"We live out repeated patterns in our lives. We experience the same story line over and over until the story has the conclusion of a change in our consciousness, a transformation of our way of Being.

Much of the Dreaming remains a mystery. But we have the opportunity to BECOME.

Ah Beloved, know thyself!

Know the weaknesses and the limitations. Know the talents and the abilities. Open to the possibilities of Magnificence. Let yourself shine in your own eyes. BE."

( from the text of “The Bud of Compassion: The Story of Love and Hope” by Ilyhana Kate Kennedy.)   

Please note with care: The images within ‘Natural Maths & Sacred Geometry’ that have been created by other authors are granted for educational purposes to this author only and should not be reproduced from this work for any purpose without permission from their respective Copyright holders acknowledged below and in the reference list.

The concepts in this work are based upon an understanding of number as related to traditional geometric symbols:

0 the circle, 1 the circle with dot in the centre, 2 the yin yang symbol, 3 the equilateral triangle, 4 the quadrated circle, 5 the pentagram, 6 the hexagram

Sacred Geometry is the mathematically precise dynamic that organises matter to take coherent form in space.

Sacred Geometry orders chaos.

Five is the number of creativity, change and energy.

The Golden Mean is inherent within the five-pointed star of the pentagram.

The ancient symbols above hold the keys to understanding the mathematical principles that apply to energy and solid form.

Each two dimensional symbol is representative of something in three dimensions.

For example, the symbol of the dot within the circle, which is the traditional symbol for the sun, is representative of the Sacred Geometry of the tube torus, the donut shaped energy field of the Earth, and the human being. The representation is a ‘top-down’ view, as though you are looking down on the north pole of the Earth or the crown of the head.

What is there in an intangible reality is this:

(diagram on right of tube torus courtesy of Bruce Rawles and Nancy Alese Bolton-Rawles on www.intent.com )

Each one of these curved lines is a spiral emerging from the central vortex, passing over the external rounded rim to enter the vortex at the southern pole, or vice verse. Where there is a spin in one direction, there is a natural spin within, operating in the opposite direction. This is understood as yin within yang, and yang within yin. You can see this concept of yin within yang represented in the Chinese yin yang symbol with the seed of one within the other. The funny shape is like looking at a cow horn type spiral side-on.

The chi energy spiral is formed from the Golden Mean mathematics of the five-pointed pentagram, symbol of infinite possibility. The arms of the pentagram cut each other in the ratio of the Golden Mean or phi, which is 1:1.618

The pentagram is the symbol of infinite possibility. Trace the continuous line of the formation of the star shape from any point and you can trace your finger around the lines that form the star perpetually, that is, to infinity. The particular angles that form the star allow these lines to have a continuous motion to infinity, over and over and over.

Follow the arrows on the star below:

The Golden Mean is the ratio that is derived from the proportions formed by the lines cutting each other. The cut is in the ratio of 1.618: 1.

When you cut a line in the Golden Mean Section, the ratio of the whole line to the longer section is the same as the ratio of the longer section to the shorter section. You can keep cutting in this ratio to infinity because one section is always longer than the other and where do we reach the infinitesimal end point of cutting?

The length of the whole line AC is in proportion to the longest part AB in the same ratio as the length of the longest part AB is to the shortest part BC.

The ratio is 1.618:1, that is, AC is 1.618 times the length of AB and AB is 1.618 times the length of BC.

(The figure 1.618 is a rounded off decimal. It appears from computer calculations that the end decimal can never be found and the decimal numbers change perpetually, perhaps to infinity. There is some research happening into the patterns formed by the numbers.)

( Sacred Geometry artwork courtesy of Jonathan Quintin on www.sacredgeometry.com)

It is this Golden Mean spiral that holds the mathematics of the energy of Chi, Abundance, Love, the Universal energy.

Dan Winter has done some wonderful work on the mathematics of this Golden Mean energy pattern. You’ll find his work on www.soulinvitation.com  Here are some of his diagrams:

intersecting spirals side-on and top-down

the donut shaped energy field side-on

two intersecting spirals within the field

Do you see the heart-shape of the wrapping spirals? And do you see the cup shape of the computer graphics of all the spirals? Is this The Holy Grail Cup of the Heart that runneth over with Love?

The four basic forces are yin within yang and yang within yin.

The traditional symbol for the Earth, the quadrated circle (cross within the circle), is representative of these four basic forces of spin. We might also call them active (yang) and receptive (yin), the active force moving outwards and the receptive force drawing inwards. Feminine energy is both receptive and active, the latter within the other. Masculine energy is both active and receptive.

Ancient cultures in the Americas used patterns to express the four prime forces as below:

Each of these forces of energy follows the blueprint mathematics of the Golden Mean Spiral. The energy of the flow of water and air abides by this principle, according to the research work of Theodor Schwenk, Victor Schauberger and Rudolf Steiner.

The six-pointed hexagram is related to solid form. Six is the number of form in the solid world.

 Six is a multiple of three. The basic building block of solid form is the tetrahedron, a pyramid with a triangular base and all sides the same length. The pollen grain below appears to have the same base form as the human embryo at four cells. The fourth cell of the human embryo is at the back, in the middle. Imagine joining the centres of the cells in three dimensions. Now you have a triangular pyramid called a tetrahedron.

(microscopic image of pollen grain courtesy of University Microscopy Imaging Center, University Hospital and Medical Center, State University of New York, Stony Brook, NY. www.umic.sunysb.edu/sempoll.htm )

The human embryo develops according to the mathematics known as the Binary Sequence, created by doubling the previous number. Cells split into two. When the tetrahedron form of four cells doubles to eight cells, the embryo takes up the mathematical geometry of the star tetrahedron, two intersecting tetrahedrons, as below. You cannot see the eighth cell, which again, is at the middle at the back.

You’ll find in-depth work on the geometry of the human embryo in the Flower of Life work of Drunvalo Melchisadek found at www.floweroflife.org.

The two intersecting tetrahedrons take up the geometric arrangement of the six-pointed hexagram when viewed in two dimensions.  The traditional symbol revered by the Jewish religion is the Star of David, the hexagram, which holds the blueprint Sacred Geometry to the human embryo at eight cells and to water crystals.

Water is essential to life.

Water also takes hexagonal form when frozen. The complexity of the patterning of the hexagonal crystal varies according to what is recorded by the memory of the water.

This is the work of Dr Masaru Emoto, which may be found on www.hado.net

A water crystal formed after the water was exposed to the words ‘Love, appreciation’ typed on paper and taped to the container.

The Platonic Solids are the basic building block three-dimensional shapes of life. They are six in number, being the sphere, the tetrahedron, the cube, the octahedron, the dodecahedron and the icosahedron. The geometric information within the platonic solids is like the invisible skeleton to solid forms.

(drawings courtesy of Bruce Rawles and Nancy ‘Alese’ Bolton-Rawles on www.intent.com )

All points on the five platonic solids shown above touch the inside of a sphere.

Within the mathematics of Sacred Geometry there are patterns of numbers that make up sequences, and these patterns direct the organisation of solid matter into the form that it takes. The Natural Universe has clearly observable mathematical and geometric patterns of organised matter that are repeated from microcosm to macrocosm.

We’ve seen that the binary sequence created by doubling is one of the principles that governs the development of the human embryo.

There is a branching pattern that is evident throughout the natural world. It is the basic pattern for the human nervous system, for tree growth, for many land and sea plants. A binary branching pattern ( progressively splitting in two) looks like this:

The Tree of Life is an ancient symbol common to cultures all over the world.

Theodor Schwenk discovered that seashells occupy space in the same geometric spiral patterns as flowforms within water, almost as though the forms of life that began in the sea were water flowforms becoming solid and taking a permanent shape. Isn’t that amazing!

For some beautiful images of branching patterns in nature and flowforms in water as well as some interesting theory, visit: www.j.mackenzie.dial.pipex.com

For images of the fractal pattern of chaos in the Mandelbrot set, visit Ralph Losey’s site: www.FractalWisdom.com/FractalWisdom/index.html  or take a look at www.mathjmendl.org/chaos  by Jonathan Mendelson and Elana Blumenthal.  

The single most important factor to clarify in Sacred Geometry is the misconception that the Golden Mean ratio and the ratio derived from consecutive numbers in the Fibonacci series are interchangeable. They are not!

They are two different things, and there is a reason for their difference that offers us a clue to the whole process of Creation or manifestation of solid matter.

It is a clue that helps to solve the riddle in the previous webpage, about how to move from the pentagram to the hexagram.

There is an incremental shift between the energy of the Golden Mean spiral and the organisation of a separate solid form according to the Fibonacci principle. I call it the ‘plus one’ factor. It is the emergent first solid matter of the form upon which the rest is based, the ‘seed’. You’ll find more about the Fibonacci Series on www.evolutionoftruth.com/div/fibomyst.htm  . You will find an excellent animation of the method of creating the Fibonacci spiral on that site.

A man called Fibonacci, who was a great observer of Nature and a mathematician, discovered that the numbers of flower petals and other natural forms often follow a progression of particular numbers that proceed as follows:

0   1   1   2   3   5   8   13   21   34   55   89  144  etc  to infinity

This is called the Fibonacci Series of numbers.

Each number is arrived at by the addition of the previous two. The higher the numbers, the closer the ratio between consecutive numbers approaches or mimics the Golden Mean ratio.

There is no point where it IS the Golden Mean. It approximates the Golden Mean.

It moves in a pattern of a little above and a little below consecutively, getting closer and closer to the Golden Mean, the higher the number values. To begin the process, there is a ‘plus one’ factor. It is possible to draw mathematically, the two spirals that originate from The Golden Mean Ratio and the Fibonacci series. They are not the same. They are very similar but not the same.

The Golden Mean Spiral proceeds from an infinitesimal point in the center to an infinite destination at its outer limit.

The Fibonacci Series spiral has a starting point in the center, of zero moving to plus one, and moves outwards to infinity.

What does this have to do with Creation?

The nature of the Golden Mean is to be able to create a fractal pattern, that is a pattern that can be repeated throughout Creation from the microcosm to the macrocosm, or “as above, so below” and has the potential for infinite possibility of Creation i.e. infinite diversity of existence in space and across time.  

But Creation isn’t an endless soup of atoms. It is a Universe of diverse forms, each of which has to have a physical place to begin, in order to be a separate form. Hence the knobbly bit in the center of the spiral nautilus shell. The pentagram is the geometry of infinity. It has no beginning place nor any end place. The arms are cut in a ratio that has the possibility of endless expression.

So why are the first eight cells of the human embryo not in the form defined by the pentagram, but follow the geometry of the hexagram?

The Golden Mean Section and the pentagram are the mathematics of the Dreaming of what exists. These mathematical principles define the energy of Chi. The pentagram is about energy flow, the energy that gives rise to and sustains form.

The Fibonacci Series and the hexagram are the mathematics of the reality of solid form.

If we are going to understand the nature of Creation, we must respect this difference. It holds the key to understanding the incremental shift between the Dreaming and the Creation, that is the difference between the perfect sphere of the Dreaming, and the donut-shaped, or slightly flattened sphere of the Creation, notably Earth itself.

Five and six are consecutive numbers. Might they not be symbolic representations of a Creation process?

It is impossible to find the geometry of the pentagram within the hexagram. So we have to ask, “If the Creator is infinite possibility and if the geometry of the pentagram is infinite possibility, and if the solid world has this plus one factor and the geometry of the finite world is led by the hexagram, how would or could Creation move from the pentagram to the hexagram?

There are many other clues in the patterns of Nature.

I once heard The Voice inside me say, “Everything you need to know about Creation is in the egg.”

So here is clue one: The egg has two shells, one soft and one hard, one inside the other. The human egg also has this dual exterior form, the region enclosed between the two cell ‘walls’ being called the ‘zona pellucida’.

Clue number two: the spiral shaped nautilus shell has a funny little knobbly bit where it begins in the center.

Clue number three: The builders of the pyramids in Egypt left behind another clue. They marked the places where two closely related spirals begin in relation to the architecture of the pyramids. They are not far apart.

The two spirals relate to the Golden Mean and the Fibonacci Series of numbers.

Clue number four: There’s an old saying that goes, “Turn ninety degrees and you’re out of here”, meaning that you are no longer on this Earthly plane of existence. Love with a big L. Have fun with the geometry!

The 3:4:5 ratio is the blueprint pattern that structures harmonics in the musical scale. The major chords are built on the first, third and fifth notes of the scale. There are four semitones between the first and third note, three semitones between the third and fifth note and five semitones between the fifth and eighth note. You might think about the musical scale as a 3:4:5 triangle, with doh being the common connecting point.

The 3:4:5 ratio possibly plays a role in conversion of geometry to frequency waveform.

To learn more about the 3:4:5 ratio, as well as the relationship between the pentagram and hexagram geometries in relation to energy and form, refer to the text “The Bud of Compassion: The Story of Love and Hope” by Ilyhana Kate Kennedy,  on CD-ROM from Shop Shelf and Services.

If you like visual presentations for learning, or if you'd like to introduce children to Sacred Geometry, EarthSoulScience has produced some excellent PowerPoint Slideshow presentations on CD Rom for viewing on Personal Computer, or for group viewing by data projection onto a screen. 

Just pop them in the computer and learn from all the beautiful work that people around the world are doing with Sacred Geometry as it relates to Nature.

The CD Roms include "The Sacred Geometry of Nature" with 57 slides and comprehensive notes for adult learning, and "Natural Maths" with 31 slides for self-paced or group learning for age 10 to 17 years. These quality CD Roms have been created by researcher, writer and educator, Ilyhana Kate Kennedy BA (Ed). Available from Shop Shelf And Services

For other sites that enable you to learn more about the basics of Sacred Geometry:

Visit Bruce Rawles’ excellent sites: www.intent.com/sg/index.html  and Sacred Geometry Design Source at www.intent.com/elysian/sqds.htmll and for a comprehensive guide to Sacred Geometry references, visit www.intent.com/bruce/links.html  and go exploring.

Buckminster Fuller Institute: www.bfi.org

Rudolf Steiner Archive: www.elib.com/Steiner

Fractals and chaos theory: www.mathjmendl.org/chaos/

The Barbury Castle “Mother of all Pictograms” www.greatdreams.com/plpath5,htm

History of Crop Circles

www.astramate.com/insights.htm  Also worth having a look at the retranslation of the Lord’s Prayer on this site.

Gerald Hawkins work on links between crop circles, geometry, sound and music www.lovely.clara.net

Ralph Losey’s wonderful site www.schoolofwisdom.com/index.html

Dan Winter on Heartmath www.soulinvitation.com/insideout/index.html

Dan Winter on bees www.zayra.de/soulcom/honey/honey.html

Gregg Braden’s sites are www.lauralee.com/azpbook.htm  and www.greggbraden.com/  and www.greggbraden.net

 Steve Gamble’s work on the transition from lightbody to solid body www.equilibra.uk.com

www.elucida.com  for music related to DNA codings and fractal algorithms.

Hans Cousto, Cosmic Octave Tuning Forks, Tone Acupuncture: www.planetware.de/tone/The_CoOc_Tuning.pdf

On volcanoes, a warning of overheating of the “nuclear core of earth” www.bioresonant.com/news.htm 

 Reading Resources:

On Victor Schauberger, look for these books: “Living Water” by Olof Alexandersson, translated by Kit and Charles Zweigbergk, (publisher unknown), 1990, USA and “Living Energies: Victor Schauberger’s Work With Natural Energy Explained” by Callum Coats, published by Gateway Books (Direct) ISBN: 0946551979.

Also about Schauberger’s work, videos entitled “The Extraordinary Nature of Water” parts 1 and 2 from Filmstream, PO Box 583, Byron Bay 2481, Australia.

Sproul, Barbara C, “Primal Myths: Creating the World”, 1980, Rider and Co, London. ISBN 0-09-143441-6

Schwaller de Lubicz, R A “The Temple in Man: Sacred Architecture and the Perfect Man” (Translated by Robert and Deborah Lawlor), 1949, Inner Traditions International, Rochester, Vermont. ISBN 0-89281-021-1.  

Bakhtiar, Laleh. “Sufi: Expressions of the Mystic Quest”, 1976, Thames & Hudson, London.

Ardalan, Nader and Bakhtier, Laleh. “the Sense of Unity: The Sufi Tradition in Persian Architecture”, 1973, The University of Chicago Press, Chicago and London. ISBN 0-226-02560-8 (paperback).

Melchisadek, Drunvalo. “The Ancient Secret of the Flower of Life: Volume 1”, 1990, Light Technology Publishing, Flagstaff, USA. ISBN 1-891824-17-1. Volume 2, 2000. ISBN 1-891824-21-X

Temple, Robert K G, “The Sirius Mystery”, Destiny Books, Rochester, VT. 

Braden, Gregg, “Awakening to Zero Point: The Collective Initiation”, Sacred Spaces / Ancient Wisdom Pub., Questa, NM. Also, “Walking Between the Worlds” by Gregg Braden, 1997.

Carroll, Lee and Tober, Jan, “The Indigo Children: The New Kids Have Arrived”, 1999, Hay House, Carlsbad, CA.

Churchward, Col. James. “The Lost Continent of Mu”, ISBN 0-8600-7029-8 Futura Publications Ltd, 1974. Printed in Great Britain by Hazell Watson & Viney Ltd, Aylesbury, Bucks. Also, by the same author, “The Sacred Symbols of Mu”, “The Children of Mu” and “Cosmic Forces of Mu”. James Churchward’s books are also printed by BE, Books c/- Brotherhood of Life, Inc., 110 Dartmouth 8E, Alberquerque, New Mexico, 87106, USA. ISBN 0-914732-19-6 (USA). www.brotherhoodoflife.com and the C W Daniel Company Ltd, 1 Church Path, Saffron Walden, ESSEX CB 10 1JP, England. ISBN 0-85435-293-7 (UK).

Lawlor, Robert, “Voices of the First Day: Awakening in the Aboriginal Dreamtime”, 1991, Inner Traditions, Rochester, USA. Also by the same author, “Sacred Geometry”, 1982, Thames & Hudson, London. ISBN 0-5008-10303.

Mowaljarlai, David and Malnic Jutta. “Yorro Yorro: everything standing up alive”, 1993, Magabala Books, Broome, Western Australia. ISBN 1-875641-05-X.

Lambert, Johanna. “Wise Women of the Dreamtime: Aboriginal tales of the Ancestral Powers”, Collected by K Langloh Parker, Edited with Commentary by Johanna Lambert, 1993, Inner traditions International, Vermont.

Halevi, Z’ev ben Shimon, “kabbalah: Tradition of hidden knowledge”, 1979, Thames and Hudson, London. ISBN 0-500-81023-0. Szekely, Edmond Bordeaux. “The Gospel of the Essenes: The Unknown Books of the Essenes & Lost Scrolls of the Essene Brotherhood”, the C W Daniel Co Ltd, Essex, England. ISBN 0-85207-135-3.

An excellent little book on soils which will probably need to be sourced from archives, and deserves reprinting:

“The ABC of Soils” by W.E. Shewell-Cooper, The English Universities Press Ltd, 102, Newgate St, London, E.C.I., 1959.