Those who wish to communicate with non-human species generally argue that the only possible common language is mathematics and the physical laws of the universe. I happen to agree with this. I think one of the strongest arguments in favor of this position is found by examining the actions of many different human cultures. As Professor Ascher clearly demonstrates, humans seem to have a natural propensity for mathematical thought, even when the society is pre-industrial.

Nature provides us with many natural clocks by which the passage of time can be measured in several ways. The movements of the sun, moon and stars can all be used to note the number of days and all have been used to do so. Chapters 2 and 3 deal with this topic. I was very impressed with the obscurity of some of the cultures that were described. Ascher is not bound by either distance or time in her quest for examples of computations concerning time. Some of the calendars she describes are those practiced on islands with small populations in and near Indonesia. Being different from the well-known Jewish and Gregorian calendars, they are very instructive examples of culturally unique ways calendar computations can be performed.

Modern society is long settled into the cycle of the seven-day week, but that is of course not the only possible shorter cycle of the days. With no heavenly body or other natural rhythm to follow, there is no logical reason for the seven days. Cycles of 5, 6 or 8 days are just as sensible and some cultures do use numbers other than seven. In all cases, the computation of the position in the cycle is done using modular arithmetic, which can be a relatively advanced operation. Many different cultural approaches are described, along with the computations. If you are searching for some basic problems to use in a number theory class, these computations demonstrate many different problems, all arising from different cultural contexts.

Nearly all people who travel long distances rely on maps to store the information needed for their journeys. This is especially true for those who travel over water where the islands are widely scattered. Some Polynesians constructed their charts from sticks, using the patterns of ocean waves to dictate the organization of the charts and then to navigate across the open ocean. The chapter discussing this is very interesting. It is difficult for most of us to comprehend the vastness of the central Pacific, where the nearest land may be a thousand miles away. Furthermore, it may rise at most twenty or thirty feet above the sea, which means that you have only a few miles of allowable error if you are to see it.

Other topics in the book are the descriptions of interpersonal relationships in groups as distinct as the Basque in Europe and the Tongans of Polynesia, figures drawn on the thresholds of homes in southern India and divination techniques from many regions. Here divination means using some randomization method to make a decision. The divination techniques are essentially a form of logic and modular arithmetic. All throughout the book, I was struck by how many uses human cultures have found for modular arithmetic. It appears again in the sections on divination and relationships. If there is a universal mathematical idea beyond the simple arithmetic of counting, it appears to be modular arithmetic.

The designs on the thresholds in the Tamil areas of India can be described using formal language rules. I was amazed to read the description of how the complex whorls that make up the figures can be reduced to the repeated application of a simple set of actions. In some cases, only three types of moves are necessary.

From this and other evidence, it appears that mathematics may be an essential survival skill for the human species rather than an extraneous one. The descriptions in this book describe so many different applications, that it becomes hard to deny that something more fundamental is responsible for the many ways we find to perform mathematical operations.

Charlie Ashbacher (cashbacher@yahoo.com) is the principal of Charles Ashbacher Technologies, a company that offers state of the art computer training. He is also an adjunct instructor at Mount Mercy College in Cedar Rapids, Iowa, and at the end of this academic year, he will be three courses short of having taught every class in the math and computer science majors. A co-editor of the *Journal of Recreational Mathematics*, he is the author of four books in mathematics and one in computer programming.