Legacy of the Luoshu: The 4,000 Year Search for the Meaning of the Magic Square of Order Three

In Legacy of the Luoshu, The 4,000 Year Search for the Meaning of the Magic Square of Order Three, Frank J. Swetz gives an elaborate description of the aspects, characteristics, meanings, traditions, uses and constructions of the magic square of order three, the so-called luoshu. I was astonished by the numerous ways in which the luoshu has played a role in Chinese culture and traditions and enjoyed reading about them in this well written book.

Starting with the legend of the discovery of the luoshu, Swetz describes the way the luoshu served as a cosmological explanation, a model for the people of ancient China to understand the world around them. Not only were the nine squares within the magic square related to nine physical places, regions or planets, but also to gods, wind directions, and the five ‘agents’: water, fire, metal, wood, and earth. Even yin and yang, wuxing, and fengshui turn out to have been combined with the luoshu in ancient Chinese traditions, and rituals.

You might be thinking ‘Where is the math?’ Don’t worry. Besides the cultural meanings and uses of the luoshu in China and, less elaborately, in other cultures, Swetz describes different ways of constructing the luoshu. Several algorithms, originating from different times, places, and traditions, are mentioned. Even some algorithms for magic squares of higher order are included. One paragraph focuses solely on the mathematics behind magic squares, capturing it in easy to understand formulas and posing some problems, e.g., to find a magic square consisting of prime numbers.

Altogether, I found the book very interesting and a joy to read, not least because of the many diagrams and illustrations. What makes this book especially worthwhile, however, is the attention paid to the large variation in cultural meanings, both within China as well as in other countries. The only downside is the limited discussion of intercultural similarities and differences. Since the mathematical part is short and not complicated (except for some of the problems), the book can be read by anyone. Summarizing, I find in this book a beautiful uncovering of this part of the history of mathematics.

See also the review of the first edition.

Aldine van der Ham-Aaten is a math graduate of Leiden University, the Netherlands, and a former high-school math teacher. Her fields of interest are history of mathematics and mathematics education.