# The Zen of Magic Squares, Circles, and Stars: An Exhibition of Surprising Structures across Dimensions

Clifford A. Pickover
Pages: 432
https://www.jstor.org/stable/j.ctt7s1bz

1. Front Matter
(pp. i-viii)
(pp. ix-ix)
3. Preface
(pp. xi-xviii)
4. Acknowledgments
(pp. xix-xx)
5. Introduction
(pp. 1-36)

Amagic squareis a square matrix drawn as a checkerboard filled with numbers or letters in particular arrangements. Mathematicians are most interested inarithmeticsquares consisting ofN² boxes, calledcells, filled with integers that are all different. Such an array of numbers is called a magic square if the sums of the numbers in the horizontal rows, vertical columns, and main diagonals are all equal. If the integers in a magic square are the consecutive numbers from 1 toN², the square is said to be of theNth order, and themagic number, or sum of each...

6. CHAPTER ONE Magic Construction
(pp. 37-64)

When you gaze at magic squares with their amazing properties and hidden symmetries, it’s sometimes difficult to believe that there are easy-to-remember ways to construct many of them using simple rules. In fact, handbooks of “mental magic” often give these methods as “secret” ways to impress audiences.¹ The mentalist shows a large, empty square to the enthralled onlookers and, with a flourish and a cry of “voilà!”, creates a magic square with astonishing ease. Imagine yourself being able to write down a large magic square in under a minute in front of an adoring audience. In this chapter, I’ll introduce...

7. CHAPTER TWO Classification
(pp. 65-146)

Magic squares can be classified in many different ways according to special properties they may possess. For example, squares are often placed in four major categories: Nasik, associated, simple, and semi-Nasik. As will become evident, some magic squares may have more than one classification. Several of the weird, modern magic squares in chapter 3 don’t easily fall into any of the broad classes of magic squares that have interested mathematicians for decades, and in some cases for centuries.

Thesimple magic squaremeets the minimum requirement that the sum of the integers in each row, column, and main diagonal is...

8. CHAPTER THREE Gallery 1: Squares, Cubes, and Tesseracts
(pp. 147-295)

This chapter contains an exhibition gallery of my favorite magic squares and related constructs such as magic cubes and tesseracts. The objects span the centuries, with some arrangements so exotic that they seem to defy easy classification. I suspect that many contain hidden patterns yet to be articulated. Let’s start with a beautiful square from the eighteenth century.

Excluding George Washington, Benjamin Franklin (1706–1790) was the most famous eighteenth-century American. He was a scientist, inventor, statesman, printer, philosopher, musician, and economist. It is easy to see how such a curious person could create one of the most fascinating squares...

9. CHAPTER FOUR Gallery 2: Circles and Spheres
(pp. 297-324)

This chapter contains an exhibition gallery of my favorite magic objects involving circles and spheres. Although circles and spheres have been discovered with incredible beauty and complexity, these kinds of objects have been studied much less than magic squares and cubes. Perhaps one reason for the relative paucity of research is that the circles and spheres are more difficult to represent on paper than the squares and cubes consisting of tables of numbers. Perhaps the twenty-first century, with its increasing use of virtual-reality computer tools and 3-D graphics, will be a renaissance in our understanding of magic circles, spheres, and...

10. CHAPTER FIVE Gallery 3: Stars, Hexagons, and Other Beauties
(pp. 325-368)

This chapter contains an exhibition gallery of my favorite magic stars, hexagons, and other related geometrical constructs.

A magic star is a variation of a magic square. The numbers are arranged in a star formation such that the sum of the numbers in each of the straight lines formed by the star’s points and intersections yields a constant sum.

Figure 5.1 is a magic star containing the numbers 1 through 12, with 7 and 11 omitted. The magic constant for each straight line is 24, which is the smallest possible sum for this range of integers. There is no solution...

11. Some Final Thoughts
(pp. 369-374)

Picture in your mind an infinite floor tiled by adjacent marble slabs, one foot on a side. On each tile is a number, and each 12 × 12 tile set forms a pandiagonal square. As you learned in this book, each pandiagonal square forms a continuous tiling of overlapping magic squares. With each step you take along the floor, you feel as if you are in an ever-changing magic square sparkling in and out of existence as you travel. No matter in which direction you travel, you feel whole, complete, surrounded by patterns stretching for as far as your eye...

12. Notes
(pp. 375-394)