Le Corbusier, Le Modulor and Modulor 2 [English edition] available at Amazon
AL's Sacred Geometry Corner by Alan S. Glassman Second in a series of articles about what should be of interest to anyone involved in the building professions. (View Part 1 of the Series) The relevance of geometry itself is obvious to those responsible for the planning, design and construction of our buildings and infrastructure. But, although it's been moving more into the spotlight for today's New Age devotees, Sacred Geometry is still little understood.
The Basics - Part 2: Meaning in Shapes and Forms --
Incorporating the Ratio and the Number 5
The Third Food of Man
In designing buildings for use by human beings, we always have to keep in mind the size and proportions of the human body so our constructions are easily related to human functioning. Almost every year, a new addition of the book Architectural Graphic Standards is published relating building materials and methods to assembly by human workers and to final use by a building's occupants.
Le Corbusier's dictum, A house is a machine for living, applies in a broader sense to all construction types and uses. However, what is sometimes forgotten is that human functioning is both external and internal. Yes, the body walks and crawls, stretches and reaches, climbs and descends, sits and lies down. But, these are only some of its outward manifestations. The designer must not forget the internal workings of the body. Not only does a healthy person use all five physical senses, but the body, mind and emotions form impressions from an aggregate intake of those senses. Just as from food and drink we nourish the body and from the air we breathe we extract oxygen and other nutrients for our living and being, so do impressions form another kind of food. In fact, without the intake and processing of impressions (even while we're sleeping), we are considered dead. In spite of artificial respiration and intravenous feeding, a physician still considers a patient clinically dead if there are no brain waves exhibited on the EEG monitor.
Just as we would wish to ingest fresh food and breathe clean air to keep the body healthy, it can be shown we thrive better with positive versus negative impressions. And so it is that designers should strive to produce buildings their users will perceive as having pleasing proportions, color, texture, and scale beyond mere operational functionality. Because we humans have physical bodies that we assess as more attractive or less attractive aesthetically, so it is with the buildings we inhabit. Interestingly, we apply many of the same criteria to our perceptions of architecture and to our judgments of human form. So much so that pleasing and even inspiring architecture can be shown to correlate to specific qualities of the human body. This is undoubtedly why many of the great visual artists have also been great architects.
Measuring Man by the Golden Mean
In order to proceed with this line of study, a good place to begin is with Leonardo da Vinci's so called Vitruvian Man.
Leonardo was concerned with seeing the harmony in human anatomy, and this sketch among his many anatomical studies is perhaps the best example. It sets the proper proportions for all subsequent artists and sculptors of the human form. And, as evidence of his wisdom, it also shows us the relationship between anatomy and pure geometry. The saying, Man is the measure of all things, starts to become evident. Also, in keeping with our guide phrase, As Above, So Below, we see the parts of the body expressing the same ratios as the whole.
The ancient Egyptians, Greeks, Japanese and other even older cultures around the world long ago identified the ratio of the circumference (C) of a circle to its diameter (D) as C/D = 3.14159 . Most of us know from high school geometry class that we represent this number by the Greek letter (Pi). It's called an irrational number because it can be carried out to an infinite number of decimal places, depending upon how exact we want to be. Another important ratio, but one we are usually not taught in high school, is represented by the Greek letter (Phi), where = 1.6180339887 (also an irrational number). It is derived from what is alternately called the Golden Mean, Golden Proportion, or Golden Section---a relationship that satisfies the classical definition of a proportion whose whole is to the greater part as the greater part is to the lesser. For those of us who are graphically inclined, we can see this proportion by drawing a line segment and dividing it into two parts, a and b, with the ratio of 1: as shown below (not to scale):
a = 1 b = = 1.6180339887
In algebraic terms, it is represented by the equation: (a + b)/b = b/a. This relationship can only occur if b/a = 1.6180339887 , the numeric value of . We can easily see this if we assign a value of 1 to a and of 1.6180339887 to b. Take a pocket calculator and do the simple math yourself. The more decimal places you include, the closer you will approach the value of . (As a convenience, I will from now on in this text carry out to only three decimal places.)
We remember being shown in school that the ancients used this Golden Mean proportion in designing their buildings. The Parthenon in Athens and the Great Pyramid at Giza immediately come to mind. Aside from the numerical value itself and ignoring for a moment that this ratio is supposed to represent an ideally pleasing proportion (so we've been told), what is its relationship to man/woman in general and, specifically, to Leonardo's sketch of the Vitruvian Man?
Five, the Number of Man
Without getting into any more calculations at this point, let me say that the number 5 plays a significant role in the Golden Mean. To easily illustrate this, let's generate a regular 5-sided figure, a pentagon, with each side measuring one unit as shown here
so that AB = BC = 1. Now, if we connect any two opposite points with a straight line (for instance, AC), that line will measure 1.618 , the numerical value of . Of course, if we look at man/woman in frontal outline form, we see 5 appendages extending from the central body trunk: two arms, two legs and a head. If we take Leonardo's Man and slightly rotate the position of the arms downward, we can superscribe a perfect pentagon around the figure. Within that pentagon we can inscribe a perfect pentagram---a five-pointed star, as shown below in a sketch attributed to Agrippa. It goes without saying that both the pentagon and the five-pointed star have great significance, especially here in the United States.
Man (the Microcosmos) inscribed in a pentagram from Libri tres de occulta philosophia by Heinrich Cornelius Agrippa von Nettesheim (1486-1535)
We have in five-foldness a number and geometric shapes with profound and multi-layered meaning. Man himself and woman herself are symbolically represented overall by that number and by those shapes. And, once more, As Above, So Below applies: we have five fingers on each hand and five toes on each foot. The whole is echoed in the parts. But, that's just the beginning.
Below is an overlay drawing (not to scale here) that can be seen more accurately at: www.gnosissolutions.com. If we assign a value of 1 to the distance from the line AD measured vertically to the top of the man's head, then, measuring vertically downward from AD to the bottom two points of the pentagram at the man's toes, we find that distance is 1.618 . Also, we find the following ratios: AD / AC = , and AC / AB =. Again, the parts are in proportion to the whole.
Center of Gravity
And, finally, we have Albrecht Dürer's sketch showing the human body divided by at the navel and divided in half by the sex organs:
When a baby is born, the navel divides the body in half. As the child grows up, the bodily proportions change so that the legs are longer in relation to the trunk. By adulthood, the navel has moved up (relative to the person's total height) to the position of the ratio as shown in both the da Vinci and Dürer drawings. Some say this represents the principle of a human being maturing from a nature of course duality to a nature of asymmetrical but harmonious proportionality. It is interesting to note that the level of the navel is seen in some Eastern traditions as a person's Center of Gravity and is used as such in certain meditative practices.
In next month's issue, The Basics-Part 3, we will further explore the Golden Mean and its expression in Nature as we continue to look at Meaning in Shapes and Forms.