Chapter 3: Opening the Box

“Some symbols always mean the same thing.”

--Alan Moore, Promethea

In architecture, city planning, furniture, pavements, game boards, wall tiles, plaid designs, graphs, grids, and weaves--the geometric square is everywhere. One speculates that knowledge of the square form goes all the way back to some ancient ancestor who first wove rushes or scored lines in the sand. Certainly by 20,0 BCE or so a Solutrean cave artist had inscribed square figures on the wall of the Cosquer Cave in southern France (see fig. 6.4, p. 64).

The square as a mathematical function is everywhere too, as the multiplication of a number by itself and, in square root form, as the unique factor that, multiplied by itself, produces a given number. Squaring is a feature of formulas for areas of circles and spheres; for geometric theorems; for coordinate systems; in the inverse square law; in laws concerning vacuums, gases, gravitation, radiation, acceleration, and more. Laws of mathematical logic and laws of nature are full of squares. Einstein’s famous E = mc2 is only the most famous equation that involves squaring. In that connection it was once commented to me that whereas some people are most interested in E (energy), others in m (solid matter), others in c (speed and light), I seem to have got stuck with the bit that most people hardly notice--the little number “2 ”. Squaring, however, is the “power” of a number, as defined by mathematicians. The ancient world, too, had it right: “qualitatively” the square denotes the manifest result that comes from the creative dynamism of the triad. Squaring adds magnitude to simple length. Whenever one sees the symbol √ or a superscript number (e.g., √31 or 1237) the squaring function is implied. A line of unit length becomes a plane of unit area when squared.

Figure 3.2 is a unit square, that is, the side length is 1. It has some immediate implications. Simply in the drawing of a square, another length, that of the diagonals, is suggested by the relationship of the opposite corners with each other. The diagonal of the unit square is a line that reveals its nature when it is itself squared; it becomes the side of a square twice the area of the original unit square. This new square has an area of 2, as is shown by counting up the component triangles. The line of its side is thus “the root of a square of value 2”--therefore “the square root of 2” (√2).

There is an immediate numerical mystery. The actual length of this diagonal produces the decimal 1.4142135 . . . , a number that goes on interminably and never falls into a repeating pattern. This means that the decimal part cannot be expressed as an exact fraction because it never settles down. A fraction is a ratio between the number above the line and that below. It is said to have been Pythagoras himself who first realized that √2 was neither a whole number nor a fractional ratio and was therefore “irrational,” meaning literally “without ratio.” It is reported also that Pythagoras only taught his private group (the mathematikoi) about irrationals and the related qualities of the pentagon--which, as we shall see, fairly seethes with another irrational: √5. Apparently one of the private group talked indiscreetly to some of the public group (the akousmatikoi or “hearers”). Versions of the story have Hippasus, the indiscreet student, being put to sea in an open boat, banished, or even murdered--which one doubts--by Pythagoras himself.

Though the Pythagoreans may not have known it, that diagonal and its awkward length was known more than a thousand years before in Mesopotamia, from where we even have a clay tablet that gives a solution of √2 correct to five decimal places!

Returning to basics, what actually is a square? Can it be dismantled beyond its obvious familiar and rather bland face? It certainly can, and this book is the outcome of just that.

Despite my grumbles about the way geometry was taught in school, we had a mathematics teacher, Mr. Ronald Hunt, who taught us something very important. At the top of his voice he would shout the word “WOMEK!” The word was an acronym from the initial letters of “write or mark everything known.” It is a key to finding fresh information. When we do this we are much more likely to pick out potential developments latent in the geometry. When, for instance, in order to get a perpendicular bisector, we draw a full vesica, rather than mean little arcs, we are “womeking.”

We can apply this to our main topic, which is simply an ordinary square--a flat box with four equal sides. Drawing such a square we may give each side some chosen length--1 being the simplest. Then having marked the right angle and put in the diagonals it might seem that we have finished. But in fact we know far more than this because a square is itself a construction. It has an internal rationale that unfolds in simple steps from the vesica that is the root/womb of all geometry. In its unfolding, it carries information with it. The sand reckoner’s square is generated by seven similar circles that establish nine points. One only notices this if one “womeks” the process by fully drawing everything in as it becomes relevant. It can be drawn very simply, by rotating a forked stick in the sand and stretching and snapping down a cord for the straight lines; such methods have been with us from very ancient times.

It is now as though we have lifted the surface of our square, like a hinged lid, to see its ancestral inner workings. In actually making it one has learned far more than is immediately obvious from just looking at it as an equal-sided box shape. The lattice triangulates the square in a way that is the key to all that follows. The triangle is the basic straight-line plane. Two points establish a line. Three points--unless all on the same line--establish a triangular plane. A square, of course, can be cut into triangles in an infinite number of ways; but here using the guide points from the square’s own generation, we get a natural, economic, and uniquely elegant triangulation that is truly loaded with information. There are two forms of the lattice (fig. 3.6 and 3.7); sometimes it is useful also to retain the central circle that was part of the construction (fig. 3.7).

It works as a mnemonic device for a vast swath of lore from the metric and sacred geometry of India, Mesopotamia, Egypt, China, Greece, Judea, Islamic culture, and Renaissance Europe. What is more, if one forgets its content, one has only to observe its measures, numbers, and proportions and everything reveals itself again.

I first came across it by chance as a possible grid underlying an Indian temple ground plan, of which I will tell you more in the next chapter.