When I design visually expressive art work for businesses, I use transformations and symmetry to create interesting geometric patterns. Simple shapes such as triangles, rectangles, and circles can be arranged in spectacular ways just by reflecting shapes over an invisible axis or rotating them around a point.
Scale and Proportion
I often create scale models of cities, parks, and bridges. To make sure my models are accurate, I have to find the scale factor. For example, if a building is 120 ft tall in real life but my model is only 2 ft tall, the scale factor is 120:2 or 60:1. If a tree near the building is 30 ft tall, I would have to make my tree model 0.5 ft tall to keep everything in proportion.
I create a lot of art work using mathematical equations. The designs that are made are very beautiful and can be easily reproduced with the right parameters. Lately, I've been experimenting with fractals. Fractals begin with something as simple as a triangle or a square and become a picture of infinitely intricate patterns. If you could examine the fractal under a microscope, you would still see the same level of complexity. One single fractal image may require millions of repetitive mathematical computations but the end result is well worth it.
Tessellations are created by tiling a plane figure without any overlap or spaces. The simplest tessellations are made with squares, equilateral triangles, or hexagons. Things become more interesting when more than one type of shape is used. An example would be combining triangles with hexagons or triangles with squares. A famous artist named MC Escher created tesselations with many unusual shapes, including birds and fish.