Back to Chapter 5b


At some time in the sixth century B.C. when Greek trade with Egypt intensified, the Greeks obtained from Egypt knowledge of their manner of correlating elements of design. The Greeks called this Egyptian "rope-stretcher" system harpendonapate, which is the Greek word for "rope-stretchers."

In Greek hands, the Egyptian system was highly perfected as a practical geometry, and for about three hundred years it provided the basic principle of design for the finest art of the Greek Classical Period. Euclidean geometry gives us the Greek development of the idea in pure mathematics; but the secret of its artistic application that it once had had completely disappeared. Its later recovery has given us dynamic symmetry-a method of establishing the relationship of areas in design/ composition.

The Greeks used this Egyptian method measurement and design for their pottery vases, their sculpture, their friezes and their temple architecture. Strange as it may seem, there is no essential difference except in scale, between the plan of a Greek vase and the plan of a Greek temple, either in general aspect or in detail (See Figure V-8 below.)

Figure V-8. Schematic composition of a Greek vase

as measured and drawn by L. D. Caskey, Boston Museum. Design form and dynamic symmetry analysis.


The above vase, as measured and analyzed by L. D. Caskey, has a ratio of 1.472, derived from the base measurements of: the diagonal JK (.618), the diagonal EF (.618) and the diagonal 3F (.236) which total 1.472. These main areas are further subdivided for repeated design needs.

The Greek sculptor, Roecus of Samos, learned his art in Egypt in the sixth century B.C. The traveller Diodorus Siculus, a contemporary of Roecus , tells us that one of Roecus' sons lived on the island of Samos and the other son lived in Ephesos. They each sculpted one-half of a statue using the Egyptian method of measuring and when the two halves were put together they matched so perfectly that you could not see where they were joined.

Another Greek traveller claimed that he had been everywhere in the world and that his mathematical system was better "even than the Egyptians," and that he'd lived in Egypt for five years.

The Greek philosopher Pythagorus brought the knowledge of geometry to Greece using odd (versus even) numbers. Later Plato supplied a rule beginning with even numbers. Plato's students fixed areas up to the "root-seventeen" rectangle but the entire dynamic system ideal is contained in the "root-five" rectangle, so it is not really necessary for rhythmic needs to use extensions of the root higher than "root-five."

As mentioned above, when reading and researching you will often come across Plato's "Golden Section," "Golden Rectangle," "Golden Mean," and even "Divine Section." These are all related to dynamic symmetry's whirling square rectangle (also termed the 1.618 rectangle) and the ratio 1.618.

It is interesting that, also in the sixth century B.C., the Hindus of India also adopted the Egyptian method of measurement. They used it primarily for construction of their sacrificial altars. The instructions for this method are contained in their Sulvasutras, meaning "rules of the cord." There is no indication that they ever developed the system beyond "root-six" rectangles and there is no indication that they knew anything of the special properties of the "root-five" rectangles.

The Hindus soon fell into inaccuracies in their use of this system and in their art in general.

The use of the Egyptian measurement system, the "rules of the cord," and the "rope-stretchers," descriptive names show that this system was a well-established profession in the ancient world thousands of years before there is historical reference to it in either India or Greece.

The Greeks, as seen from a written passage of Democritus of Abdera (450-360 B.C.), the first philosopher who seems to have used the expression Macrocosmos and Microcosmos, borrowed this method from the Egyptian ritual land-surveyors, haredonapts. Phythagorus generalized this special case into the theorum applying to all right-angle triangles. The theorum is: 3squared x 4squared = 5squared (this is: 3 x 3 = 9 plus 4 x 4 = 16 = 5 x 5 = 25, which has the square root of 5) The followers of Pythagorus called this right-angle triangle the "angle of equity."

This measuring system eventually fell into disuse in Greece after a devastating plague in Athens that killed tens of thousands of her inhabitants and after the death of Alexander the Great in 323 B.C. However, there were many "rope-stretchers" and artisans and architects who had the knowledge and continued the practice using this system. I believe that it is possible that some of this knowledge was kept alive by artists whose function was to decorate walls and panels, even though its use for temple plans and sculpture had ceased. The reason this system was not adopted by the Romans, although they valued Greek sculpture highly, was that they misread the Greek word for area as line and therefore changed the formula and method, and hence were unable to copy Greek works correctly.

In looking at the historical record for the development of Byzantine sacred arts and ultimately its spread throughout the ancient world, there are some clues as to the use of dynamic symmetry. I believe that it was possible for the Greek icon-painters to have learned their craft from their master-teachers who used this proportional system, and then to have gone out into the Byzantine Empire decorating churches and iconostases and to have passed their method on to their apprentices - this system, as said before, having no special name - it was "just done that way."

It needs remembering that our understanding of dynamic symmetry has been obtained mostly from the storehouse of Greek design from the Classical Period. This Classical Period probably covers two or three centuries and the material consists of very many examples. So many, in fact, that it would seem almost impossible for any one person to learn them all. Indeed, one person could not really learn them well at all without an arithmetical background. This is the reason for using simple numbers in the analyses we find. It is improbable that the average Greek artist knew more than a few of the areas although there must have been some who had a rather profound knowledge of the elements of their design. The design of the Parthenon in Athens (448 - 427 B.C.), for example, is a perfect "root-five" rectangle design system and must have been done by experts in dynamic symmetry and mathematics both (See Figure V-9 below).


Figure V-9. The Parthenon at Athens

A schematic of the dynamic symmetry analysis.