G E O M A N A T O M Y The new Vitruvian Man I articulated, which I called Geoman, adds two of the three fundamental shapes in plane geometry, the equilateral triangle, and the pentagram, both in the golden mean relationship to the circle represented with the square in Leonardo daVinci's Vitruvian Man. Leonardo's drawing combines two of the four simplest shapes in plane geometry. These are the circle representing unity, or one, and the square representing four. I had noticed that the relationship between the circle and the square in Leonardo's drawing is the golden mean ratio. Thus the length of a side of the square is 1.618 times the radius of the circle. I found this by measuring Leonardo's drawing, but D'Arcy Thompson had also demonstrated by statistical analysis that on average the distance from the ground to a person's navel is the golden mean of their overall height. I also knew that the navel, our center, the center that we grow from, is also our center of mass, the point which we would spin about weightless in space, which is also the center that we find when we meditate and find our "center" (yellow in the diagram). At the time I was working on the problem I had been studying the division of circles. As a result I had noticed that there are essentially three families of symmetry in plane geometry, which may be represented by the numbers 3, 4 & 5. Each of these numbers represents one of the ways in which a circle may be divided using only a string, or the equivalent compass and straight edge. These numbers correspond to the shapes equilateral triangle, square and pentagram. I had an intuition that if the circle and the square were present, then the equilateral triangle and the pentagram should also be. I had no good rational reason to believe this, but the intuition was so strong that I worked on the problem for months, tracing over a large poster of Leonardo's drawing. I got nowhere. Then one night, after returning from the studio, I was lying in bed and the solution came to me. I took out my notebook and drew the diagram connecting the three fundamental shapes representing the three Pythagorean numbers, 3, 4 & 5. When I looked at my watch to record the time, it was 3:45 a.m. on April 28, 1983. Each of the three shapes shares the same edge length as the square, and thus each is in the golden mean relationship to the radius of the circle. The key to the success of the diagram is that the tip of the equilateral triangle protrudes through the base of the square, counter to the medieval teaching that the arm span is equal to the diagonal to the feet. Instead, the two points where the triangle meets the ground represent the centers of the feet, and the tip of the triangle represents the pointed toes as if in a pirouette. The square provides the location of the elbows, knees and hips just as indicated in Leonardo's drawing, and the center of the square gives the location of the first chakra (blue in the diagram). The center of the triangle locates the sexual center, "triangle" or second chakra (red in the diagram). But it is the breakdown of the pentagonal system into smaller and smaller shapes which provides all of the detailed proportional pattern for the torso and even the head. This was no surprise to me as one need look no further than our own hands to see that five is most intimately related to the proportions of organic growth and symmetry. The pentagram expresses the golden mean in all its proportions. But, there still remained one question, one element of arbitrary ambiguity which prevented me from claiming that the puzzle was finally and completely solved. How high should the line formed by the common edge of the triangle and the pentagram be placed? This cross piece clearly represented the arms, but in terms of the purely geometric construction what defined its location? It took me another seven years of meditating on the question to realize the solution, but it finally came to me. If you take half of the edge length of the square (which is also the edge length of the triangle and pentagram) and multiply it by the golden mean, it gives you the correct height of the line through the center of the outstretched arms between the longest finger tips. This completed the puzzle and caused the whole system to "sync" together such that multiple systems of geometry meet on the central axis at the three key points. Diagonals of the pentagonal system crisscross the central axis at the center of the square. The center of the triangle meets a cross piece of the pentagonal system at the base of the spine, and the pentagonal system meets the top of the square at the top of the head at the crown chakra (yellow in the diagram). Thus, by arranging the four most fundamental shapes in plane geometry using the most fundamental proportional ratio twice, the complete GEOMANATOMY diagram describes both the human skeletal system, and the chakras, in precise detail. I believe that this diagram raises serious ontological questions about the nature of the design of human form. It seems to demonstrate that we did not just evolve in response to random forces, we evolved toward an attractor represented by this diagram. One more note: If one draws just the vertical and horizontal axis of this diagram to the tips of the polygons, one arrives at a figure of a human on a crucifix - perhaps helping to explain why that particular archetypal image has held such a grip on our collective subconscious. Jim Fournier Link to a translation of the full text of Vitruvius accompanying Leonardo DaVinci's Vitruvian Man © 1997 James L. Fournier - All rights reserved