For Mother May I hold you in memory SACRED GEOMETRY FOR ARTISTS, DREAMERS, AND PHILOSOPHERS “From a drop of water to the expanse of the solar system and beyond, this book reveals an unseen harmony pervading the natural world.” MARK VIDLER AND CATHERINE YOUNG, AUTHORS OF SACRED GEOMETRY OF THE EARTH “John Oscar Lieben has developed a simple framework with which to view the whole world of geometric formation and proportion. His visual language maintains wholeness while giving detailed examples in architecture, musical proportion, the human form, cosmic time, and the elements. This art and craft can form sacred images which, like traditional mandalas and yantras, find a balanced order in the world.” RICHARD HEATH, AUTHOR OF THE HARMONIC ORIGINS OF THE WORLD, SACRED NUMBER AND THE ORIGINS OF CIVILIZATION, AND MATRIX OF CREATION CONTENTS Cover Image Title Page Dedication Epigraph Introduction. A Harmony That Structures the Universe Chapter 1. The Field of Harmony THE RING FORM THE VESICA-TONE CONSTRUCTION TONE Chapter 2. The Geometry of Time THE CENTER OF TIME THE TRITONE HARMONIOUS RECTANGLES Chapter 3. Moon and Earth MEASURES OF TIME AND SPACE THE THREE MASTERS (PART 1) CONSTRUCTING A SEVEN-TONE SCALE THE THREE MASTERS (PART 2) EARTH AND MOON THE INNER EARTH Chapter 4. The Temple of Water LAND AND WATER THE SHAPE OF RAINDROPS THE TEMPLE GARDEN A TEMPLE OF WATER Chapter 5. Musical Vessels PLANETARY POTTERY THE WORLD IN A GRAIN OF SAND Chapter 6. The Spiraling Octave THE SKY SPEAKS OCTAVE SPIRALS MUSICAL SPACES Chapter 7. Harmonic Structures THE HARMONIC SERIES THE MEAN THE RECIPROCAL MUSICAL RECTANGLES THE VESICA-HARMONIC CONSTRUCTION THE MINOR THIRD Chapter 8. The Circle of Resonance RESONANCE TWO FROM ONE ONE FROM TWO THE BIRTH OF THE MOON CROSSINGS HARMONIC SCIENCE Chapter 9. The Musical Universe AS ABOVE, SO BELOW THE WATER OF LIFE THE GREAT CENTRAL SUN THE GALACTIC CORE Chapter 10. The Tree of Time A SPIRAL OF TIME THE VESICA OF TIME THE ERA BEFORE TIME THE TREE OF TIME WITHIN THE ROOT OF TIME Chapter 11. The Cosmic Flower THE SYMBOL IN THE SKY SACRED COSMOLOGY THE SYMBOL SPEAKS CEREMONY Appendix 1. The Vessel of Ceremony CEREMONY WITHIN GEOMETRIC SPACE CLASSICAL GEOMETRY CREATING SACRED SPACE AN EARTH-HONORING CEREMONY A BIRTH OF THE MOON CEREMONY A WATER-HONORING CEREMONY A BIRTH OF THE COSMOS CEREMONY A SUN-HONORING CEREMONY Appendix 2. A Miscellany of Numbers and Forms Endnotes Bibliography Illustration Credits About the Author About Inner Traditions • Bear & Company Books of Related Interest Copyright & Permissions Index INTRODUCTION A Harmony That Structures the Universe WE ARE IN THE MIDST OF A REVIVAL of an ancient way of looking at the world as an interlocking system of harmonious numbers. The revelations that result from this perception awaken feelings of wonder in those who care about such things. Why should it surprise us that this is the way things are? Marie-Louise von Franz writes in Number and Time: We have long known that numbers did not always possess the dry-as- dust abstract quality that characterizes them today. Formerly they were not only something “godlike” but possessed—curiously enough —an all-embracing significance. They did not divide but united two worlds.1 When we begin to see in this way it becomes clear that the concept of a meaningless universe in which evolution proceeds blindly from random event to random event is a delusion. This new way of seeing is really very old. It has been called harmonics, which includes the disciplines of number, geometry, music, and cosmology—the quadrivium of the ancient mystery schools. Of great importance to the ancients was the discovery of the numerical ratios that were the basis of musical intervals. While musicians had intuitively employed these ratios since the beginnings of their art, the practitioners of the harmonic arts realized that such intervals could also be obtained through the integral divisions of the musical string. This discovery of a numerical basis for sensory experience was the beginning of exact science. This science of sound as embodied in the musical scales became the foundation of a musical-numerical conception of nature, which included ideas as diverse as astronomical cycles and the nature of the human soul. The art of geometry was part of this exploration since it could visually portray numbers, just as music could portray them through sound. In the process of their geometric work it became apparent to the harmonicists that the generated forms had symbolic qualities—that is, they could somehow awaken perceptions about higher realities. For example, the circle seemed to have an affinity with universal cycles and with the infinite, and the square with the earthly and the finite. Thus the circle-in-the-square could represent the human being who was seen as an earthly vessel containing a divine essence. This symbolic character of geometry provides a simple way of contemplating complex ideas and processes, and as such it became an indispensable foundation in the training of both artists and philosophers in the ancient world. Today, as in the past, when architects and designers of spaces who are working with sacred intention wish to place a firm foundation under their work they must turn to the laws of creation, which include the laws that determine cosmic time cycles and the shapes that arise from natural forces. While these laws can be complex, there are simple ways of making use of them, and this is what harmonics offers. The study of harmonics is an entry into a world of cosmic meaning. This ancient art is now rising from the dark sea of the unconscious like a lost continent where once a great, golden civilization flourished. As we unearth the artifacts, we wonder why they should appear now. Is there a reason? Isn’t everything random and meaningless as so many of us have been taught? The harmonicist knows that this is untrue, and the reason the mountains are rising from the dark sea is that—it is time. The wheel has turned, the flower is opening, and the Law must be fulfilled. * This book demonstrates specific ways of creating harmonious forms using geometry and numbers derived from the musical scale. In this approach we are working with tools that are as ancient as time yet perfectly suited to be impeccable instruments for the creation of new forms. The reader will find that the discipline of harmonics naturally weaves together symbolic form (geometric, numerical, and musical) and the practical. The practical applications, the human creations, will be seen to arise naturally from the energy contained within primordial number and geometry. This process resonates with nature’s way of creation in which complex forms arise from the simple geometric shapes of atomic structures. I have found that there is a creative matrix in which these energies can interact. I have called this the vesica construction. Its central figure is the vesica, the leaf- shaped product of the intersection of two circles. Most geometers are familiar with the vesica piscis, the vesica that is formed when the two circles intersect each other’s centers. The vesica piscis has been an important form generator in sacred geometry. An example of its power is demonstrated in chapter 11 in which it generates the foundational polygons. I have explored some other vesicas, which can take infinite forms ranging from a thin sliver, when its intersecting circles are barely penetrating one another, to wider and wider figures resembling fish and eyes and then to broad figures shaped like fruit until it becomes a single circle as the intersecting circles join and become congruent. While this figure may seem simplistic, it actually has profound symbolic and generative properties. The process of intersection of the vesica-forming circles exactly corresponds to the intersections of waves, which are perhaps the most universal of dynamic processes in nature. The infinite variety of vesica shapes corresponds to the innumerable height-to-width ratios of created beings. And the vesica ratios that we will use are of great interest in their intimate association with tonal ratios. This will be discussed further in chapter 1. The establishment of the vesica-construction matrix is also described in the first chapter. We should note that almost all the constructions in the book can be made using classical geometry, which is geometry created using only compass and straightedge—that is, without the need for measurement. While this simplifies the drawing process, there are deeper reasons for this restriction. It requires one to proceed one step at a time, building each new part on a previous part. This is the way of nature: from seed to sprout to leaf to flower. The sequence must be followed. One cannot hurry through, and in this process there is room for quiet contemplation that invites the unknown, the new, to enter. Also in the first chapter we introduce the seed form of our vesica constructions, the circle-in-the-square, a simple form with great generative potential. This is the field, the matrix within which our vesica geometry will be generated. We briefly mention an invaluable technique of generating tonal ratios from the intersection of the many kinds of diagonals that the square generates. This method is described in appendix 2, drawing A2.2. This technique makes it possible to create a direct correspondence between the vesica and the ratios found on the musical string. Simply put, we have created a harmonic workshop in which we will work for the remainder of the book. In chapter 2 we journey further into the realm of tone-geometry and how it can be used to reveal connections between the Mayan calendar, an ancient Greek temple of Apollo (whose ground-plan ratio will occur in different contexts throughout the book), and the lunation cycle. Two other themes that will recur throughout the book are introduced: the golden proportion and the tritone interval, the mysterious center of the musical scale. In this chapter we begin to get a sense of the nonrandom nature of cosmic space-time—that there seems to be a musical-geometric harmony that structures the cosmos. Chapter 3 explores, among other things, the harmony of the Earth-Moon system. The significance of the 366 rotations of the Earth on its yearly voyage around the Sun is noted as well as its relation to the elusive huang chung note of the ancient Chinese. This number 366 will play an important role in future chapters. Next, a simple explanation of the formation of a diatonic musical scale is joined with an imaginary dialogue among three ancient masters who are selecting the tonally charged ratios that form the basis of our foot-and-mile measuring system. We will realize that this system reflects the musical structure of cosmic space-time and takes an honored place alongside the ubiquitous decimal metric systems, which have their own virtues but seem somewhat dry when used to describe an enchanted cosmos. Next we use our vesica-tone construction to accurately generate a model of the inner Earth. Chapter 4 explores the substance of water. Surprisingly, the water-to-land ratio of the Earth generates the same vesica construction as that of the Apollo temple previously mentioned. The shape of the waterdrop is considered and geometrically constructed and compared to certain Neolithic flattened standing- stone circles in Britain. The intersecting vesical pattern formed by raindrops on the surface of a pond is related to the Flower of Life, a symbol of universal creation. An accurate model of the molecule of water is generated from the vesica construction at the interval of the major third. This model becomes the center of a design for a water temple garden. Chapter 5 demonstrates how the vesica-tone construction can be used to generate the sizes of the inner planets and the Moon in relation to Earth. We take this further and generate pottery shapes from these planetary circles whose sizes are then compared, resulting in several interesting ratios. We briefly look at three