trees.

 

[CellCoree]

Fractal geometry is a relatively new and exciting field of mathematics that has numerous real-world applications. One of the main uses of fractals is in computer graphics, where they are used to create realistic and visually stunning images. Fractal patterns are also found in nature, such as in the branching of trees, the formation of coastlines, and the structure of snowflakes. Understanding fractal geometry allows us to better understand the complexity and beauty of the natural world.

Mathematically, fractals are described as self-similar objects, meaning that they have the same pattern at different scales. This self-similarity is achieved through the use of recursion, where smaller parts of the fractal are created using the same rules as the larger whole. This recursive process can create infinitely complex and detailed structures, making fractals a fascinating subject for mathematicians and scientists alike.

One of the most intriguing aspects of fractal geometry is the idea of fractional dimension. Unlike traditional geometry, where objects have integer dimensions (e.g. a line has one dimension, a square has two dimensions), fractals can have non-integer dimensions. This means that fractals can have a dimension that falls between two whole numbers, giving them a unique and complex nature.

In conclusion, fractal geometry has a wide range of uses and is a fascinating subject in mathematics. It allows us to better understand and describe the patterns and structures found in nature and has practical applications in fields such as computer graphics. The study of fractals continues to expand and reveal new insights into the complexity and beauty of our world.