Pythagoras showed that the golden ratio, phi ( φ) – 1.61814, used as the relationship of parts in a structure is amazingly pleasing. Physical dimensions in this relationship produce an order that is compelling and beautiful. The relationship of the successive chambers in a nautilus, the relative length of the tip of a finger to the tip to first knuckle, then to the length of the first two segments to the entire finger, then the hand, the relationship of successive vein length segments in a leaf all merge to the limit of φ . People know that symmetry is a characteristic of beauty, as you can see in my portrait, The Vitruvian Man by Leonardo da Vinci. φ is the “constant” of proportional symmetry in nature. Recognizing this constant and its derivation can bring order to a practical assessment of what makes things beautiful.

Although the use of phi by nature is not exact, the proportion of growth at a constant compounding rate develops physical relationships at the golden ratio. φ, or the golden ratio is the relationship between two quantities (or the length of two lines) a and b such that (a+b) is to a as a is to b.

In nature, compounded constant growth means that the organism has grown under consistent conditions. If conditions were constantly unfavorable to the organism, it would probably die or whither. If the conditions were variable, the organism might survive through the bad times to succeed and replenish reserves during the good times. If conditions were favorable and constant, then the organism would flourish. The latter organism would be likely to have a growth rate proportional to φ, and appear more proportionally balanced.

Dr. Stephen Marquardt, a plastic surgeon, researched the numerical relationships of the dimensions of the human face, and found that visages considered beautiful generally have a common factor – φ !
Phi is the ratio of the width of a beautiful persons’s nose to the width of their mouth, the width of the tip of the nose to the maximum width, etc. He found that beautiful people’s faces were filled with golden ratios. He termed this discovery the “Human Mask”. His site, beautyanalysis.com, shows some of his findings.

In architecture, golden ratios help to form a room that is good acoustically. A room built with walls with an integer relationship, ie. 1:1, 1:2, 1:3, 2:3 in length generally sounds boomy and echo filled. As one of the wall lengths approaches 1.62 of the other, from a ratio of 1.5 or 2.0 the room starts to sound “nice” and the amount of sound absorbing material needed to be pleasant decreases. The “reverb time” of the room can be substantially longer without annoying artifacts. If the room is golden squared – for example 3 meter ceilings 4.85 meter walls a and c, and 7.85 meter walls b and d, you have an amazing space for performance or a studio! The room will also look nice.

When scientists are looking for patterns in data on natural processes, I suspect that searches through data seeking relationships of approximately the factor φ in addition to conventional harmonic relationships and other regular searches would be worthwhile. I see value in establishing the statistics of the variation about φ for the various relationships that are found.