Plato's Wacky Sacred Geometry

30 November 2024

Strolling through Temple Bar, the other day,

 I chanced on a Book Fair, 

an annual event, featuring mainly self-published books, and well attended by punters. I went in to observe, not expecting to buy, because I have too many books at home.  However, I came across an unusual book that quite took my attention and I had to buy it:

How the World Was Made (the story of  creation according to Sacred Geometry) by John Michell and Allan Brown

This was full of geometric designs illustrating the Platonic view that the universe was created in geometric patterns and in accordance with unchanging mathematical relationships, with extensive discussion within.

The geometric designs in this book sparked a memory. They brought me back to when I was 11 years old, and had decided to sit for the Dublin Corporation Scholarship examination, taking Art as one of the subjects, even though I was not studying it in school. One of the exam questions would be to reproduce a given geometric design using compass and ruler.

There is an infinite (that is, unlimited) number of such designs, examples here taken from the Web: 

The designs in the examination were never too complicated or difficult. By reviewing previous exam papers, I knew this would be no problem to me.

You always begin by drawing a circle, using the compass. Then, using the ruler, you draw a straight line through the centre point, giving you a diameter. Drawing a second diameter at right-angles to the first gives you four points on the circumference, joining which, "squares the circle," i.e., gives you a square within the circle. It is easy to add other guide-lines, leading to properly proportioned eight-sided or sixteen-sided figures.

Somewhat more interesting than squaring the circle is to draw a hexagon within the circle. This is done by placing the point of the compass successively at each end of one diagonal and, with the length of your radius, marking off two points above and two points below on the circumference. You now have six points, which, joined, give you a hexagon within the circle. 

By using the six hexagonal corners as the compass point, we can  now draw additional arcs that give us the twelve corner points of a dodecagon (12-sided figure). 

We find that 6 radius-length hexagon sides (nearly) fit perfectly within the circle, but the perimeter, being curved, is longer than that.

I had not yet heard of Pi (π), the mathematical ratio between the circumference and diameter of a circle, approximately 3.142. When this came up in math class, I was not too surprised, for it corresponded to my own finding that the circumference was somewhat larger than 6 radiuses, or 3 diameters.

I was surprised, however, when the teacher suggested that our stone-age predecessors must have known the value of π in order to build their stone circles, and monuments like Newgrange. They might, indeed, have known the value of π, but this knowledge was not required for construction of their stone circles. Like me, they could construct their circles, square them, and mark out the peripheral points, using a rope in the way I would use a compass.

Knowing Pi would actually be of no assistance in drawing or laying out their circles. It would be more a hindrance than a help. (You can visualise the scholarly professors in a huddle trying to figure out multiplications and divisions, while the practical workers just walk around with their rope marking out the perimeter).

You simply got a rope, looped one end around a stake at the centre of your proposed circle, and walked the periphery with the other end of the rope in your hand. 

To mark the position of the rising sun, one person stood at the centre (or chosen inside point) of the Stone Circle at sunrise and another stood on the periphery. "Left, left, a bit; stop, a bit right now; hold it," the one at the centre calls, until the person at the periphery stands exactly in line with the rising sun. Repeating the exercise every morning, they finally mark the point where the sun turns around, (the solstice) and repeat the exercise six months later to mark the other solstice. The Equinox is marked as the half way point between the solstices.

The stone age gurus would, in the beginning, observe their perimeters to be approximately Six times the radius, (as I did), but would soon enough observe that six was not quite enough. By walking and measuring their circles, they eventually would figure that, if the circle had a diameter seven sticks long, the circumference would be exactly (or as near as you like to) 22 sticks long. The relationship would, then, be 22/7, which the Greeks called Pi. (Modern mathematicians have a more exact measurement).

Even if monuments like that at Newgrange were built to a pre-drawn architectural plan, the designers still would not have required to work out the mathematics of the relationships between the parts; they would be drawn by some compass-like instrument and a straight edge. However, since Newgrange is more kidney-shaped than circular, it looks likely that it's builders were not that interested in getting their circles perfect.

Pythagoras, of ancient Greece, (real or mythical) expected to find that the creator built this world using whole numbers. Well, Pi, as it appeared to him, was made of two whole numbers, (22 and 7), so it was all right.

However, he was baffled to find that there were relationships that were not whole numbers, such as ✓2 (found in the hypothenuse of an equilateral right-angled triangle) and ✓3 (in the height of a rhombus). These were constant, never-changing ratios, not capable of being expressed in whole numbers, that were used in geometric patterns and in the construction of the universe, and there were many more such.

Note, however, my friend, that the equilateral, right-angled triangle, and the properly proportioned rhombus, can both be constructed using your compass and ruler, or equivalent instruments, without having any knowledge of Pi or square roots.

Far from being a master of advanced mathematics, as Einstein speculated, perhaps the creator never bothered with Pi and Square Roots, but just threw matter, with great energy, into the great spin we call the Universe, where all the mathematical relationships naturally emerge.

Perhaps the Creator was not even conscious of what he was doing, but that consciousness and intelligence are consequences of evolution within the universe, as philosophers like French Jesuit Theilhard de Chardin and psychologist Carl Jung hint.

There (The Athai plain outside Nairobi) the cosmic meaning of consciousness became overwhelmingly clear to me.

“What nature leaves imperfect, the art perfects,” say the alchemists. 

Man, I, in an invisible act of creation put the stamp of perfection on the world by giving it objective existence.

This act we usually ascribe to the Creator alone, without considering that in so doing we view life as a machine calculated down to the last detail, which, along with the human psyche, runs on senselessly, obeying foreknown and predetermined rules. In such a cheerless clockwork fantasy there is no drama of man, world, and God; there is no “new day” leading to “new shores” but only the dreariness of calculated processes.

My old Pueblo friend came to my mind. He thought that the raison d’etre of his pueblo had been to help their father, the sun, to cross the sky each day. I had envied him for the fullness of meaning in that belief, and had been looking about without hope for a myth of our own. Now I knew what it was, and knew even more: that man is indispensable for the completion of creation; that, in fact, he himself is the second creator of the world, who alone has given to the world its objective existence without which, unheard, unseen, silently eating, giving birth, dying, heads nodding through hundreds of millions of years, it would have gone on in the profoundest night of non-being down to its unknown end. Human consciousness created objective existence and meaning, and man found his indispensable place in the great process of being. 

(Carl Jung, "Memories, Dreams, Reflections" German version 1961; English translation 1995) 

My summary: "There was no consciousness of existence until mankind became conscious."

This, then, brings me to Plato. He proposed that the entire universe was constructed on the basis of (intelligent design using) geometric shapes and their mathematical ratios. The universe itself, he says, must be a sphere, as would be many of its constituent parts, such as the earth, the moon and the sun, as well as all the stars, and, he speculates, the atoms from which matter is formed, which must be three-dimensional shapes that fit in a sphere like my square fits in its circle. (Well, modern science bears Plato out here, for the sphere is the best image we have not only of the atom but of every sub-atomic particle).

The creator, in the middle of nothingness, threw the universe into existence in a spherical form all around Him.

We can observe that Planet Earth, basically spherical in shape, spins on its axis. However, the axis has no substance; it is nothing. Similarly, the point around which the entire universe spins has no substance. Unlike the Christian God, who cavorts in glory amid the angels and saints in some glorious place in the sky, Plato's creator has no substance and sits, a veritable dimensionless ball of nothing, at the very centre of reality, with the universe spinning round Him.

What does He do all day? He does not have to do anything; the universe continues, without any further effort from him, to spin.

Nature, perhaps, unconsciously throws out mathematically valid patterns.

Those Greeks loved numbers, and each number had an associated geometry, as this book I bought recounts; the One and Only (sphere and symbol of the universe),  Duality and Paradox, Three, Triangle, Vesica Piscis and Marriage Made in Heaven, Four and the Square, Reason, Stability and Order, Five and Ten, numbers of life and growth, Six and the Hexagon, the perfect number ...

Hexagons and related shapes

All the numbers and their associated shapes and their ratios are found throughout the universe, and in carpentry, construction and every kind of endeavour, including music. Everything in nature is in sacred proportion.  A building will be beautiful if, with its windows and doors, it is built in proper proportions. If, however, it is out of proportion, it will be ugly.

Plato had a notice printed on the door of his academy, which stated, "Nobody ignorant of geometry may enter here." 

When it comes to planning and laying out cities, according to Plato, the ideal shape is the circle, and that circle should be divided radially into twelve equal sectors. There are twelve gods on Parnassus, twelve sectors in the universe, twelve months in a year, and twelve sectors in a well-designed city. Hexagons (six-sided figures) neatly fit in with and feature in the dodecagon (12-sided) structure. 

For Plato, the twelve gods operate within nature, but the Creator is outside of nature.

We in Ireland used to look back proudly at our Golden Age, 1,500 years ago, and Europe used to look back longingly at the Roman Empire. Well Plato used to look back from the Greece of his time to a glorious Greece of even more Ancient times. And he had good reason, for the folklore of his time told of the glories of the ancient past. 

Well, 2,000 years before his time, in the Bronze Age, there had actually been a golden age (celebrated, indeed, in Ireland's National Museum as Ireland's First Golden Age). The Bronze Age, in Europe, had seen international peace and wealth, with trade stretching all the way from Ireland to Afghanistan, together with widespread use of writing and account-keeping.  

The Bronze Age came to an end when a spate of Global Warming brought famine, mass migration, invasions and revolutions, the collapse of governments, disruption of trade and a shortage of tin, the collapse of the civilisation, the break up of nations, violent times, warfare and ignorance. The ancient writing was totally lost, except in Egypt, leaving no remnant, unlike even more ancient Mesopotamia, which left remnants when some of their writings, on clay tablets, were accidentally baked by fire.

By Plato's time, of course, the cities of Greece had recovered wealth and a new literacy had come, via Phoenician influence, but there were many folk memories, incorporated in stories, of glorious ancient times, and Plato accepted that cities like Athens had a glorious, lost past.

Most of Plato's teaching is expressed through his dialogues in the mouth of Socrates. Socrates, in Plato's telling, on one occasion invited his pupils to recount folk-memories they had inherited. Socrates had advised his pupils to observe the world, believe what they see and question what they are told. Plato, however, observed that you can't believe what you learn through the senses, since we live in a world of decay and change, but we can use our intelligence to figure things out. There was often nothing to justify his elaborate geometric ideas except his own imagination. (His own pupil, Aristotle, brought us back to scientific enquiry).

One of Socrates' pupils, Critias, in Plato's telling, tells of the visit to Egypt of his own ancestor, Solon, 600 years before, where the Egyptian priests decried the Greeks' lack of memory of things past. "You don't even remember your country's great victory over the Atlanteans!" 

The Egyptians were boasting of their ancient writings that recorded this great event, apparently of 9,000 years before. Atlantis had been a country, or, indeed, a continent, in the Atlantic Ocean, in between Europe and America, and nine thousand years before (even way before the Bronze Age) had apparently conquered and enslaved most of Europe, but the Athenians, though much smaller in number, had defeated them and taken Atlantis, after which Atlantis was deserted by its gods and sank beneath the ocean.

Now Plato gave full rein to his imagination to speculate why the Atlanteans were defeated and why their country sank beneath the waves. It was, he said, because of a fundamental error in the construction of their city.

O, the Atlanteans did give their city a circular peripheral wall, but, instead of the sacred dodecagon (12 sided figure), they converted their circle into a decagon (10 sided figure). Though their city was magnificent, redolent with pentagons, this mistake meant that nothing was ever really right about Atlantis. 

Atlantis City, according to Plato, was constructed per Pentagon-geometry, providing a ten-sided perimeter wall, instead of the proper 12-sided wall, and, so, was doomed from the start.

The blue rings are filled with water and ships could sail right up to the city docks. The great white outside ring is an agricultural area of 100,000 acres, divided into one-acre-sized plots, capable of keeping the city well supplied with food. 

Atlantis, wrongly built to Decagon/ Pentagon geometry, failed, and Athens, built to Dodecagon/ Hexagon geometry survived and prospered.

Now, if we look at our own Irish Stone-age (Neolithic) structures, we see that the circle was, indeed, the preferred shape, for residences and temples or monuments, though we can't boast of cities. The designs engraved onto rock tell of the spiral of life, and the arrangement of great stones tell of the cyclical nature of the year and the planetary cycles. Their explanation of Atlantis would be different to Plato's.

Here, I show a map of the North Atlantic sea-bed, taken from National Geographic magazine:

There are mountain ranges rising from the Atlantic bed, but now covered by water, leaving only groups of islands, such as the Azores, Bermuda, Canary Islands, Hebrides. Clearly, when the sea-level was much lower, and Ireland was joined onto England, and England to France, there was a lot more dry land out there in in the Atlantic Ocean. 

Ireland and Portugal both have folklore about a great land in the middle of the ocean called Brazil. In Irish it was "Uí Bhreasail," meaning "The Land of a man named Brazil," (and after which the American Brazil was named in a later time).

Now, Ireland did not suffer the Biblical flood, although this story was imposed on our tradition by the Christian clerics in the Middle Ages. What we had was an Ice Age. After the Ice Age, mammals and birds travelled across the land-bridge from Europe, followed by men. 

The humans lived more densely by the coast than inland, and left no trace except middens, i.e., piles of  shells of devoured shellfish. These Mesolithic people were the most successful settlement of Ireland ever, living in harmony with nature for 3,000 years (comparable, for example, to the Christian era, which has, so far, lasted 2,000 years). When temperatures rose again, the sea consequently rose higher and it swallowed up many of the sea-side middens. The humans retreated up-hill to the new shore-line. Many of their piles of sea-shells have been found now under the sea, evidence that the sea-level was previously so much lower.

Just as the Irish Continental Shelf disappeared under the sea at this time, so also did the land of Atlantis, and other countries between here and America. This was all part of the cycle of life, where sea levels rise and fall with the rising and falling of world temperatures. There is no chance of Atlantis rising again any time soon, but plenty of chance that low-lying lands in Ireland and Europe will sink under the sea.