Geometry Essay
Very brief, cursory, summarized and truncated essay on Sacred Geometry and some of its underpinnings to organic life, time-space and music. This is an ongoing work in progress, updated and expanded periodically. It sort of dwindles off near the end.
(originally included as an appendix in my book “Evergreen”) , since extracted and redone
For thousands of years geometry was an intuitive science. It was taught orally first-off, then used in the arts, architecture, pottery and so on, now written in books. Geometry was and still is Sacred, it has a philosophical component. Only until relatively recently was it entirely utilized for its analytical aspect, only. Through a historical period of successive stages, we, overall, some groups and individuals more than others, lost direct knowing with our intrinsic knowing of our interconnectedness with all which surrounds us. It is a vast topic where any of these points below one may find an entire book covering it with in-depth detail from experts who dedicated their creative lives to the study. And they all admit that all of the research people have conducted, all put together even, are only scratching the surface. What is of vital importance to know first off is that the entirety of physical existence is motivated by a very few very simple archetypal ordering principles. And also, that all these have a spiritual counterpart – or more deeply, are, is a spiritual motivation – a non-manifestation which we do not ever see, made manifest in all that we do see. We, in this tangible realm go through process, cycles, they are somewhat periodic, ordered and may be studied and known, to the degree one is open to receiving and being a caretaker to some knowledge beyond what is presently understood by them. This is a gift to know that all peoples the world over share this very same intrinsic knowing. Humility is the first step forward. Humans in the past were in complete immersion with nature, Earth and all its happenings was known as a reflection and reification of the cosmos encompassing. We once had a geomatic orientation to our daily habits which gave us a reassurance that we were taken care of and belonged here. But through much time and many changing spaces, and for reasons to be stated elsewhere, we lost direct correspondence with the worlds’ other beings, and in a special way – with ourselves. We required certain sciences to re-adhere ourselves, our minds, bodies, our spirit, our soul, back into that once unified state. We all long for it in many diverse ways. This here writing on the subject, it is an ongoing work. Another aspect on this topic is put into my book “Evergreen”, and more to be added herein when I think of it, when the purpose comes up. Also look into my essay on this website concerning the “apocalypse” And again, as said, this little essay is just a glance. If you can follow it through…..
Sacred geometry cannot be divorced from its inherent philosophy, so it is approached from a balancing standpoint of both practicality and spirituality. One of the first postulates in classical geometry is the arising of all being though the inherent properties of the Vesica Piscis: Two equal circles overlapping, where the radius of both circles will meet at the center of the other, which is at the edge of the others circumference or perimeter. The first and simplest shape- polygon to be created – drawn, from this ‘eye’ shape or opening is the equilateral triangle. Out of that arise all the base geometric symmetries we can see in nature along with their respective mathematical symbolism. From the cube-4 to the pentagon-5, to the hexagon-6, heptagon-7, octagon-8, enneagon- 9, to the decagon-10; the Decad is the receptacle of the individual number-shapes 1-10. With their respective corresponding geometries these simple numerological entities serves to us a constant lesson of Nature’s prototypical alphabet concerning its most base archetypal intelligence. Numbers are voiced through geometric form. The symbology is the same fundamentally for all human cultures. The 10 base numbers are a Unity on their own and the central geometric point of each one contains the next proceeding number and also the previous- mathematics comes in here. They all contain the possibility of the prior and post; addition and subtraction simultaneously. They appear all at once in evolution along with their 3 dimensional counterparts. Numbers co-evolve with form or shape: the point- 1, to the line- 2, to the surface- 3 – the first polygon the equal angled triangle and its counterpart in form – the prime solid most simple shape – tetrahedron.
At the depths of sacred geometry and mathematics there are three sacred “roots”- √. They individually have equal significance in the general totality of their relatedness to life. These are, geometrically, life’s form giving functions. All three of these special roots growing from the same general root here called Unity. These three are also imaged as growing within unity or dividing unity within, each in its own particular way and all simultaneously. They are personified as the square, equilateral triangle and pentagon. Respectively these roots are threefold, fourfold and fivefold. They are sacred symmetries of the physical world wherein form with its functions is derived; themselves based on three essential archetypal states of the universe. Each has a fundamental process or mind-frame associated with it. The unifying root trinity of the physical/ generative; the mental/ formative; vital/ regenerative are mathematically represented here by the geometric roots 2, 3, and 5. They are relative to root 1- Unity, which is their source and destination, the encompassing infinite circle or sphere, the universal symbol of The Absolute Spirit, Great Soul. The measurements of the internal diagonal of the figure in ratio to its side is a comparison to its progenitor Unity=1. These three basic root shapes (polygons) of the cosmos- pentagon, equilateral triangle, and square are at the foundation of the virtually endless complexity of forms that we see which are all imperfections based on those 3 perfect ideals, which enigmatically are never seen. 
A “square root” √ in mathematics is, where the number when multiplied by its self will give the whole number required: Such as √ 2 = 1.414 x 1.414… All the square roots in mathematics can be seen to grow, or spiral one from the next. They are incommensurable, mysteriously never coming to end yet each somehow factor a unity unto itself. The √ 2 and √ 3 are widely found in the mineral, plant and animal realms while the √ 5 is pre-dominant in organic life, eminent in botany and rarely seen in the minerals. √ 5 is associated with the golden division – proportion, and Fibonacci sequence discussed below.
Here are their ideal regular shapes with associated internal ‘root’ number division seen on a two dimensional plane.
The triangle and square can expand indefinitely, tessellated and ‘close packed’ on a two dimensional plane while the pentagon cannot without other simple polygonal shapes filling in the spaces where the angles create gaps. Symbolically the √ 5 represents the living, the life that is all things and their movement; individual life must die because no being can be perfectly imitated. The former two √ s’ generate and give the materialized form to the living as it uses and befriends them in evolution, a base and substructure to live.
Now, one way to look at how these three correspond is this:
Root √2 of the flat two dimensional square plane which forms the foundation of the root √ 3 cube, is lifted, and raised up into it (the cube). This √ 3, the solid cube, is defined by and is containing both those sacred roots along with Unity. The cube is another unity. The next in this progressing line is the double square rectangle ( 2 dimensional) or double cube( 3 dimensional) which is another ‘rising up’, doubling and expansion of both roots 2 and 3 creating and forming the √ 5- another Unity, and simultaneously the √ 5 is doing the same to them, drawing them in. This solid is intimately related to the pentagon with its ‘’fiveness’’ opening up to its relation with the ‘golden proportion’. Again, we see a growing from within.
The three prime ‘sacred’ geometric orders performing as the trinity of formative, mental, vital forces in nature can also be defined and illustrated each with a particular spiral delineating an expansion of the same basic shape/ idea into a larger or depending, smaller area/ mind-frame. The meaning is inward as well as outward. This automatically signifies two motions within one motion whereas physical and spiritual balance is maintained. Each respective spiral can be described within its own rectangle explaining further the same principle of similitude of structure’s order and function. These 3 special examples increase by the same ratio at each successive stage or point of measure, replicating in a self-similar pattern. Overall, the spiraling principle elucidates an ongoing growth from within such as seen in the whirling of a plants phyllotaxis, electrons in a bubble chamber, vortex motion in liquids or gases, galaxies, hurricanes, the human ear, muscles of our heart and the spiraling tendencies of sound as it rises and dissipates.
It could be said that anybody, of any kind, or body part with its associated geometric proportions is representative of a certain life process and its inherent purpose which it stands for, the examples are endless. The proportion divided from unity has an exact process in unity with a purpose in relation to all others. Geometric patterns are manifested through the matrix of the cosmos; coinciding with it invisibly, made visible by it.
All things and their movements can be described geometrically. Many thousands of species display obvious geometric principles of various orders, so we will mainly deal hereon for purposes of brevity with the pentagonal properties of ‘five-ness’ with its implications to us, as it gives continual comparative parallels within the organic world of growth. The mathematics coupled with geometry show the pentagonal properties are both static and flux- it evolves its details through particulars while it’s principle-self stays the same. The specifics specify the general which encompasses them. The base is here as our body and is where we being our viewing from. Our own anatomy reflects and reiterates the 5; we count the world with our five and ten fingers, we have a five-fold physiology of our basic system of limbs and head which fits neatly into a pentagram; 5 base physical senses. We are representatives of the Pentads special properties, a microcosm in relation to the Decads’ macrocosm.
The Pentad, fiveness, the root 5 has intimacy with another supremely special geo-mathematical principle. Among many traditional names this relationship has been called the Golden proportion or ratio, the Divine Section, Golden section. Golden mean is another. It is intimately associated with the √ 5. The golden proportion is numerically defined in a ratio of 1 to 1.618 or .618 to 1. It is symbolized by the Greek letter PHI …
The Golden ratio can described as an association, it is not a number, but an relationship between accumulation; it works with many series of numbers, say 5, 7, 12, 19, 31, etc. When 5 and 7 are added they equal 12; then 12 plus 7 =19; 19+12=31; 31+19= 43 and so on. The ratio between successive terms will forever come closer and closer to producing true Phi- 1.618…..never ever closing at or stopping at an ended whole number. Rounded down it is usually written as 1.618. But the quickest way to find PHI is using a certain ideal series of numbers to illustrate this in a progression. We can see it like this: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, etc. Each number is the sum of the previous two digits. Where beginning with 2 which results from the addition of the two one’s preceding then into the 3=1+2: divide 3 by 2 we get 1.5. Then the 3 divide into 5 we get 1.66666., the 5 into 8 we get 1.6. Doing this mathematical operation over and over again we come closer to the irrational 1.618033988749895…, yet never ever reaching an end, it goes on forever, so far it is thus discovered. When graphed out: The higher the numbers, the closer the approximation to the un-reachable ‘Ideal’ whole number. It is logarithmic growth, derived from a geometric relationship of the small to the large as is the large is to the whole. It is a harmonic relationship of keeping the parts in balance with the entirety, which life continually works for. This regenerative PHI proportion is the perfect dividing of unity- cleaving apart and it is the bonding- cleaving together.
The flowers of geometry are prior to the geometry of flowers. All the cosmos is pre-proportioned to fit into whatever ecosystem while perfectly positioned to others and is structured in an order giving it a function which describes its purposeful placement. The entirety communicates through itself. No one thing can be alone, really. Organic growth twirls into form through the unified dual forces of the spiral coupled with the natural resistance surrounding its manifestation in the environment. Seen like this, all life has within and is a pattern of moving energy kept in check by itself and its surroundings. Every plant displays geometry in some way which depicts its specific intelligence relating to its environment. Animals do to. We may understand it as we are made up in many like images. All beings are representative of nature’s fractal powers spread through the course of existence. Look at the cap of an oak’s acorn, see the double spiraling pattern, look at a maple leaf, they always have five-part structure, as do its winged seeds measuring in line with the golden proportion; as does an alder flower/ catkin, for millions of years the same pattern; same with ferns as the young frond spirals out to maturity. All edible berries for us have 5 petals, as do most of our other fruits. Look at a strawberry, a pineapple, an artichoke, a cactus, a sunflower or daisy or any flower in the Aster family, a dandelion seed head, or plantain, or the any of the Brassicas- a cauliflower; cut open a cabbage, it is there as a spiral, as so in a celery bunch, as it is with leaf arrangement on many, many plant stems. Look at the spiral of a seashell, a hurricane, a galaxy. Look at the five-fold shape of a starfish, a sea cucumber or the sand dollar. Seen in the stems of all pines and the like as the pre-needles twist spiral-like around the stem before coming out to the leaves, then seed cone, where it is obvious, exploding out like a flower; these and other like phenomenon upon inspection reveal the spiral related to the mathematics of the pentagon. We know that the basic design of every pinecone has a double spiral of bracts twisting around its axis at its periphery and has this underlying archetypal proportion. We, as all living beings, are similar to and differentiated by these root patterns.
This esoteric principle of five-ness is made exoteric in so many dead lives in the form of fossils; in many that are preserved from deep antiquity, giving wonderful evidence to the fact that this is a certain prescript which has a special unchanging presence in our evolving universal order. Often made manifest through spiral formations such as in the DNA molecule; this in itself displays the inherent connectivity among all living creatures, another proof as to the innate intelligence common and shared and within all beings. When viewed from a cross-section cut away ‘inwards’ and down into its axis, its geometric pattern is seen to be a pentagonal spiraling structure creating a ten-fold symmetry. The designs’ function is known; as life is born, grows and evolves with this genetic, spiraling helix formation and has so for billions of years; so thus all share in this type of ordering. It is noted as obvious and ever present within the anatomy of the basest primitive organisms that there are fossil records of going back to where the past diminishes to unreadable. So we make inference. It is in archaic microfossils, also in their many uncountable descendants alive today: In the genetics of cyanobacteria- the blue-green algae, ever-present in the shapes of diatom plankton and the wonderfully symmetrical radiolarian sea creatures, it is present in primordial non-vascular plants such as the mosses and liverworts and is evident in the structure of their spores and the pollen of many vascular plants, and most obvious in the shells of snails and many sea shells. It is the physique of amphibians, in fish, both modern and ancient, in their scales like lizards and serpents, and in the dinosaurs. It is seen in insects, in butterfly’s, in birds. A fivefold patterning is seen in our own physiology on so many levels from our bone structural relations to in the design of our face. The human body, every animal in fact has a five-fold symmetry to its skeletal structure and is related by this in part. Dividing the human at the naval, our body is divided into 7 PHI sections; our skeletal growth contains and is a physical equivalent of the musical root harmonies in music. When measuring details of growth through time it is seen in the placement of the naval in its relation to the overall height of the body – the mark of memory to our natal life. This is, through our maturing stages ever moving up and down and is somewhat different between genders. Humans are but one of the many personifications of this principle of five-ness, we may be its penultimate, so far as we can see. The Golden proportion and the pentads regenerative properties of self-replication where the whole and the parts simultaneously relate to each other in the same way is the archetype of logarithmic mutual replication. It has been doing so through the agency of a Divine proportion from our world’s incipient beginnings. “From this never-changing proportion is enacted the play of infinite differences.” It is like a code-key, a talisman, a living doorway through which the spiritual formative forces work to continuously re-create our world.
Above are the 5 regular solids with their respective total number of plane angles:
Tetrahedron, Octahedron, Cube, Icosahedron and Dodecahedron. It can be said that the physical world is found to be reduced to these 5 ideals and reciprocally springing or built from them- in the archetypal ideal sense. A natural extension or evolution of these is the 13 Archimedean solids which are ‘truncations’ of the original 5, expanding the geometric surface into further subdivisions. These simple numbers we can see are symbolically related to the musical scales and the corresponding astrological time-space cycles. The regular 5 constituted a sort of ‘university’ in some ancient traditions which taught of a cosmic order, a cosmography easily perceptible through geometry and its subsequent orders throughout the world.
All the musical scales around the world broadly contain the same intervals in some form, however many slight variations, through instruments, language and region. Built into the modern piano keyboard in a kind of truncated, reduced order is this ancient conception of the spectral hierarchy of sound, form and color related through the Fibonacci sequence. The Chromatic hierarchy may stand for the music of photosynthesis played as form through the grace of the vegetal realm. We are composed of roughly 2/3rds water, as is the surface covering of earths oceans which has very near the same ratio of salt, as does our blood. The 2/3 proportion is a musical 5th. To illustrate further the unity of the Macrocosm and the Microcosm we will see here how they connect and reflect through number and archetypal form. Sacred geometry, in many instances is scale invariant in space and time.
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There are three revolutions of our celestial body we go by; one is the daily rotation of the earth around its axis, another is the yearly revolution around the sun and the third is circle of precession or the ‘Great Year’. The angles of each the regular solids correspond to an aspect or natural division of the “Great Year’, in astrology. The great year is also called the ‘Precession of the Equinox’s’. This cyclical phenomenon also called axial precession whereas the earth’s axis of rotation gradually, continually shifts, like a spinning top, creating a double cone describing a great circle through the traditional 12 constellations throughout a period of 25,920 years. This precession will oscillate one way for 12,960 years then the other way for the same time to round off a complete cycle. It is a retrograde motion based on a natural tilt and wobble of the earth’s rotation relative to the stars seen on the plane of the ecliptic. It will, through this time change our respective view of the ‘pole star’ which is directly straight up from the polar north as of now. This of course will occur at the southernmost pole as well. 12 constellations = 12 months; each is 2160 yrs.. Some ancient cultures used this Great Year as another calendar coincident with the yearly revolution of the earth and the moon cycle calendar. The vernal ’spring’ equinox is where everything in this great cycle is measured from, where the sun is perfectly east west everywhere on earth. Considering the Great year in a perfect circle of 360° where each 1 degree of arc = 72 years, the number of total plane angles in a tetrahedron is 720, and x 2= 144 years, is the octahedron = 2°. 3° = 2160 years, the number of total plane angles in a cube. The icosahedron contains 3600°, or 5 of those 720 year periods. The dodecahedron, the fifth regular solid has 6480° total. That is 9 of those 720 year periods. 6480 x 2 = 12,960 which is one half of that great year or 1/4th of a full cycle – this is one ‘season’ of the Great Year. Many ancient systems of measure were derived from this cosmography.
Keeping in mind the numbers above: There are 1,296 inches in a square yard. There are 144 sq. inches in a square ft. and 1,440 minutes in a day. This is just barely touching upon an ancient science of earth measure which really does go on and on in its corresponding details; encircling, covering the ‘natural’ method of measuring time and space. It shows an ageless and gentle proof that we live in an ordered Cosmos in coincidence with itself. Many our ancient forebears knew this. Our science is old. We can, only come to the same thing as they, if seeking an understanding, a comprehension with what it means internally to us in existence.
Nearing a half of one of these great years – roughly 11,600 years ago the earth, with humans of course, experienced a catastrophe. The Younger Dryas period is a climatic boundary between the Pleistocene epoch and the current warmer Holocene. It is a 1300 year period of intense cold, snow and ice, where temperatures initially dropped quickly, by many degrees- in a decade or so- mainly in the northern hemisphere; less-so in the south and near the equator. Temperatures stayed cold for the duration. It affected all things considerable, causing the extinction a great number of mega-fauna species and probably mega-floral which the animals depend upon. Other plants compensated to the new climate. At the end of the Younger Dryas another abrupt climate change occurred where temperatures went up nearly as fast as they went down, so its said. Ice caps and glaciers melted quickly, sea levels around the world rose, the jet stream and oceanic conveyor belts were affected and even more disturbance of ways of life incurred. Much physical evidence of this is found around the world, there is much in North America, in Europe and the Near East. The cultural evidence is found in the coincidence that at the end of this time period went parallel with the ‘Neolithic Revolution’; in the adoption of cereal grains for food as a steady form of civilization – due to abrupt climate change which brought on the need of drastic lifestyle change. Extreme social hierarchy was implemented as patriarchy supplanted matriarchy and a generally more egalitarian organic life-way. Territorial disputes and wars began, and have not ceased. Massive environmental destruction at the hand of humans went with these changes. This affected every way of life for every person on earth, though as said mainly in the north. We are approaching the other side of the Great Year now, and we can see in the waves of events rising, something new coming in. We’re being told of this juncture by the fact that this very information is now present for the common masses. It is in the media of art, all arts bespeak of it.
The fruits of philosophy, the natural outcome of science confirm to us that that which is primal is present and pertinent forever. Here is why: Evolution’s blossoming has cyclical tendencies; through our current co-mingling phase of experience we now know that its movement entails transformative sequential changes through time; Expositions are semi-ever-repeating in like-manner through great Aeons; Life develops and themes are recapitulated, yet never the same as before; A finale is variation of a new start, where the tail meets the head in the apparently endless circuity of existence; yet the spiraling tendency of consciousness moves on beyond to a renewed stage of being, however its path laid prior – Spirals in circles raying forth and reflecting to and from a center, which is everywhere.
There is a natural geometrical relationship of fundamental proportions within the present size of earth (interior circle) and moon (smaller circle on top). The moons diameter is 2160 while the earths is 7900 at the poles and 7926 at the equator. So for philosophical cosmology 7,920 will suffice as average to construct this diagram. When seen side to side, lined up together at their perimeters, their total diameter is 10,080. The radius of Moon is 1080, or in numerology its the same as 108. One of those near-ubiquitous numbers we notice in many phenomenon of time and space. This figure also lays out the ancient philosophical exercise of squaring the circle. The circle is female rounded intuition, while the square is male straightened rational. It is the spiritual work of the ancient cosmographer to blend and balance the two within the soul, thus the community, and the earth. This circle-squared is where both have equal area, equal importance. Also in this layout we can see an additional design fortifying our world’s cosmic relationships in the two 3-4-5 triangles (Pythagorean theorem) at the top; how these enclose the moon sphere resting on the earth square. The moon is 3 which would mean the earth a measurement of 11. The knowledge of these sacred numbers and geometry has been known since time immemorial. These in particular (and others) have been codified in the great pyramid of Egypt (the triangle within) and shows the PHI relationship intrinsic to the physiology of and in between Earth and Moon. Also, set into this Pyramids proportions is another incommensurable PI = 3.141592653589793238……….. =
As the height of the structure is taken as the radius of the circle, then the circumference of that circle is equal to the total measurement around the four sides of the pyramid. this the value of 22/7 = . This relationship was not always as now, nor will be in the future, it is for now, as we are, as evolution evolves.
Another example that coincides with PHI: The relation of Venus in its retrograde motion to earth as seen from the earth is a curving pentagram of conjunctions recreated every 8 earth years (13 Venus years) like a flower perpetually growing. Every 584 days the Sun, Venus and Earth line up 2/5ths of a circle farther around the zodiacal circle. We have 5, 8 and 13, all Fibonacci #’s. All the planets have there own geometric- numerical coincidences which relate to the earth, to other planets and all are harmonic and can be plotted and graphed out to study and remind us there is a music of the spheres. This study may bring us nearer to the supra-natural intelligence which has formulated this universal coincidence.
This is the ‘lunation’ triangle. We often allot 13 moons to a 12 month solar year and this figure which determines very near to the true number of months in a year- 12.368. This diagram divides the side of the triangle 5 by 3:2- a musical fifth. This is a further illustration of the Fibonacci sequence and PHI in natural patterns which has been known since antiquity. Through number we can see the pre-scripted interrelationship of how the macro and microcosm’s intermingle in time and space, how we may reflect on these truths and come to see how we fit in, and stay in. This triangle when doubled to form a rectangle is displayed in the central structure of Stonehenge. It also maps out two of the location points of two kinds of the stones which were used in its building; thus triangulates its location on a large scale. It is another proof of an ancient geomancy far exceeding that of today.
All these special instances and examples are overlapping, and interpenetrating. They rely upon each other, as we all do.The whole of our world and its universe, its structure, function and order is simple; thus our life-ways might be, if we center around the fundamental principle essences; then we may stay adhered to those archetypal pathways. Ancient, sacred geometry can again be a practice for this as it was in other times.
“Geometry is an essential condition of existence…representing as it does order in space.” Keith Critchlow
“The laws which govern the creation of things are the same laws which allow for their comprehension.” Robert Lawlor
“The structure of numbers is replicated in the structure of creation” Michael S. Schneider“
“We can, in a sense gaze into the secrets of the universe in so far as we are shaped by them” Rudolf Steiner
Books of recommendation for a thorough IN-DEPTH treatment of sacred and philosophical and practical geometry:
The Geometry of the End of Time and Sacred Geometry; Robert Lawlor
Islamic Patterns, Order in Space, Time Stands Still, and The Hidden Geometry of Flowers; Keith Critchlow
A Beginners Guide to Constructing the Universe; Michael S. Schneider
How the Universe is Made, and A New View Over Atlantis, and City of Revelation, John Michell
The Geometry of Art and Life; Matila Ghyka
The Power of Limits; Gyorgy Doczi
The Geometrical Foundation of Natural Structure; Robert Williams
The Little Book of Coincidence; John Martineau
Randal Carlson
