Power of Limits

This beautifully illustrated and diagrammed book attempts to show the harmony that exists in nature and all good art and architecture. Not only that, Doczi attempts to weave into this picture, (with some success) Pythagorean concepts of harmony and it's relation to growth in nature.The essential concept in this book is the 'power of limits.' Doczi shows that this limiting factor is the golden section. And he does it using almost no math! The golden section has the powerful quality that division or expansion by this proportion always leads to harmonious growth. No matter how small or large is the division, there is never anything "left over" to create disharmony. This limiting factor is of transcendental power, thus "The Power of Limits."Unregulated growth could never achieve anything but randomness, which is not what we observe in nature.Of course in nature and in life it is impossible to achieve perfection. Yet Doczi elegantly explains how nature compensates for this inability by using the Fibonnaci sequence instead. Profusely illustrated with many detailed, easy-to-understand diagrams, this book is a must for those who wish to understand more deeply how our world is constructed, without wading through a lot of math.

Any serious observer of nature can tell you there are rhythms in nature, patterns that repeat themselves. Far from chaos, there is harmony. The Golden Section, A : B = B : A + B, is part of that harmony. From Fibonacci numbers to the Golden Rectangle, this book will awaken the reader to a sense of perfection, an incredible order, found virtually everywhere in nature. Beyond the interesting phenomena, there lies a larger message. This, all of this, did not happen by some cosmic accident.

For millenia, naturalists and artists have observed repeating geometrical patterns in natural objects. For example, many two- and three- dimensional structures (such as leaves and seashells) have dimensions that reflect the phi ratio, which is (approximately) 1 to 1.618034. This ratio, known amongst the Greeks as the Golden Mean, and used extensively in their sculpture and architecture, lends structural stability, as well as aesthetic balance, to objects into which it is incorporated. In The Power of Limits, Doczi, an architect, explores some of these basic ratios underlying structures found in the universe-both natural and manmade. He provides beautiful graphics on each page--photographs, drawings, and diagrams, all in black and white. The images illustrate the concepts presented in the text. There is a lot here about dynamic symmetry and Fibonacci number relationships (which gave us the dimensions for the 8 1/2 x 11 inch page and the 3 x 5 index card). But Doczi also searches for the deeper principle which organizes the harmonic ratios of the Fibonacci numbers. His term for this principle is "dinergy." It is a combination of the Greek word "dia," meaning opposite, and "energy. " Dinergy refers to the generative power of the union of opposites, and according to Doczi, it organizes all the harmonic patterns which combine into physical forms. Each chapter explores dinergy in a different context, such as plants, crafts, animals, art, and philosophy. While it is true that there are indeed repeating patterns that we can discern in nature, and which lend beauty to our man-made objects, there are those who have criticized the work of Doczi and his predecessors for engaging in a sort of "number mysticism" without scientific support. The concept of dinergy resembles the Taoists' Yin and Yang, the two opposite principles which animate each other and the cosmos. In this, Doczi's work does more resemble mysticism than science. The confusion arises because he uses natural forms to "prove" his theory, thus blurring the line between science and mysticism. However, this is only a problem if you believe there should always be a line between those two fields. Sometimes, it's true, the twain ought not to meet, but there are other times--for instance, when the imagination needs activating-- when the union of those two opposites can be very fruitful. I find that the author's quest for the geometry of harmony takes me on a compelling journey into the beauty of the natural world, and within the confines of the book his arguments work. Another criticism of Doczi's work is based on the fact that much of nature is chaotic, that is, having a deep organization that is not readily apprehended by the human mind. It's certainly true that we cannot reduce everything to these geometrical principles, without vastly oversimplifying the universe, nor should we want to do so. However, there is certainly much to be gained by appreciating harmony and order where we fi

Dozci demonstrates with great clarity one of the the great comsmic mathematical cords that runs through the universe. The concepts he articulates are valuable tools- not only for deeper thinking, but for practical design.

Doczi shows the commonality and connectedness of all things in the universe through the perspective of the golden mean. He does so in such a manner that the expert or layman, the physicist or metaphysicist can readily understand. This is by far the most readable book of its kind.