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Schneider, an education writer and computer consultant, combines science, philosophy, art, and common sense to reaffirm what the ancients observed: that a consistent language of geometric design underpins every level of the universe, from atoms to galaxies, cucumbers to cathedrals. He discusses numerical and geometric symbolism through the ages, and concepts such as periodic renewal and resonance. Contains numerous b&w photos and illustrations. Lacks an index and a bibliography. Annotation copyright Book News, Inc. Portland, Or.
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Michael S. Schneider is an educator developing new perceptions of nature, science, art, and mathematics, holding workshops for teachers, artists, architects, and children concerning nature's numerical language. He has a Bachelor of Science degree in Mathematics from the Polytechnic Institute of Brooklyn and a Master's Degree in Math Education from the University of Florida. He was a Fulbright-Hayes Scholar in India and taught in public schools for eleven years. An education writer and computer consultant, he designed the geometry harmonizing the statues at the entrance to the Cathedral of St. John the Divine in New York City, where he lives.
The Circle Draws Us
In the fourteenth century Pope Benedictus XII was selecting artists to work for the Vatican, requesting from each applicant a sample of his ability. Although the Florentines painter Giotto (1266-1337) was known as a master of design and composition, he submitted only a circle drawn freehand, the famous "0 of Giotto." Yet he was awarded the commission. Why? What's so impressive about a simple circle?
Give a young child a crayon and paper and observe what he draws. At the earliest ages children scrawl lines and zigzags. There comes a time when they discover that a line's end can meet its beginning, and they take delight in the loop. It continues endlessly around and creates an inside separate from an outside. Eventually, they come upon the circle. The circle brings the loop to perfection, so round in every direction. Children love to trace circular objects like cups and cans to achieve the fascinating perfection so hard for a young hand to draw.
The discovery and appreciation of the circle is our early glimpse into the wholeness, unity, and divine order of the universe. Some psychologists say that the discovery of the circle arrives as the child discovers the self and distinguishes himself from another. Even as adults our attention remains hypnotically drawn to circles, toward their centers, in objects we create and those we see. We draw circles and they draw us.
Looking at a circle is like looking into a mirror. We create and respond irresistibly to circles, cylinders, and spheres because we recognize ourselves in them. The message of the shape bypasses our conscious mental circuitry and speaks directly to the quiet intelligence of our deepest being. The circle is a reflection of the world's--and our own--deep perfection, unity, design excellence, wholeness, and divine nature.
To ancient mathematical philosophers, the circle symbolized the number one. They knew it as the source of all subsequent shapes, the womb in which all geometric patterns develop. The Greek term for the principles represented by the circle was Monad, from the root menein, "to be stable," and monas, or "Oneness." In the alphanumeric correspondences of the Greeks, the letters of the word monas add up to 361. The system allows a difference of one, so the word for "oneness" becomes 360, not coincidentally, as that is also the number of degrees around a full circle. Ancient mathematical philosophers referred to the Monad as The First, The Seed, The Essence, The Builder, The Foundation, The SpaceProducer and, most dramatic, The Immutable Truth and Destiny. It was also called Atlas because like the Titan upholding the heavens it supports, it connects and separates the numbers it produces.
The ancient philosophers conceived that the Monad breathes in the void and creates all subsequent numbers (111111111 X 111111111 = 12345678987654321). Numbers only express different qualities of the Monad. The ancients didn't consider unity to be a "number" but rather a parent of numbers. They noted that unity exists in all things yet remains inapparent. They saw the relation of the Monad to all numbers in a metaphor of simple arithmetic: any number when multiplied by unity remains itself (3 X 1 = 3). The same 5 is true when unity divides into any number (5/1 = 5). Unity always preserves the identity of all it encounters. We might say that "one" waits quietly within each form without stirring, motionless, never mingling yet supporting all. The Monad is the universe's common denominator. The ancient Gnostics called it the "silence force." The universe was carved of this primeval silence. Everything strives in one way or another toward unity.
There's more to a circle than just a curved line. It's a wonderful first glyph of nature's alphabet. Every circle is identical. They only differ in size. Each circle you see or create is a profound statement about the transcendental nature of the uni-verse. Expanding from the "nowhere" of its dimensionless center to the infinitely many points of its circumference, a circle implies the mysterious generation from nothing to everything. Its radius and circumference are never both measurable at the same time in similar units due to their mutual relation to the transcendental value known as "pi" = 3.1415926 . . . When either the radius or circumference is measurable in whole, rational units, the other is an endless, irrational decimal. Thus, a circle represents the limited and unlimited in one body.
Our deepest awareness, the power that motivates all awareness, which we can call the "Power to Be Conscious," of which we are not ordinarily cognizant, recognizes its own transcendental nature in the geometry of the circle. For this reason the circle has been a universal symbol of an ideal perfection and divine state that always exists around and within us whether we acknowledge it or not. Religious art has traditionally turned to the circle to symbolize thisstate of divinity as "heaven," "paradise," "eternity," and"enlightenment."
Giotto's perfectly drawn circle communicated this universal ideal.
Important Tips on Using the Geometer's Tools
Before we use the three classic tools of the geometer, here are some points you should consider:
1. These three tools--compass, straightedge, and pencil--are ancient, found in various forms in most cultures. Used by artists, architects, and craftspeople, they are both practical and symbolic.
2. Whether you use a metal compass or a string tied to a stick in the earth, these tools represent divine attributes. Treat them with respect.
3. Do nothing unconsciously. Be aware of each action you perform with them. No act in a geometric construction is trivial or without profound symbolism and correspondence to the world's creating process.
4. Don't erase mistakes. Just as we cannot undo life's missteps, leave all marks where you make them and live with them until you can do the construction differently.
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