| Symmetry: part of our appreciation of beautiful objects lies in their symmetry: they are approximately unchanged under translation, reflection, or rotation. We present another type of symmetry, under magnification. | |
| Self-similarity: a key to understanding fractals is the characterization that they are shapes made of smaller copies of themselves. Here we note some of the simple consequences of this property. | |
| Fractals: are shapes made of smaller copies of themselves. Here we sample some of the variety allowed by this concept. | |
| Scale invariance: self-similarity has an interesting visual conswquence. Fractals do not have a preserred size. Distinguishing a small fractal viewed nearby from a large fractal seen far away is not so easy. | |
| Mountains: small children at play use crumpled rugs for mountains. Children often have a good eye for natural patterns, so we should not dismiss their ideas without consideration. They have produced a crude representation of the fractal form of mountains. | |
| Synthesis of natural images: because fractals are an efficient language for encoding nature, both appearance and process, a good understanding of fractal geometry is a powerful tool in generating convincing forgeries of natural landscapes. | |
| The Mandelbrot set: one of the most complicated shapes in mathematics is generated by a one-line formula involving only multiplications and additions. | |
| Fractals and the stock market: folklore asserts that a five year plot of weekly prices looks like a one year plot of daily prices. This simple observation was the motivation for a fractal theory of price variation. | |