What’s under and behind some places of power? And what makes certain pieces of art or the unfolding structures of plants and trees suddenly show us their hidden order? We see and hear beautiful patterns, ratios and repetitions in nature and music.
The huge array of different crystals are all formed by only seven simple geometric forms. And at the core of life spins a double helix always rotating around a consistent DNA pattern that consistently forms a double pentagon. How can this be?
It's called sacred geometry. Holy measuring. A mysterious unfolding.
We recognize Geo-metrics, the measurements of the earth. But how can we measure the sacred? And we certainly can’t “form” it or “contain” it—with numbers and shapes, squares and cubes and triangles, circles and dodecahedrons. Can we? But if, as Walt Whitman believed, spirit and soul are sacred, might the “spirit and soul of the earth” also be sacred? And can it somehow be measured? Can we rationalize the irrational?
The ancient Egyptians and Greeks thought so. And the Arabic world preserved this knowledge for its reappearance in the 12th century. Plato thought geometry and number formed an ideal philosophical language and geometry was often personified by an elegant, refined woman. She represented this intuitive, synthesizing, creative yet exact mental understanding of a higher order of things.
David N. Elkins, a psychology professor, says he believes in the sacred “because I simply no longer have the strength to sustain disbelief.” He wrote about eight alternative paths to the sacred in Beyond Religion: the feminine, the arts, the body, psychology, mythology, nature, relationships and dark nights. Of course no one, not even Elkins,can know the true nature of the sacred. It’s like trying to define a kiss. A kiss is much more than lips meeting. It’s more than faces brushing and skin touching. It’s a mysterious union that can't be measured. Still, our minds search for ways to understand.
Some claim sacred geometry is a very murky field; others are convinced human constructions such as the great pyramid and alignment of early stone works as well as later buildings that have lasted and continue to call to us, were governed by ancient mathematical and astronomical principles. Consider for a minute the claim made by Peter Tompkins and others that the base of the Great Pyramid of Cheops equals the circumference of a circle with radius equal to the height of the pyramid. Very precise. Very predictable. It's fun to think about why people are drawn to certain places, certain alignments, certain structures. to places such as Chaco Canyon in the Four Corners area of the U.S. where, it is said, heaven and earth meet.
We are drawn to ancient sites like iron filings to a magnetic sacred something. Perhaps we feel a pull to certain structures or selected art work because they have been constructed using careful geometric proportions--ratios, sequences, progressions.
Measuring with Compass and Squares
Early builders used simple shapes. Circles. Squares. Triangles. Rectangles. And they measured these shapes with straight sticks, string, and compasses to create spaces that felt suitably proportioned, located on the “right” spot, facing the “right” direction. They learned how to square the circle.
Vitruvius, the Italian architect who lived from 70-25 B.C. wrote the manual many early sacred-space architects called their bible. He said, “Geometry…teaches us the use of the rule and compasses…and to rightly apply the square, the level, and the plummet.”
Those were the only tools early masons needed to build cathedrals, and some calling themselves Masons today, still find these principles to be symbolically important. Our own capital city, thanks to Thomas Jefferson, is built on a Pythagorean square based on the number 10, ironically enough given our recent Iraq history, identical to the one that Babylon was also built on. A 10x10 square contains 36 squares around a perimeter of 360˚. In other words, Washington D.C. squares the circle!
Vitruvius' Ten Books of Architecture were rediscovered about 1000 A.D. In 1521, Cesare Cesariano translated Vitruvius into Italian and added this familiar illustration.
And here is Leonardo da Vinci's later rendition.
Vitruvius called this the perfect human body--male, of course, since female bodies at that time were considered much less perfect. The central point is the naval; arms and legs form a compass.
so too a square figure may be found from it.
For if we measure the distance form the soles of the feet to the top of the head,
And then apply that measure to the outstretched arms,
The breadth will be found to be the same as the height.”
(Vitruvius, The Ten Books on Architecture)
Geometric principles inspired Bernard of Clairvaux, who founded the Cistercian order and sought to build “perfectly” proportioned Christian architecture, to ask “What is God? He is length, width, height and depth.”
In order to discover King Solomon's old building secrets, he sent nine men, who eventually grew into the mighty Templars, down to Jerusalem to scope out what was left of Solomon’s Temple and to rediscover the secrets of sacred geometry. How was that early temple constructed? Bring me back the numbers, he instructed, as Pythagoras taught, because they are the sources of form and energy in the world. Soon after their return, magnificent cathedrals began springing up all over Europe, such as Chartres.
What sacred patterns returned with those Templars, we might ask.
The biblical proportions of Solomon's temple are all described in careful cubits; you can read about them in I Kings: 5 and the following chapters. Solomon’s father, David, was a good friend of Hiram, an architect-builder in a nearby country—Tyre. So they pooled their resources and built a stone-pillared building in the style of the Phoenecian goddess temples Hiram knew so well. They lined the cedar paneling with gold. Even the subsequent temple rebuilding is all gone now except for one wall—the Wailing Wall in Jerusalem. The Dome of the Rock mosque is likely built over the original temple courtyard.
Pythagoras, The Father of Geometry
Pythagoras, who lived in the late 500’s B.C., bequeathed us geometry and called it the measurement of earth and the heavens. Numbers aren’t merely counting sticks, he said. He spent 20 years learning ancient arts from the Egyptians. And then he studied with the Persians. But who taught the Egyptians and Persians? I prefer to think the Atlanteans. And before them…?
What we know of Pythagoras comes only from his disciples and later people such as Aristotle, because, unlike the Platonists, Pythagoras saw no need to write things down. He established ways of living that some monks and ascetics still emulate. For instance, he taught people to value silence, to be vegetarians, to wear linen. He practiced self-denial, used incense, music, and rigid cleanliness. In his Timaeus, Plato built on what he had learned from the Pythagorians and brought us the term world soul. Lovelock later picked it up and reinterpreted Gaia. And now we look to green czars and czarinas to help save our planet. That’s how our world evolves. We are all dwarves stepping up onto the shoulders of prior giants.
We, too, can learn meanings hidden in music and signs, as the early Druids and Pythagoras taught. We are more than just earth, air, fire and water. We are, Pythagoras argued, justice, soul, reason and opportunity. He said whole numbers are the numbers to pay attention to. Zeroes and irrational numbers didn’t hold much meaning for these early mathematicians. After all, odd numbers were feminine, irrational, and even numbers, rational and masculine. Except for One. The monad was the source of all numbers. The possibility that “One” could be feminine didn’t get much traction from the Pythagoreans. Or from most later aesthetics and male writers, for that matter. But perhaps "Her" time has now come.
The Golden Ratio
We sense unity and wholeness when we look at art formed by what is now called the “golden ratio.” Euclid, in Elements, first put it to us this way, “A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less.” O.K. So what does that mean for us mathematically challenged?
The golden ratio, expressed in Greek by their 21st letter, ø, divides something that is whole, like a wall, a room, sculpture or a picture into two parts. One larger, one smaller. But the division is not random. It is carefully planned, thank you Euclid, so that the ratio of the larger to the smaller is perfectly proportioned roughly one to 1/3 to 2/3. Actually, it’s 1.618 033 988 749 894 848 etc. In most pleasing paintings, the sky will come down about 1/3 of the way into the composition. This proportional divide is called the golden mean, the golden section, the golden number, the divine proportion.
In his famous Last Supper painting, Leonardo da Vinci carefully used perspective and built in 13 “rays”, if you overlay lines radiating out form Jesus’ forehead showing 13 vanishing point lines defined by the room’s architectural features. The disciples are grouped in four groups of three each.
This gridded image shows us how to visualize the 2/3, 1/3 ration and provides an illustration of the Fibonacci sequence. Add the two prior numbers together for the next number: 1, 2, 3, 5, 8, 13, 21, 34 and so on.
Stephen Skinner in Sacred Geometry: Deciphering the Code, says the Fibonacci series also seems to determine how many petals a plant will have on its flowers, growing out from the stem at geometrically predictable intervals. Without actually counting them, we can take his word for none ever exceeding 144.
Here’s how he sets that possibility out on page 63:
3 Lilies, irises, trilliums
5 Columbines, primroses
8 Delphiniums, sanguinarias, cosmos
13 Cinearia, corn marigold
21 Chicory, black-eyed Susans
34 Plantain, pyrethrum
55 Aster novi-belgii
89 Michaelmas daisies
Fibbanaci’s proportions please us—they make us feel whole and at ease when we enter a space or view a work of art. When they’re not there, we feel more fragmented and sometimes, even nervous.
Squares, circles, spirals. No wonder our kids, and our “inner children” love spirographs and computer programs that build fractals and wonderful complex figures from lines and circles. Nature shows us how. Take the chambered nautilus growing in a logarithmic spiral. Or a fossilized ammonite.
As it grows, each mother of pearl segment is exactly proportional to the previous smaller ones. And this beautifully shelled animal has been doing this for millions of years.
Or take the spiraling horns of a Dall sheep with their perfectly spaced right hand twists for the right side, and left hand twists for the left.
Hexagonal honeybee cells, fractals of broccoli, sunflower seeds--they all display patterns of intricate beauty following a sacred pattern.
Geometric Crop Circles
In August of 1980 a mysterious design appeared in a farmer’s field in Wiltshire, England. Based on Euclidian geometry, it was 60 feet in diameter. Then three more circles appeared. 120 were recorded between 1980 and 1987. In 1990, a peak year for the crop circles, over 1,000 were reported. Who creates them (aside from the few confirmed hoax sites)? What do they mean? What are they for? People discover physical phenomena around the circle sites: grain twisted in consistent patterns; interferences with electrical equipment; high pitched sounds; alterations at the grain’s cellular levels; seeds that germinate more vigorously.
Over the past twenty years or so, crop circles show us even more of these beautiful shapes, teasing and begging us to figure out what they are telling us. Most of the ones below appeared in Britain's August grainfields.
Sacred Geometry is not a topic we all can or even need to completely understand. But if you're seeking more, I’ll point you to two books I have found very helpful. Like the golden mean itself, one is bigger, and one smaller. The big one is actually a textbook my friend Paul Calter wrote, available from Wiley: Squaring the Circle: Geometry in Art and Architecture. The other slimmer book is by Robert Lawlor called simply Sacred Geometry. In Calter’s book, you’ll find a fascinating appendix: “The Art-Math Tourist” with a list of 52 places he illustrates in his book—not an exhaustive list, but certainly an intriguing one, from Agra, India and the Taj Mahal to Wimbourne St. Giles, England with a polyhedron atop a parish church.
What’s behind and under the places that tug at us—old or more recent ones—where you sense beauty, peace and wholeness? Have you had any experiences of geometry becoming music? Or music becoming geometric symbols? Of space opening up to something much larger and wholly beyond us? Could it be "holy math"? Sacred geometry?
What a fascinating piece on Mother Nature's geometric phenomena. Those many crop circles were truly works of art, hoaxes or not. Thanks for an intriguing look @ sacred geometry.
You don't know me but we have a common friend, Bonnie Miller. Please keep up the good work & I hope you conquer your illness.
Posted by: Larry Retzack | June 10, 2009 at 01:10 PM
Thank you for posting this summary of sacred geometry!
See my website.
Best regards,
Vincent
Posted by: Vincent | November 19, 2009 at 04:02 AM