FEARFUL SYMMETRY: William Blake and Sacred Geometry, by Rod Tweedy

The Human Form Divine: Sacred geometry and its relationship to our physiology

 

Section 1: The Nature of Sacred Geometry

Measuring Urizen: The geometry of geometry

This first section explores what is meant by “sacred geometry”, studying and measuring its terms in relation to the study of physiology, the ‘science of life’. It therefore provides a sort of “geometry of geometry”. This seems apposite: the very idea of measuring is after all embedded in the word “geometry”, which comes from the ancient Greek words Geos, meaning “Earth”, and Metron, meaning “to measure”. The act or assumption of measurement is therefore contained within the system that is used to measure reality. Urizen thereby inscribes itself in the very utensils it uses to explore the deep: as Neil Postman acutely observed, “within every technology there is embedded an ideology” (Technopoly: The Surrender of Culture to Technology). These sorts of isomorphic (or “fractal”) repetitions and self-reflections constitute one of the defining characteristics of sacred geometry.

Sacred geometry is usually understood as the science and study of the fundamental patterns, shapes, forms, proportions, and ratios that constitute the basic nature of physical, physiological, and psychological reality. In ancient traditions, these geometries were considered ‘sacred’ because they recurred with such remarkable frequency and on so many different levels, thereby seeming to suggest a ‘hidden order’ to the world. As Skinner notes, “geometry and numbers are sacred because they codify the hidden order behind creation”. As such, they were sometimes considered to reveal the “mind” of God: as Galileo succinctly put it, “Mathematics is the alphabet with which God has written the universe.”

Repetitions of the mind of God?

For Pythagoras, sacred geometry was the most important science and underwrote not only mathematics and physics, but also the forms and structures of society, biology, and thought itself. Sacred geometric forms included the circle, the spiral, the ‘Vesica Pisces’ (the intersection of two circles, which forms a sort of lens or opening), the ‘Flower of Life (which consists of 19 interlocking circles); and the ‘Five Platonic Solids’, which are geometrical forms said to act as a “template” from which all life springs. The forms that constitute the original five Platonic solids (the tetrahedron or ‘pyramid’, the cube or hexahedron, and the octahedron, dodecahedron, and icosahedron) occur naturally in nature and throughout the crystalline world. The symmetry, structural integrity, and beauty of these solids have inspired architects, artists, and artisans from ancient Greece and Egypt to the present.

The intersection of two as Opening

 

The Secret Life of Phi

One of the most fascinating and striking examples of sacred geometry is known as the “golden ratio” (also known as Φ or “phi”, or the “divine proportion”). In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. For example, it exists when a line is divided into two parts and the longer part (a) divided by the smaller part (b) is equal to the sum of (a) + (b) divided by (a), which both equal 1.618. It’s perhaps easier to recognise visually than to describe:

The ratio has been used in painting and architecture, from the Pyramids and the Parthenon to the Mona Lisa and the Sistine Chapel (indeed such is its remarkable aesthetic beauty and attraction for us that our standard “portrait” and “landscape” frames themselves, within which most pictures are hung, and text files written, are based on its proportions).

The golden ratio describes predictable patterns in everything from atoms to huge stars in the sky – as with spheres, circles, solids, and spirals, it’s another fundamental measure that seems to crop up almost everywhere, including crops. This beautiful principle recurs throughout the universe, from ferns and flowers to seashells and even hurricanes. As such, it mathematically and aesthetically links each new generation to those preceding it, preserving the continuity of relationship as the means for retracing its line and lineage.

Numerically, the ratio is usually written as 1.618 – though like its famous cousin pi, phi is an irrational number, which means that its decimal digits carry on forever without repeating a pattern. It’s interesting that the principle responsible for so much repeating in the world is in itself one that goes on without ever repeating.

Meisner notes that the golden number wasn’t actually ‘golden’ until the 1800s. It’s believed that the German mathematician Martin Ohm (1792-1872) was the first person to use the term, and the first known use in English was in an 1875 Encyclopedia Britannica article by James Suller, on aesthetics. This is significant because, as with “sacred” geometry, calling a ratio “golden” immediately changes and indeed charges its intellectual and ideological nature, aligning it with ideas of magic, mysticism, and the divine. As we shall see, this association between ratios (or “ratio-nality”) and the divine, is one that is embedded deeply within our ways of thinking, not always positively. But there is no denying the remarkable beauty of the golden ratio. Phi is also embedded within the human body, including the visual system itself. It’s perhaps unsurprising that we see its patterns and repetitions as beautiful since its ratios and proportions lie within the very physiological structures that perceive beauty.

These fascinating geometries are also considered  “sacred” not only because of their recurrent use in nature but also because they are found in many religious and spiritual contexts, such as ancient sacred spaces, sites, and buildings. They are used, for example, to define the shapes and patterns carved into altars and cathedrals, and were considered to be essential in the building of many sacred structures such as temples, mosques, cathedrals, megaliths, and churches. These same patterns and ratios are also found in religious art and iconography.

Indeed, harnessing such “divine” geometry may be understood as a sophisticated and deliberate form of pattern recognition within these communities – a complex system of religious symbols and structures that both invoke and involve issues of space, time, and form, and illustrate the relation of the part to the whole. By studying the nature of these patterns, forms, and relationships and their connections, it was felt that insight may be gained into the mysteries. From the Gothic cathedrals of the late medieval era and Wren’s St Paul’s Cathedral in London, to the mosaics of Islamic art, the pyramids of Ancient Egypt, the mandalas of Vedic Philosophy, and the calendars of the Aztecs, sacred geometry has long been held to bridge the worlds of spirit and matter, through their energetic evocation of shape, vibration and symmetry.

In some ways, these ancient understandings about the importance of underlying patterns and ratios seem very prescient, resembling in some striking ways the ideas of contemporary physics, which hold that the fundamental nature of the material world is knowable only through its underlying patterns of wave forms.

As Lawlor notes, “the point of view of modern force-field theory and wave mechanics corresponds to the ancient geometric-harmonic vision of universal order as being an interwoven configuration of wave patterns.”  And as the great philosopher and mathematician Bertrand Russell observed, “What we perceive as various qualities of matter are actually differences in periodicity.” Perhaps nowhere are these patterns, structures, shapes, and symmetries more evident than in the case of the body itself.

 

Section 2. Sacred Geometry and Physiology

As below, so above

We’ve seen how sacred geometry underlies, and underwrites, many forms – from the geometry of cathedrals to the spiralling of galaxies and the golden ratios of aesthetics. But nowhere is its appearance more pervasive, or powerful, than in the forms and functioning of living things.

Physiology is the study of these living forms, functions, and mechanisms (from the Ancient Greek φύσις or physis, meaning ‘nature, origin’ – note the embedding of “phi” again), and sacred geometry can help illuminate our understanding of this in two important ways: first, in revealing the underlying proportions, shapes, and processes of living beings and their relationships; and secondly, in emphasising ideas of biological and physiological integration and interconnectedness, on multiple systems (molecular, cellular, structural, physical, chemical, biological, and formal). It therefore allows us to recognise how the body works in health, focussing on issues of integration, wholeness, and wellbeing, and how living things grow, respond and adapt to their wider environments and systems.

In nature, we find these recurring patterns, designs, symmetries, and structures everywhere, from the most minuscule particles to the greater cosmos. Biological life is inextricably interwoven with geometric forms, from the angles of atomic bonds in the molecules of the amino acids, to the helical spirals of DNA, to the spherical prototype of the cell, to the first cellular division of an organism’s life. Plants, for example, can carry out the process of photosynthesis only because the carbon, hydrogen, nitrogen and magnesium of the chlorophyll molecule are arranged in a complex twelvefold symmetrical pattern, rather like that of a daisy. Similarly, the specialization of cells in the body’s tissue is determined by the spatial position of each cell in relation to other cells in its region, as well as by an informational image of the totality to which it belongs. This spatial awareness on a cellular level may be thought of as the innate geometry of life.

In the field of botany, Adolf Zeising, in the nineteenth century, discovered the golden ratio in the arrangement of branches along the stem of plants, and of veins in leaves. From this starting point he extended his researches to the skeletons of animals and the branchings of their veins and nerves, to the proportions of chemical compounds and the geometry of crystals, and finally to human and artistic proportions.

One famous example of this is the pattern of seeds in a sunflower. The spiralling of the plants, and the unfurling of the leaves, follow the geometry of the Fibonacci sequence. And there’s a very close relationship between the Golden Ratio and Fibonacci Numbers (i.e., 0, 1, 1, 2, 3, 5, 8, 13, 21, … etc – each number is the sum of the two numbers before it), whose ratios echo that of the “golden ratio” (both being around 1.61).

Living plants seem to follow these deep symmetries not because they are “obeying” some transcendental or abstract “law”, but because they spontaneously deliver extremely efficient as well as very beautiful modes or forms of survival and being: the Fibonacci sequence, for example, occurs in many forms that have spirals – leaves, branches, and petals, as well as pinecones and pineapples. These spirals mean that new leaves don’t block the sun from older leaves (the rotating pattern facilitates this), thereby ensuring that the maximum amount of rain or dew also gets directed down to the roots. The Fibonacci Series is therefore linked to ‘branching’ processes, and to relational situations or spaces (e.g., leaf size relative to surrounding ones), and the optimization of light.

The same ratio sequence is also found in the spiral pattern and progress of storms, tornadoes, hurricanes – these all spin in this golden sequence of 1:1.6 – as well as in the shell-like spirals of galaxies, the classic shape of the chambered nautilus shell itself (the creature building this shell uses the same proportions for each expanded chamber that is added) – even the shape of the early human foetus follows the spiral-like forms, again suggesting their relevance to processes of growth, unfolding, and development, in both time and space. Fibonacci ratios appear in the ratio of the number of spiral arms in daisies, in the breeding pattern of rabbits (a classic symbol for fecundity), the ratio of males to females in honey bee hives, in the sequence of leaf patterns as they twist around a branch, and a myriad of places in nature where self-generating patterns are in effect. That is to say, these numerical sequences seem rooted in issues of reproduction, or “multiplication”, within living systems.

The Fibonacci sequence is the rational progression towards the irrational number embodied in the quintessential “golden ratio”. This rational “progression” towards an “irrational” number is perhaps metaphysically suggestive, pointing us to a dimension beyond rational numbers and ratios, in a deep physiological and psychological process that the German philosopher Hegel called aufgehoben, in which something is taken into the organism and transformed, or integrated (indeed, Hegel illustrates this key philosophical concept in The Phenomenology of Mind by reference to the development of a plant).

Unfolding: we are embodied in the wider world, and the wider world is embedded in us

This process, Hegel remarks, works on multiple levels – within consciousness (dialectics), in the way we learn (education), and in the way plants develop and change (physiology) – i.e., it is fractal. That is to say, we are embodied in the wider world, and the wider world is embedded in us: the differentiation between “human” and “cosmic” or “natural” breaks down, through these repetitions: this is the source of the power of da Vinci’s image of Vitruvian Man, which contains within the human body the entire process of the manifest unfolded world, which constitutes its (alienated) reflection.

These patterns also reveal how we are deeply embedded in, and participate in, much wider fields and contexts: we are trees, within forests, within landscapes, each repeating and belonging to the other. A striking example of this was found in a recent study examining the fractal geometries of a South American forest, in which it was discovered that the branching algorithm of a single tree directly mapped onto the larger surrounding area of the forest. As one commentator noted, “In simple terms the pattern of the entire area was fractally mapped onto one tree, showing that the part contains the information necessary to describe the whole. This amount of context reference is astounding, and points towards the ability of nature and natural forms to store and distribute information effortlessly at any scale.”

The implications of such holistic mirroring for the health and functioning of other kinds of organism and organisation are evident: “when we design and create a space that is based on the fractal geometry of nature, nesting shapes, forms and ratios in a physical construct, our senses are activated in a holistic integrated way which our brains and bodies find pleasing. This sensory ease facilitates the generation and experience of feelings that are the foundation of health and happiness.”

“If we extend this principle into other realms of sensory perception”, notes Michael Rice, “such as the overall shape of the space, the quality of light and use of colour, the texture of the materials, the sounds etc., we can increase and enhance the physiological and psychological effect dramatically” (The Science of Seeing). That is, these shapes and spaces aren’t “just” shapes and spaces: they embody and evoke deep relational responses and relationships, which is precisely why they were used so often in sacred architectures and practices.

To see a world in a grain of sand: isomorphic perception

 

Section 3. Sacred geometry and human physiology

From the moment that we’re born, we’re born into a world of geometries – into a world of shapes, spaces, fields, and relationships. Before conception, the ovum is a sphere, and exists inside another sphere, the female pronucleus or germinal vesicle. A zygote then splits into four cells and forms a tetrahedron inside a sphere. The next division yields eight cells and a star tetrahedron, which is also a cube. At this point it is the egg of life. The eight cells appear to be identical in all ways. The location of these eight cells is in the geometrical centre of the body, at the perineum. All of the energy fields around a human body are centred on these eight cells. Humans grow radially out from this form.

The golden ratio, just as we saw in plants and animals, is the foundational geometric ratio upon which the human body itself is built, and constitutes the geometry of the structure of the human skeletal system (da Vinci, Vitruvian Man). The musculature of the human body spirals up and around the body like a double helix, mirroring DNA and many other natural spirals found in nature. The DNA molecule itself is based on the golden section. It is 34 angstroms long and 21 angstroms wide. The ratio closely approximates phi. It is composed of pentagons around a central dodecagon and is built from rotating dodecahedra stacked on top of one another.

   

These principles and sets of relationships are fundamental in terms of health and well-being, and the potential of bioarchitecture lies in its ability to help explain how the brain and nervous system respond to stimuli from our environment.

Research has shown, for example, that when people view a natural and beautiful scene or geometrical presentation, the nerve cells in the opiate-rich pathway that connects the visual cortex to the parahippocampal gyrus become very active – that is to say, it is experienced as deeply “satisfying” and even calming. (We have a similar response when engaging with satisfying symmetries, structures, and sequences, as well as in processing satisfying mathematical and “rational” thinking activities). Some wonderful work done by researchers in Kyoto University in Japan demonstrates how the geometry of a space, place, or scene can suggest a pleasing picture to our subconscious which creates measurable effects in the brain resulting in feelings of connection, groundedness and well-being.

 

In part, this is due to the profound but implicit sense of relatedness and integration that recognising cascading patterns of repeated geometries or ratios evokes in us (and it seems in all living creatures).

Over the centuries, sacred art and architecture have utilised this effect, for example in mandalas (Sanskrit for “sacred circle”). These are complicated geometrical forms that represent or narrate the stories of various phenomena of human history, from our relationship with deities to the design of the cosmos. These colourful diagrams—which Carl Gustav Jung introduced to the West for therapeutic purposes—are notable for constructing allegories through a nearly impeccable segmentation of space. He found mandalas to be “a safe refuge of inner reconciliation and wholeness,” a pathway for productive self-reflection. Mandala symbolism historically has strong links to Jung’s individuation process and transpersonal psychology and are well known in the realms of spirituality, mindfulness, and wellness, again suggesting the powerful and subtle links between sacred geometry and psychological health and integration.

“the sense of relatedness and integration that recognising cascading patterns of repeated geometries or ratios evokes in us”

Contemplation and study of these special geometries therefore seems to have significant healing effects on both human physiology and psychology, suggesting their interconnecting and integrative nature. As one commentator notes, “The fractally harmonic context-rich visual experience creates a perception process which releases bio chemicals that manifest feelings of health, happiness and well-being. When the mind, which is in essence the process of regulating the flow of energy and information, is freed from the rigid boundaries created by boxed scaling, it can move through the doors of perception at will.”

Such forms of regulation and response do not have to be only visual: there is a powerful connection between healing and hearing which, for example, music evokes (music was one of the sacred practices of Pythagorean philosophy and mathematics). Pythagoras “heard” ratios: they were, and are, relational, emotive, sets of relationships and perspectives – our word “harmony” perhaps captures something of this qualitative aspect to sacred geometry.

in-built harmony: the music of the spheres, vibrational string theory

 

Ratios as relations

These are not just therefore lines and symmetries – but relationships, the formation and structure of underlying energies, in accordance with a deep sense of relationship between parts and wholes.

That is to say, sacred geometry is about energising space – not abstracting it (as so often done in ‘left brain’ systems of thought) to some lifeless, ‘eternal’ geometrical Realm of Pure Ideas – to the “Museum of Eternal Forms”, as Mark Johnson aptly calls it (The Meaning of the Body: Aesthetics of Human Understanding), which are then simply “imposed” onto being, like some medieval king issuing dictates. That would be the opposite of these emerging, evolving, dynamic processes and networks of nodal flow and fractal movement, which are all about the here and now. To truly understand sacred geometry we have to understand that everything in the universe is made up of energy and is in a continual state of transformation. What we call “geometry” and “pattern” is the effect, not the cause, of these underlying living forms. As Lawlor notes, “Functioning then at the archetypal level, Geometry and Number describe fundamental, causal energies in their interwoven, eternal dance”.

“sacred geometry is about energising space”

In his fascinating exploration of how the sense of beauty (which Plato had identified as a fundamental quality or expression of the Ideal Forms) has evolved, contemporary biologist Rupert Sheldrake shows how there is a striking resonance between the thing perceived and the structure of the perceiving consciousness, noting that every phenomenon has an intimate and necessary relationship to the whole of being (“holarchy”). This sense of resonance and relationship, he suggests, is what we understand as “beauty”. Holarchy also seems to emerge from the harmonic organisation of the universe, an organisation that allows the beings within it to be aware of the recurring patterns and ordering principles within all things. It also resonates strongly of course with the idea of sacred geometry. It’s deeply moving to observe this resonance, and even more deeply moving to participate in it.

“Ideal geometries … pervade organic form because natural law favours such simplicity as an optimal representation of forces”, observes Stephen Jay Gould. Every self-organising system has to coordinate and regulate its whole and parts, and every self-organising system exists in direct living relation to other such systems. Beauty ‘is’ this relation of the parts and wholes – of petals – to sepals – to stamens – to the unfolding flower itself. 

These recurring structures and habits are actually very simple, and very efficient, practical ways for living system to emerge and grow. The ‘beauty’ of a flower for example is usually only to do with repetitions of 3, 4, or 5 patterns (eg petals), or those involving simple radial symmetry: its elegance, as with any mathematical form, lies in its simplicity. Rather as with Friston’s “free energy principle”, these formations or formulations seem to rely on and be rooted in a sort of conservation of energy principle: if one path tends to work, it tends to be repeated. (For example, if you decide to cross a field, there’s probably only one or two routes that you are likely to take – the unfurling of fern leaves or sunflower seeds is rather the same. You are not blindly or mechanically obeying some “law”: the typical misapplication of the machine metaphor onto living being).

“These recurring structures and habits are actually very simple, and very efficient, practical ways for living system to emerge and grow”

These living symmetries and patterns of morphic resonance work on a basis of similarity – any pattern of activity that’s similar to a later pattern of activity in a self-organising system influences it across space and time. They are not the result of some “ideal” or abstracted, static pattern or template that has simply been imposed “onto” living beings (of which, in Platonic thinking, they are merely “imperfect” reflections or shadows). They are emerging, dynamic properties and ways of relating of living beings.

Like the Fibonacci sequence, one of nature’s most common forms of growth is growth by accretion or accumulative increase, in which the old form is “contained” within the new. This is called “gnomonic” expansion or growth, and is found in the spiralling trunks of huge eucalyptus trees, the horns of rams and reindeer, and the way the more permanent tissues of the animal body (such as bones, teeth, horns and shells) develop, in contrast to the soft tissue, which is discarded and replaced. It’s also found in mollusc shells, skeletal bones, and the cochlea of the inner ear. Even the human brain itself – the organ used to understand gnomonic expansion – seems to have evolved through gnomonic expansion, as Lawlor suggests.

“gnomonic” expansion: growth by accretion or accumulative increase, in which the old form is “contained” within the new

What all these geometries and symmetries, ratios and measurements, seem to point us to is the organ or instrument that is doing the measuring. The golden ratio, just as we saw in plants and animals, is the foundational geometric ratio upon which the human body itself is built.

Indeed, our whole body is a symphony of the golden ratio – as Leonardo da Vinci revealed in his famous design of the ‘Proportions of the Human Body’, showing its multiple occurrence and incarnation within the human form: in the ratio from shoulder to finger tips; fingertips to elbow; span of fingers to span of hand; span of fingers themselves; ratio of body to belly-button; the spirals of the ear and the ratios of the eye itself – the very instrument seeing all these ratios. Indeed, scientists have suggested that the golden ratio is to be found within the human brain, the neural system, the lung system, and our sense organs. It is even found in the helix of our DNA, and it forms the very rhythm of our heart-beat pattern. Given that these ratios are embedded into our very life, and into our own heart-beat, it’s perhaps natural that they are going to profoundly affect our aesthetics and the arts, and even our perception of the divine. As the Greek philosopher Protagoras observed, “Man is the measure of all things”, meaning that reality seems to exist in relation (or ratio) to the perceiver – that is, to the organs of the perceiver. Another golden ratio.

Indeed, many of the the measurements that we use to describe the “external” universe are actually rooted in the dimensions and ratios of the human body. As Lawlor observes, “the human body contains in its proportions all of the important geometric and geodesic measures and functions. The ancient Egyptian cubit, which is a time-space commensurate measure (from Latin cubitum ‘elbow, forearm, cubit’) is also 1/1000th of the distance that the earth rotates at the equator in one second of time, the foot, the fathom (a unit of measurement based on the span of the outstretched arms) – all these measures are commensurate with the size or movements of the earth”. And commensurate with the size and movements of the human body, both its embodiment and its measurer. This is true Geo-metry: Earth-measuring.

We measure the world in terms of our own embodiment within it, our own bodily relationship with it – in our “feet”, and “cubits”, and “spans” (the distance between the tip of the thumb and the little finger of an extended hand), and “fathoms”, and “digits”. And all these are of course relationships and ratios of the universe itself, as found in our form of it. As da Vinci observed, “the workings of the human body are an analogy of the workings of our universe”, and we can see these deep correlations in his celebrated drawing, in which all these modes of spatial being converge in the human mind-body. That is, the mind’s perception of this underlying pattern is itself part of the pattern. To understand what gives rise to these structures and patterns, these repeated ratios and numbers, we need to go deeper, into the nature of Being itself – which, as Blake saw, is relational rather than rational.

The Measure

 

Section 4. Sacred Geometry and the Left Hemisphere

Abstracting and imposing the Measure

We have seen how the study of sacred geometry, as well as regular geometry and formalised systems of abstract measurement, arose in the notably rationalising (left-hemisphere) cultures of early Greece (typified by Plato, Pythagoras, Euclid) and the equally rationalising cultures of the Enlightenment (Galileo, Kepler, Newton). We have also seen how the concept of “measurement” –  that is, of dividing and describing the world in terms of numbers and quantity – is itself contained within and assumed by the word “geometry”, and how these measurements, digits, and numbers are in fact rooted in the proportions and structures of the human body, including the brain. In other words, this has been a journey from external abstraction (the static ‘Ideal’ world of Plato’s Forms), to living embodiment (the embodied world of emergent processes). But in many ways, what we are seeing now in the work of Sheldrake, Johnson, Merleau-Ponty, and McGilchrist is a “return” to the pre-Socratic understanding of numbers as qualities rather than simply quantities, and of ratios as expressions of even deeper underlying relationships, which are as felt and affective as they are statistical.

The psychology of numbers

As McGilchrist notes, “numbers can either signify absolutes – a quantifiable amount, as in statistics – which would suggest an affinity with the left hemisphere, or signify relations, which would suggest an affinity with the right hemisphere. For Pythagoras, it was this regularity of proportion or relationship, rather than number in any absolute sense, that underpinned music and beauty – the music of the spheres, the natural harmony of the universe”.

Thus, when Pythagoras said, “All is arranged according to Number”, he was not thinking of numbers in the ordinary, enumerative sense. In addition to simple quantity, numbers on this deeper level are possessed of quality, so that ‘twoness’, ‘threeness’ or ‘fourness’, for example, are not merely composed of 2, 3, or 4 units, but are wholes or unities in themselves, each having related powers. The number ‘One’ for example, represents or signifies the principle of absolute Unity (and is often therefore related to God), ‘Two’ is seen as the original essence from which the power of duality proceeds and derives its reality (as in cell mitosis, or schizoid thinking), and ‘Three’ acts as germinative symbol: in India the triangle was called the Mother – that is to say, the triangle (dialectical Aufhebung) acts as the ‘mother’ of form.

The underlying relationships within the number Three

The relation of geometry and numbers to our hemispheres is of particular interest, given the Vitruvian nature of our form. It is the left hemisphere which is skilled at tasks like counting and reciting multiplication tables, and which typically converts unique things into abstract categories and types (such as Plato’s Forms), whereas the right hemisphere deals more with spatial abilities, and recognizes holistic patterns.

“Numbers can either signify absolutes – a quantifiable amount, as in statistics – which would suggest an affinity with the left hemisphere, or signify relations, which would suggest an affinity with the right hemisphere” (McGilchrist)

As Lawlor observes, “it absorbs spatial and simultaneous orders while the ‘left’ rational faculty is best suited to grasp temporal, sequential organization. The esoteric, functional aspect of Number, for instance, would be apprehended through the ‘right hemisphere’ faculty, while the exoteric, enumerative aspect of Number is apprehended by the ‘left’ ” (Sacred Geometry: Philosophy and Practice). Since Plato and Galileo, we live largely in a world sadly shorn and bleached of this deeper, qualitative and relational aspect of “Number”, and therefore of “geometry”.

Remember that, for the left hemisphere, space is not something lived, experienced through the body, and articulated by personal concerns as it is for the right hemisphere, but something symmetrical, measured and positioned according to abstract measures. (Lawlor)

Space as something lived, and experienced  this goes to the heart of what is truly ‘sacred’ about ‘sacred geometry’. It is to do with how space, and different sorts of space, make us feel. (Compare, for example, the expansive golden ratios of a Gothic cathedral to the box-like cubicles of office culture). Ratios themselves have an affective or relational, or intuitive, nature. The essence of the circle, for example, exists in a dimension that transcends the linear rationality that it contains.

The feeling of space: how space makes us feel. Blake believed that those who think that God is a geometrical shape (like a triangle, or circle), or exists in a geometrical shape, actually have a very cold and dissociated (left brain) idea of what God actually is. In his short poem ‘To God’, he addresses this sort of abstract, Gnostic, Platonic deity: “If you have formd a Circle to go into/ Go into it yourself & see how you would do”. He tries to get us to imagine what these spaces, these shapes, feel like. To live forever inside a triangle would be Hell. To ‘be’ a Triangle would be even worse. Jung’s more modern concept or image of the deep inner self or sense of identity as an abstract geometrical “mandala” shape is equally affectively de-centred and dissociated: it fails to see that ‘God’, or indeed the ‘Self’ is actually a personal Being. As Blake pointedly noted in The Laocoön, “God is not a Mathematical Diagram”.

In this, the issue of symmetry is particularly significant. “The universe is built on a plan, the profound symmetry of which is somehow present in the inner structure of our intellect”, the French poet Paul Valéry once remarked. But as McGilchrist has noted, this observation “is at one and the same time a brilliant insight into the nature of reality, and about as wrong as it is possible to be. In fact the universe has no ‘profound symmetry’ – rather, a profound asymmetry.” Few living creatures are actually symmetrical – indeed, the deader something is – mineral, crystals, snowflakes – the more symmetrical it becomes (this is also true of human faces – the more “symmetrical, regular, crystalline” faces become, the more frozen and inert they seem; and it is true also, as McGilchrist notes, of art and aesthetics – for example, the rather lifeless symmetries of Augustan verse: the universe frozen into a perfect heroic couplet, “with its closed, static, self-involved structure”).

Fearful Symmetry

The human brain and body themselves are of course profoundly asymmetrical. Symmetries are in part appealing because they have a sort of hypnotic, self-enclosed quality (particularly appealing to, and reflective of, the left hemisphere’s mode of operating), and it’s no surprise to learn that therefore symmetry  “was perhaps the ultimately guiding aesthetic principle of the Enlightenment.” Oddly though, McGilchrist adds, “symmetry does not appear in the phenomenal world, although it is approximated by living things, which on closer inspection are, however, like the brain, not truly symmetrical, and are constantly moving and changing.”

“Symmetries are in part appealing because they have a sort of hypnotic, self-enclosed quality”

What the phenomenal world delights in, and excels at, is variation, and evolution, and emergence, and asymmetry. It is not an imperfect “reflection” of some other world, and its sacred geometries are perhaps best seen and understood in this post-Darwinian, phenomenological framework. In this, they are a beautiful and “continuous reminder of our relationship to the whole,”, as Bruce Rawles nicely puts it (Sacred Geometry).

That is to say, these deep ratios actually arise and are rooted in even deeper processes. Within every ratio there is embedded a relationship, whether dyadic or parental, or comparative, or affective, or circular. Sacred geometry is not the cause of these repeated patterns and forms that we see all around us, but an effect of them – of the underlying dynamics, which are perhaps best understood as affective and dynamic, organic – the expressions (from Latin expressio, ‘press out, express’: literally ‘ex-pressions’) of living forms and beings, rather than mathematical processes and repetitions that they dutifully “obey”. 

π: relationship within ratio

The repeating patterns and forms found in nature, such as the helix and logarithmic spiral, the geometry of plant growth and the fractal, are products of the internal geometry of growth. The so-called “laws” of nature, as Sheldrake has suggested, are more like habits, that can be modified, altered, transformed, evolved. Indeed, this seems to be the very nature of Form: not to be eternal and ideal and static, but to continually form and trans-form. As Mark Johnson acutely notes, “If logic doesn’t fall down from the Platonic heavens above, then it must surely rise up from our embodied experience as functioning organisms within changing environments.”

Even the brain itself, as neuropsychologist Mark Solms has shown, has evolved – and is structured – as “the feeling brain” (the title of his seminal book on this), and this feelingness operates on both a neurological and a psychological level. This seems to be a return not only to the insights and values of the “right hemisphere” – the hemisphere which, as McGilchist has demonstrated, is the one that is actually more in touch with reality, however useful the left brain’s abstractions are – but also to the early roots of geometry itself. As Lawlor notes, “ancient geometry rests on no a priori axioms or assumptions. Unlike Euclidian and the more recent geometries, the starting point of ancient geometric thought is not a network of intellectual definitions or abstractions.”

Numbers are intrinsically kinds of relationship, as Pythagoras understood. “The irrational functions (which we will consider rather as supra-rational) are a key opening a door to a higher reality of Number. They demonstrate that Number is above all a relationship” (Lawlor, Sacred Geometry).  And what is a ratio? “A ratio is a comparison of two different sizes, quantities, qualities or ideas” (e.g., “a : b”), and as such is not only “the fundamental notion for all activities of perception, but also signals one of the most basic processes of intelligence in that it symbolizes a comparison between two things, and is thus the elementary basis for conceptual judgement.” And indeed for metaphor. 

Even angles are affective: in ancient symbolism we often find that the angle was used to designate a group of stable relationships controlling interacting complexes or patterns, such as in astronomy – where it was the angle which specified the influences of celestial patterns on earthly events. The ancient astronomers therefore designated the movement and position of celestial bodies through angular notation. The varied angular positions of the sun, moon, planets and stars were related to the cyclic changes in the natural world, such as moon phases, seasons, tides, plant growth, human and animal fertility, etc. It was the angle which specified the influences of celestial patterns on earthly events.

This is an idea which recurs in the contemporary science of heliobiology, as Lawlor notes, in which “the angular position of the moon and planets does affect the electromagnetic and cosmic radiations which impact with the earth”. In this way, he nicely observes, “we can appreciate the similar root of the words angle and angel”. 

“Even angles are affective”

“Numbers are the sources of form and energy in the world”, wrote Theon of Smyrna, “they are dynamic and active even among themselves almost human in their capacity for mutual influence.”

Numbers, Pythagoras asserted, can be androgynous or sexual, procreators or progeny, active or passive, heterogeneous or promiscuous, generous or miserly, undefined or individualized. “They have their attractions, repulsions, families, friends; they make marriage contracts. They are in fact the very elements of nature.” Indeed, Goethe’s novel Elective Affinities is based on the metaphor of human passions being governed or regulated by the laws of chemical affinity, and explores whether or not the science and laws of chemistry undermine or uphold the institution of marriage, as well as other human social relations – we often, for example, speak of people’s “chemistry” together. Pythagoras believed that numbers were themselves sacred and existed in their own right, and his esoteric way of thinking about them continues to profoundly shape the way we see them today: he divided numbers into “odd” and “even”, for example, and also into “masculine” and “feminine” – through ideas of “division” and “multiplication”.

This way of understanding sacred geometry restores us the lived universe, as a home, rather than a graph chart. From Plato to Galileo, we have been led to believe that God is a Geometer, that the “real” world exists in some dissociated and abstracted, disembodied space. This is the result of seeing God as the projection of the rationalising program itself (“Logos”), a projection shared both by the Christian theologians such as St. Augustine (who proclaimed that “Numbers are the thoughts of God”) and by Enlightenment mathematicians such as Kepler (“Geometry provided God with a model for the Creation”), who both present and portray God in their own image: the image of their own dominant (left hemisphere) cognitive circuits. Such ‘Gods’ do not actually “create” the world, as their iconography reveals: they merely divide and measure it (hence all their golden compasses). This is geometry as ideology: trying to convince us that this universe is actually composed merely of numbers and patterns, and that God is – as Blake contemptuously put it – merely “a Mathematical Diagram”.

God the Geometer: “Such ‘Gods’ do not actually ‘create’ the world, as their iconography reveals: they merely divide and measure it”

It is small wonder that so many humans, in such a dissociated, disembodied, and disconnected “ratio-nalised” universe, feel so alienated and adrift – a universe which resembles more a flow chart or Excel spreadsheet than a relational dwelling place, than a lived “space”.

“The damage that this hyper-rationalising, estranged reasoning has done to human subjectivity and meaning has been incalculable”. This is not God, but merely the projection of the Left Brain’s processes and programs back onto reality in the form of numbers and measurements.

The left hemisphere’s idea of a perfect being would probably be an isosceles triangle; indeed Plato’s “rationality” gets so extreme, or pure, in the Timaeus  that he actually maintains that the human body is made out of triangles. Similarly, Galileo might claim that the book of Nature “is written in the mathematical language, and the symbols are triangles, circles, and other geometrical figures”, but this language is also one that is always only retrospectively read back into nature: an after-effect, not a cause.

The damage that this form of hyper-rationalising, estranged reasoning has done to human subjectivity and meaning has been incalculable. But it is also based on a fundamental mis-reading of that “book”: the “book of Nature” is not written in numbers and abstract shapes, pace Galileo, but in relationships and living forms: in “betweennesses” as McGilchrist aptly calls them. And these betweennesses – these recurring and familiar ratios and resemblances – are what the sacred geometries continually remind us of, and lead us towards, a world that is only experienced through our relationships, through the lived human body.

 

Coda

Liberation: the Living Form, not the false Urizenic Mathematic Form of Plato and Galileo, it’s digital shadow

The recognition of these deeper underlying patterns lies at the heart of sacred geometry, and their ratios and relationships recur in many fields – including language. The term ‘meter’, for example, is found both in geographical space and in poetry: in kilometers and iambic pentameters (‘five-measures”) — the metrical “feet” of Shakespeare’s plays (“iambic” referring to their asymmetrical, “heart-beat”-like rhythm: one unaccented syllable followed by one accented syllable). Meter, and measure, and movement, are found in multiple systems, uniting perceiver with perceived, the system measured with the measurer of the system. To help illustrate this, and participate in its relationship, the structure of this present piece also follows the ratio of Φ: both in its form (the ratio of the page height to the breadth, which closely replicates Phi), and in its structure (the ratio of all the sections to the first three is 1: 1.618, signifying the relation of the basis of geometry to its elaboration).

As Lakoff and Núñez observe, “the connection between mathematical ideas and the world as human beings experience it occurs within human minds. It is human beings who have created logarithmic spirals and fractals and who can ‘see’ logarithmic spirals in snails and palm leaves.”  “Meaning is relational”, notes Johnson in his compelling study of the fundamental importance of embodiment, The Meaning of the Body, which has perhaps done more than any other book to help return us to the source of that meaning: “From the day we are brought kicking and screaming into the world, what and how anything is meaningful to us is shaped by our specific form of incarnation.” It is the body’s own tendency toward self-concealment that prevents us so often from seeing this – the ultimate ratio, perhaps, which lies in our own relationship to the world we see as beautiful.

the book of Nature is written in the language of betweennesses

Rod Tweedy, PhD, is the author of The God of the Left Hemisphere: Blake, Bolte Taylor and the Myth of Creation (Routledge, 2013), a study of William Blake’s works in the light of contemporary neuroscience, and the editor of The Political Self: Understanding the Social Context for Mental Illness (Routledge, 2017) and The Divided Therapist: Hemispheric Difference and Contemporary Psychotherapy (Routledge, 2020)

‘The Annunciation to the Shepherds’, from William Blake’s illustrations of Milton’s On the Morning of Christ’s Nativity Hymn, Stanzas 8-12 “At last surrounds their sight A globe of circular light, That with long beams the shame-fac’d Night array’d…” Note how Blake beautifully humanises the “globe of circular light”. For him, ‘God’ is not an abstraction, a geometrical shape (“circle”), or an impersonal electromagnetic force (“light”), but is both the source and embodiment of “the human form divine”, its “Image”. Amongst post-Platonic spiritual and esoteric thinkers he is almost unique in this way, and this essentially right-hemispheric, integrated perception lies at the beating heart of his whole work. “The Gods of Greece and Egypt were Mathematical Diagrams”, he once observed. He was not to make the same mistake: “He who sees the Ratio only sees himself only. He who sees the Infinite in all things sees God. Therefore God becomes as we are, that we may be as he is”.

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