‘Nature geometrizes universally in all her manifestations.’ – H.P. Blavatsky
Humans have always looked upon the beauty, majesty, and ingenuity of the world around them with reverent wonder. The grandeur of the heavens, the regular rhythms of the sun and moon, the myriads of living creatures in all their splendid diversity, the magnificence of a rose flower, a butterfly’s wings or a snow crystal – the idea that all this might be the product of a mindless accident strikes many people as far-fetched. The ancient Greeks aptly called the universe kosmos, a word denoting order and harmony. This underlying order is reflected in many intriguing patterns in nature.
Consider the line ABC in the following diagram:
Point B divides the line in such a way that the ratio between the longer segment (AB) and the shorter segment (BC) is the same as that between the whole line (AC) and the longer segment (AB). This proportion is known by various names: the golden section, the golden mean, the golden ratio, the extreme and mean ratio, the divine proportion, or phi (φ). If the distance AB equals 1 unit, then BC = 0.6180339887… and AC = 1.6180339887… . The second of these two numbers is the golden section, or phi (sometimes this name is also given to the first number). Many designs in nature are related to the golden section, and it has been widely used in sacred architecture and artwork throughout the ages.
The golden section is an irrational, or transcendental, number, meaning that it never repeats and never ends. It is unique in that its square is produced by adding the number 1 (φ² = φ+1), and its reciprocal by subtracting the number 1 (1/φ = φ-1). (Phi equals (√5+1)/2, the positive root of x² = x+1.) The golden section is the only irrational number that approaches more closely to rationality the higher the power to which it is raised. For instance, φ3000 = 1.0000000000 x 10500.
The golden section is part of an endless series of numbers in which any number multiplied by 1.618 gives the next higher number, and any number multiplied by 0.618 gives the next lower number:
Like phi itself, this series of numbers has many interesting properties:
• each number is equal to the sum of the two preceding numbers;
• the square of any number is equal to the product of any two numbers at equal distances to the left and right (e.g. 1.618² = 0.618 x 4.236);
• the reciprocal of any number to the left of 1.000 is equal to the number the same distance to the right of 1.000 (e.g. 1/0.382 = 2.618).
(To obtain perfectly exact results, the numbers would have to be extended to an infinite number of decimals.)
The large rectangle below is a golden rectangle, meaning that its sides are in the proportion 1.000:1.618. If a square is removed from this rectangle, the remaining rectangle is also a golden rectangle. Continuing this process produces a series of nested golden rectangles. Connecting the successive points where the ‘whirling squares’ divide the sides of the rectangles in golden ratios produces a logarithmic spiral, which is found in many natural forms (see next section). A similar spiral can be generated from a golden triangle (an isosceles triangle whose sides are in the golden ratio), by repeatedly bisecting one of the angles to generate a smaller golden triangle.
A number series closely related to the golden section is the Fibonacci sequence: it begins with 0 and 1, and each subsequent number is generated by adding the two preceding numbers:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 …
If we take these numbers two at a time, and divide the larger one by the smaller one, the value will oscillate alternately above and below the golden section, while gradually converging on it. If we divide the smaller number by the larger number, the value converges on 1/φ (0.618). (Note that in any series of numbers in which we start with any two ascending numbers and add each consecutive pair to produce the next term, the ratio of successive terms will always tend towards φ.)
The Fibonacci sequence has many curious features. For example, the sum of any 10 consecutive Fibonacci numbers is always divisible by 11, being equal in fact to 11 times the seventh number. The sum of all Fibonacci numbers from the first to the nth is equal to the (n+2)th number minus 1. Thus the sum of the first 15 numbers (986) is equal to the 17th number (987) minus 1. The square of any term differs by 1 from the product of the two adjacent terms in the sequence: e.g. 3² = 9, 2×5 = 10; 13² = 169, 8×21 = 168. The last digit of any number in the sequence repeats itself with a periodicity of 60: e.g. the 14th number and the 74th number both end in 7. The last two digits (01, 01, 02, 03, 05, 08, 13, 21, etc.) repeat in the sequence with a periodicity of 300. For any number of last digits from three upwards, the periodicity is 15 times ten to a power that is one less than the number of digits, (e.g. for 7 digits it is 15 x 106, or 15 million).
Many of nature’s patterns are related to the golden section and the Fibonacci numbers. For instance, the golden spiral is a logarithmic or equiangular spiral – a type of spiral found in unicellular foraminifera, sunflowers, seashells, animal horns and tusks, beaks and claws, whirlpools, hurricanes, and spiral galaxies. An equiangular spiral does not alter its shape as its size increases. Because of this remarkable property (known as self-similarity),* it was known in earlier times as the ‘miraculous spiral’.
*Shapes and forms that look similar under any magnification are known as fractals. Numerous apparently irregular natural phenomena display approximate self-similarity, meaning that similar patterns and details recur at smaller and smaller scales. This applies, for example, to the branching of lightning, rivers, trees, and human lungs. Other fractal objects include the chambered nautilus, a head of cauliflower, coastlines, clouds, snowflakes, and rocks (a magnified rock can look like an entire mountain).
Fig. 2.1 The Whirlpool Galaxy (M51).
Fig. 2.2 Each increment in the length of the nautilus shell is accompanied by a proportional increase in its radius, so that the nautilus does not need to adjust its balance as it matures.
Fig. 2.3 A ram’s heavy corkscrewed horns keep a stable centre of gravity as they grow.
Fig. 2.4 Owing to the structure of their compound eyes, insects such as moths follow an equiangular spiral when drawn towards a candle flame. Peregrine falcons, which have eyes on either side of their heads, follow a similar spiral path when flying at their prey.
Phi can be found in the proportions of the bodies of humans and other creatures, including birds, flying insects, frogs, fish, and horses. The height of a newborn child is divided by the navel into two equal parts (1:1), whereas in adults, the division of the body’s height at the navel yields two parts in the ratio 1:φ, though the navel is usually a little higher in females and a little lower in males. Phi is found in the proportion between the hand and the forearm, and between the upper arm and the hand plus forearm.
Fig. 2.5 Each section of an index finger, from the tip to the base of the wrist, is larger than the preceding one by about 1.618, fitting the Fibonacci numbers 2, 3, 5 and 8.
The DNA molecule measures 34 angstroms long by 21 angstroms wide for each full cycle of the double helix spiral. 34 and 21 are Fibonacci numbers and their ratio closely approximates phi. DNA has two grooves in its spirals, with a phi ratio between the major groove and the minor groove (roughly 21 angstroms to 13 angstroms).
The Fibonacci numbers and golden section are widely found in the plant kingdom. In nearly all flowers, the number of petals is a Fibonacci number. For instance, lilies have 3 petals, buttercups have 5, many delphiniums have 8, marigolds have 13, asters have 21, and daisies commonly have 13, 21, 34, 55 or 89. Non-Fibonacci numbers don’t occur anything like as often. For instance, very few plants have 4 petals, an exception being the fuchsia. Some plant species are very precise about the number of petals they have, e.g. buttercups, but with others only the average number of petals is a Fibonacci number.
Fig. 2.7 Shasta daisy with 21 petals.
Fibonacci numbers are often found in the arrangement of branches, leaves, and seeds (phyllotaxis). If we look at a plant from above, the leaves are not arranged directly above one another, but in a way that optimizes their exposure to sun and rain. The Fibonacci numbers occur when counting both the number of times we go around the stem from one leaf to the next, and when counting the number of leaves we meet until we encounter one directly above the starting one. The number of turns in each direction and the number of leaves are usually three consecutive Fibonacci numbers.
In the diagram below, to get from the topmost leaf to the last of the 5 leaves of the plant on the left takes 2 anticlockwise turns or 3 clockwise turns. 2, 3 and 5 are three consecutive Fibonacci numbers. For the plant on the right, it takes 3 anticlockwise rotations or 5 clockwise rotations to pass 8 leaves. Again, 3, 5 and 8 are consecutive Fibonacci numbers. An estimated 90% of all plants display this pattern.
Apple, cherry, apricot,
oak, cypress, poplar
Holly, pear, spruce,
Fig. 2.8 (by Michael S. Schneider from A Beginner’s Guide To Constructing The Universe)
Similar arrangements can be found in the scales of a pinecone or the seeds of a sunflower. The florets in the head of a sunflower form two intersecting sets of spirals, one winding clockwise, the other anticlockwise. In some species the number of clockwise spirals is 34, and the number of counterclockwise spirals is 55. Other possibilities are 55 and 89, or 89 and 144. Pineapples have 8 rows of scales sloping to the left, and 13 to the right. Again, these are successive Fibonacci numbers.
Fig. 2.9 Sunflower head.
Fig. 2.10 This pine cone has 13 spirals to the right and 8 to the left.
The fact that in the arrangement of seeds on flower heads, the numbers of spirals in each direction are nearly always neighbouring Fibonacci numbers means that each seed is roughly 0.618 of a turn from the last one, i.e. there are roughly 1.618 seeds per turn. This results in optimal packing of the seeds, because no matter how big the seed head gets, the seeds are always equally spaced. Similarly, trees and plants tend to have 0.618 leaves or petals per turn. In terms of degrees this is 0.618034 of 360°, which is 222.5°. If we measure the angle going in the opposite direction around the circle, we get: 360 – 222.5 = 137.5°, which is known as the golden angle. The Fibonacci numbers appear in leaf arrangements and the number of spirals on seedheads because they form the best whole-number approximations to the golden section. The higher the numbers are on the Fibonacci sequence, the more closely the golden ratio and the golden angle are approached and the more complex the plant.
Fig. 3.1 Each successive pair of heavy lines in this pentagram is in the golden ratio.
The golden ratio appears in pentagonal forms of symmetry, notably in the five-pointed star (or pentagram), which was the emblem of the Pythagorean brotherhood. Pentagonal symmetry abounds in living organisms, especially plants and marine animals (e.g. starfishes, jellyfishes, sea urchins). Flowers with 5 petals (or multiples of 5) include all fruit blossoms, water lilies, roses, honeysuckles, carnations, geraniums, primroses, orchids, and passionflowers.
Fig. 3.2 Left: Sea cucumber (cross section). Right: Sand dollar (sea urchin).
(by Michael S. Schneider from A Beginner’s Guide To Constructing The Universe)
Fig. 3.3 Pentagonal diatom.
Fig. 3.4 Apple blossoms have five petals, pentagonal indentations are seen at the bottom of the fruit, and cutting an apple in half reveals a star pattern of seeds.
Fig. 3.5 The human body clearly expresses the pentad’s symmetry in its five senses and five extensions from the torso, each limb ending in five fingers or toes. (by Michael S. Schneider from A Beginner’s Guide To Constructing The Universe)
While pentagonal patterns abound in living forms, the mineral world favours twofold, threefold, fourfold, and sixfold symmetry. The hexagon is a ‘close-packing’ shape that allows for maximum structural efficiency. It is very common in the realm of molecules and crystals, in which pentagonal forms are almost never found. Steroids, cholesterol, benzene, TNT, vitamins C and D, aspirin, sugar, graphite – all show sixfold symmetry. The most famous hexagonal architecture is built by bees, wasps, and hornets.
Fig. 3.6 Six water molecules form the core of each snow crystal.
Fig. 3.7 Snow crystals (courtesy of Kenneth G. Libbrecht).
Fig. 3.8 Benzene (C6H6).
Fig. 3.9 Honeycomb.
Fig. 3.10 The facets of a fly’s eyes form a close-packed hexagonal arrangement.
Fig. 3.11 A hexagonal lattice, or honeycomb, of convection
cells (Bénard cells) in a heated liquid.
The atmosphere of Saturn, one of the four gas giants, circulates around its axis in striped bands. Seen from the poles, the bands generally appear to be circular. But the band nearest the north pole is hexagonal, with sides about 13,800 km long. Unlike other clouds in its atmosphere, it rotates slowly, if at all, relative to the planet. It is one of Saturn’s most enigmatic features.
Fig. 3.12 Saturn’s northernmost hexagonal cloud band. The dark radial lines are artefacts of the way the above images were produced.
A regular polyhedron is a three-dimensional shape whose edges are all of equal length, whose faces are all identical and equilateral, and whose corners all touch the surface of a circumscribing sphere. There are only five regular polyhedra: the tetrahedron (4 triangular faces, 4 vertices, 6 edges); the cube/hexahedron (6 square faces, 8 vertices, 12 edges); the octahedron (8 triangular faces, 6 vertices, 12 edges); the dodecahedron (12 pentagonal faces, 20 vertices, 30 edges); and the icosahedron (20 triangular faces, 12 vertices, 30 edges). In each case, the number of faces plus corners equals the number of edges plus 2. These five polyhedra are also known as the platonic or pythagorean solids, and are intimately connected with the golden section. Their striking beauty is derived from the symmetries and equalities in their relations.
Fig. 4.1 The five platonic solids. The ancients considered the tetrahedron to represent the element of fire; the octahedron, air; the icosahedron, water; and the cube, earth. The dodecahedron symbolized the harmony of the entire cosmos.
Fig. 4.2 An icosahedron’s 12 vertices are defined by three perpendicular golden rectangles. With an edge length of one unit, its volume is 5φ5/6.
The symmetry of the platonic solids leads to other interesting properties. For instance, the cube and octahedron both have 12 edges, but the numbers of their faces and vertices are interchanged (cube: 6 faces and 8 vertices; octahedron: 8 faces and 6 vertices). Similarly, the dodecahedron and icosahedron both have 30 edges, but the dodecahedron has 12 faces and 20 vertices, while for the icosahedron it is the other way round. This allows one solid to be mapped into its dual or reciprocal solid. If we connect the centres of all the faces of a cube, we obtain an octahedron, and if we connect the centres of the faces of an octahedron, we obtain a cube. The same procedure can be used to map an icosahedron into a dodecahedron, and vice versa. (In the Hindu tradition, the icosahedron represents purusha, the male, spiritual principle, and generates the dodecahedron, representing prakriti, the female, material principle.) The tetrahedron is self-dual – joining the four centres of its faces produces another, inverted tetrahedron.
There are two other ways of passing from dodecahedron to icosahedron, and icosahedron to dodecahedron. If we join interiorly all the vertices of an icosahedron, the lines will intersect at 20 points defining the vertices of a dodecahedron. If we then do the same with the resulting dodecahedron, a smaller icosahedron is generated within it, and so on, ad infinitum. Likewise, if we lengthen the sides of an icosahedron, an enveloping dodecahedron will be formed. Lengthening the sides of the dodecahedron will generate an enveloping icosahedron. Again, this operation can be repeated indefinitely – a fitting symbol of the theosophical teaching of worlds within worlds.
The sum of the angles of the platonic solids is 3600 degrees for the icosahedron, 6480 for the dodecahedron, 1440 for the octahedron, 2160 for the cube, and 720 for the tetrahedron. Each of these numbers is divisible by 9, which means that the sum of the digits is 9 (e.g. 6+4+8=18, 1+8=9). They are also divisible by the canonical numbers 12, 60, 72 and 360. As we shall see, the numerical elements of the five regular polyhedra frequently recur in nature’s cycles.
The platonic solids (especially the tetrahedron, octahedron, and cube) form the basis for the orderly arrangement of atoms in crystals, though the regular dodecahedron and icosahedron are never found.
Fig. 4.4 In salt crystals, the sodium chloride atoms tightly pack along cubic lines of force.
Tetrahedral geometry commonly occurs in organic and inorganic chemistry and in submicroscopic structures. For example, the methane molecule (CH4) is a tetrahedron, with a carbon atom at its centre and a hydrogen atom at each of its four corners.
Carbon exists in three pure forms. In graphite crystals, the carbon atoms lie in hexagonal sheets, which readily slide off a pencil as we write. In diamond, the hardest substance known, each carbon atom is bonded to four others in a super-strong tetrahedral arrangement. Buckminsterfullerene, the third, highly stable allotrope of carbon, consists of 60 carbon atoms, arranged at the vertices of a truncated icosahedron (i.e. one with its corners cut off).
Fig. 4.5 Graphite; diamond; buckminsterfullerene.
The great majority of viruses are icosahedral, including the polio virus and the 200 kinds of viruses responsible for the common cold. Icosahedral symmetry is believed to allow for the lowest-energy configuration of particles interacting on the surface of a sphere. The five platonic solids are also found in radiolarian skeletons.
Fig. 4.6 The platonic solids have been found living in the sea. The tetrahedron, somewhat rounded as if from internal pressure, is embodied in a protozoan called Callimitra agnesae, the cube is Lithocubus geometricus, the octahedron Circoporus octahedrus, the dodecahedron Circorrhegma dodecahedrus, and the icosahedron Circognia icosahedrus.
Due to a slow gyration of the earth’s axis, the spring equinox occurs about 20 minutes earlier each year, when the earth is still about 1/72 of a degree (50 arc-seconds) from the point in its orbit where the previous spring equinox occurred. The vernal equinoctial point therefore moves slowly through the constellations of the zodiac. This key astronomical cycle is known as the precession of the equinoxes. The ancients called it the ‘great year’.
At the average rate of 1/72 of a degree per year, the earth enters a new constellation about every 2160 years, and takes 25,920 years to make a complete circuit of the zodiac. The entry into each new constellation marks the beginning of what is called in theosophy the Messianic cycle, which is said to be marked by the appearance on earth of a spiritual teacher or avatar (lit. a ‘descent’ of divinity); for instance, the commencement of the Piscean age was marked by the appearance of Christ. Interestingly, 2160 is the total number of degrees in the angles of a cube, and if a cube is opened out it forms the shape of a cross, a universal symbol representing the descent of spirit into matter, or the ‘crucifixion’ of spirit in the ‘tomb’ of matter.
The ancient chronological charts of the Brahmans refer to four great cycles or yugas. The krita- or satya-yuga lasts 4000 divine years, the treta-yuga 3000, the dvapara-yuga 2000, and the kali-yuga 1000, a ‘divine year’ being equal to 360 earth-years. Each yuga is introduced by a ‘dawn’ and concluded by a ‘twilight’, each equal to one-tenth of the period of the yuga. The total length of the four yugas in earth-years is therefore 1,728,000 for the satya-yuga, 1,296,000 for the treta-yuga, 864,000 for the dvapara-yuga, and 432,000 for the kali-yuga. The lengths of these four yugas are related in the ratio 4:3:2:1, i.e. they are all multiples of 432,000. If the individual digits of each yuga are added together, the result is always 9 (e.g. 1+2+9+6=18, 1+8=9).
The four yugas together make up one maha-yuga, lasting 4,320,000 years, which in theosophy is said to equal half the evolutionary period of a root-race or ‘humanity’, of which we are now in the fifth. The total life-period of earth, or a day of Brahma, is said to last 1000 maha-yugas or 4,320,000,000 years, and is followed by a rest period, or night of Brahma, of the same length, making a total of 8,640,000,000 years. 360 days and nights make up one year of Brahma, equal to 3,110,400,000,000 ordinary years. 100 years of Brahma make up an age of Brahma, this being the total life-period of our universal solar system, equal to 311,040,000,000,000 years. If we take the precessional cycle of 25,920 years and add to it a dawn and a twilight equal to one-tenth of its length, we get 31,104 – the initial digits of a year and age of Brahma.
4320 is twice 2160, the total number of degrees in a cube, the length of the Messianic cycle, and the average time it takes the equinoctial point to pass through one constellation of the zodiac. 432,000 is equal to 4 x 108,000, and 108,000 years is the average length of an astronomical cycle known as the revolution of the line of apsides. The line of apsides is the line joining that point in the earth’s elliptical orbit where it is closest to the sun (perihelion) and that point where it is furthest from the sun (aphelion). This line turns extremely slowly – by an average of 12 arc-seconds a year – from west to east, and therefore rotates through the whole zodiac in 108,000 years.
The sun has a radius of approximately 432,000 miles (432,475) and a diameter of about 864,000 miles, while the moon has a radius of 1080 miles and a diameter of 2160 miles. Silver, the metal associated with the moon, has an atomic weight of 107.9. In addition, 108 is roughly the average distance between the sun and earth in terms of solar diameters, the average distance between the surfaces of the moon and earth in terms of lunar diameters, and the diameter of the sun in terms of earth diameters (actual figures: 107.5, 108.3, and 109.1 respectively). As a result of these remarkable ‘coincidences’, the moon has the same apparent size as the sun, as seen from earth, and almost exactly covers the disk of the sun during a total solar eclipse.
According to conventional astronomy, in each precessional cycle the earth’s north pole slowly traces an approximate circle, with an average radius of about 23.5 degrees (the current inclination of its axis), around the north pole of the ecliptic – a point in the constellation Draco perpendicular to the plane of the earth’s orbit around the sun. (According to theosophy, the earth’s axis does not trace a circle but a spiral around the ecliptic poles, as the tilt of the earth’s axis is said to change by four degrees every precessional cycle.)
The ‘circle’ traced by the earth’s axis around the north ecliptic pole is not smooth but wavy, as the moon’s gravitational pull causes the earth to ‘nod’ about once every 18 years (currently 18.6 years), a movement known as nutation. The circle therefore contains about 1440 waves, since 18 x 1440 = 25,920. Note that the average human heartbeat is equal to 72 beats a minute, and on average we breathe about 18 times a minute. 72 years (= 6+60+6 or 6×12) is said to be the ‘ideal’ lifespan of a human being, during which time the sun moves through one degree of the zodiac in the precessional cycle. A human breathes 72 times in 4 minutes, the time required for the earth to turn 1 degree on its axis. In 24 hours (86,400 seconds) we breathe 18 x 1440 = 25,920 times, equal to the number of years in the precessional cycle.
Each human heartbeat takes about 8 tenths of a second. The time-periods occupied by each of the five cardiac phases, reckoned on the basis of one hour, are as follows: auricular systole, 432 seconds; ventricular systole, 1296 seconds; repose of the entire heart, 1728 seconds; diastole, 3024 seconds; and ventricular diastole, 2160 seconds. There are on average 4320 heartbeats in one hour, 8640 beats in 2 hours, 12,960 beats in 3 hours, 17,280 beats in 4 hours, 21,600 beats in 5 hours, and 25,920 beats in 6 hours.* In these figures we can recognize the digits of the four yugas, the Messianic cycle, and the precessional cycle (whose lengths are all multiples of 12, 60, 72, and 360). Thus, again and again we find correspondences between what takes place in the small (the microcosm) and what takes place in the great (the macrocosm) – as above, so below.
*G. de Purucker, Esoteric Teachings, San Diego, California, Point Loma Publications, 1987, 7:20-1.
6. Formative power of sound
In the late 18th century, German physicist Ernst Chladni demonstrated the organizing power of sound and vibration in a visually striking manner. He showed that when sand is scattered on metal plates, and a violin bow is drawn across them, the resulting vibrations cause the particles to move to the places where the plate is almost motionless, producing a variety of beautiful, regular, intricate patterns.*
*Joscelyn Godwin makes an interesting comment on this phenomenon: ‘Once, passing by a crowded dance hall where rock was being played, I could not help perceiving the floor of the hall in terms of a Chladni plate, and the dancers appeared for all the world like the jumping, helplessly manipulated grains of sand’ (1995, p. 246).
Fig. 6.1 Chladni figures. For a video, click here.
A century after Chladni, Margaret Watts-Hughes created images by placing a powder or liquid on a disk then letting it vibrate to the sound of a sustained musical note. She experimented with several musical instruments but had most success using her voice. The particles arranged themselves into geometric shapes, flower patterns (such as pansies, primroses, geraniums, and roses), or the shape of a fern or a tree. The higher the pitch, the more complex the patterns produced; a powerful sustained note produced an imprint of a head of wheat.
Fig. 6.2 Figures generated by the voice of Margaret Watts-Hughes.
In the 1950s the study of wave phenomena was continued by Swiss scientist and anthroposophist Hans Jenny (1904-1972), who named the field ‘cymatics’. Using crystal oscillators (which allow precise frequencies and amplitudes to be used), he vibrated various powders, pastes, and liquids, and succeeded in making visible the three-dimensional effects of sound. He produced an astonishing variety of awe-inspiring geometrical and harmonic shapes, including life-like flowing patterns, which he documented in photographs and films.
Jenny, too, found that higher frequencies produced more complex shapes. A low frequency produced a simple central circle surrounded by rings, while a higher frequency increased the number of concentric rings. Even higher frequencies created shapes resembling petals, butterflies, or crustaceans, zebra patterns, mandala-like patterns, and images of the five platonic solids. As the frequency rises, the dissolution of one pattern may be followed by a short chaotic phase before a new, more intricate, stable structure emerges. If the amplitude is increased, the motions become all the more rapid and turbulent, sometimes producing small eruptions. Under certain conditions Jenny was able to make the shapes change continuously, despite altering neither frequency nor amplitude.
Fig. 6.3 Hexagonal pattern produced by light refracting through a small sample of water (about 1.5 cm in diameter) under the influence of vibration. The figure is in constant dynamic motion. (Jenny, 2001, p. 112; courtesy of Jeff Volk)
Fig. 6.4 A round heap of lycopodium powder (4 cm in diameter) is made to circulate by vibration. At the same time two centres of eruption rotate at diametrically opposed points. (Jenny, 2001, p. 108; courtesy of Jeff Volk)
Fig. 6.5 Jenny built a tonoscope to translate the human voice into visual patterns in sand.
Left to right: ‘oh’ sound, ‘ah’ sound, ‘oo’ sound. (Jenny, 2001, p. 65; courtesy of Jeff Volk)
More recently, Peter Guy Manners found that an acoustic recording of the Crab Nebula caused sand to form a pattern strikingly like the nebula itself. As the near-infrasonic sound was played, the two swirling arms pulled in, forming a tight ball. Towards the end of the audiotape, the sand became very highly compacted and then suddenly exploded, throwing sand off the table.
John’s Gospel begins: ‘In the beginning was the Word, and the Word was with God, and the Word was God.’ The Egyptian Book of the Dead, contains a parallel passage: ‘I am the Eternal, I am Ra … I am that which created the Word … I am the Word …’ The Hindu tradition teaches that ‘Nada Brahma’ (the world is sound). The underlying idea is that everything we see is a divine word – or vibration – that has solidified and become manifest, the vibrations originating in the inner, more ethereal realms. All of nature is essentially rhythmic vibration. Everything from subatomic particles to the most intricate lifeforms, from planets to galaxies, comprises resonating fields of pulsating energy in constant interaction. In the poetic words of Cathie Guzzetta:
The forms of snowflakes and faces of flowers may take on their shape because they are responding to some sounds in nature. Likewise, it is possible that crystals, plants, and human beings may be, in some way, music that has taken on visible form. (D. Campbell, ed., Music: Physician for times to come, Quest, 1991, p. 149)
If we draw a circle representing the earth – which has a mean radius (in round numbers) of 3960 miles – and then draw a square around it, the square will have a perimeter equivalent to 31,680 miles. If we then draw a second circle with a circumference equal to the perimeter of the square, its radius will be 5040 miles (using 22/7 as a good approximation to pi (π), as the ancients often did) – or 1080 miles more than the smaller circle. Just as 3960 miles is the radius of the earth, 1080 miles is the radius of the moon. In other words, the relative dimensions of the earth and moon square the circle!
Fig. 7.1 Earth and moon square the circle. Note that 5040 (the radius of the outer circle) = 1x2x3x4x5x6x7 (known as ‘factorial 7’, also written: 7!) = 7x8x9x10 (or 10!/6!). A quarter of its circumference (also equal to the diameter of the earth-circle) = 7920 = 8x9x10x11 (or 11!/7!), and the area of each semicircle = 11!.
Exactly the same proportions and digits (expressed in feet rather than miles) can be found at Stonehenge (see Michell, 1995, 2001). The outer (sarsen) circle has a mean radius of 50.4 ft and a circumference of 316.8 ft. This is equal to the perimeter of a square drawn round the smaller (bluestone) circle, which has a radius of 39.6 ft. This radius is also equal to the diameter of the circle defined by the inner U-shaped structure. This is clear evidence that the ‘English’ foot and mile are at least as ancient as Stonehenge and, like many other ancient systems of measures, are closely related to the dimensions of earth, moon, and sun.
Fig. 7.2 Ground plan of Stonehenge. The lintels on top of the stones of the outer (sarsen) circle were mortised to the uprights and jointed at their ends, forming what was once a precision-made, perfectly level platform.
Fig. 7.3 Given its slope angle of 51.83°, the Great Pyramid, too, squares the circle. The length of each base side divided by the height equals π/2. In addition, the apothem divided by half the base side equals φ.
If a tetrahedron is inscribed in a sphere with the apex placed at either pole, the three corners of the base will touch the sphere at a latitude of 19.47 degrees in the opposite hemisphere. This latitude marks the approximate location of major vorticular upwellings of planetary and solar energy. The primary focus of sunspot activity is about 19.5° N and S. On Venus, there are volcanic regions at 19.5° N and 25.0° S. Mauna Loa and Kilauea (Hawaii), Earth’s largest volcanoes, are located at 19.5° and 19.4° N respectively. On the moon there is a mare-like lava extrusion at 19.6° S. On Mars, Olympus Mons, possibly the largest volcano in the solar system, is located at 19.3° N. The Great Red Spot of Jupiter is located at 21.0° S. On Saturn there are storm belts at 20.0° N and S. On Uranus, there are upwellings causing cooler temperatures at 20.0° N and S, and a dark spot at about 22.5° S. The Great Dark Spot of Neptune, photographed by Voyager 2 in 1989, was located at 20.0° S, but when the Hubble Space Telescope viewed the planet in 1994, the spot had vanished – only to be replaced by a dark spot at a similar location in the northern hemisphere.
Fig. 7.5 Jupiter’s Great Red Spot.
Are the planets of our solar system located at random distances from the sun? The Titius-Bode law, discovered 1766, suggests they are not. The law is obtained by writing down first 0, then 3, and then doubling the previous number: 6, 12, 24, etc. If 4 is added to each number and the sum divided by 10, the resulting numbers give the mean distances of the orbits of the planets in astronomical units (1 AU = the earth’s mean distance from the sun). Uranus, discovered in 1781, fitted the law, as did Ceres, the largest asteroid between Mars and Jupiter, discovered in 1801. However, the law breaks down completely for Neptune and Pluto, which were discovered later. Various efforts have been made to modify the Titius-Bode law to make it more accurate.*
*William R. Corliss, The Sun and Solar System Debris, Sourcebook Project, 1986, pp. 34-42.
What the Titius-Bode law essentially means is that planetary orbits become progressively greater by a ratio of approximately 2:1 (the ratio of the octave) with increasing distance from sun. This is brought out in columns 3 and 4 of the table below, in which half the distance between Mercury and Earth is taken as the unit of measurement. Uranus and Pluto have mean orbits close to the exact distances necessary to complete two further octaves. Neptune is located almost exactly half-way between Uranus and Pluto, as though to fill in the half-octave position. This may indicate that it was not an original member of the solar system (theosophy says it was captured from outside our solar system). Likewise, the Titius-Bode law works for Pluto if we ignore Neptune.
Mean distance of the planets from the sun Planet Calculated
in units of
Mercury 0.4 0.387 0 0 Venus 0.7 0.723 1 1.1 Earth 1.0 1.000 2 2 Mars 1.6 1.524 4 3.7 Asteroid belt
2.8 2.767 8 7.8 Jupiter 5.2 5.203 16 15.7 Saturn 10.0 9.539 32 29.9 Uranus 19.6 19.191 64 61.4 Neptune 38.8 30.061 (64×1.5=96) (96.8) Pluto 77.2 39.529 128 127.7
Due to its ad-hoc nature, the Titius-Bode law is usually dismissed as a numerical coincidence that has no physical basis. However, the fact that the planets’ distances from the sun follow a pattern can easily be demonstrated by plotting the logarithm of the mean distance of the planets (including the asteroid belt) against their sequential number (1 to 10). The fact that all the points lie very nearly on a straight line proves that gravitation is ‘quantized’. However, there is currently no detailed mainstream theory that explains how gravity works and why it should be quantized. The orbits of satellites around moons show the same quantized spacing, as do the orbits of electrons around an atomic nucleus (in the Bohr model of the atom).
Figs. 8.1 to 8.6 courtesy of John Martineau (A Little Book of Coincidence, Wooden Books, 2001)
Curiously, the mean orbital radii of the four inner and four outer planets reflect about the asteroid belt. For instance, if we multiply together the orbital radii of Venus, Mars, Jupiter, and Uranus, we get virtually the same value as multiplying the orbital radii of Mercury, Earth, Saturn, and Neptune (5.51 x 1034 km as against 5.56 x 1034 km).
The spacing of the planets shows many geometrical regularities. For example:
Fig. 8.2 Three circles touching: if Mercury’s mean orbit passes through the centres of the three circles, Venus’ orbit encloses the figure (99.86% accuracy).
Fig. 8.3 Left: The mean orbits of Mars and Jupiter can be drawn from four touching circles or a square (99.995%). Right: A related pattern spaces Earth’s and Mars’ orbits (99.8%).
Fig. 8.4 In this diagram, the smaller and larger circles represent not only the relative sizes but also the orbits of Mercury and Earth; they are related by a pentagram (99.1%).
Fig. 8.5 This diagram shows Earth’s and Saturn’s relative sizes and orbits;
they are related by a 15-pointed star (99.3%).
Fig. 8.6 Earth’s and Jupiter’s mean orbits can be created by spherically nesting three cubes, or three octahedra, or any threefold combination of them (99.89%).
The 17th-century astronomer Johannes Kepler discovered a remarkable relationship between a planet’s mean distance from the sun and the time it takes to orbit the sun: the ratio of the square of a planet’s period of revolution (T) to the cube of its mean distance (r) from the sun is always the same number (T²/r³ = constant). For instance, measuring T in earth-years and r in astronomical units, we get:
Venus: 0.61521²/0.7233³ = 1.0002 Earth: 1.0000²/1.0000³ = 1.0000 Mars: 1.88089²/1.5237³ = 1.0000 Jupiter: 11.8623²/5.2028³ = 0.9991
Orthodox science has no real explanation for this, or for the many ‘resonances’ in solar system dynamics. For instance, the periods of Jupiter and Saturn show a 2:5 ratio. The periods of Uranus, Neptune, and Pluto stand in a 1:2:3 ratio. Mars and Jupiter are locked into a 1:12 resonance, Saturn and Uranus are in a 3:1 resonance, and there is a 2:3 resonance between Mercury’s rotational and orbital periods.
What is the nature of such ‘resonances’? Abstract mathematical concepts such as ‘curved spacetime’ shed no light on the matter. A concrete explanation must be sought in the behaviour of the dynamic ether filling space, whose vorticular motions cause the planets and stars to rotate and carry them along in their respective orbits. According to the ether-science model known as aetherometry, T²/r³ is a constant for all the planets because it refers to the constant flux of energy that the solar system as a whole extracts by its primary gravitational interaction with the ether, entailing a nearly constant energy supply for each of its members.
The speeds of the planets in their orbits represent their pitch-frequencies. Cosmic ‘chords’ are produced when planets come into conjunction (or ‘kiss’), i.e. stand in a straight line with the earth and sun. A certain number of regular planetary conjunctions occur over particular periods of time, and the ratios between these numbers reflect with considerable accuracy the length ratios necessary to produce the diatonic notes of an octave, i.e. the seven notes of a musical scale.
Fig. 9.1 Planetary conjunctions as ‘chords’ (Tame, 1984, p. 239).
The line in this diagram represents an octave, divided into seven intervals by eight notes. The line could represent the string of a one-stringed musical instrument. The numbers above the line are the numbers of conjunctions of each planet with the sun and earth, and those below the line are the numbers of years involved.
The planets’ orbits are ellipses with the sun at one of their foci. Their speeds are therefore variable: they speed up as they approach perihelion (the point in their orbit nearest the sun) and slow down as they approach aphelion (their greatest distance from the sun). Following Kepler, Francis Warrain expressed the ratio between the minimum and maximum speeds of one planet and those of different planets as a musical interval. The results (only those for the inner planets are given below) rule out chance altogether, and constitute ‘a powerful argument for the harmonic arrangement of the solar system’ (Godwin, 1995, pp. 132-6). Of 74 tones, as many as 58 belong to the major triad CEG.
Harmonies of the planets’ angular velocities, as seen from the sun Harmonic number 1 9 5 3 25 27 15 2 Mars aphelion (l)
l:m = 2:3
l:n = 9:20
l:o = 5:12
m:n = 2:3
m:o = 3:5
Earth aphelion (n)
n:o = 15:16
n:p = 3:5
n:q = 3:5
o:p = 5:8
o:q = 5:8
Venus aphelion (p)
p:q = 24:25
p:r = 5:9
p:s = 1:4
D E G G#
Mercury aphelion (r)
q:r = 16:27
q:s = 81:320
r:s = 9:20
The time taken by Venus to seemingly orbit the Earth (i.e. a Venus synod) is currently 584 days, so that 5 Venus synods are equivalent to 8 ‘practical’ earth-years (of 365 days). Venus has a sidereal orbital period of 225 days, and 13 of these periods equal 8 practical earth-years. In both cases, the numbers composing these ratios are consecutive Fibonacci numbers, and therefore give approximations to the golden section: 8/5 = 1.6, and 13/8 = 1.625. Venus rotates extremely slowly on its axis: its day lasts 243 earth-days, or 2/3 of an earth-year (the same ratio as a musical fifth). Every time Venus and earth ‘kiss’, Venus does so with the same face looking at earth. Over the 8 years of the 5 kisses, Venus will have spun on its own axis 12 times in 13 of its years.
Thus, in 8 years Venus has 5 inferior conjunctions (when it lies between earth and sun) and 5 superior conjunctions (when it lies on the opposite side of the sun). Plotting either of these sets of 5 conjunctions in relation to the zodiac produces a five-pointed star or pentagram, the segments of the constituent lines being related according to the golden section. There is a slight irregularity, for the pentagram is not completely closed, there being a difference of two days at the top. This irregularity generates a further cycle, as it means that the pentagram will rotate through the whole zodiac in a period of about 1200 years. It is interesting to note that the pentagram was associated with the Babylonian goddess Ishtar-Venus, and that depictions of Venus as a five-pointed star have also been found at Teotihuacan in Mexico. In theosophy, Venus is said to be closely connected with our higher mind (manas), the fifth principle of the septenary human constitution.
Fig. 9.2 Teotihuacan: stellar symbol of Venus dispensing its influence
downwards towards the earth.
Fig. 9.3 The Venus pentagram.
According to theosophy, the key numbers to the solar system lie in a combination of the year of Saturn and the year of Jupiter, expressed in earth-years (Purucker, 1973, pp. 3-15). About 12 earth-years (11.86) make 1 year of Jupiter, and about 30 earth-years (29.46) make 1 year of Saturn: 12 x 30 = 360, the number of degrees in a circle and the number of days in an ideal earth-year. (Theosophy says that an earth-year oscillates above and below 360 days over very long periods of time.)
The whole process of evolution can be summed up as a descent of divine consciousness-centres or monads into matter, and their subsequent re-ascent to spirit, enriched by the experience gained on their aeons-long evolutionary journey. This process can be symbolized by two interlaced triangles, known as Solomon’s seal or the sign of Vishnu, the upward-pointing triangle representing spirit, and the downward-pointing triangle representing matter. Significantly, as Saturn and Jupiter revolve around the sun, they mark out two interlaced triangles around us every 60 years! The upward triangle is formed by their conjunctions and the downward triangle by their oppositions. Once again, there is a slight irregularity: after 60 years the conjunction does not take place at exactly the same point; there is a gap of 8 degrees, so that the interlaced triangles slowly rotate through the entire zodiac in a period of 2640 years. There are 432 of these 60-year Jupiter/Saturn cycles in a precessional cycle of 25,920 years.
Fig. 9.4 Conjunctions and oppositions of Jupiter and Saturn.
The sun is the heart and brain of the solar kingdom and the regular sunspot cycle is akin to a solar heartbeat. The sunspot cycle has a major impact on earth, especially terrestrial magnetism and the climate. Over the past 250 years its length has varied irregularly between 9 and 14 years, averaging 11.05 years. Sunspots peak shortly after Jupiter passes the point in its orbit closest to the sun. (The ‘ideal’ sunspot cycle is said in theosophy to be 12 years, so there would be one such cycle for each year of Jupiter.) The sunspot maximum does not occur exactly in the middle of the sunspot cycle. The ascending part of the cycle has a mean length of 4.3 years – very close to the figure of 4.22 years that would divide the 11.05-year cycle exactly according to the golden section.*
*Theodor Landscheidt, ‘Solar activity: a dominant factor in climate dynamics’, http://www.john-daly.com/solar/solar.htm.
To attribute the order, harmonious proportions, and marvellous recurring patterns in nature to pure chance is absurd. The pagan philosopher Cicero wrote:
If anyone cannot feel the power of God when he looks upon the stars, then I doubt whether he is capable of feeling at all. From the enduring wonder of the heavens flows all grace and power. If anyone thinks it is mindless then he himself must be out of his mind. (On the Nature of the Gods, Penguin Classics, 1972, 2.55)
However, the traditional theological picture of ‘God’ as a supreme selfconscious being who thinks, plans, and creates – even managing to make the entire universe out of nothing – is untenable. If ‘he’ is a being, he must be finite and limited, and have a relative beginning and end. Whereas if the divine is infinite, it cannot be a thinking being, separate from the universe, but must be one with it, as taught by pantheism. The divine essence would then be synonymous with boundless space, or infinite consciousness-life-substance. And since nothing can come from nothing, it must always have existed.
Materialistic scientists prefer to attribute the patterned order of the cosmos to ‘laws of nature’, with new ones emerging ‘spontaneously’ as evolution proceeds. But this explains nothing, for the word ‘law’ simply denotes the regular operations of nature – the very regularities that the term is supposed to explain! Patterns in the living world are sometimes attributed to genetic programmes, but this is merely a declaration of faith, since all that genes are known to do is provide the code for making proteins – not for arranging them into complex structures. Moreover, it is hard to swallow the conventional claim that genetic programmes themselves originated by chance.
If some patterns arose through random genetic mutations and natural selection, it would mean, for example, that the reason the golden angle is widely found in leaf arrangements throughout the plant kingdom is because it contributed to their survival. This implies that at one time most plant species did not embody the golden angle – but such a claim can never be tested. What we do know is that life emerged on earth incredibly quickly, that new and fully functional types of organisms have tended to appear on earth incredibly quickly, and that there is no evidence whatsoever for vast periods of trial-and-error experimentation along darwinian lines.
The theosophic tradition, or ancient wisdom, teaches that nature’s patterns and regularities are better seen as habits, an expression of its ingrained instinctual behaviour, of the tendency for natural processes to follow grooves of action carved in countless past cycles of evolution. For it teaches that all worlds, on every conceivable scale, reembody again and again. And behind these habits lies an all-pervading consciousness, the universe being composed of interworking hierarchies of intelligent and semi-intelligent ‘beings’ or energy-forms, from elemental to relatively divine. The order of nature also reflects the essential interconnectedness of all things, and the fact that the same basic patterns and processes recur on widely different scales.
All monads, or units of consciousness, are said to progress through a series of kingdoms towards a state of relative perfection in the system of worlds in which they are then evolving, before passing, after a long rest, into other world systems, on other planes. Humans, in their present stage of rebellious selfconsciousness, often succumb to the temptation to misuse their free will for selfish and shortsighted ends, creating discord and suffering. But it lies within our power to attune ourselves to the fundamental harmony of our inner, spiritual selves – sparks of the universal ‘Self’ – and to become voluntary coworkers with nature in the great cosmic adventure of evolution.