Clifford A. Pickover - The Zen of Magic Squares, Circles, and Stars: An Exhibition of Surprising Structures Across Dimensions

Art and science will eventually be seen to be as closely connected as arms to the body. Both are vital elements of order and its discovery. The word art derives from the Indo-European base ar, meaning to join or fit together. In this sense, science, in the attempt to learn how and why things fit, becomes art. And when art is seen as the ability to do, make, apply or portray in a way that withstands the test of time, its connection with science becomes more clear.

Sven Carlson, Science News

If we wish to understand the nature of the Universe we have an inner hidden advantage: we are ourselves little portions of the universe and so carry the answer within us.

Jacques Boivin, The Single Heart Field Theory

Sometimes you are a butterfly collector, a person with a net. You dont always know what the meadows will yield, but you know when the breeze is right, where the meadows are fertile, and what type of net to use. Often the specific catch is a surprise, and this is the enjoyment of the quest. There are no guarantees. There are often unexpected pleasures.

Follow me as we search through unknown forests. Each structure we uncover contains magnificent patterns as beautiful as any swallowtails wing. Hopefully you will enjoy looking at the catches or magnifying them further to learn more about their internal structures. Sometimes you may feel as if you are walking through an exhibit of butterflies in a gallery. At other times you may feel as if you are standing on the edge of a gigantic crystal, surrounded by thousands of butterfly wings beating in unison, producing a hum that reminds you of the chanting of monks.

Magic squares have fascinated humans since the dawn of civilization. Even in ancient Babylonian times, people considered these squares to have magical powers, and in the eighth century A.D., some squares were considered useful for turning ordinary metal into gold. The patterns have also been used as religious symbols, protective charms, and tools for divination. When the squares lost their mystical meanings, laypeople continued to use them as fascinating puzzles, while seasoned mathematicians studied them as problems in number theory. Albrecht Drer, the fourteenth-century painter and printmaker, used them in his artworks, and today magic squares continue to intrigue us with their elegant, beautiful, and strange symmetries.

A magic square is a square array of integers in which the rows, columns, and diagonals have the same sums. However, in this book we will go far beyond ordinary magic squares and consider many unusual variations, some in higher dimensions, all with mind-boggling patterns. Many of the squares possess an intrinsic beauty and complexity hard to describe or understand without careful study. For example, some squares remind me of fractals, geometric patterns with similar structures contained within larger structures. The numerical diagrams tantalize us with hidden patterns as if some mathematician-God played a role in their design. In this book, you will find squares within squares; cubes within cubes; four-dimensional objects; patterns based on ancient, sacred geometry; and structures that defy classification. Mathematician Benoit Mandlebrot likens the creation of nested structures to Zen philosophy: