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Part VI

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On Size and Growth

Mathematics is the study of patterns and relationships. It can explain why there are no Godzillas in reality, point out unexpected similarities in ancient pottery, and suggest new and beautiful artistic designs. Mathematicians search for and classify numerical, geometric, and even abstract patterns. In these chapters, we follow some of those searches. We concentrate on geometric patterns but find that those lead to numerical considerations, too.

In Chapter 18, Growth and Form, we look at how the sizes of objects influence their forms. Godzilla, very tall trees, mile-high buildings, and stratospheric mountains: Are they possible? If not, why not? Seeing the underlying principles of scaling will help you to appreciate why objects in the world have the shapes and sizes that they do.

We start with a simple numerical pattern in Chapter 19, Symmetry and Patterns, which leads to questions such as: What proportions make a pattern esthetically pleasing? How important is bilateral symmetry? We expand our notion of symmetry but discover surprising limitations that even broader notions of symmetry face. We examine the ingredients of the beauty of fractal patterns, ones that resemble themselves at finer and finer scales, in nature and in traditional art from Africa and elsewhere.

Chapter 20, Tilings, asks how to arrange objects symmetrically on a surface. What shapes can we use? What patterns can arise if all the objects are the same? What if the objects themselves are symmetrical, or if we allow irregular shapes but demand that they all face the same way? Can you arrange shapes in a pattern that does not repeat but is nevertheless systematic?