The dimensions of the Great Pyramid reference the size of our planet. Various units of measure are derived also from the size of the earth, such as the metre, the inch, the foot, the mile, the digit, and the nautical mile. What if the very dimensions of our planet, as measured in miles, were based on the quadrature of the circle?

Expressing the earth's equatorial circumference in inches, and feet

The equatorial circumference of the earth is 40 075,017 km, which works out as 24 901.4611 miles, or 1 577 756 573.193 inches. A yuga of 4 320 000 sidereal years of 365.25868 gives a total of 1 577 917 497.6 days. A yuga is a period of time in Hindu cosmology. 4 320 000 tropical years of 365.242199 are equivalent to 1 577 846 299.64 days. 

If we think of these time periods expressed in days as expressions of distance, in space, with each day converted to inches, we can see that the equatorial circumference of the earth and the period of 4 320 000 years in days are very close. Indeed, these two time periods converted to inches and then miles give 24 904.0004 and 24 902.8772 miles respectively, a difference of only a couple of miles from today's estimate.

The use of inches to express days is also found at Giza, where, for example, the outer casing of the Great Pyramid, the mean socket sides, measure 9125.9 inches each. 5 years of 365 days are 9125 inches. Or another example at Giza is the width of the rectangle which encompasses the three main pyramids, estimated by Petrie as 29 227.2 inches. 80 solar years of 365.25 days are 29 220 days. The system of using inches to count days is used elsewhere at Giza, and in megalithic structures, as the work of Richard and Robin Heath, Howard Crowhurst, David Kenworthy, Dennis Payne, and others has shown.

Another way of expressing the equatorial circumference in terms of the year is in feet. Robin Heath, has suggested the equatorial circumference was divided by the number of days in a solar year, and that figure was then divided by 360,000, to produce one small unit of measurement, the English foot. Hence, the equatorial circumference of the earth is 365.242199 x 360,000 =131,487,191.64 feet, or 24,902.877 miles. (1)

There is another way of interpreting this circumference in feet which also merges spatial and temporal measure. With 1 lunation as 29.53059 days, and the difference in days between the solar and lunar years as 10.87512 days, a mile, in feet, can be defined as approximately 70,000 lunations / (36 x the difference in days between solar and lunar years).

The earth’s equatorial circumference in feet can be expressed as 70,000 lunations x √(π³ x 20,000,000) / ( 36 x the difference in days between solar and lunar years) = √(10,000,000 π) x √2 x π x 5,280. This is equal to √(10,000,000)×√(2)×π×√(π) miles.

 We can also consider the length of a year in days as approximately √20 000 000 × π × √π × 12 × 5280 ÷ 4 320 000 ≈ 365.234. If we look at it this way, we can see the square root of pi, which suggests perhaps a squaring of the circle.

We can see that at some point in the ancient world, someone began to think of space and time as connected. Measurements of time, such as a year or a day, or indeed a yuga, become linked to spatial measurements, such as the earth's circumference. This integration reflects a holistic view whereby time and space are interconnected aspects of cosmic order and natural cycles.

  In modern physics, particularly in the framework of Einstein's theory of relativity, time and space are intimately connected. Space and time are not separate entities but are unified into a single, four dimensional continuum known as spacetime. The use of time periods, measured in days for example, expressed in spatial measures, inches for example, shows a similar understanding of time and space

Knowledge of the size of the earth

It can seem fanciful to attribute any kind of knowledge of the dimensions of the earth to any civilisation or people before the ancient Greeks. This is probably because, traditionally, the ancient Greeks were regarded in such high esteem that the beginnings of many scientific, mathematical and philosophic achievements was attributed to them, to the expense of other people from different times and places. We today are of course greatly indebted to the ancient Greeks, and thanks to their enthusiasm for the written word, we know a lot of what they did achieve. There is also a tendency to confuse the earliest known example of something with the first ever instance of that thing. An archaeological discovery of a temple is made, older than any other yet known, and then declarations of the "first" temple usually ensue. This is ridiculous, by anyone's logic, of course. All these factors contribute to a general tendency to believe that before the earliest known piece of evidence for something, there was absolutely nothing at all.

And yet, if we consider the evidence of the numerical coincidences given in the reading of the earth's equatorial circumference in feet or in inches, we should consider this as some kind of evidence or a very old accurate measurement of the earth. Equally, we should consider the units involved, here being the inch and the foot, as ancient artefacts.

There is also evidence from Julius Cesar that the druids of Gaul and Britain discussed the size of the earth, as well as the size of the universe, and the motion of the stars and planets.

This description by Julius Cesar implies that they had some kind of data to discuss in the first place, even though he doesn't tell us what it was.

We can at least ask the question of just how the English inch and the foot were designed, as it is surely beyond coincidence that a year in days expressed in inches can so closely approximate the equatorial circumference. The use of inches perhaps symbolise a bridging of the macrocosm and the microcosm, the vastness of the earth with human scale measurements.

Expressing the earth's equatorial circumference in miles

We can also find intriguing coincidences when we consider the measure of the equatorial circumference in miles. Hugh Franklin found intriguing connections between the mile and the earth and linked the value of the circumference in miles to pi, the ratio between a circle’s circumference and diameter.

In Hugh Franklin's article “Earth, Pi, Miles and the Barleycorn” (2), he points out that the equation √(π³ x 20 000 000) = 24 902.3198 is very close to the contemporary figure for the equatorial circumference of the earth in miles, estimated as 24 901.461 miles (Wikipedia).

These units, the mile, the foot and the inch, and the system that produced them, possibly come from ancient India originally. It is likely that the inch, the foot, and the mile are remnants of an ancient system that was used in ancient Egypt, ancient India, and in Europe also. The mile is linked very precisely to ancient Indian units of length. In traditional Indian measures, a Yojana is 14.484096 Km, which works out at exactly 9 miles. (See this earlier post)

How can we understand √(π³ x 20 000 000) = 24 902.3198 geometrically? A circle with a circumference of 24 902.31984 miles will have a diameter which can also be the diagonal of a square. The square will have a side of 24 902.31984 /(√2x π) miles, and the area of this square will be 10 000 000 π square miles. This square would be equal in area to a circle with a diameter of 10 000 000 miles. This would suggest that the equatorial circumference of our planet, via the geometries of the circle and the square, gave rise to the mile itself, as a unit of measure. The circumference of the earth as a value in miles is equated to a square with an area of 10 000 000 π square miles.

In this model, the approximation given for pi by a calculator is used, not a rough approximation such as 22/7. Since we cannot know what value of pi was used in applied geometry at whatever time the inch, foot and mile were designed, as there are no historical records, and since approximations such as 22/7, 25/8 and 864/275 don't work as well in fitting this equation to the size of the earth, I have chosen to use the approximation of pi given by my calculator, as it fits the data best. I believe it is no more anachronistic than using one of the approximations just mentioned, as there are absolutely no records either way, and to use a value that may have been considered useful at a different time and place in history makes no sense here. At this stage, there is no need to ask ourselves whether the ancient mathematicians knew pi was irrational or transcendental, because whether they did or not, it is about applied geometry, so we have to use some sort of approximation. In fact, in the world of practical geometry generally, we are only ever going to be using approximations, and that applies not only to pi, or the square roots, but also to the very geometrical objects themselves, from the line to the circle to the square.

John Michell wrote that the mean earth diameter was 7 920 miles. He showed that reading the earth's size in miles was important and that it could be found in the New Jerusalem measures:

He also showed that measuring the same thing in two different units could be important, just as we see when we read the circumference of the earth, for example, in inches, feet, and miles - or in metres, if we are looking at the polar circumference. Michell goes on to write:

John Michell's description of the holy city emphasise the importance of understanding its dimensions through the lens of geometry and number, suggesting that these dimensions hold ancient secrets that can be deciphered through mathematical study. The idea that the dimensions of the holy city are critical and encoded with ancient wisdom underscores the belief that geometry is not just a practical tool but a sacred language.

Squaring the Circle

"Squaring the circle" refers to an ancient geometric problem of constructing a square with the same area as a given circle using only a finite number of steps with compass and straightedge (a ruler, but without the use of its markings). A proof of the impossibility of this task was widely accepted in 1882, due to the transcendental nature of π, it was for many centuries an important part of mathematics, and perhaps important too on a philosophical level. It may seem like a pointless game now, but perhaps it had a use once, or even a metaphysical role. It is curious that we find hints of squared circles in Hugh Franklin's equation of the equatorial circumference in miles. Can the equatorial circumference be understood as the outcome of a squaring of a circle?

Hugh Franklin suggested √(π³ x 20 000 000) = 24 902.3198 miles provided a good estimate for the equatorial circumference, which is a brilliant observation. What if we tweak the equation slightly and instead write it as  √(π×20 000 000)×π = 24 902.3198?

The Earth's equatorial circumference is approximately 24,901.4611 miles.

√(π × 20 000 000) × π = 24 902.3198. The difference is just over a mile.

Dennis Payne has independently found this connection to the equatorial circumference.

How can √(π × 20 000 000) × π be understood, and represented geometrically?

If we start with a circle which has an area of 20 000 000 square miles, the radius will be √(π × 20 000 000) miles. A square with the same area as the circle will have a side length of √20 000 000 miles. And if we start with a circle whose area is 20 000 000 π square miles, the square of equal area will have sides of √ (20 000 000 π) = 7 926.6546 miles. The equatorial radius of the earth is 6 378.137 km, or 3 963.1906 miles, so the diameter is 7 926.3812 miles. This is almost exactly the value of the sides of the square we just saw:

√ (20 000 000 π) = 7 926.6546 miles. If we multiply this diameter by π, we obtain the circumference of a circle. √ (20 000 000 π) x π = 24 902.3198. This is very close to the value of the earth's equatorial circumference in miles.

Alternatively we can start with a square with sides of 2 000 miles, and then double it, to make a 1:2 rectangle, with sides of 2 000 and 4 000 miles, and the diagonal will be of

2 000√5  miles. This diagonal becomes the radius of a circle, whose area is 20 000 000 π square miles. A square of equal area will have sides of √ (20 000 000 π) = 7 926.6546 miles. This is the equatorial diameter: √(20 000 000 π) = 7926.6546. This is then multiplied by pi to obtain a circumference, which closely matches the equatorial circumference of earth in miles. √(20 000 000 π) x π = 24 902.3198.

The double square is a very important in ancient geometry, it is found at Giza for example, in the King's Chamber, and it is found in megalithic Brittany, as Howard Crowhurst has demonstrated.

In the diagram below, the starting point is a double square of 2 000 x 4 000 miles. The diagonal of this rectangle is 2 000 x √5 miles, because the diagonal of a 1 x 2 rectangle is √5. This diagonal is the radius of a circle, which has an area of 20 000 000 π square miles. The next step is the square the circle by area: to produce a square which has the same are as the circle. So the new square has an area of 20 000 000 π square miles also, and sides of √(20 000 000 π) miles. Finally the side of the square becomes the diameter of a new circle. The radius is half the diameter: √(20 000 000 π)/2 = 3 963.3273 miles. The circumference is √(20 000 000 π) π = 24 902.3198 miles. The areas of the two circles multiplied together are π³ x 10¹⁴. Area of the first circle: 20 000 000 π sq miles, multiplied by the area of second circle: (√(20 000 000 π)/2)² x π = 5 000 π² sq miles, gives π³ x 10¹⁴ sq miles.

The area of the white circle, which represents the equatorial circumference of the earth, is 5 000 000π² square miles, and a square of the same area will have sides of √(5 000 000π²) miles. We can imagine an alternative way of arriving at the equatorial circumference from a double square, which has a diagonal of √5 if the sides of the square are 1. With this method we also have to deal with the problem of a square and a circle of equal area, though here it is in reverse. Rather than squaring a circle, it is circling a square.

The process in the diagram above is as follows: start with a square of 1 000 square miles (1 000 x 1 000 miles). Then double it, to form a 2 x 1 rectangle. The diagonal of this rectangle will be 1 000 x √5 miles, because the diagonal of a 1 x 2 rectangle is √5. The diagonal of this rectangle becomes the diameter of a circle. the circumference of a circle is diameter x 𝜋 (pi). So the circumference of this circle will be 1 000 x √5 x 𝜋 miles. Next we are going to create a very large square, with each side equal to the circumference of the circle. So each side is 1 000 x √5 x 𝜋 miles long. The total area of the square will be 1 000 x √5 x 𝜋 x 1 000 x √5 x 𝜋, or (1 000 x √5 x 𝜋)² square miles. The final step is to "circle the square", by area, so to create a circle which has the same area as the square. The area of a circle is the radius squared x pi. So to calculate the radius, we divide the area by pi and then take the square root. The radius is therefore √((1 000 x √5 x 𝜋)² / 𝜋) miles. This is equivalent to 3 963.3273 miles. From here we can calculate the circumference, by multiplying the radius by 2𝜋. The circumference is √((1 000 x √5 x 𝜋)² / 𝜋) x 2𝜋 = 24 902.3198 miles. This can also be written as √20 000 000 x 𝜋 x √𝜋.

The earliest known person to have attempted to square a circle is Anaxagoras, the Greek philosopher, circa 500 BC. Plato (circa 428-348 BCE) and his Academy placed great emphasis on geometric problems, including squaring the circle, which in turn inspired many others to study the problem. Why do we seem to find it in the equatorial circumference of the earth as expressed in miles? Is it about bridging the world of the divine and the material, the idea and the form? If so, it would be fitting that pi should be part of that bridge, being impossible for humans to entirely define, or truly comprehend. It would be relevant to the process if it had been known, long ago, if and when the earth's circumference was defined in miles, that pi was irrational, and transcendental. The integration of philosophical concepts into the expression of the earth's size would suggest a harmonious relationship between the divine (symbolised by the circle and pi, and the relationship to the square) and the material (represented by the square and human measurement). The Greeks saw the heavens as a perfect, unchanging sphere, while the earth was considered imperfect and finite. This can be found in Plato's dialogues, where he describes the cosmos as a series of nested spheres.

Squared circles and the number 3168 are particularly significant in the ancient canon of numbers, and in John Michell's work. It is associated with the perimeter of a square at Jerusalem, Glastonbury and Stonehenge. This square is part of a process of squaring the circle, in this case not by area but by circumference and perimeter. To begin with a circle is drawn with a diameter of 7 920 000 feet. This diameter is of course linked to the size of the earth, albeit in another unit, the mile. The mean circumference of the earth is close to 7 920 x π = 24 891.42857 miles (with the current estimates, the mean is 24 881.026 miles, or as John Michell put it, 12⁵ / 10 = 24 883.2 miles, which, with pi as 864/275, the diameter becomes 7 920 exactly). The circle is inscribed within a square with sides of 7 920 feet, and a perimeter of 31 680 feet. (Or the square is drawn fist and the circle then drawn inside it.) The squaring of the circle consists in then drawing another circle which has the same circumference as the square. This process is somewhat mysterious if it stays within the realm of thought, because pi is irrational, and so it is impossible to exactly match the perimeter of the new square to the perimeter of the square. In an applied way, however, where approximation is necessary, this can be done, but it takes on a symbolic importance in that it transfers the mysterious world of the irrational, ineffable, and incomprehensible to this world.

The mean diameter of the earth, with pi as 864 / 275, is 7 920 miles, but there is an additional layer of meaning to the squaring of the circle process, which involves the moon. This is probably one of John Michell's most famous insights. The square drawn round the circumference of the earth which then produces a second circle with a circumference equal to the square's perimeter is just the right size to include the radius of the moon. By a strange natural coincidence, the square that is drawn around the circle representing the earth has the same perimeter as the circumference of a circle that has a diameter equal to the diameter of the earth plus the diameter of the moon.

The moon's equatorial diameter is exactly 2 160 miles, or 1.738.1 km (Wikipedia). With the mean earth's diameter as 7 920 miles, this gives a total of 10 080 miles. All these numbers are multiples of 9, 2160 is 240 x 9, 7 920 is 880 x 9 , and 10 080 is 2 x 7 x 80 x 9. We previously used the 864 / 275 approximation for pi to obtain the mean diameter of the earth in relation to a circumference of 24 883.2 miles (which is just over two miles of the current estimate). Here we now use 22/7 for pi. A circle with a diameter of 10 080 miles, representing the joint diameters of the earth (mean) and moon (equatorial), and pi as 22 / 7, the circumference is 31 680 miles. This is the significant number 3168 multiplied by 10.

   If we were to use today's value for the earth and moon sizes, and the approximation for pi given by a calculator, we would obtain 31 666.866 miles. The mean radius is given as 6371.0 km or 3 958.75586 miles (Wikipedia), so the diameter is twice that.

(3 958.75586 + 2 160) x π = 31 659.43680

This is also very close to 190 000 / 6 = 31 666.66667, and interesting possible connection to the Metonic cycle of 19 years, and 6, the all important and perfect number of ancient metrology. The number 31 680 is in fact 190 080 / 6. The number 190 080 is also important and a multiple of 9, being 10 x 33 x 64 x 9. This number 190 080 is also linked to the sun: the diameter of the sun can be roughly approximated by 19 008 000 / 7 miles, and the diameter by 19 008 000 / (7π).

   The sun's equatorial radius is given as 6.957 x 10⁸ m today (Wikipedia), which is 432 287.9384 miles. It was once thought of as 432 000 miles, another important number in the ancient system. The circumference then becomes 2 716 145.2232 miles (with today's values) or 2 715 428.571428 miles with the ancient value, which is also 19 008 000 / 7 miles.

   In the system described by Michell, we can see that it is important to round up or down to the closest multiple of 9, and that variations on approximations of pi can achieve this.

The square and the circle

The earliest indications of the heavens being symbolised by a circle and the earth by a square come from diverse cultures such as ancient China, Egypt, Mesopotamia, Greece, India, and Native American traditions. These symbols reflect their understanding of cosmology and the natural order, with the circle representing the infinite and divine nature of the heavens, and the square representing the finite and orderly nature of the earth. If the circle represented the infinite, it is possible that the infinite nature of pi as a number was known.

In ancient China, there were two opposing concepts, of Heavenly Roundness and Earthly Squareness. In ancient Chinese cosmology, the concept of "Tian yuan di fang" (天圆地方) literally translates to "heavenly round, earthly square." This idea is evident in texts dating back to at least the Zhou Dynasty (1046–256 BCE). The circle symbolises the heavens, which are seen as boundless and infinite, while the square represents the earth, which is perceived as finite and orderly. This philosophy has been incorporated into traditional architecture. For example in Beijing there are two temples, the Temple of Heaven and the Temple of Earth. The Temple of Heaven is round, symbolising the heavens and sky, and the Temple of Earth has a square base and many square walls and altars.

The sun, the moon and the stars were widely considered deities in the ancient world. This can be seen in the ancient Egyptian, Greek, Roman, Mesopotamian and Aztec religions, as well as in Hinduism and Shinto. We can see this in the yoga practice of sun salutations. We see this was true also in ancient Greece, in a passage of the Symposium, describing how Socrates stays standing all day, one day, from dawn, thinking about a difficult problem, and remains standing all night long. "He stood till dawn came and the sun rose; then walked away, after offering a prayer to the Sun."(4)

In the contemporary world, the principle of the square and circle representing the earthly and the divine remains important, for example in China. For the 2008 Olympic Games, the Water Cube, based on the square, and the Bird Nest Stadium, based on the circle, were built side by side.

In Hindu cosmology, mandalas and yantras illustrate the importance of squares and circles, in various combinations, to represent the universe. Here also, the circle symbolises the infinite nature of the cosmos, while the square represents the material world. These symbols are evident in Vedic texts and temple architecture.

In various Native American traditions, the medicine wheel, which is a circular symbol, represents the universe and the cycles of life. The earth is often symbolised by the square or the four directions, indicating the tangible world.

The world of forms

For the Greek philosopher Plato, the ideal world of forms is considered separate from the world of physical things, and is more perfect from the world of experience, though just as real. In Plato's view, objects of pure geometry, such as straight lines, circles and squares, are only ever approximated in the finite world of our existence. Precise mathematical truths and geometrical objects exist only in a separate world, an ideal world of concepts. The only way we can access this other world is via the intellect. In this other realm, the process of squaring the circle becomes possible, because it is conceptual. And as a result not only does pi's transcendentality cease to be a problem in squaring the circle, but it gives the process of equating a square and a circle to each other through their areas meaning.

Indeed, it is possible that the ratio between a diameter and a circumference, pi, which is impossible for us humans to accurately define, confers upon the world of the material, in this case the square, an element of the divine. It is this interplay between the material and the divine which gives the process of squaring the circle it's importance, not the practical difficulties in constructing such a process geometrically, which mathematicians have wanted to try to overcome over the centuries. Perhaps this idea of the material world and the ideal world had existed before Plato. This would fit with the mathematical system which produced the mile.

The exploration of concepts like squaring the circle and measuring the earth's dimensions encompasses a profound mix of philosophical, cosmological and practical considerations.

Squaring the circle symbolises the attempt to translate the divine, the irrational and the transcendent (the circle, pi) into the tangible and finite (the square) and in a sense bridges the worlds of the material and the divine. It is a process that remains possible only in the ideal world if we take accuracy seriously. However, if we allow for approximation, as we must anyway in practical geometry, it is also possible, but in a different way.

Perhaps squaring the circle reflects the human desire to understand and embody universal truths in material forms. This interplay invites philosophical inquiry into the nature of reality, the limitations of human knowledge, and a balance between the ideal and the empirical. It highlights the tensions between the infinite and the finite, the divine and the human. Yet it also seems to suggest that while they are separate, they can exist one within the other, and that there is in infinitely small gap between the human and the truly divine, which is beyond our comprehension. Pi can be approximated, and must be in any geometry which uses pencil and paper, in architecture, in engineering. But ultimately, pi remains a mystery, a constant which is impossible to completely define or understand.

Abstract geometric constructions, like drawing the diagonal of a square, or the diameter of a circle, or indeed squaring the circle, embody ideal forms and mathematical truths that are not possible in the physical world. Indeed, there is no number which, squared, gives 2 exactly, and while it is easy to draw the diagonal of a square on paper, this diagonal is impossible to truly define, in terms of number, in relation to the sides of the square. This is the problem of incommensurability between a square's sides and diagonal. This problem could represent the boundary between the different worlds of the divine and the material, or between two different dimensions in spacetime. The square, and the octagon made up of two squares on on top of the other, are associated with life and death. Many baptistries and mausoleums use the octagon as a central feature. So it is possible that one square symbolises what Mother Earth gives and the other what she takes away.

Architecture, astronomy, engineering have no need to dwell on this, as practical applications of irrational numbers are used. Squaring the circle, in ancient cultures, transcended mere mathematical curiosity. It may have been linked to an exploration of the universe's structure, of human understanding, and spiritual aspirations. Knowledge of the irrationality of pi, if it existed, might have meant such an exercise was about uniting the two worlds, the ineffable divine (the circle), and the material world (the square), knowing that it was impossible to truly understand the area of a circle in terms of the area of a square in the material world, in an applied way. Yet in the world of the intellect, of thought, constructing perfect shapes, a circle, a square, and by extension, a square with the same area as a circle, is not problematic. By integrating divine symbolism with practical measurement, ancient civilisations interested in the idea of a square and a circle of the same area sought not only to quantify physical and temporal dimensions but also to explore deeper philosophical truths about the nature of existence and the cosmos. Mathematics and geometry become tools for exploring the mysteries of nature, and expressing profound insights through measurement.

While the realisation of pi as a transcendental number implied the impossibility of exactitude in squaring the circle in applied mathematics, today we mostly take the problem to be a meaningless one. Yet we think nothing of tracing a circle with a compass, and calling it a circle, even though pi is transcendental. Applied mathematical constructions are different to conceptual mathematics. Since the mid 19th century, attempts to square circles have been mostly ignored or ridiculed. The challenge of squaring a circle is an applied maths one, and the task is to construct a square which has the area of a specific circle, by using only a finite number of steps with a compass and a straightedge. Why it is always specified that the number of steps has to be finite is strange, as presumably, anyone who attempts this construction is mortal, and will eventually run out of time. Also strange is the debate around precision, as it is difficult to find which parameters, or margins of error are acceptable in such constructions. While it is impossible to trace a perfect circle to begin with, if we take the nature of pi seriously, the concern shown for precision often seems to skip this aspect of applied mathematics and focus on an unquantified level of precision for the square that is constructed from the circle. It would be more consistent to make a requirement at the outset for the level of precision in the construction of the circle, before attempts to square it have even begun. Or else it would be just as consistent to declare that tracing a perfect circle is impossible, therefore squaring it is impossible.

If we do concede that it is possible to construct a circle with a compass, then it is possible to construct a square with the same area as a circle, as long a we agree at the outset that these constructions will be approximate. This has been done by outstanding mathematicians, such as Ramanujan, with pi as 355 / 113. By using a specific rational value (355/113) for π, he provides a clear and precise framework for practical constructions, ensuring that the limitations of the approximation are understood and accounted for.

This approach contrasts with the implicit approximations often made when drawing geometric shapes, where the inaccuracies are not explicitly quantified. Ramanujan’s method demonstrates mathematical honesty and rigour by working within the known limits of approximation and providing a highly accurate solution within those bounds.

As Petrie put it:

The impossibility of drawing a circle

Let's assume that it is possible to draw a circle starting from a diameter. Let's use a compass and give it a diameter of 1. A true circle with diameter 1 has a circumference exactly equal to π. Let's assume that we can measure the circumference of this true circle precisely. So the circumference of our circle should be π, according to our assumption.

It is known that π is an irrational and transcendental number, meaning it cannot be expressed as a finite decimal or fraction. It is inherently non-repeating and infinite. If the circumference of the circle is measured and found to be π, this implies an exact definition of π. However, since π is transcendental and cannot be exactly measured or expressed in finite terms, this measurement is impossible. Therefore we cannot draw a true circle.

The exact value of π cannot be measured or represented in the material world due to its transcendental nature. It is impossible to draw a true circle with a diameter of 1 unit and a circumference of exactly π in the material world. Any circle drawn or measured will always involve an approximation of π, thus proving that a true, exact circle, as defined by having a circumference exactly equal to π, cannot be constructed or measured. Any attempt to construct or measure a circle must necessarily involve approximation, thus reinforcing the conclusion that a perfect circle with these properties cannot exist in the material world.

The impossibility of drawing a true circle with a diameter of 1 unit and a circumference of exactly π serves as a vivid illustration of Plato's theory of forms and the philosophical division between the material and ideal worlds. It reinforces the idea that while ideal mathematical forms exist perfectly in the abstract realm, their exact realisation in the physical world is inherently limited by practical constraints. This interplay reflects a broader philosophical and mathematical inquiry into the nature of reality, the limits of human understanding, and the quest to bridge the gap between the ideal and the material.

The allegory of the cave

Plato's theory suggests that abstract entities (Forms) such as perfect circles and mathematical truths exist in a non-material realm. Pi would be part of this realm, as it defines the relationship in the ideal form of a circle. According to Plato, we access these Forms through intellectual reasoning, not through sensory experience. The existence of π as an abstract concept supports the idea that there is a realm of perfect, immutable entities.

One of the most relevant dialogues by Plato that discusses the nature of forms and the distinction between the material world and the world of ideal forms is the Republic. In particular, the "Allegory of the Cave" and the discussion of the "Divided Line" provide insight into Plato's theory of forms.

   In "Republic" (Book VII, 514a–520a), Plato describes a group of people who have lived chained to the wall of a cave all their lives, facing a blank wall. They watch shadows projected on the wall by things passing in front of a fire behind them and begin to ascribe forms to these shadows. According to Plato, the shadows are as close as the prisoners get to viewing reality. The philosopher is like a prisoner who is freed from the cave and comes to understand that the shadows on the wall are not reality at all.

   This passage illustrates the idea that the physical world is a mere shadow of the true reality of forms. In the "Republic" (Book VI, 509d–511e), Plato introduces the analogy of the divided line, which is divided into two sections, representing the visible world and the intelligible world. The visible world is the world of change and uncertainty, while the intelligible world is the world of unchanging forms and mathematical truths.

   In "Phaedo," Plato discusses the theory of forms in the context of the immortality of the soul. He emphasizes that true knowledge comes from the understanding of these eternal forms, rather than the sensory experiences of the material world.

   For Plato, certain realities exist beyond our experience, but if this is the case, do these realities exist independently of human thought? Does the Existence of π Prove a Non-Material World? The existence of π as a well-defined concept that cannot be fully captured in the material world suggests that there is a realm where such perfect concepts exist. The implication of π's existence depends on how we interpret the existence of mathematical objects. If we take a Platonist view, then π supports the existence of a non-material realm. If we take a nominalist view, π is simply a useful human construct. While π may not exist in the physical sense, it points to a realm of perfect, immutable mathematical truths, be it imagined, or independent of human minds.

The idea that there might be a difference between physical reality and the world of the intellect and concepts continues to be important. Roger Penrose's framework of the three worlds is a philosophical and conceptual model that aims to explain the relationship between mathematics, the physical universe, and human consciousness. (11) The three worlds Penrose identifies are the physical world, comprising everything that exists physically, including all matter and energy, from subatomic particles to galaxies, the mental world, which is the realm of human consciousness, thoughts, perceptions, and mental experiences, and the Platonic world, which is the abstract realm of mathematical forms, concepts, and truths. The Platonic world includes numbers, geometrical shapes, and all mathematical structures that exist independently of human thought and the physical universe, whereas the mental world is about subjective experiences, including emotions, creativity, and the sense of self. Human consciousness allows access to the Platonic world of mathematical truths, and mathematics provides a precise language to describe the physical universe. Penrose's three worlds framework emphasises the profound interconnectedness between mathematics, physical reality, and human consciousness.

The impossibility of exact π in physical world constructions

Pi is an irrational and transcendental number, meaning it cannot be expressed exactly as a finite or repeating decimal, nor can it be the root of any non-zero polynomial with rational coefficients. Any physical construction or measurement involving π will necessarily be an approximation. For instance, drawing a circle with a specific diameter and attempting to measure its circumference will always yield an approximation of π. In the material world, exact constructions of geometric figures involving π are impossible due to the inherent nature of π. In the ideal world of forms, however, a true circle with an exact circumference of π is perfectly realised. When we draw a circle with a compass, we approximate π. Even though we can get extremely close, the act is fundamentally an approximation because π cannot be represented exactly in any physical form. Similarly, the process of squaring the circle—constructing a square with the same area as a given circle—can only be achieved approximately in the material world. Given that any practical use of π is an approximation, the act of squaring the circle falls into the same category of approximate constructions.

Existence of true circles and squaring the circle in the ideal realm

In the realm of Platonic forms, geometric constructions are perfect. A circle with a diameter of 1 unit has a circumference of exactly π, and a square can be constructed to have the same area as this circle. In the abstract, mathematical realm, where exact forms and precise relationships exist, both conceiving of a true circle and squaring the circle are possible. The limitations that prevent exact constructions in the physical world do not apply in this ideal realm.  In practical terms, both drawing a circle and squaring the circle involve approximations of π. Since we accept the approximate nature of drawing a circle, we should similarly accept the approximate nature of squaring the circle. If we accept that drawing a true circle is possible in the ideal, non-material world (as per Plato and Penrose), then squaring the circle is also possible in this realm. Both are ideal geometric constructs that exist perfectly in the world of forms.

Requiring π to be exact in physical constructions is inherently impossible due to its irrational and transcendental nature. Both drawing a circle and squaring the circle in the material world are approximate activities. However, in the ideal world of Platonic forms, exact constructions are possible. Therefore, if it is possible to draw a true circle in the ideal realm, it is also possible to square a circle in that same realm. In practical terms, accepting the approximate nature of geometric constructions means that squaring the circle is as feasible as drawing a circle, both being approximate representations of their ideal forms.

Understanding the distinction between the material and ideal realms clarifies why certain mathematical constructs are seen as impossible in practice but possible in theory.

Does Pi exist?

The question of whether π exists touches on philosophical issues about the nature of mathematical entities and the existence of abstract objects. In the physical world, π is always approximated. We can measure it to many decimal places, but we never capture its full, infinite decimal expansion. Every physical circle is only an approximation of an ideal circle. If π, as an irrational and transcendental number, cannot be fully expressed in the material world, then it either does not exist at all, or else it exists as a concept only, or else it exists in some other, non-material world. It can be considered an abstract object, existing in the same sense as other mathematical entities like numbers, functions, and sets. It doesn't exist in space and time but rather in the realm of mathematical truths.

A paradox: the cosmic temple

Squaring the circle could symbolise a process of travelling between the material world, and this other realm, in which perfect concepts such as pi can exist, and as a result, where perfect circles, and squares can exist, and within which squaring a circle is possible. The paradox is that it is within this perfect world of forms, that the bridge between the circle, representing this perfect, or divine world, and the square, representing the material world, can happen. If the circle symbolises the divine, the infinite, and the perfect, it can also be said to represent some perfect world of forms, within which pi exists, in a way which cannot be fully grasped or measured by finite means. On the other hand, the square represents the material world, the finite, and the measurable, and perhaps the more tangible aspects of human existence.

In the realm of forms (perhaps we could also call it the divine realm), exact squaring of the circle is possible because this realm contains perfect concepts and entities. In this realm, π is not just an irrational number but a fundamental and exact ratio that defines the circle perfectly. Thus, squaring the circle here symbolises the unity and harmony of divine principles. In the material world, any attempt to square the circle is inherently approximate due to the limitations of physical reality and our tools. This symbolises the human striving to comprehend and bridge the divine with the material, acknowledging our limitations.

The paradox lies in the fact that the act of squaring the circle—a bridge between the divine (circle) and the material (square)—is itself bound by the nature of the realm in which it is performed. If done exactly, it belongs to the divine realm, yet it represents a connection to the material world. If done approximately, it acknowledges the material world's limitations while striving towards the divine.

The paradox symbolises the human quest to understand and reach the the world of concepts, the ideal, the infinite, the divine. While we can conceptualise perfect truths and entities, our physical realisation of these truths is always approximate. Human endeavours to bridge this gap, even if never fully successful, are meaningful and symbolic of our connection to the divine. If the concepts of π and the perfect circle suggest that there is a realm where these ideals are real and exact, it is a realm which is also a foundation for the material world, influencing and giving meaning to our attempts to understand and replicate divine principles. It would make sense to use the squaring of a circle as a basis for measurement, and perhaps also music and architecture, inviting us to appreciate the symbolic meaning of mathematical and geometrical pursuits as reflections of deeper philosophical and metaphysical ideas.

Ultimately, at the level of the individual, the square and the circle represent something of human nature, and at the level of society, the square and the circle create a space within which to live within a sense of order, derived from the workings of the universe. The squared circle is the basis for a cosmic temple.

As John Michell put it in City of Revelation:

So why is squaring a circle part of the calculation of the earth's equatorial circumference in miles?

Let's return to the equatorial circumference as  √(π×20 000 000)×π = 24 902.3198 miles. Why not simply divide up the estimated length of the equator into as many units as seem fit, for example to correspond to a significant astronomical number, as we see with the number of inches in the equator: a yuga's worth of days. With the mile, there is an elaborate system involving squares and circles, which might seem over complicated, if it weren't for the fact that the square, the circle, and the process of squaring the circle, are highly symbolic.

In the realm of Platonic forms, geometric constructions are perfect. A circle with a diameter of 1 unit has a circumference of exactly π, and a square can be constructed to have the same area as this circle.

If we accept Hugh Franklin's equation √(π³ x 20 000 000) = 24 902.3198 as an accurate estimate of the length of the equator in miles, or this alternative, √(π × 20 000 000) × π = 24 902.3198, there are several assumptions we can make. Firstly, the people who came up with the mile and this system of measuring the equator were excellent mathematicians, and excellent geometers. Secondly, they knew the earth was round. Thirdly, they had an accurate measure of it, very similar to today's estimate. Fourthly, pi was important to them.

I can think of no other reason for the presence of the square root of pi in the equation other than as a product of a squaring of the circle. If the area of a circle is π, and the circle is squared to produce a square of equal area, the square will have sides of √π. The equation connecting the equatorial circumference to its measurement in miles can be interpreted as not just a mathematical relationship but a reflection of deeper truths about the nature of the Earth and its place in the cosmos.

If indeed it was known that the earth was a sphere (or oblate spheroid, even), it would have been important to translate its shape, associated with circles are the heavenly and divine, into a square, to acknowledge its being in the finite world. The square is a symbol of the finite, and it would make perfect sense to measure a square representing earth, rather than a circle, which represents infinity, and infinity cannot be measured, even though the equator is a circle. Squaring the circle represents something beyond human comprehension, and specifically within the process it is pi that is at the root of our troubles in understanding. I would suggest that pi is a symbol of the divine and of infinity, and that squaring the circle is about acknowledging this mystery, preserving it even. A square which is based on a prior circle, and depends on pi, or the square root of pi, has a special status, in that it represents the finite, material world, but has an undissolvable element of the divine within it, that is quite literally insoluble.

This does not mean that circles were considered too divine to ever measure, or to use in architecture. Rather, it is about the earth, which belongs to the world of the heavens at the level of the macrocosm, but to the world of the material and mundane, at the microcosm. At the level of the inch, which symbolises a day, the equator is represented by a long period of time in days. But at the level of the mile, which is 63360 inches, the earth is represented by its being at the intersection of two domains, the heavenly and the earthly, the infinite and the finite, the equator representing a sort of liminal friction between what we can understand and what we can't. Squaring the circle is a symbolic transfer of the infinite and unknowable into the domain of the human, not an exercise in impossible practical mathematics. It is an acknowledgement that when we measure something at the edge of the finite material world, we must approximate, which is a sort of concession. But we can also conserve and celebrate the element of divine.

According to this system, the mile becomes a unit which is well suited to measuring the finite aspects of the heavenly world, such as the dimensions of the sun and moon, and even the mean diameter and circumference of the earth, and there is no need to go through the squaring of the circle. Indeed the diameters of the sun and moon are close to 864 000 and 2 160 miles respectively, simple multiples of 6 (4 000 x 6 and 10 x 6 x 6 x 6). And the mean diameter of the earth can be interpreted as 7920 miles.

The number 6, like 28, is a perfect number, in that 1+2+3 = 1 x 2 x 3, and this could be why the sun, earth and moon are defined in miles with multiples of this number. John Michell described the mile as a unit which "measures the cosmic intervals in terms of the number 6". (7)

But as Hugh Franklin has shown, with his interpretation of the equator, the mile can also be used in more complex ways. The mean diameter of earth fits with the way the sun and moon diameters are measured, but the equatorial circumference of earth can be seen as having been treated differently. This could be because it functions as a boundary between the heavens and the world of plants, animals and humans. On the one hand it is another heavenly body, and on the other it is the outer limit of our material world. And so it is appropriate to combine the geometries of the circle and the square to measure it. We can give the equation which corresponds closely to the equatorial circumference of the earth in miles a philosophical, perhaps even religious interpretation. Accordingly, we can think of the circumference of the earth which is at a right angle to the axis upon which the earth spins both a geographical, or physical, and a metaphysical dimension. While the polar circumference is linked to the axis of the earth's daily rotation, and is aligned with the polar axis, linking the terrestrial to the celestial, the equator is different in symbolic character, and acts more as a boundary between these two worlds. Perhaps the presence of the number 2 in the equation √(π × 20 000 000) × π signifies the duality of day and night, as it is the equator, as well as all lines to latitude, which follow the motion which gives rise to this daily dualism of light and darkness. Perhaps, then, it is more appropriate to divide the polar circumference into four parts rather than two, as on top of the night and day duality, there is the duality of north and south. So it is fitting that the metre should be a 10 000 000th part of a quadrant, or fourth part, of this polar circumference.

  The duality of light and darkness reflects the dual nature of the equator as both a geographical and metaphysical boundary. The measure of the equatorial circumference in miles suggests an integration of mathematical, philosophical, and symbolic perspectives. Seeing the circumference as a boundary between the heavens and the material world, reveals a connection between geometry and cosmology. The process of squaring the circle to measure the equatorial circumference serves as a powerful metaphor for the interplay between the divine and the material, and a striving to understand and bridge these realms.

There is another possible reason for the squaring of the circle when measuring the circumference of the earth,. If we look at the sphere in only two dimensions, as a circle, it is divided in two horizontally and then again vertically by the equatorial and polar circumferences, which meet at a right angle, which hints at the square.

This meeting of two cross section lines forming a right angle is what Meton does in The Birds, Aristophanes's play. Meton declares (1004–5):

He places his straightedge across the centre point of the circle to bisect it, twice, declaring he has squared a circle. Meton, who was a real astronomer, and whose name is associated with the 19 year cycle, is made a figure of ridicule in the play. He is shown as a sophistic

scientist who describes the world in esoteric terms, and uses obscure language, thus putting others at a disadvantage. This is the reality of an elite which controls knowledge. While the methods used may be sophisticated, the knowledge acquired is completely out of reach to most people. One possible reading of Aristophanes's play is an attack on this completely undemocratic system, whereby only a very few people get to learn about and research into the workings of the natural world, mathematics and astronomy. In Aristophanes's play, Meton's ridiculous squaring of the circle is perhaps not as ridiculous as may seem, as it is an important symbol, and can be said to represent the earth, with it's polar and equatorial circumferences. It is similar for example to the Lakota medecine wheel pictured above, and to the celtic cross, pictured below.

A metre of sorts, the polar circumference, and calculating pi

   It is intriguing that while √(π³x 20 000 000) = 24 902.3198, if we remove a zero and replace 20 000 000 with 2 000 000, we get √(π³ x 2 000 000) = 7874.80497, which, divided by 200 gives 39.37402. As a value in inches, this is approximately a metre. It can also be expressed in inches as √(π³ x 2) x 5, or √(2π) x 5π or √(2π) x 10π / 2. Again here we find the square root of pi, and the hint of a squaring of the circle. It's possible to think of the metre as linked to the inch via a squaring of the circle.

 If we start with a square with sides of 2 inches, the diagonal will be 2√2 inches. This diagonal can become the diameter of a circle, which will have an area of 2π square inches. The circle is squared, in that a square is drawn which has an area equal to the circle. And this square will have to have sides of √(2π) inches. If we take a side of this square, and turn it into the diameter of a circle, the circumference will be √(2π)π = 7.87480497 inches. This is almost exactly 20 centimetres (2 x 39.374025), and is close to the 39.375 inch value, which is linked to the polar circumference.

If we take the value of this particular metre, √(2π) x 10π / 2 = 39.37403 inches, and see if it fits in the polar circumference of the earth, it fits very well.

Expressed in inches: √(2π) x π / 2 x 400 000 000 = 1 574 960 994.572

Expressed in miles: √(2π) x π / 2 x 400 000 000 / 63 360 = 24 857.3389

Expressed in actual km: √(2π) x π / 2 x 400 000 000 x 25/ 10 000 000 = 40 004.0093

The current estimate is 24,859.734 miles, or 40,007.863 km, so while this value for the metre doesn't exactly fit, it fits well, and slightly better than the official metre. So perhaps the squaring of the circle also works for the polar circumference, in a slightly different way than for the equatorial circumference.

A squaring of the circle process, starting with a square of 100 000 inches, works for the polar circumference. The equatorial circumference squaring of the circle started with a square of 2 000 miles which was then doubled.

The mile can be used to express the polar circumference in terms of time:

The polar circumference as 1 575 000 000 inches, or 25 857.954545 miles is key to the work of Jim Alison, Stephen Dail, David Kenworthy, and others, and indeed an Egyptian digit can be precisely written as 0.7291666667 inches, which is 1 575 000 000 / (6³ x 10 000 000). Sixteen such digits make a Roman foot of 11.666667 inches, eighteen make a Saxon foot of 13.125 inches, twenty make a remen of 14.5833333 inches, which multiplied by 99/70, as an approximation of the square root of 2, give 20.625 inches, for the Royal Egyptian cubit., and 54 make a metre. So this approach, squaring a circle to give the polar circumference, fits well within this system. A digit as a 1 / (6³ x 10 000 000) the division of the polar circumference derived from the squaring of the circle process would be 0.7291486 inches, a Roman foot 11.666378 inches, a Saxon foot 13.124675 inches, a remen 14.582972 inches, an Egyptian royal cubit 20.623437 inches (with √2), a metre 39.3740286 inches.

   If the metre is supposed to be 39.3700787402 inches, what would the value for pi be, in this equation: √(2π) x 10π / 2 = 39.3700787402 ?

39.3700787402 is √1 550

Replacing π with a, and simplifying to:

√(2a) x 5a =  √1 550

Squaring both sides:

(√(2a) x 5a)² =  (√1 550)²

Simplifies to:

2a x 25a² =  √1 550²

50a³ = 1550

Dividing both sides by 50

 a³ = 31

a = ∛31 ≈ 3.1414

So using this equation, and with the current value for the metre, pi is a ≈ 3.1414.

And indeed, π³ ≈ 31.00627668.

If we use the same method but with the value for the metre as 39.375 inches, we get a value for pi ≈ 3.1416. And with the value for the metre as √(2π) x 10π / 2 = 39.37403 inches, the value of pi here already depends on 'calculator pi' having been used.

   A square with an area of π³ will have sides of √ (π³). If one of these sides is the diagonal of another square, this second square will have sides of √ (π³) / √ 2 = 3.9374025 inches, which is about 10 cm.

Both values for the metre, 39.3700787402 inches (the actual official value) and 39.375 inches, an important value in historical metrology, can be shown to be linked to approximations of pi that are very close to our own today, 3.1414 and 3.1416 respectively.

It's possible that the value of pi was at some point in the distant past understood as the cube root of 31. It's possible that the metre was conceived as √(31 x 50) inches. The metre can be interpreted as a derived unit connected to π, especially when considering historical and geometric contexts.

   This alternative conception aligns the metre with ancient mathematical practices of relating circles and squares (or cubes), embedding the process of π approximation into the unit of measure. Considering the metre as √(31 x 50) ​ inches embeds the process of π calculation into the unit of measure, connecting it to the historical context of geometry and measurement

Indeed, why stop at the squaring the circle, when we can cube it? A circle with a diameter of π has an area of π². Next comes the squaring of the circle. A square with an area of π² also, has a side length of π. Next we cube it. A cube with a side length of π has a volume of π³. We can relate this back to Hugh Franklin's equation, √(π³ x 20 000 000) = 24 902.3198.

The polar circumference can also be expressed as follows, inspired by a find by Dennis Payne:  50 000 / 3 x √2 x π x √π / 5 280 = 24 857.3389.

Great Pyramid Dimensions

At Giza, there are no circles made of stone, as for example at the Temple of Heaven in Beijing, but mainly squares, rectangles, and of course pyramids, and triangles. Yet, the base of the Great Pyramid can be defined in terms of the height by pi (π), the ratio between a diameter and a circumference, in a circle. The height, of 5776 inches, multiplied by 2π, gives the base, that is the total perimeter, with each side measuring 9068.8 inches, as per Petrie. The height multiplied by π / 2 gives the mean side. The height of the Great Pyramid relates to the square perimeter as a radius relates to a circumference.

Does the Great Pyramid, with its square base, represent earth? Or does it represent heaven, with its reference to the circle, via the ratio between the height and the base? The height of the Great Pyramid already represents something of a heavenly nature in that in inches, it corresponds to the number of sidereal lunar months in a very long, but important time period called a yuga in ancient Indian tradition, which is 4 320 000 sidereal years. The Indian mathematician Aryabhata, though from a different time and place to the Great Pyramid, says that in a yuga of 4 320 000 sidereal years there are 57 753 336 revolution of the moon. While he doesn't mention the Great Pyramid of Giza, it's clear that there's a connection to the height of the pyramid in inches, given by Petrie as 5776. At Giza, an inch represents a (sidereal) day, not just in this measure of the height, but in many other instances.

We can think of the ratio between a yuga of 4 320 000 sidereal years and the sidereal lunar month as being expressed in the geometry of an equilateral triangle. If the triangle has sides of 2 x 10⁸ / 3, the height will be 10⁸ / √3. And 4 320 000 sidereal years divided by a sidereal month of 27.321661 days are almost exactly 10⁸ / √3 also. The height of the Great Pyramid is very close to 10 000 / √3 inches and can be interpreted as illustrating this connection.

So we can think of the square base of the Great Pyramid as related to the earth, being square, but also channelling something of the heavens through this astronomical ratio, combined with pi and the mysteries it brings. Through pi, which creates a circle from a straight line, we get a square, but a square that is linked to the circle through its perimeter and its ratio to the height.

We've seen that the equatorial circumference of the earth expressed in inches can be approximated by the equation 4 320 000 x 365.242199. This is strikingly similar to the height of the Great Pyramid being 4 320 000 / 27.321661 inches. As before, an inch representing a day, if we take a yuga of 4 320 000 sidereal years again, but this time rather than dividing by a sidereal month, as we did to obtain the height of the Great Pyramid, this time we multiply it by a year in days. It is possibly the same system at work, in one instance the mile is derived from a yuga of tropical years and the length of the equator, and in the other, the height of what was for a long time the tallest building in the world the same system, the same method.

If an architect, trained in a tradition whereby the heavens are represented by circles and earth by a square, were going to design a building to represent the earth, he or she would probably come up with some kind of square design. Indeed, the square base does represent the circumference of the earth, but it is the polar circumference which is converted into day inches into the main base, with the mean sides of 9068.8 inches.

This is an aspect of the Giza complex that has been written about and talked about a lot: that the equatorial circumference of the earth is represented by the perimeter of the Great Pyramid, and this, to a significant scale, related to precession. This is most famously put forward by Graham Hancock. The mean base side being 9068.8 inches, and using the factor 43 200, we would obtain an equatorial circumference of 43 200 x 9068.8 x 4 / 63360 = 24 733.0909 miles, which is way off the current estimate, by about 163 miles. The socket base perimeter is according to Petrie 9125.9 inches, and this part of the base would correspond to the mean circumference of the earth, that is, the average of the polar and the equatorial.

Perhaps the exterior platform corresponds to the equatorial circumference, but going by Flinders Petrie's measurements, the outer casing perimeter is the largest part of the base of the actual pyramid, and it is too short to match the equatorial circumference. Perhaps some aspect of the basalt paving around the pyramid can be measured to about 9130 inches on each side, which would provide a perfect match, using the same 43 200 scale.

Graham Hancock also makes use of the number 43 200, the scale at which the dimensions of the Great Pyramid correspond to the size of the earth, to link it to precession. Indeed, the traditional value for the precession of the equinoxes cycle is 25 920 years, and is very easily connected to 43 200 by multiplying by 10 / 6. But perhaps it is simpler to keep the number 43 200 and see what it itself represents or is connected to, rather than another number's connections. There are 12 x 60 x 60 = 43 200 seconds in 12 hours, so it is easily linked to the sexagesimal system. And of course a yuga cycle in Hindu cosmology is 4,320,000 years long. The polar, or meridional circumference is linked to the height of the Great Pyramid. If we divide the circumference into 40 000 parts, and then by the average number of lunations per year, which is 12.368266, and multiply by π / √3, we get the height of the Great Pyramid.

The Great Pyramid height relates to the polar or meridional circumference via π / √3, the average number of lunations per year, and the number 40 000. And it relates to the equatorial circumference via the sidereal month in days and the number 10 000. In both these cases, the unit used is irrelevant. However, when we express the height in inches we can see the geometric equivalence in an equilateral triangle appear. And when we express the height in metres we can see a connection to the definition of the metre itself, as it is (approximately) a 40 000 000th part of the polar circumference.

We can interpret the side and height of the Great Pyramid, and the polar circumference, expressed in inches, in terms of the numbers 11, 9, 7, 5 and 3.

This diagram was inspired by two brilliant finds, one by Dennis Payne and one by Quentin Leplat, in relation to the dimensions of the earth and Giza measures.

Squaring the circle and astronomy

A lunar year, and a lunation, can very simply be approximated by a squaring of the circle based on a circle with a diameter of 400, or 100, or 100/3, each creating a square with approximately each side equal to a lunar year, or a perimeter equal to a lunar year, or to a lunar month, respectively.

More generally, an approximation of pi can unite various astronomical cycles, as this example shows.

The importance of the circle at Giza

Here are just a few examples of circle equivalences at Giza. The first one links the Great Giza Rectangle, i.e. the perimeter of the outer north-east corner of the Great Pyramid and the south-west corner of the third pyramid, to the side of the third pyramid. This side can also be approximately linked to the equatorial circumference of the earth in fact. If we take  √(20 000 000 π) x π = 24 902.3198 as the length in miles of the equatorial circumference, one side of the third pyramid, given by Petrie as 4 153.3 inches, can be interpreted as approximately  √(20 000 000 π) x π / 6 = 4 150.38664 inches.

At Giza, 29.53059 x 2π / ( 3 x √3), and more simply, π /√3, occur frequently. For example the side of the Great Pyramid is equivalent to π /√3 x 5 000 = 9 068.9968 inches, which is almost exactly Petrie's measure (9 068.8 inches). And the length of the Great Giza Rectangle, so the north-south distance ebtween the north side of the Great Pyramid and the south sideof the third pyramid, can be interpreted as π /√3 x 5 00 x 29.53059 x 4 / 3 = 35 708.3769 inches, just a few inches under Petrie's estimate of 35 713.2 inches.

π /√3 links key astronomical cycles, and implies a triangle (the height of an equilateral triangle being √3 if the sides are each 2), and a circle (π being the ratio bewteen a diameter and a circumference).

Pi and the square root of three are often found combined in the Giza dimensions, and can even be understood as part of the Egyptian royal cubit.

There are many instances of circles being implied, if not explicitly drawn out, such as with the dimensions of the King's Chamber.

In the diagram below, the large red square and the large black circle share a perimeter, instead of an area, and the square corresponds to the base of the Great Pyramid, while the radius of the black circle corresponds to the height of the Great Pyramid. The smaller green circle has a circumference equal to one side of the Great Pyramid. The circles and the square are drawn around an equilateral triangle with sides of 10 000 inches.

In this example, a square and a circle are compared one to the other, but not in terms of area, in perimeter instead. The perimeters of the red square and the black circle are of equal length.

The diagram above, on the left, is similar to John Michell's diagrams below, to the right, numbers 3 ands 4, and to the left, which is his interpretation of Stonehenge's layout. The perimeter of the large square is equal to the circumference of the large circle. Compared to the diagram above, Michell's diagram has an extra circle, inscribed within the square, and an extra triangle, inscribed within this circle. The dimensions are different. Other than that, there is a similarity in the approaches, and this is another hint, among many, of a similar way of thinking at Stonehenge and at Giza. The radius of the large circle corresponds to the height of the Great Pyramid, and the square corresponds to the base of the Great Pyramid. Perhaps the two triangles and the inner circle in Michell's diagram can be interpreted as the union of the male and female principles.

If we take a slightly reduced side of the Great Pyramid, 9065.8 inches instead of 9068.8 inches, a circle with the same area as the square base would have a radius of 29.530589 x √3 x 100 inches.

9068.8² = 82 243 133.44

 (29.530589 x √3 x 100)² π = 82 189 317.7309

√82 189 317.7309 = 9065.8324

If we think of the side of the Great Pyramid as the side of an equilateral triangle, using 9069.13 inches, a value close to Petrie's 9068.8, a circle drawn within this triangle would have a radius of 2618.03232, which is Phi² x 1 000 inches. A circle drawn around the triangle would have a value of 5 236.06465 inches, which is Phi² x 2 000 inches. It is also 10 000 Egyptian cubits expressed in metres. This confirms the importance of the inch at Giza, and links it to the mile by extension, the mile being 9 x 64 x 110 inches.

Conclusion

The system which produced the inch and the metre has close links to the metrology behind the pyramids of Giza. Both highlight the importance of pi, and equating circles and squares, and both rely on an accurate measure of the earth. The connection to the number 4 320 000 is also clear.

This highlights the importance of attempting to read the dimensions of the pyramids of Giza in inches, feet, miles, and metres, as well as other units, as they are all linked.

Both drawing a true circle and squaring the circle are impossible in the physical world because π is irrational and transcendental. Physical constructions and measurements can only approximate π. In another realm of concepts, of the intellect, or Plato's forms, perfect geometric constructions are possible. In this ideal world, a true circle can exist with an exact circumference of π, and squaring the circle is achievable. In the material world, practical geometry always involves approximations. Just as we accept that we can only draw approximate circles, we should accept that we can only square the circle approximately.

If we acknowledge the existence of perfect geometric forms in the ideal realm, then squaring the circle is not fundamentally different from drawing a circle. Both are perfectly achievable in the non-material world of pure mathematics and ideal forms.

 The claim that squaring the circle is impossible is valid only within the constraints of the material world, where exact constructions involving π cannot be performed due to its irrationality and transcendence. The impossibility does not extend to the ideal world of forms, where exact geometric constructs, including the squaring of the circle, are conceptually possible. This distinction highlights that the perceived impossibility is tied to physical limitations, not mathematical or philosophical ones.

There is a paradox in basing a measurement, such as the equator, or a building, such as the Great Pyramid of Giza, on a squaring of the circle, either by area of by perimeter length. Generally, in fields like metrology, architecture, astronomy, and engineering, irrational numbers such as π are used in practical computations despite their infinite nature. Approximations are necessary and accepted because they provide sufficiently accurate results for real-world applications. The Great Pyramid of Giza, with its height-to-base ratio that approximates π\piπ, demonstrates an ancient understanding and application of these principles. However, the precision in such constructions often goes beyond mere practicality, suggesting a deeper symbolic intent. It appears that in the measurement of the earth, and in the design of certain structures such as the Great Pyramid of Giza, which has a height to base ratio which can be interpreted as linked to pi, but expressed as a square, the squaring of a circle is apparent. Can we infer that somehow this is an attempt to atone for approximation within the real world, or to link what is necessarily imperfect to a perfect form beyond our world?

The use of the squaring of a circle as a basis in measure and in construction in ancient measurements and designs can be interpreted as one aspect of a sophisticated philosophy and system, which reconciles the finite with the infinite. This reconciliation might not be purely mathematical but deeply symbolic, representing a bridge between the earthly and the divine. It is a grounding of human work in the perfection of the ideal forms.

The mile is an incredible artefact from the past, a product of a sophisticated world view, which places mathematics and philosophy to the fore, and according to which all things which matter should be designed. It stands as a testament to the advanced understanding and holistic approach of ancient civilisations, influencing how we design and measure the world even today.

Notes

1. Heath, Robin, and Michell, John, 2005, The Lost Science of Measuring the Earth, Adventures Unlimited Press

2. Franklin, Hugh, 2000, “Earth, Pi, Miles and the Barleycorn”,  http://hew_frank.tripod.com/epmb2.htm 

3. https://haifeng-piao.blogspot.com/2008/09/what-is-square-earth-round-heaven.html

4. Plato, Symposium, 220d, from Plato. Plato in Twelve Volumes, Vol. 9 translated by Harold N. Fowler. Cambridge, MA, Harvard University Press; London, William Heinemann Ltd. 1925. https://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.01.0174%3Atext%3DSym.%3Asection%3D220d

5. Michell, John, 1973, City of revelation : on the proportion and symbolic numbers of the cosmic temple, London: Abacus p 37, City of revelation : on the proportion and symbolic numbers of the cosmic temple : Michell, John F : Free Download, Borrow, and Streaming : Internet Archive

6. Ibid, p 40

7. Ibid, p 42

8. Plato, Republic, Book VII, 515e, translation by Benjamin Jowett, Oxford Clarendon Press, Jowett’s translation of Plato’s Republic, 3rd ed.—A Project Gutenberg eBook

9. Plato, Republic, Book VI, 509d–511e, translation by Benjamin Jowett, Oxford Clarendon Press, Jowett’s translation of Plato’s Republic, 3rd ed.—A Project Gutenberg eBook

10. Plato, Phaedo, 74a–75d, translation by Benjamin Jowett, Oxford Clarendon Press, Phaedo, by Plato (gutenberg.org)

11. Penrose, Roger. The Road to Reality: A Complete Guide to the Laws of the Universe. Jonathan Cape, 2004

12. Caesar, Julius, ‘The Conquest of Gaul’, translated by S.A. Handford (1951: 31-33),

quoted in MacKie, Euan W.. Professor Challenger and His Lost Neolithic World: the Compelling Story of Alexander Thom and British Archaeoastronomy, Archaeopress, 2021. ProQuest Ebook Central, http://ebookcentral.proquest.com/lib/dcu/detail.action?docID=6728884.

13. Michell, John, 1973, City of revelation : on the proportion and symbolic numbers of the cosmic temple, London: Abacus, City of revelation : on the proportion and symbolic numbers of the cosmic temple : Michell, John F : Free Download, Borrow, and Streaming : Internet Archive

14. Petrie, William M. Flinders, 1877, Inductive Metrology; or, the recovery of ancient measures from the monuments, London : Hargrove Saunders, https://www.digitale-sammlungen.de/en/view/bsb11353790?page=44,45

Bibliography

Amati, Matthew, “Meton’s Star-City: Geometry and Utopia in Aristophanes’ Birds.” The Classical Journal, vol. 105, no. 3, 2010, pp. 213–27. JSTOR, https://doi.org/10.5184/classicalj.105.3.213. Accessed 26 July 2024.

Franklin, Hugh, 2000, “Earth, Pi, Miles and the Barleycorn”,  http://hew_frank.tripod.com/epmb2.htm 

Heath, Robin, and Michell, John, 2005, The Lost Science of Measuring the Earth, Adventures Unlimited Press

MacKie, Euan W.. Professor Challenger and His Lost Neolithic World: the Compelling Story of Alexander Thom and British Archaeoastronomy, Archaeopress, 2021. 

Michell, John, 1973, City of revelation : on the proportion and symbolic numbers of the cosmic temple, London: Abacus, City of revelation : on the proportion and symbolic numbers of the cosmic temple : Michell, John F : Free Download, Borrow, and Streaming : Internet Archive

Petrie, William M. Flinders, 1877, Inductive Metrology; or, the recovery of ancient measures from the monuments, London : Hargrove Saunders, https://www.digitale-sammlungen.de/en/view/bsb11353790?page=44,45

Plato, Phaedo, translation by Benjamin Jowett, Oxford Clarendon Press, Phaedo, by Plato (gutenberg.org)

Plato, Republic, translation by Benjamin Jowett, Oxford Clarendon Press, Jowett’s translation of Plato’s Republic, 3rd ed.—A Project Gutenberg eBook

Plato, Symposium, 220d, from Plato. Plato in Twelve Volumes, Vol. 9 translated by Harold N. Fowler. Cambridge, MA, Harvard University Press; London, William Heinemann Ltd. 1925. https://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.01.0174%3Atext%3DSym.%3Asection%3D220d

Thank you to Norman Wildberger for his many maths videos, including this one : https://www.youtube.com/watch?v=REeaT2mWj6Y