In this article we will continue our discussion of the geometry of the human body. In the last article we saw that the golden ratio is the foundational geometry of the structure of the human skeletal system. We also saw how the musculature of the human body spirals up and around the body like a double helix.
We will now look at some other ways the golden ratio and other geometries show up in human anatomy and physiology.
Professor Hanno Ulmer from Innsbruck Medical University in Austria took blood pressure readings of over 160,000 people and published his report in the British Medical Journal in 2009.1
He found those with a ratio of 1.618 between systolic and diastolic blood pressure counts were less likely to have a fatal heart attack.
He found those who had a higher ratio, or 1.7459, were the ones who suffered fatal heart attacks.
In the next article we will focus specifically on the human heart, as it is full of unique geometric aspects.
The trachea divides into two main bronchi. The left one is long and the right one is short.
This asymmetrical division continues into subsequent divisions of the bronchii (Fractal branching).
All these divisions follow the golden ratio. The short division to the long = 1/1.618.
Plato writes in Timaeus of the human body: “When frightening events are anticipated and when passion is being stirred, the heart leaps, but the gods knew in advance that it was fire that was going to be responsible whenever any part of the body that is subject to passion swells like this, and so, as a remedy, they had implanted the lung in the body. For the lung is not only soft and bloodless, but it also has pores drilled inside it, like a sponge, and so, as the recipient of breath and drink, it cools the heart down and provides relief and comfort from the heat…
This is why they wrapped the lung around the heart like a kind of cushion, so that when passion is most active there, the heart has something soft and cooling to leap against. For the calmer the heart is, the more it can join passion in serving reason.” 70 c-d
Dr. Jasper Verguts, gynecologist at the University Hospital Leuven in Belgium measured 5000 women to determine average ratio of a uterus’s length to its width for different age bands.2
The ratio was about 2 at birth.
It decreases steadily to 1.46 in old age.
Between ages 16 and 20 the ratio is 1.6 when women are most fertile.
We extensively discussed the geometry of DNA in articles 168-171.
We saw clear evidence that the DNA molecule 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.
Credit: Dr. Mark White
We have discussed microtubules several times in Cosmic Core, specifically in relation to consciousness.
It was Sir Roger Penrose & Stuart Hameroff who suggested that consciousness may emerge through the quantum mechanics of microtubules.
Microtubules are the scaffolding of the cell. They help maintain its structure and shape.
Each is composed of microscopic hexagonal lattices of fine filaments of protein called tubulins.
These tubulins form tiny hollow cylinders of indefinite length.
Each microtubule is composed of 13 tubulin that spiral around the hollow core. They exhibit 8:5 phyllotaxis.
Clathrins are located at the tips of microtubules. They are truncated icosahedra. These contain many golden ratios.
A clathrin cage
All microtubules in a cell radiate outward from the center of the cell membrane.
They act as tracks in transporting various products along cells, particularly nerve cells and are vital for pulling apart chromosomes during cell division.
They constantly remake themselves, assembling and disassembling.
They appear to be excellent conductors of pulses – pulses in one end traveled through pockets of protein and arrived unchanged at the other end.
Hameroff discovered a great degree of coherence among neighboring tubules – a vibration in one would tend to resonate in unison through its neighbors.
He theorized that microtubules within the cells of dendrites and neurons might be ‘light pipes’ acting as ‘waveguides’ for photons, sending these waves from cell to cell throughout the brain without any loss of energy.
They’d also shown that this focusing of waves would produce beams 15 nanometers in diameter – precisely the size of the microtubules inner core.
All of this led to a heretical thought – Consciousness was a global phenomenon that occurred everywhere in the body, and not simply in our brain.
Consciousness, at its most basic, was coherent light.
The Golden Ratio and the Vitruvian Man
The golden ratio appears prominently in Leonardo DaVinci’s famous Vitruvian man (canon).
Da Vinci wrote, “The workings of the human body are an analogy of the workings of our universe.”
Da Vinci’s Vitruvian Man is “based on the correlations of ideal proportions with geometry described by the ancient Roman architect Vitruvius. Vitruvius described the human figure as being the principal source of proportion among the classical orders of architecture.”3
“The drawing itself is often used as an implied symbol of the essential symmetry of the human body, and by extension, the symmetry of the universe as a whole.”4
Drunvalo Melchizadek pointed out the following in his book The Ancient Secret of the Flower of Life:
Using a grid of 8 x 8 = 64 squares, overlay the Vitruvian man onto the grid (page 198). This square encloses the man with arms raised up and legs straight down.
Draw a circle around the body so that the body fits inside. This circle encloses the man with arms straight out and legs spread apart.
The center of the circle fits the navel.
The eight squares in the center represent the first 8 cells or the Egg of Life.
If you draw a horizontal line through the navel the lower portion is a golden rectangle.
If you draw a line from the top corner of the large square to the middle point at his feet, the semi-diagonal will pass through the exact center of a Fibonacci spiral (page 205).
8 Fibonacci spirals can be drawn around the body just as the one spiral is shown on page 205.
“Those outer eight squares happen to be where the eight semi-diagonals intersect the body and where the eight spirals begin. We have eight places around the body and a central pattern of four squares in the middle centered exactly around the original eight cells.”5 (page 206)
If you move the centers of the square and circle together you will see the ‘squaring of the circle’.
Five-fold Man exists between six-fold Heaven and four-fold Earth.
Da Vinci shows how a man’s span equals his height.
This measure also equals seven of his feet.
The bellybutton is the exact center of the circle.
The bellybutton symbolizes the 3rd chakra – seat of will power. This chakra is the springboard to spiritual evolution – the 4th chakra of love/understanding.
Therefore, the circle represents the arena of spiritual evolution.
The groin is the exact center of the square.
The groin represents the reproduction center – the 2nd charka – the reproduction of physical life.
Therefore, the square represents the arena of material life and evolution.
When the feet are together and arms are held up the body forms a Cross. A square fits around the body in this case. This body position mirrors the position of one who is crucified. Crucifixion, as exemplified by Jesus, refers to the process of forgiveness. Only forgiveness can break the bonds of karma and set one free from the ‘prison’ of physical reality in which they can transform, be ‘born again’ and continue their cosmic evolution unfettered by the shackles of grudges, despair, fear, anger and attachment to physical possessions.
When the feet are apart and the arms are held horizontal the body forms a Star. A circle fits around it in this case. The star or pentagram symbolizes the Cosmic nature of each human. When the bonds of physical reality are broken (through forgiveness, acceptance, compassion and understanding) one is finally free to evolve to higher and higher heights in which the human consciousness expands to reintegrate into the cosmic consciousness from which it came. This results in the ability to move throughout the universe unhindered by the physical body. To put it poetically human consciousness and star consciousness merge. A human is a star in the making.
Even Dante knew this, as he ended each of the three books in the Divine Comedy with the word stars:
“He first, I second, without thought of rest,
We climbed the dark until we reached the point,
Where a round opening brought in sight the blest
And beauteous shining of the Heavenly cars.
And we walked out once more beneath the Stars.”
“I came back from those holiest waters anew,
Remade, reborn, like a Sun-wakened tree
That spreads new foliage to the Spring dew
In sweetest freshness, healed of Winter’s scars;
Perfect, pure, and ready for the Stars.”
“Here my powers rest from their high fantasy,
But already I could feel my being turned –
Instinct and intellect balanced equally
As in a wheel whose motion nothing jars –
By the Love that moves the Sun and the other Stars.”
Gyorgy Doczi tells us, “The outward pilgrimage of the Divine Comedy is really the inward journey of all of us. We all fail to share ourselves in love. We all know the frozen lake of lovelessness. We all scramble over the craggy rocks of egoism, and if we are fortunate we may eventually see the light of ‘love that moves the Sun and the other Stars’.”
As we can see our human body is a squared circle in its proportions.
“Leonardo da Vinci’s Vitruvian man…stands in a square and circle – uniting our two primary polar forms.”
“Instinct and intellect balanced equally…”
Other Geometries & Proportions in the Human Body:
The Dyad (2) and the Human Body
The height of the human body divided into the harmonic division of √2:
√2 is located at the vital center in the belly (hara), just above the navel at the solar plexus.
The body will measure 2 – √2 from the soles of the feet to the hara.
And √2 – 1 from the navel center to the top of the head.
In reverse √2 is located at the knees.
The in-and-out of our breathing symbolizes the radiating and absorbing or contracting and expanding qualities of the Dyad.
The heartbeat also represents a foundational aspect of the Dyad.
Systolic – the pressure when the heart beats while pumping blood.
Diastolic – the pressure when the heart is at rest between beats.
These two intertwining actions are opposites, yet they are two parts of a greater whole. Like the in-out of the breath, we cannot have life without these two opposing energies. When combined in an oscillatory fashion they create a greater whole that allows life.
Human sexuality is perhaps one of the most obvious aspects of the Dyad.
The masculine (sperm) and feminine (egg) combine to form new human life.
Just as we see with the in-out breath and the in-out beat of the heart, we also see with the masculine-feminine of humanity: neither are more important than the other. They are different but equal aspects of life and parts of a greater whole. One without the other could not exist. They are not opposites, but merging tendencies.
Gyorgy doczi writes, “We have two eyes and we see two images, which are united in the brain into a single, three-dimensional stereoscopic vision. We have two ears, receiving signals from two opposite directions, which are transmitted through the spiral-shaped cochlea of the inner ear to unite in the brain as stereophonic sound.
The entire nervous system is a dinergic double structure, consisting of peripheral and central component systems, again united by the brain. The brain itself consists of two hemispheres, their integration being performed by centrally located organs within the brain.”
Dinergy refers to the “energy-creating process that transforms discrepancies into harmonies by allowing differences to complement each other.”
The human body, as well as the entire universe, is a dinergic symphony of opposing yet complementary parts.
The Triad (3) and the Human Body
There are many examples of triads in structure and function of the human body. These and more can be found on Dr. John A. McNulty’s List of Three’s in Anatomy.6
The sacrum of the pelvis is a triangle. This allows for a stable, seated posture. It is also the keystone supporting our body when we stand. It is the center of gravity of the entire body.
“The ballet dancer’s visible grace is a symbol of the other, invisible grace: the potential of harmony and beauty that exists within every human beings, for indeed all of us, not only accomplished dancers, have an inner center, a sacrum, both in a physical and a spiritual sense.” ~Gyorgy Doczi
The scapula (shoulder blade) is roughly triangular. There are three borders.
There are three types of bone cells:
- osteoblast – creates new bone and repairs old bone by extracting calcium from the blood and laying down new bone matrix
- osteocytes – maintain and repair bone’s substance and strength
- osteoclasts – dissolves and re-assimilates old bone materials by dissolving bone matrix into the blood
There are three body planes: frontal, sagittal, transverse.
There are three aspects of a human being: body, mind, spirit.
There are three germ layers: endoderm, mesoderm, ectoderm. Each gives rise to specific organs/structures. Sometimes these three from separate parts of the same structure (your ear for example).
There are three life support systems of the human body – brain, heart, lungs (nervous, circulatory, and respiratory).
There are three layers of skin: Epidermis, Dermis, Subcutaneous layer.
There are three phalanges for each finger/toe.
There are three semicircular canals in the vestibular apparatus. These are responsible for balancing, as well as some unpleasant experiences like motion sickness. The three of these canals are each for one axis of 3D space.
There are many muscle groups of threes: triceps; deltoids (anterior, lateral, posterior); hamstrings; gluteals (maximus, medius, minimus); erector spinae
There are also three muscles of the anterior abdominal wall: External oblique, Internal oblique and the Transversus Abdominis.
There are three major nerves in the forearm: Radial, Ulnar, and Median. These are responsible for various motor actions and sensations.
There are three sections of the small intestine: duodenum, jejunum, and ileum.
There are three types of muscle: smooth, cardiac, skeletal.
And there are many, many examples of threes in the circulatory and nervous systems.
The Tetrad (4) & the Human Tetrachord
“Since the whole earth was symbolized by the human microcosm, it is we who are referred to as the “four corners of the globe” containing the “seven seas” (chakras), as well as the four stages of a spiritual journey and four levels of a temple or society.”7
This is the most dense and slowest to change.
It represents the Body Area of the Pelvis.
Its Action is Digestion.
The Body Area symbolized by water is the Belly.
Its Actions are Emotions. Emotions are ‘mood’s like the movement of water: surging, rising and falling, turbulent, stormy, calm, sometimes a laughing mountain stream, sometimes a ferocious torrent.
Like water, emotions take the form of their container.
They are shaped by our worldview, traditions, beliefs, opinions and thoughts.
This is key! Emotions follow beliefs. Beliefs do not follow emotions.
The Body Area represented by Air is the Chest.
Its Action is our Thoughts.
Like clouds and wind, they are a pollinating source, spreading ideas over our inner landscape.
The Body Area represented by Fire is the Head.
Its Action is our Intuitions.
Fire (like intuition) is the most ephemeral, the most electric, least dense and most difficult to discern.
Fire is a symbol of purpose.
The Four Elements of Transformation
- “If you can think so deeply that you can rediscover the spiritual roots of thought, if you can recognize thoughts as living, spiritual beings …(Thoughts)
- If you can develop a strong enough sense of your own individuality that you can become aware of your interaction with the Thought-Beings that weave in and out of yourself, yet not be overwhelmed by this reality …(Emotions)
- If you can recreate the ancient sense of wonder and use this sense of wonder to help awaken the willpower that lies sleeping in your deep, dark recesses …(Will Power)
- If the fire of love for your fellow human beings rises from your heart and causes you to weep tears of compassion …(Love/Compassion)
… Then you have been working on the Four Elements. You have begun the process of their transformation.”8
The Pentad (5), Phi & Natural Fractals in the Human Body
There are obvious examples of the Pentad in the human body, such as the body structure of one head, two arms and two legs. The golden ratio is seen extensively throughout the human body, as we have been discussing. Yet fractal branching is also an aspect of the Pentad.
Fractal branching is seen in many aspects of the human body. Some of these include:
Neurons in the brain
Lungs – trachea to bronchus to bronchii to alveoli contains 11 orders of branching
- Similar to a tree (branches and roots)
- Both use their surface areas to exchange oxygen and CO2
The Hexad (6) and the Human Body
Human lung alveoli are seen as a fish net pattern of tessellating hexagons. This maximizes the passageways for oxygen-carbon dioxide transfer.
“The skin surface – known as the epidermis – is made up of sheets of cells that are stacked up in layers…These cells resemble a flattened version of a shape called Kelvin’s tetrakaidecahedron: a 14-sided solid with six rectangular and eight hexagonal sides.”9
This shape allows them to achieve homeostasis of the tight junction barrier. This is classical example of ‘form follows function’. The cell shape provides a fundamental basis for the barrier homeostasis and physical strength of cornified stratified epithelia.
Red Blood Cells
The human blood cell membrane skeleton is a network of roughly 33,000 protein hexagons that looks like a microscopic geodesic dome.8
This mesh-like skeleton gives a healthy human red blood cell both its rubbery ability to stretch without breaking and potential mechanism to facilitate diffusion of oxygen across its membrane.
At the center of each hexagon is a rod-shaped protein complex called the proto-filament. The proto-filament is 37 nanometers long and made of a protein called actin.
“Elsewhere in the human body, bundles of actin form contractile muscles, and matrices of actin are responsible for the gel-like properties of various cells’ cytoplasm. However, the foreshortened actin fibers in the proto-filaments act as rigid rods held in suspension by six precisely positioned fibers made of the actin-binding protein spectrin.”10
Simple Squamous Epithelium
This lines the blood and lymph vessels. They are composed of a single layer of flat, hexagonal cells. They act in diffusion, filtration, secretion and absorption.
Corneal endothelial cells
These cells in the eyes are “terminally differentiated cells, specialized in regulating corneal hydration and transparency. They are highly polarized flat cells that separate the cornea from the aqueous humor. Their apical surface, in contact with aqueous humor is hexagonal, whereas their basal surface is irregular.”11
The liver is organized into lobules which take the shape of irregular polygonal prisms. However, lobules do not have distinct boundaries and they are seldom but neatly in cross section. Yet the idealized representation of the ‘classical’ liver lobule is a six-sided prism (hexagonal) about 2 mm long and 1 mm in diameter.
Cell Shapes in the Human Body
There are many different cell shapes in the human body yet the geometry of the cells mirrors shapes we have seen in bacteria and other single-celled organisms.
Some of these shapes are:
- squamous (flattened Vesica or eye shaped- like smooth muscle cells)
- cuboidal (cube)
- polygonal (often hexagonal, like hapatocytes)
- stellate (star-shaped – like bone or nerve cell)
- spheroidal (spherical)
- discoid (like a disc)
- Fusiform (spindle-shaped)
- Fibrous (worm-like)
Credit:Mr. A.C Andersen’s Classes
The Heptad (7) in the Human Body
There are 7 chakras – root, sacral, solar plexus, heart, throat, brow, and crown.
There are 7 endocrine glands – pineal, pituitary, thyroid, thymus, adrenal, pancreas, ovaries/gonads.
There are 7 levels of self – physical, etheric, astral, ego, manas, buddhi, atma.
We discuss the Heptad in the human body in great detail in Articles 200-203.
The Octad (8) in the Human Body
On average: The thumb, wrist, neck & waist form a series of doubling octaves:
- thumb base circumference x 2 = wrist circumference
- wrist circumference x 2 = neck base circumference
- neck base x 2 = waist circumference
The Ennead (9) in the Human Body
Nine represents the ultimate expression of birth. Ponder on the fact that after 9 months in the womb we reach the horizon, unto which we will be born into a whole new world in order to learn, grow and spiritually evolve.
There are several Nines in the Human body:
Cross section of tentacle-like cilia
- “Cilia are projections from the cell. They are made up of microtubules and are covered by an extension of the plasma membrane. They are motile and designed either to move the cell itself or to move substances over or around the cell.”12
- A cross-section shows a circle of nine doublets, each of which have one complete (A Tubule) and one incomplete (B Tubule) microtubules.
Bundles of microtubules in centrioles, essential for cell division
Tail of a sperm half-cell made of 9 twisted threads
9 months in the womb (9 month gestation)
9 openings on the body – 2 eyes, 2 ears, 2 nostrils, 1 mouth, anus, urethra
The tenth hole of the body is only present in the female – the vaginal canal – the opening that brings new life/new beginnings (symbol of the Decad) into the world.
Platonic Solids & the Geometry of Human Movement (Kinesiology)
Laban Movement Analysis is a method and language for describing, visualizing, interpreting and documenting human movement.
It is based on the work of Rudolf von Laban (1879-1958), a dance artist and theorist and one of the pioneers of modern dance in Europe.
Laban described a complex system of geometry based on crystalline forms, Platonic solids, and the structure of the human body.
Specific forms such as the cube, octahedron and icosahedron served as the basis for geometrical observations of where movement is being done in terms of emphasis of directions, places in space, planar movement, etc.
He used the five regular polyhedra as “Kinespheric Scaffolding”. They represent the most symmetrical and regular division of 3D space and so used most often in choreutics (Space Harmony).
He saw spatial patterns in human movement and recognized the shapes of the Platonic solids within these patterns.
“He applied the ideal patterns of the Platonic solids as forms to the actualized movement of humans – aligning with and closely approximating the space of these forms.
Linking the directions of vertices of a shape, following the natural spatial pulls to move along all directions within this shape, he came to specific movement Scales: patterned movement sequences that can be repeated, in which one moves through a Platonic Solid in a predefined way.”13
Robert Lawlor points out in his book Sacred Geometry: Philosophy & Practice how classical Indian dance is based upon Platonic solids and regular polygons.
Credit: Robert Lawlor
“The positions of Hindu classical dance (Bharat Natyam) describe geometric angular relationships from the axis of the body’s center of gravity just below the navel. These positions, while defining principle angles, are also often attributed to various deities and are meant to convey their characteristic powers.”
Nikolai Bernstein (1896-1966) was a Soviet neurophysiologist who studied human movement.
Nikolai Bernstein & son in the 1960s
“His research showed that most movements, like hitting a chisel with a hammer, are composed of smaller movements. Any one of these smaller movements, if altered, affects the movement as a whole.”14
Later he made films of human subjects dressed in black with white dots to mark the limbs.
The participants danced against a black background while being filmed.
When processed, the film showed a series of white dots moving in a continuous pattern in wave form.
When he analyzed the waves he found all the rhythmic movements could be represented in Fourier trigonometric sums.
Bernstein found he could predict the next movements of his dancers “to an accuracy of within a few millimeters”.
Bernstein “discovered that even our physical movements may be encoded in our brains in a language of Fourier wave forms.”15
Scientists such as Karl Pribram used this information to support his theory that the brain stores movements as wave patterns holographically and explains our rapid learning of complex physical tasks.
The Geometry of Perception
Robert Lawlor writes in Sacred Geometry: Philosophy & Practice, “Our different perceptual faculties such as sight, hearing, touch and smell are a result of various proportioned reductions of one vast spectrum of vibratory frequencies. We can understand these proportional relationships as a sort of geometry of perception.”
Lawlor then adds, “The specialization of cells in the body’s tissue is determined in part 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.”
All our sense organs function in response to the geometrical or proportional – not quantitative – difference inherent in the stimuli they receive.
“When we smell a rose we are not responding to the chemical substances of its perfume, but instead to the geometry of their molecular construction. That is to say, any chemical substance that is bonded together in the same geometry as that of the rose will smell as sweet.”16
Recall our discussion in Articles 166-167 how important the molecular structure is to a substance and how the structure determines the function and qualities of a substance, such as its smell, or its healing/harming effects.
Also recall how it is geometry that structures molecules.
“We do not hear simple quantitative difference in sound wave frequencies, but rather the logarithmic, proportional differences between frequencies, logarithmic expansion being the basis of the geometry of spirals.”17
The cochlea of a cow, guinea pig, mouse, and rat are spirals that register different levels of frequency.
Credit: Michael Schneider – A Beginner’s Guide to Constructing the Universe
Discerning the Difference between Music and Noise:
The ear analyzes the interval or spacing or ratio between the adjacent notes.
This happens automatically every time the hair-lengths use a golden spiral in your inner ear (cochlea) to measure (Spectrum analyze) the ratio between sounds and thus determine if the incoming pressure waves are harmonious music or chaotic noise.
“Our visual sense differs from our sense of touch only because the nerves of the retina are not tuned to the same range of frequencies as are the nerves embedded in our skin.”18
Recall from our series on the geometry of light (Article 124) how the shape of our visual receptor (the eye) mirrors the geometric shape of light as the tetrahedral photon spins creating a Vesica Piscis with 90° axes. One axis is electric, the crossing axis is magnetic.
The shape of the receptor matches that which is being received (the shape of the light).
In this article we delved deeper into the geometry of human beings, showing not just how geometry structures the body, but how it affects function and physiology, playing an important role in everything from blood pressure, fertility, smell, hearing, vision and movement.
In the next article we focus specifically on the human heart and associated geometry.
- Devlin, Kate, Human Heart follows the ‘golden ratio’ for beauty, The Telegraph, 14 December 2009, http://www.telegraph.co.uk/news/health/news/6790529/Human-heart-follows-the-golden-ratio-for-beauty.html
- Bellos, Alex, Golden Ratio discovered in uterus, The Guardian, 14 August 2012, https://www.theguardian.com/science/alexs-adventures-in-numberland/2012/aug/14/golden-ratio-uterus
- Melchizadek, Drunvalo, The Ancient Secret of the Flower of Life, Volume 1, Light Technology Publishing, 1998
- McNulty, John A, List of Three’s in Anatomy, http://www.meddean.luc.edu/lumen/meded/grossanatomy/threes.html
- Schneider, Michael, A Beginner’s Guide to Constructing the Universe, HarperPerennial, 1994
- Booth, Mark, The Secret History of the World, The Overlook Press, 2010
- Yokouchi, Atsugi, Logtestijn et al, Epidermal cell turnover across tight junctions based on Kelvin’s tetrakaidecahedron cell shape, eLife 2016;5:e19593, 29 November 2016 https://elifesciences.org/articles/19593
- UC San Diego, Jacobs School of Engineering, Scientists Discover Secret Behind Human Red Blood Cell’s Amazing Flexibility, 21 October 2005, http://jacobsschool.ucsd.edu/news/news_releases/release.sfe?id=484
- He, Zhiguo et al, 3D map of the human corneal endothelial cell, Scientific Reports 6, Article number 29047, 6 July 2016https://www.nature.com/articles/srep29047
- Ramesh, Chidambaram, Thought Forms and Hallucinations: Some Curious Effects of the Holographic Mind Process, Notion Press, 2014
- Lawlor, Robert, Sacred Geometry: Philosophy & Practice, Thames & Hudson, 1982