THE JOY OF PURE SPIRIT
Sacred Geometry – Class 12 – Main Lesson
We all know – both teachers and Class 12 students! – that the word ‘geometry’ means earth measure. WE also know that ‘sacred’ means holy. Together these two seem oxymoronical, holy being thought of more in relation to Unearthly phenomena. Alas, life often confounds human reason with spiritual reality. Rudolf Steiner repeatedly stressed the metaphysical nature of matter; of nature-created matter at least. For instance he described the geometric-formative principles of crystal creation as originating from the starry heavens – whence the God of Polystyrene!?
The sacred mission of this – sadly – last geometry lesson in twelve years of education is to lead the 18-year-olds across the bridge leading from the clamorous suburbs of Materialism to the luminous Mansion of Divine Creative Forces. We take our first step in the acknowledgement of higher worlds as inspiring our efforts by considering Mathematics itself.
Of the twelve members of the also sacred Subject Zodiac, maths lives in the House of Taurus, with its Steiner nominated Sense of Thought. The Master insists that sense-free thinking is one of the only paths we can psychically tread to comprehend – behold indeed! – higher number principles. In confirmation he ascribes Rationalism to The Bull in this 12 Philosophical Points of View. To quote one of his apposite aphorisms: “A great deal of philosophizing goes on without the remotest understanding of mathematical thinking. Philosophy is fundamentally impossible without a grasp of at least the spirit of mathematical thinking”.
So the supersensible vision of number must be preceded by a path of discipline in rational thinking. Taurus was the vernal sign of Ancients Egypt-Babylon-Chaldea-Assyria-Phoenicia-Mycenae-Crete-Etruria – et al!
Formal geometry was born in Egypt; the earth measure mysteries of the Hebrew Cabbala were also received in this period – 2907 to 747BC.
Much of the Sacred Geometry of today had its genesis – as it were – in Cabbalistic lore. This 2160-year Taurus (northern hemisphere) period was an Age of the Ten, counting from Ancient Saturn 1, Sun 2 and so on. Our ten fingers and the Ten Commandments, both of which relate us to the earth, stem from the same creative geometry-earth evolutionary principle.
In whatever diverse areas, the above peoples, including their Ancient Eastern equivalent, the Han Chinese (not the later dynasty, the term being generic, referring to the – mainly – southern archetypal Chinese to distinguish them from the northern Mongols), expressed in mathematics, there was always one thing in common; number was sacred – an expression of the pure mind of the Divine indeed.
Class 12 students are at last, with their incipient Ego forces, mature enough to comprehend the cosmological implications of the many concepts and revelations explored in these potentially luminous three weeks.
One medium of increasing the appreciation of the oft arcane phenomena is Art. As this is a geometry lesson, drawing is the art form most commonly called upon. Even the so-called “talentless” can produce beautiful works. As Goethe so perceptively stated:
“S/he to whom Nature begins to reveal her open secret, will feel an irresistible yearning for her most worthy interpreter – Art.”
Of the three high school math streams of Geometry, Finance and Numeracy, the first is a Will expression of the 3-fold soul ordering. Even the most elementary geometric principles must be actually drawn. This is preferably done by hand – in a school context at least – rather than by computer. Divine Will is incapable of manifesting electronically!
But there is a sagacious counterpoint to this geometric-will expression, in so far as, of the three Visual Arts of painting, sculpture and drawing, the last is an aesthetic expression of Thinking – Rationalism again. The veil of the mystery of drawing as a ‘thinking’ art is slightly parted in my high school Visual Arts book, Birth of Venus.
So we really have the whole human being involved in the creation of artistic and colorful geometric drawing – the subject is geometry, will; the artistry is the feeling; and the content, thinking. The fact that Sacred Geometry is programmed as a main lesson, with its head emphasis (middles heart, afternoon blocks, hand), reinforces the essential ‘rationality’ the teacher must bring to this Sense of Thought subject.
There have been four geometry high school mains prior to this Class 12 unit. These began in Class 8 with Shape and Space (see my book What is X? Why is Y?); then Trigonometry 9; Navigation 10 (both in Of Pine and Palm); and Projective Geometry in Class 11 (earlier in this book).
If a Steiner maths teacher is true to his/her craft, there will be aspects of Sacred Geometry in all the above, as there will be in the many geometry and allied units throughout the seven years of Primary.
This unit however is where we solely focus upon the evidence of Divine Intelligence in creation. For simplicity, and in an attempt to formalize the complexity of the phenomena we can potentially explore, the lesson could be divided into three segments: The Natural World, The Human Being, and Man-made. The ‘sacred’ element is the man-made being the assumption that God has moved through the head, heat and hand of the artist or artisan in the creation of the sarcophagus, temple, city or ‘other’ inspired construction. In a 3-week main lesson it would seem to be convenient to present one of the three each week; but only in the spirit of focus or emphasis, not rigidly or restrictingly so.
In every year of the students’ life, they have unconsciously revisited an epoch, era or civilization on the long and winding path of evolution; beginning as Ancient Saturn beings in their first year, Sun in the second, and so on. Class 12 is an exception, as these now young adults, already with distinctive and individualized Ego faculties, truly live in their own time, the 20th-21st Century. Actually Rudolf Steiner never actually recommended Sacred Geometry in his first Stuttgart curriculum; he rather culminated the ‘earth measure’ stream with Projective Geometry.
This was probably because a no-hold-barred revelation of the esoteric was inappropriate in the early 20thCentury, recognized as it now is as merely a preparation for the 21st Century. Steiner’s school was both a pedagogic climax to the 2nd Millennium, and a forerunner of the 3rd; the New Moses Age, as he called it. This was a reflection of the 2nd Millennium BC, the original Age of Moses, an era of Natural Science. Moses was a ‘master of weights and measures’; by extension, of geometry.
As we move from the 2nd to the 3rd Millennium, we cross the Rubicon from a Moses, to the New Age of Abraham, a thousand-year epoch of rather Spiritual than natural science. Sacred Geometry is an ideal representative of this New Science imperative, and 18-year-olds, living exclusively in their own era as they are, are perfectly placed to educationally manifest it.
Most pre-industrial age civilizations expressed Sacred Geometry in their structures, artifacts and even political geography – some more perfectly than other! Cold intellectualism, and later materialism, exiled the spirit from our modern world; its dominant form, the right-angle, being an expression of matter only. The square exists in many ‘holy’ geometrical – and even natural (especially minerals) – principles, but usually in a subordinate role; as an element in one of the popular constructions of a logarithmic spiral for example. Being inspired by the numerical profundity of the past, I attempted to ensoul my own house by designing Sacred Geometry components throughout. An example was a central triangular window in the Eurythmy Room, which, instead of the two base angles being arbitrary, I made them 23 ½ °. This accords with the angle of inclination the earth presents to the sun – a divine relationship between the terrestrial physical and the cosmic etheric, as Eurythmy surely is.
The top angle of the triangle is 133°, with its Christ-numeracy implication – the number of the Etheric Christ. This can be considered a marriage of 1 and 3, of Unity and Trinity; or 10 and 33, the Son of Man and the Son of God, of earth and spirit. As the earth presents the two solstice angles on its annual orbit round Ol’ Sol, the 133° is created perpendicularly above the sun. A ‘window’ to even higher worlds?
23 ½ ° is also the angle formed between the pituitary and the pineal glands – another earth-sun relationship; the 133° being at the fontanelle – the ‘fountain’ of the Spirit. These divine head angles are the ‘earth measures’ active in the evolution of consciousness.
Whether this Scared Geometry element made any difference to the sublime art studied and performed in the Eurythmy Room, I cannot know. (Perhaps it helped protect it from the malignant chaos of the Mordor-like neighbor without?!) However it certainly felt good – self-fulfilling spiritually perhaps, but better than no spirit at all! There were many other examples of deliberate, and otherwise, number principles in the house that Alan built, like various wall panels being on a scale of 1:1.616 – the famous and so-satisfying Golden Section. Then there were 12 verandah posts (not 11, not 13), representing the 12 Apostles, holding up the 3-fold roof, itself symbolizing Thinking, Feeling and Will! (See more of these in my book Many Mansions; especially in the chapter “The Spiritual House”.) While on the Golden Mean; a few examples can be given of its expression in the natural world, such as a Black Skate structured on multiples of Golden rectangles:
If it’s possible to consider a house as a Being, one can reduce a being, a human being, to an abstract, numerical even, principle. Both the body and head are essentially geometrical. This, in Steiner terms, is based on the hexagram – the Star of David. The divine scale of a person can be determined by drawing over a photo or painting the 6-pointed star.
This begins with the top point of the ascending triangle being the larynx, the two triangle arms descending to both hip bones – join these two points to create the baseline of the first triangle. The descending triangle starts with the shoulders, the two-triangle arms meeting for the lower point at the genitalia. The six points for the had hexagram are crown and mandibles for the ascending triangle, and temples and chin for the descending.
Further back still, we have not just incarnate, but mineral man indeed. This was in the form of Noah’s Ark; not just an improbable animal transport, but the carnal human being itself. This mineralizing process was complete only in the final ‘days’ of the Flood – Atlantis. The Ark’s (L. ‘box’) dimensions was 300 cubits (units) long by 50 wide by 30 high.
Even a cubit is body-based, being the length of a man’s forearm, about a foot! The Ark was 3 stories high (head, chest, abdomen), with a window at the top (the aforementioned fontanelle); allowing old Noah to retain contact with his divine protector, The Lord. After all, this is the purpose of the baby’s fontanelle, the lifeline to the spirit; which is of course dimmed (as was Noah’s) as the holy gateway closes.
This 30;5’3 scale is that of a coffin, the final resting place of the material body; the soul and spirit having quite different – moon and sun – dimensions. From the square or cube, to the circle or sphere; from earth-matter to heaven – or to the sun actually. In geometry, but especially so with Sacred Geometry, we begin with the perfect, the circle-sphere, with its sublimely undifferentiated numerical structure.
From this we may proceed by drawing out an obolid, or egg-shape, seen expressed in so many ways in the natural world. This can be from eggs (of course!) to the silhouettes of many pine trees, and their cones – and even in the flower buds of the aptly-named Sacred Lotus. The earth ‘glove’ itself is a subtle oblate sphere, having an equatorial diameter of greater length than that of the polar.
Oblate, as in ‘oblation’, means “to bring forward” (lengthen); its spiritual implications being seen in the Christian Eucharist, the Oblation being a ‘bringing forward’ of the wine and bread. Needless to say all drawings of the selected phenomena, must be beautiful and be constructed using precise mathematical formulas. The egg of course is ubiquitous in creation legends – Sacred Geometry truly is the Alphabet of the Adept.
A modern Spiritual Science researcher, Robert Gilbert, an ex-Instructor for the US Marines in Nuclear-Biological-Chemical Warfare Defense, asserts that all these ‘weapons of mass destruction’ are based on – and thankfully can be countered by! – geometric principles. So Sacred Geometry actually has practical applications, for both good and evil. The former is certainly widely applied in Anthroposophical endeavors; from architecture to water purification to Biodynamics to health care.
A path curve surface is one in which an object’s surface contours remain stable and unchanged under conditions of numerical transformations. The archetypal path curve form is the egg. These numerical sequences channel living spirals through and around the object. Laurence Edwards, Steiner geometrician supremo (see Projective Geometry earlier) initiated research which revealed that the obolid is interpenetrated by three sets of cortices; respectively aligned to one of the three X, Y, Z, 3-dimensional axes. Each vortex comprises ten concentric nested vortexes – nine contained by one. Ten, in occult numerology, is known as the ‘gnomic number’, that of mineralization, as occurs in the shell of an egg.
Strangely, the orientation of the three axes is independent of the actual alignment of the material egg! Nine is rather the ‘number of cosmology’, as found in the Nine Hierarchies. These Nine Choirs of Angels sing into life the initially chaotic (albumen) content of the egg. The egg is interpenetrated by supersensible vortexes, which create form fields for the three axes. This principle occurs in ancient representations as the (etheric) serpent winding around the cosmic egg, and other similar images. Our study naturally unfolds with the number principles found in the vortices itself. Another Steiner geometrician, Theodore Schwenk, has achieved impressive breakthroughs in this sphere, even importantly contributing to the runaway global consciousness-awakening book, Chaos, by James Gleick. The following is an example:
Schwenk’s book is the more beautifully and accurately titled, not merely ‘Chaos’, but Sensitive Chaos. Reference material of this nature, and even of the more desiccated mathematical descriptions, should be accumulated begore launching the good ship Sacred Geometry.
The vortex is the natural pathway of the etheric, of life energy in man, the world and the universe. This spiral form, in all its manifold diversity, is also omnipresent in a more spiritually perceptive ancient world. It did take the modern power of rationalization however to reveal the number laws underlying the many spiral forms. Indeed most vortexes are represented – accurately – as two counter-rotation spirals, a 2-dimensional depiction of a 3-dimensional form. This is easily seen in the pinecone, and pineapple even. These spirals are rivers of life; energized conduits from the supersensible to the sensible – and back again!
The clockwise form invests matter with life, the counter-clock draws life out of matter. The Buddhist clockwise swastika (meaning ‘well-being’ – or alive!) is a minimalist symbol of the first. The Nazi ‘Agathi’ (wrongly called a swastika), meaning ‘cave’, with its sub-earth or anti-light implications, is the second.
This is actually a death process, as when the ether body departs (anti-clockwise from below, clockwise from above!) from the physical at death. This has been described by many who have had ‘return from the edge’ experiences as a dark spiral tunnel leading to a land of light.
Rudolf Steiner alerted researchers to the potential of the spiral, its symbolic and practical significance, with descriptions of the German wirbel. This is a counter-rotating form of spiral arms which – almost – join at a central point. It is commonly found in everything from the energy fields emitted by atoms, to spiral galaxies. It is the symbol of Cancer, the first “Initiative” (creation, not cultural) sign of the Zodiac.
A former NASA atmospheric physics researcher, Barbara Brennan, discovered seven ‘energy centers’ of the human body. These were consistent with the chakras (‘wheels’) – the Lotus Flowers of Eastern mysticism. Not only that, these life vortices were the same in number as ascribed by ancient teaching. For instance, the 12-petal Anaharta or Sun-Heart chakra was energized by 12 vortices; the same with the 2-petal Jupiter chakra of the face – and the other five as well. Um, I’m not sure how she measured the cortices in the 1000-petal (meaning ‘illimitable’) Crown of Saturn chakra?! Brennan found that the single chakra containing these numbers was – naturally – a spirit-suctional clockwise vortex.
There was found to be both a front and back vortices for each energy center, evidence of the counter-rotating double spiral. Of course the celebrated DNA double helix is constructed thus; though it is far more beautifully free-form than depicted in those rigid cast-plastic models and drawings in science journals As is the case so often in science, they have to kill something, recuing it from the majestic to the mundane, before they can imprison it in thought. Brennan’s illuminating book, excellent as a resource for the Sacred Geometry lesson, is called Hands of Light.
Other biology researchers have found that the precise geometry of vortices express pathologically or passively, benignly or malignly. This has he implications for medical science; so variant an approach from the ‘drug and bug’ mentality of mainstream allopathy!
From the vortex, this lesson can range freely, exploring geometric curiosities wherever they be found; such as the synchronicity of the following: Japan and New Zealand are north-south complements. They are both island chains of about the same size, shape and latitude. Both are eastwardly adjacent to their western neighbor continents; they are both justly proud of their highest mountain, both volcanic and both forming an axis, a spiritual-physical fulcrum, to their countries and cultures.
How easy it is to overlook the fact that those two famous prominences are a tiny 12-meter difference in height – Fujiyama 3776, Cook (Aorangi) 3764! A few hours work with a work-for-the-dole team would – either leveling Fuji of mounding Cook – make them exactly the same height!
This of course satisfies the human hunger for absolutes; nature is, thankfully, more subtle, allowing the Spirit room to move in her timeless creation. The above is an example only; see how many geometrical correspondences first yourself, then the students, can find. An obvious one is the Arctic and Antarctic (‘bear’ and ‘non-bear’ – or in more esoteric understanding, physical and metaphysical).
The first is an ocean, the second a continent; the first is about 3000 meters deeps, the second the same in eight – they are both about the same size and shape – and so on. For this amazing globally significant balance, did mighty Atlas scoop out a hole o the top of the world, and dump and fill on the bottom!?
Another area where art and mathematics entwine is in Charters Cathedral, an object of Steiner research for decades. Almost every element of this magnificent building, particularly its cryptic ratios, express higher principles of some kind. One example is the 3-tiered nave-choir-vault. Details can be found in Louis Charpentier’s wonderful book The Mysteries of Charters Cathedral. The scale of the architectural elements is based – aptly – on another scale, the musical; in the Gregorian the three ‘good notes’ are RE-FA-LA. The ogive is constructed around the points of a pentagram; symbol of the etheric archetype of man.
A fun digression in a Sacred Geometry lesson is the mysterious phenomenon of Crop Circles (though many are not circular). Who or whatever create these – mostly near sacred sites in Britain, like Stonehenge and Avebury – expend an enormous amount of energy and talent for absolutely no profit – or recognition even. The idea of crop circles being of the nature of mere pranks has been discounted in the minds of all but the most myopic of sceptics. The main reason for a more credible explanation (?) is the precise and imaginative geometry. One attempt by a large newspaper to discredit crop circles by making one of their own was laughable in comparison to the legion truly exquisite designs that regularly and enigmatically appear – to the local farmers’ consternation – each year.
The sacred is so often ‘enigmatic’, this is certainly the case with the Five Platonic Solids, first formally unveiled to an uncomprehending world in Ancient Greece. These are thought to be the archetypes of material Creation. The Octahedron is the gaseous or ‘air’ representative of the Famous Five; with the tetrahedron, fire; hexahedron, earth; icosahedron, water; and dodecahedron, ether. A perfect example of the octahedron is the fluorite crystal. Have some of these beauties around for the students to play with to experience their mystery. Of the seven soul-planetary substances, yellow fluorine is of Mercury; this being the most easily bonded – most capable of movement – of all 92 chemical elements. Mercury is also of course an air or winged divinity. Steiner recommends that the students create interpenetrative drawings of these classical forms…
Half of the octahedron, an 8-surfaced triangular form, is of course found in the Giza pyramids. Actually pyramids (akin ‘pyre’, fire) are found in ancient cultures right around the world. There was even a huge stepped pyramid in Australia, the controversial Gympie Pyramid. (Esoteric content like this may seem a far cry from conventional maths teaching, but does provide the spice for the often-dull fare of soul-less number.) This protuberance, albeit tree covered, severely eroded and desecrated by souvenir hunters, was 60 meters high with sides of 125 meters.
Books on the numerology of the Egyptian pyramids are numerous; the most awe-inspiring feature of these eternal icons being the precise mathematics employed by their supposedly ‘primitive’ creators. This is also a good opportunity to explore another ancient mystery, the connection between the Northern Hemisphere Taurus (2907 to 747BC) Egyptian and Han Chinese civilizations. Han here is a generic term, referring to the original people of southern China, not the later Han Dynasty.
The Han built great pyramids too; but in a country as big – and inscrutable! – as China, their scale, in an endless, featureless plain, is less impressive than their Giza counterparts. One giant of seven pyramids at remote Shensi is reputedly the largest on earth, with a height of 300 meters and sides of 500 meters. This is twice as big as the Great Pyramid at Giza. The Shensi Pyramid is made of earth, not stone; but was originally stone capped. Like the Gympie Pyramid (unlike arid Egypt), the Shensi Pyramid has been subject to human, foliage and water erosion; hence has lost some of the crisp integrity of form.
One of the mathematical mysteries is a number found inscribed inside the Great Pyramid – 16944. This improbably happens to be the number of minutes of arc of longitude between the Great Pyramid and the tallest at Shensi! Both have complex astronomical alignments. Numerologically this number reduces to 6, the Number of Nature.
The half tetrahedron of a pyramid is based on the vision that the other half – the secret, spiritual half – is below ground. Most pyramids have shafts to sacred places penetrating this no-go zone. In this spirit of the mathematically sacrosanct, I conclude this last of a multitude of meaningful and exciting geometry lessons over 12 years, with Rudolf Steiner’s golden words on one of this favorite subject. In speaking of happening upon a geometry book at school, she said: “It brought me an inward joy, to be able to grasp something purely in spirit. It was in geometry that I first learned to know this joy.”
The Great Shensi Pyramid
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