103. Can the Great Pyramid Encode the Earth?

Aug 7

37 min read

Updated: Aug 11

At Giza, we see geometry acting as a bridge between lengths, areas, circles, squares and astronomical cycles. A natural question therefore arises. If the Great Pyramid can be read as a geometrical system, might it also preserve something of the size of the Earth itself?

This possibility may at first seem improbable. We tend to associate knowledge of the earliest knowledge of the Earth's dimensions with the ancient Greeks, and especially with Eratosthenes, a Greek employed in Egypt long after the pyramids were built. The achievements of Greek mathematics and astronomy were undoubtedly extraordinary, and thanks to the survival of so many Greek texts they occupy a central place in the history of science. But could the earth have been accurately measured long before the time of Eratosthenes?

Greek Interest in the Size of the Earth

Cleomedes stated that Eratosthenes found the circumference of the Earth to be 250 000 stadia, though other ancient authorities attributed to him a circumference of 252 000 stadia, equivalent to 700 stadia per degree. Aristotle, in the Meteorologica, records a circumference of 400 000 stadia. The source of this figure remains uncertain and may derive from earlier authorities such as Eudoxus. Archimedes, in the Sand-Reckoner, remarked that "some" estimated the Earth's circumference at 300 000 stadia, before proposing, purely for the purposes of calculation, to assume a vastly larger Earth with a circumference of 3 000 000 stadia. Cleomedes attributed the figure of 300 000 stadia to Aristarchus and Dicearchus. Posidonius later proposed 240 000 stadia, apparently reasoning that the meridian arc between Alexandria and Rhodes measured 5 000 stadia and represented one forty-eighth of the circumference. Strabo reported that Posidonius gave a revised value of 180 000 stadia, the figure later adopted by Ptolemy. Whatever these various estimates represent, they demonstrate that determining the Earth's size was already an established problem in Greek scientific geography, before and after Eratosthenes, whose work belongs not to the beginning of geodesy, but to an ongoing scientific conversation about the dimensions of the world. Clearly, the measurement of the Earth belonged to a long and contested tradition.

The Greeks themselves frequently acknowledged earlier traditions of learning, particularly in Egypt and Mesopotamia. It’s reasonable to celebrate the achievements of ancient Greece not so much as not an ex-nihilo miracle, but within a wider geographical and longer historical context. Indeed, a study of ancient metrology will strongly suggest that indeed the earth was measured in great antiquity, long before Eratosthenes. The ancient Greeks didn’t necessarily realise that the units they used were geodetic, which could imply that they were the product of an earlier culture which did have an accurate value for the size of the earth.

Today it is often assumed that attempts to relate the Great Pyramid to the dimensions of the Earth belong exclusively to the fringes of scholarship. Historically, however, the question arose within mainstream scientific enquiry. Livio Stecchini argued that the common source underlying the pyramid descriptions of Diodorus, Strabo, Pliny and Philon was the second-century BC geographer Agatharchides of Cnidus. From these later authors he reconstructed a lost account in which the dimensions of the Great Pyramid were interpreted geodetically, relating its perimeter to the length of a degree of latitude.(6) Since Agatharchides' original text has not survived, this interpretation remains Stecchini's reconstruction rather than a directly attested ancient testimony, but is very interesting. During the seventeenth, eighteenth and nineteenth centuries, determining the size and figure of the Earth became one of the central scientific problems of Europe. The Great Pyramid was repeatedly drawn into this discussion, because many believed it might preserve an ancient geodetic standard whose precision had subsequently been lost.

Greaves, Jomard and Taylor

There has never really been a period, in historical times, in which the Great Pyramid was not an object of measurement and speculation. Interest in the dimensions of the Great Pyramid is almost as old as the surviving literature about the monument itself. Greek and Roman authors recorded its measurements in feet, plethra and stades; medieval Arabic writers proposed their own dimensions and units; and by the early nineteenth century Jomard was able to devote several pages simply to reviewing these earlier attempts. The question was never whether the pyramid had measurable proportions, but how those proportions should be understood and which system of measurement they embodied.

A significant change had already occurred in the seventeenth century, when in 1638 John Greaves travelled from England to Egypt to measure the monument himself. He hoped the monument might help to establish the dimensions of the planet. Dissatisfied with existing estimates of the Earth's circumference, he also travelled to Rome to measure the foot of the statue of Statilius Aper, believing it preserved an ancient standard ultimately related to terrestrial measurement. His Pyramidographia marked the beginning of the modern tradition of instrumental survey, a tradition continued by the French expedition under Napoleon and brought to an unprecedented level of precision by Flinders Petrie. Isaac Newton looked to Greaves's work for answers on the question of the earth's size. As Peter Tomkins wrote:

Newton's preoccupation with establishing the ancient cubit "was no idle curiosity... his general theory of gravitation... was dependent on an accurate knowledge of the circumference of the earth."(6)

Modern discussions often dismiss nineteenth-century pyramid studies as speculative or "pyramidological". Such judgements risk obscuring the genuine scientific questions that motivated many of these investigations. Could ancient civilisations measure with remarkable precision? Could architecture preserve standards of length? Could geometry record astronomical or geodetic knowledge? These were legitimate questions within the history of science, even if many of the answers proposed at the time no longer command universal acceptance. Rather than dismissing these authors wholesale, it is more fruitful to distinguish between the quality of their observations, the accuracy of their measurements and the interpretations they drew from them.

John Taylor was the first modern writer to argue explicitly that the Great Pyramid intentionally embodied the ratio between the circumference of a circle and its radius. Yet for Taylor this was not merely an elegant mathematical curiosity. He regarded the geometrical relationship as evidence that the builders possessed accurate knowledge of the size of the Earth. The perimeter of the pyramid corresponded to the Earth's circumference at a fixed scale, while the height corresponded to the Earth's radius. In Taylor's interpretation, the geometry served a geodetic purpose: the pyramid was conceived as a permanent monument preserving the dimensions of the planet itself. Taylor wrote:

It was to make a record of the measure of the Earth that it was built... They knew the Earth was a sphere... had ascertained its circumference, and were desirous of leaving behind them a record of the circumference as correct and imperishable as it was possible for them to construct. (7)

During Napoleon's expedition to Egypt (1798–1801), Edme-François Jomard and his colleagues produced the monumental Description de l'Égypte. Among its many achievements was a careful survey of the Great Pyramid, considerably more accurate than the one published by John Greaves almost two centuries earlier, aided in part by the removal of large quantities of sand from around the base. Greaves, Newton and Jomard deserve recognition as pioneers of historical metrology in modern times. They regarded the Great Pyramid not simply as an architectural monument but as a potential archive of ancient systems of measurement.

Jomard interpreted the Great Pyramid as important from both a geodetic and metrological viewpoint. From a measured base of 230.902 metres and a sloping height of 184.722 metres, he proposed dimensions of 500 by 400 Egyptian cubits, using a cubit of approximately 0.462 metres. (1) His broader suggestion was that the pyramid embodied knowledge of the size of the Earth, and this was remarkably far-sighted.

Working from his own survey of the monument and the best geodetic values then available, he observed that the reconstructed sloping height of the pyramid, approximately 184.722 m, was very close to one six-hundredth of the length of one degree of latitude in Egypt. Likewise, the side of the base, approximately 230.902 m, corresponded closely to one four-hundred-and-eightieth of the same terrestrial degree. He therefore concluded that the pyramid preserved fractional parts of the Earth's circumference and functioned as a permanent archive of the Egyptian system of measurement.

Nearly two centuries later, both the dimensions of the Earth and those of the Great Pyramid are known with considerably greater precision. Petrie's survey gives an original base side of approximately 9068.8 inches (230.347 m). Using a reconstructed height of 5776 inches (146.710 m), the calculated sloping height is 7343.2 inches (186.517 m). Modern geodesy places the length of one degree of latitude at Giza at approximately 110,945 m, giving 110 945 / 600 = 184.91 m for the slope, and 110 945 / 480 = 231.14 m for the base side. These calculated values differ from the measurements taken by Petrie. The modern sloping height exceeds one six-hundredth of the local degree by approximately 1.61 m, while the reconstructed base is around 0.79 m shorter than one four-hundred-and-eightieth of the degree. Similar discrepancies arise when the modern mean terrestrial degree is employed. Neither value now coincides particularly closely with the reconstructed dimensions of the monument.

Jomard's reconstruction illustrates an enduring methodological difficulty. The chosen cubit and the proposed whole-number dimensions partly confirmed one another, while the sloping height itself had been measured on a monument whose original casing and apex were already missing. This does not diminish the historical importance of Jomard's proposal. His insight was to recognise that the Great Pyramid might preserve geodetic knowledge and that its dimensions should be compared with those of the Earth itself rather than interpreted solely as architectural measures. Nevertheless, when tested against the more accurate surveys of Petrie and modern geodesy, the specific numerical relationship appears insufficiently precise to account for the extraordinary level of exactitude encountered elsewhere at Giza. Jomard's hypothesis therefore remains of considerable historical significance, even if the particular correspondence he proposed no longer appears compelling.

Measurements do not speak for themselves. Every survey is interpreted through a conceptual framework that determines which dimensions matter, which should be reconstructed and which units appear most significant. Greaves searched for ancient standards of length. Jomard interpreted his survey through the Egyptian cubit. Taylor emphasised geometry. Petrie sought the greatest possible observational precision. The measurements remain indispensable, but their significance depends upon the questions brought to them. Recognising this distinction between observation and interpretation is essential to any history of metrology.

The Great Pyramid, the Mean Circumference, and 43 200

If Jomard was correct in seeking the Earth within the pyramid, perhaps the relationship may be expressed differently from the one he proposed. Petrie’s mean socket side of 9125.9 inches gives a perimeter of 36 503.6 inches. Enlarging this perimeter by 43 200 gives 24 888.8 miles, approximately 8.2 miles, or 0.033 per cent, greater than the arithmetic mean of the modern equatorial and polar circumferences. Reversing the calculation, the modern mean circumference produces a socket side of approximately 9122.9 inches, around three inches shorter than Petrie’s measured mean. The relationship is close rather than exact, but substantially closer than Jomard’s proposed fractions of the local terrestrial degree.

Whether intentional or not, this correspondence is considerably closer than Jomard's original proposal and illustrates how improved surveys of both the Earth and the Great Pyramid can refine, rather than simply overturn, the intuition that the monument embodies a relationship to the dimensions of the Earth.

Figure 1: The upper calculation compares a yuga expressed in days with the equatorial circumference expressed in inches; the lower reduces the mean terrestrial circumference by 43,200 and compares the result with the Great Pyramid’s socket perimeter. The two are related by the recurrence of 432 and its powers, but they should be understood as distinct correspondences.

John Michell and the Heavenly City

If the Great Pyramid can indeed be understood as preserving aspects of the Earth's dimensions, then it belongs to a much wider tradition in which architecture functions as a geometrical image of the cosmos. This idea did not begin with Egypt, nor did it end there. Throughout history, sacred buildings have often been conceived not merely as places of worship but as microcosms: ordered representations of the relationship between heaven, Earth and humanity.

Few modern writers explored this tradition more thoroughly than John Michell. His vision of the "Heavenly City", inspired by the New Jerusalem of the Book of Revelation, medieval cosmology, Stonehenge and other sacred sites, was not simply symbolic. It proposed that sacred architecture expresses a common geometrical language in which terrestrial measurement, astronomical order and architectural proportion become different expressions of the same underlying reality.

As Michell wrote,

The macrocosmic city of 12,000 furlongs square and the microcosmic citadel wall of 144 cubits differ in scale but belong to one geometric figure. When they are brought to commensurable proportions it is found that a square of 12 furlongs contains a circle of 24,890 feet or 14,400 cubits round. The nucleus of St John's New Jerusalem can thus be identified as a cube containing a sphere which is in fact a model of the earth on a scale of one foot to one mile, for the diameter of the sphere is 7,920 feet and the earth's mean diameter is 7,920 miles.(2)

This passage introduces the idea of the same geometrical figure being read simultaneously in different units and at different scales. A length measured in cubits may equally be expressed in feet or miles without altering the underlying geometry. The unit changes, but the mathematical relationship does not.

Michell therefore regarded measurement as far more than a practical necessity. It was itself a language through which ancient knowledge could be preserved. As he wrote,

In every account of the holy city, the importance of measuring its dimensions is emphasised; and this is meant literally, for the fabric of the temple contains the secrets of the ancient world set out in such a way that they may be read by anyone in whatever age who cares to undertake the study of the language in which they are written, which is the language of geometry and number. (3)

His best-known illustration of the Heavenly City develops this idea further by placing the Earth and Moon within a single geometrical construction. Taking the Earth's mean diameter as 7 920 miles, the circumference is first transformed into the perimeter of a square. A second circle, whose circumference equals that perimeter, then has a diameter of approximately 10 080 miles, the combined diameters of the Earth and Moon. Whether regarded as symbolic, metrological or geometrical, the construction expresses a striking principle: celestial dimensions become architectural geometry through a sequence of simple transformations between circles and squares.

Michell's diagram is often regarded simply as an example of sacred geometry. It may equally be understood as an exercise in metrology. Physical dimensions are translated into geometry; geometry is translated into architecture. This movement between Earth, measure and building is the foundation of many interpretations of the Great Pyramid.

Seen in this light, Michell's Heavenly City provides less a historical explanation of the Great Pyramid than a conceptual framework within which it may be understood. Although separated by millennia and belonging to very different cultural and religious traditions, both suggest that architecture can function as a geometrical image of the cosmos. The monument is not merely situated upon the Earth at a significant place; it may itself become an image of the Earth, expressing geographical, astronomical and cosmological relationships simultaneously.

The Great Pyramid, the Yuga, the Earth and 43 200

An aspect of the Giza complex that has been written and talked about at length is 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 was most famously put forward by Graham Hancock.(4) Hancock connects the scale factor 43 200 with precessional symbolism, since one zodiacal age lasts about 2 160 years, a full precessional cycle lasts about 25 920 years (2160 x 12), and 43200 is 20 × 2 160, using harmonic numbers. Livio Stecchini is however the earliest source I know of that explicitly formulated the pyramid as a 1 : 43 200 representation of the northern hemisphere and links the scale to the 86 400 seconds in a day. (5)

However, 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. This falls short of the modern equatorial circumference by approximately 168 miles and therefore does not provide a close correspondence. 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. No measured outer base, to my knowledge, currently cited here produces the required value of approximately 9130 inches per side. The claim should therefore not be attached to the equatorial circumference without further survey evidence.

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 4320 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.

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.

It’s interesting to follow this idea through with inches. 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.

Figure 2: Equatorial circumference of the earth in inches and yuga. Two closely related comparisons can be made, depending on the definition of the year. A Mahāyuga of 4 320 000 sidereal years, converted into days and then treated as inches, corresponds to approximately 24 904 miles. Using tropical years gives approximately 24 902.88 miles, closer to the modern equatorial circumference of 24 901.46 miles. These are not identical calculations, and the distinction between sidereal and tropical time should be retained. In both cases, however, the same principle is at work: a long duration expressed in days is translated directly into a terrestrial distance expressed in inches.

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. 25 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.

Hugh Franklin

One of the most extraordinary observations in relation to the size of the earth comes from Hugh Franklin, who found intriguing connections between the circumference in miles and pi, the ratio between a circle’s circumference and diameter. He noticed that the Earth's equatorial circumference in miles can be approximated by a surprisingly elegant equation involving π raised to the third power. In his article “Earth, Pi, Miles and the Barleycorn”, 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, estimated as 24 901.461 miles (Wikipedia). The difference is just over a mile. David Kenworthy introduced me to Franklin’s article shortly after its author’s death, so I never had the opportunity to discuss the equation with him. Its significance was immediately apparent, because it suggested that the equatorial circumference might be generated not merely numerically but through an underlying geometry of circle and square. I knew that the mile was linked very precisely to ancient Indian units of length. In traditional Indian measures, the mile is equivalent to the sakrakosa, a thousand dhanus, and a yojana is exactly 9 miles. I had no issues with the antiquity of the mile.

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)

And as it happened, my friend, researcher Dennis Payne independently found this connection to the equatorial circumference. But how to make sense of this equation, √(π³ x 20 000 000) = 24 902.3198, geometrically?

Figure 3. Hugh Franklin's Equatorial circumference of the earth, circle and sphere. Franklin’s equation gives the circumference directly. The diagram begins to unpack its geometry: the resulting circumference defines a circle whose diameter is also the diagonal of an inscribed square. The square therefore has an area of 10 000 000 π sq. miles. In the following section, this numerical expression will be developed into a sequence of explicit circle–square transformations.

Squared Circles, the Earth’s Equatorial Circumference and the Great Pyramid

The process of transforming circles into squares is itself a language. Hugh Franklin presented the relationship as a numerical identity. The purpose of the following constructions is to ask whether the equation can also be understood geometrically. Rather than treating √(π³ × 20 000 000) as an isolated expression, we can unfold it into a sequence of familiar operations involving a double square, a circle, an equal-area square and a second circle. The value remains the same, but its internal geometry becomes visible. I came up with the idea of tweaking the equation slightly, and instead writing it as √(π×20 000 000)×π = 24 902.3198. A circle with a circumference of 24 902.31984 miles will have a diameter which can also be the diagonal of a square. The resulting square has an area of 10 000 000 π square miles. The importance of the construction lies not in introducing another physical circle of that size, but in showing how the original circular quantity is successively re-expressed as a square diagonal, a square side and the diameter of a new circle.

Begin with a double square measuring 2,000 by 4,000 miles. Its diagonal is 2 000 √5 miles. If this diagonal becomes the radius of a circle, the circle has an area of π (2 000 √5)² = 20 000 000 π square miles.

Squaring this circle by area produces a square whose side is √(20 000 000π) = 7926.6546 miles, remarkably close to the modern equatorial diameter of the Earth.

If this side is then used as the diameter of a second circle, its circumference becomes √(20 000 000π) × π = 24 902.3198 miles, approximately 0.86 miles greater than the modern equatorial circumference. The double square therefore generates, through successive circle–square transformations, values close to both the Earth’s equatorial diameter and circumference.

Alternatively, we could 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 very important in ancient geometry, found in megalithic Brittany, as Howard Crowhurst has demonstrated, in the Temple of Solomon, and, at Giza, in the King's Chamber.

Figure 4: The diagram summarises the complete sequence. The double square establishes √5; its diagonal becomes the radius of the first circle; the circle is squared by area; and the side of the resulting square becomes the diameter of a second circle. The final circumference closely approaches the modern equatorial circumference of the Earth. The diagram should therefore be read from left to right as a chain of transformations, rather than as a collection of independent values.

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 to square the circle by area: to produce a square which has the same area 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.

Figure 5: When the two squares and two circles are superimposed, the construction can be seen as a single coherent figure. The small double square generates the yellow circle; the equal-area blue square generates the white circle; and the diameter of the white circle provides the value close to the Earth’s equatorial diameter. The overlay makes visible the continuity that is less obvious when the stages are shown separately.

The area of the white circle, which represents the equatorial circumference of the earth, is 5 000 000 π² square miles, and a square of the same area will have sides of √(5 000 000 π²) 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.

Figure 6: An alternative construction. The same final circumference can be generated by reversing the order of operations. Instead of first squaring a circle, we begin by allowing the circumference of a smaller circle to become the side of a large square, and then construct a circle equal in area to that square. This “circling of the square” leads to the same expression, showing that the relationship is not dependent upon a single diagrammatic route.

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 as 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.

Figure 7: The completed construction may finally be placed over the Earth itself. The side of the equal-area square becomes the approximate equatorial diameter, while the circumference of the surrounding circle becomes the approximate equatorial circumference. What began as the diagonal of a double square has been transformed into a terrestrial measure through a sequence of circle–square operations.

Two things are suggested by this correspondence. The first is 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, which is composed of 63 360 inches. And the second is the connection to squaring the circle, which we find in the structure of the Great Pyramid. 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.

The connection between the squaring of the circle and the equatorial circumference of the earth as expressed in miles may be about creating 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). 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. In this context, the mile is part of a geometric system. The equatorial circumference as a measurement is the outcome of a process of transformation. This perspective aligns with the broader pattern observed at Giza. The circle is translated into the square. 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. But what about the value of the circumference: does it align with the size of the Great Pyramid?

In fact, the best fit for the Great Pyramid encoding the equatorial circumference is through its height, and in relation to the moon. 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 x 365.25636 / (27.321661 x 10 000) 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.

A yuga of 4 320 000 years, converted into days, gives a total that is very close to the Earth’s equatorial circumference expressed in inches. In other words, one may think of the circumference as one yuga in days wrapped once around the Earth. The height of the Great Pyramid then becomes that same long cycle divided by the sidereal month, and reduced by a factor of ten thousand. The number 5773.5 inches can therefore be read in three, mutually consistent ways: as the height of an equilateral triangle with side 10 000 inches; as the number of sidereal months in a Mahāyuga scaled down by 10 000; and as the Earth’s equator, interpreted as a yuga in days, converted into an inch-per-day measure and then divided by the Moon’s sidereal period. In this interpretation, the pyramid’s height stands at the junction of three motions: the Earth’s rotation, the Earth’s orbit round the Sun, and the Moon’s orbit round the Earth relative to the stars.

Figure 8: The pyramid’s height may be read in three related ways: geometrically, as 10 000/√3 inches; astronomically, as the number of sidereal lunar months in a Mahāyuga divided by 10 000; and geodetically, as a quantity derived from the same day-count that approximates the Earth’s equatorial circumference in inches. These coincidences sit at the junction of spherical geometry (through the triangle), rotational astronomy (through sidereal periods), and number (through an irrational constant).

In that same neighbourhood of relationships lies the Earth’s equatorial circumference. Expressed in inches, it corresponds to a yuga, expressed in days, a vast cycle of years converted into the daily rotation of a point on the equator. The Moon participates as well: its sidereal month divides that same count, producing the pyramid’s height. The equatorial radius of the Moon, when expressed in miles, also joins this scheme: it is close to 4 320 000 / 4000 miles, placing the same yuga number at lunar scale. To pass between these descriptions one must use π, because circumference belongs to circles. π is therefore the necessary mediator between rotation (time), circles (geometry), and linear measure (space).

The constants π and √3 sit at the heart of this web. One measures the relation of circles to straight lines; the other measures the relation of triangles to circles. While the presence of √3 and π in the relations linking Earth, Moon, day, month, and year is mathematical, it shows that the structure imposed upon earth and the heavens is quite poetic.

Figure 9: The geometric construction produces approximately 24 902.32 miles. When converted into inches, this becomes approximately 1.5778 billion inches. Independently, 4 320 000 sidereal years contain approximately 1.5779 billion days. The two values are close, allowing the terrestrial circumference to be read simultaneously as a spatial measure and as a long period of time, provided that one inch is treated as the analogue of one day. This temporal correspondence is not required for the geometric construction itself; it is an additional relationship that emerges once the result is converted into inches.

Squaring the circle and the measure of the earth's circumference

Figure 10: At its simplest, the construction places a square of side √π around a circle of unit radius and equal area. Applied symbolically to the Earth, it expresses the terrestrial globe through the relationship between circle and square. The diagram is not a scale drawing of the preceding calculation, but a visual summary of its governing principle.

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.

A further relationship links the mile to the difference between solar and lunar time. Twelve synodic months fall short of the tropical year by approximately 10.87512 days. Using this difference, the synodic month and the factor 36 gives:

29.53059 × 70 000 / (10.87512 × 36) = 5279.996, remarkably close to the 5280 feet in a mile. Rearranging the same relationship gives:

10.87512 × 5280 x 36 / (7 × 29.53059) = 10 000.007

Multiplying this result by the tropical year and by 36 produces approximately 131 487 288 feet, within about 1.43 miles of the modern equatorial circumference of the Earth. This is not an independent derivation from the earlier expression 365.242199 × 360 000,, but a lunar decomposition of it. It suggests that the factors 10,000, 36 and 5280 may be brought into relation through the synodic month and the annual difference between solar and lunar years.

The relationship may therefore be incorporated into the circle–square construction as a metrological bridge. The geometry supplies the terrestrial circumference in miles; the lunar calculation provides an approximate internal relation between the mile, the year and the synodic month. Together they suggest that geometrical transformation and calendrical reconciliation may belong to the same metrological language.

The final calculation combines the Franklin circumference, the synodic month and the annual difference between solar and lunar time to reproduce approximately the equatorial circumference in feet. It is algebraically equivalent to multiplying the generated circumference in miles by 5,280; the lunar expression supplies an approximate derivation of the conversion factor itself. The purpose is therefore not to offer a second independent circumference, but to show how the mile-to-foot relationship may be embedded within the same lunar framework.

The Earth’s Polar Circumference and the Great Pyramid

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.

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.

The relationship between inch and metre can be represented through the same sequence of transformations. A square of side two inches generates a diagonal of 2√2 inches. Taking this diagonal as the diameter of a circle gives an area of 2π square inches. A square equal in area to this circle therefore has sides of √(2π) inches. If one such side becomes the diameter of a second circle, its circumference is √(2π)π, or approximately 7.8748 inches, almost exactly twenty centimetres. Five such lengths consequently produce a metre of approximately 39.374 inches. This does not reproduce the official metre exactly, but it suggests a geometrical route through which a value close to the metre can be related to the inch by transformations involving the square, diagonal and circle.

Figure 11: Squaring the circle to link inch and metre.

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 fit exactly, the fit is actually slightly better than the official current 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 same operation can be expanded to terrestrial scale. Beginning with a square whose side is 10810^8108 inches, its diagonal becomes the diameter of a circle. Squaring that circle by area and then taking the side of the resulting square as the radius of a second circle gives a circumference of approximately 1,574,960,995 inches, or 40,004.01 kilometres. This is around 3.85 kilometres shorter than the modern polar circumference. The agreement is close, although not exact, and the significance lies primarily in the recurrence of the same geometric procedure at a radically different scale.

Figure: 12. Squaring the circle and the polar circumference.

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. The metre, Egyptian and royal cubit and remen also fit as a triad within a circle and square framework, as one metre multiplied by pi and divided by 6 gives the Egyptian royal cubit (at least, one of several accepted versions), and this cubit then divided by √2 gives the remen. So the metre is the diameter of the circle, the circumference is made up of 6 cubits, and the remen is the side of a square with a diagonal of the cubit.

Squaring, Circling, and Geodesy

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 radius 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. A square of the same area has sides of π. If that side becomes the edge of a cube, the cube has a volume of π³. In this way, the factor π³ in Franklin’s equation can be represented as the outcome of successive transformations from circle to square and from square to cube. 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.

Figure 13: The polar circumference can also be approached in miles through a related but more elaborate construction. Here the starting square incorporates the factor 100/99, after which the same operations, diagonal, circle, equal-perimeter square and equal-area circle, produce a value of approximately 24 857.34 miles. The result is slightly shorter than the modern polar circumference, but it illustrates that the same vocabulary of transformations can operate in more than one unit and at more than one scale.

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 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.

Equatorial circumference (miles): √20 π √π x 1 000 = 24 902.3198 miles (actual value 24 901.4611 miles)
Equatorial circumference (inches): √20 π √π x 1 000 x 63 360 = 1 577 810 985.307 inches (actual value 1 577 756 575.296 inches)
Sidereal year: √20 000 000 π √π x 63 360 / 4 320 000 = √20 π √π x 44 / 3 = 365.2340 days (actual value 365.256363004 days)
Polar circumference: √2 π √π x 20 x 10⁷ = 1 574 960 994.572 inches (actual value 24 859.7316 miles)
Metre expressed in inches: √2 π √π x 5 = 39.37402 (actual value 39.3700787402 inches)
Synodic lunar month in days: √2 π √π x 30 / 8 = 29.5305186 (actual value 29.53059 days)
Lunar year expressed in days: √2 π √π x 45 = 354.36622 (actual value 354.36708 days)

In each of these cases “√2 π √π” and “√20 π √π” are at the heart of these equations.

√20 π √π is equal to π √π x 2 x √5, and so geometrically, √20 π √π can be obtained by taking the diagonal of a double square, that is a rectangle with sides of 2 x 4, making this the radius of a circle, squaring this circle by area to obtain a square with sides of 2√5 √π , one of which becomes the diameter of a circle, with a circumference of 2√5 √π π = √20 π √π.

√2 π √π can be arrived at geometrically by starting with a square of sides 8. The diagonal of the square is the diameter of a circle. Then a square is constructed with a perimeter equal to the circumference of the circle. Then a new circle is constructed with diameter equal to a side of the square. This circle is then squared by area: a square is constructed with the same areas as the circle. A side of the square will measure √2 π √π.

We can think of the equatorial circumference of the earth, or a point on the equator tracing a daily circle in space, as √20 000 000 π √π miles. And we can think of the polar circumference as the equatorial circumference multiplied by 1000 √10 / 3 168. Something close to the metre is the equatorial circumference in miles divided by 200 and by √10. (√20 π √π x 1 000 / (200 x √10) = 39.37402) And the equatorial circumference in inches divided by √10 and by 12 672 000

√20 π √π x 1 000 x 63 360 / (√10 x 12 672 000) = 39.37402

We can think of a lunation in days as the polar circumference in inches (√2 π √π x 20 x 10⁷) multiplied by 3 / 160 000 000

√2 π √π x 20 x 10⁷) x 3 / 160 000 000 = 29.530519

Earth's equatorial circumference in inches: 1 577 756 575.296 inches

Approximate geometric interpretation: √20 π √π x 1 000 x 63 360 = 1 577 810 985 inches

A sidereal year is the time that the earth takes to travel in its orbit around the sun, with respect to the background of stars, not to the sun, as viewed from the earth. It takes 365.25636 days.

This can be interpreted geometrically (approximately) as:

√20 π √π x 44 / 3 = 365.2340 days

This interpretation emerges from geometric principles and reflects how ancient cultures may have used mathematical constants to approximate celestial cycles. The formula is an approximation that illustrates geometric harmony, rather than an exact match.

The sidereal year can be related to the equatorial circumference of the earth as the equatorial circumference in inches divided by 4 320 000. This number represents the yuga, a cosmic cycle in the Indian tradition, connecting celestial measurements with spiritual and mythological frameworks. If the equatorial circumference in inches can be approximately interpreted as √20 π √π x 1 000 x 63 360 inches, then the sidereal year can be interpreted as:

√20 π √π x 1 000 x 63 360 / 4 320 000 = √20 π √π x 44 / 3 = 365.2340

Figure 14

Figure 15

Figure 16

Figure 17

Figure 18

If this interpretation is correct, the builders were not merely recording astronomical information in stone. They were constructing architecture in accordance with a cosmological order. The evidence explored in this chapter suggests that the geometry of Giza is better understood not merely as a system of numerical encoding, but as the architectural expression of a particular way of understanding the relationship between humanity, the Earth and the cosmos The geometry does not merely represent the cosmos; it participates in it.

Penrose proposes that mathematics, mind and the physical world form three interrelated domains. Without adopting his model in every respect, it offers a striking modern analogue to the ancient intuition explored throughout this chapter. Geometry belongs neither exclusively to thought nor exclusively to matter. It inhabits the boundary between them. The monuments of Giza can therefore be approached not simply as constructions in stone, but as attempts to embody mathematical relationships within the material world. Their forms become a meeting place between intelligible order, human understanding and physical reality.

As John Michell put it:

...the plan of the cosmic temple is not just an expression of the religious and scientific notions of another age, but a revealed key to the interpretation of universal change and motion, true for all times.

Each generation has approached the Great Pyramid through the scientific language of its own age. Renaissance scholars sought ancient standards of measure; Enlightenment surveyors sought the dimensions of the Earth; nineteenth-century writers explored geometry and metrology; more recent researchers have emphasised astronomy and precession. The present study continues this tradition, but shifts the emphasis once again. Rather than seeking a single hidden key, it asks whether geometry itself provides a language through which astronomy, metrology, architecture and cosmology may be translated into one another. In this sense, the history of pyramid research is itself an ongoing act of interpretation, continually renewing the dialogue between the monument and those

who seek to understand it.

For much of the modern history of pyramid studies, the Great Pyramid was approached primarily as a problem in geometry, astronomy and metrology rather than archaeology. Scholars measured its angles, surveyed its dimensions, compared its proportions with those of the Earth and the heavens, and searched for ancient systems of measure preserved within its architecture. Questions concerning chronology, construction techniques and funerary practice, which dominate much contemporary Egyptology, occupied a comparatively smaller place in these earlier investigations. This is not to suggest that one approach is superior to the other, but simply to recognise that the monument has long been studied within more than one intellectual tradition.

For almost four centuries, scholars have looked to the Great Pyramid as a possible repository of ancient geodetic knowledge. Greaves sought in its measurements evidence for the true dimensions of the Earth. Newton investigated ancient standards of length because terrestrial measurement lay at the heart of his natural philosophy. Jomard proposed that the monument embodied a relationship to the Earth's dimensions. Taylor argued that its geometrical properties, particularly the relationship between height and perimeter, were intended to preserve the measure of the Earth itself. Piazzi Smyth refined these ideas through more precise survey and metrological analysis, while Stecchini, Tompkins, Michell and Hancock expanded them within broader interpretations of ancient science and cosmology. Although these authors often differed in their methods and conclusions, they were united by a common question: did the builders of the Great Pyramid intentionally preserve knowledge of the dimensions of the Earth?

One of the most striking features of the history of metrology is that, for centuries, many scholars assumed that units of length should ultimately derive from the Earth itself. The Earth was not merely something to be measured; it was regarded as the natural foundation of measurement. This conviction lay behind some of the greatest scientific enterprises of the early modern period. Astronomers and surveyors sought increasingly accurate determinations of the terrestrial circumference, while proposals for universal units repeatedly returned to the dimensions of the planet. The French metre, defined as one ten-millionth of the meridian quadrant, is perhaps the best-known example of this tradition.

The same assumption appears repeatedly in studies of the Great Pyramid. Greaves travelled to Egypt hoping that the monument preserved an ancient standard of measurement. Newton investigated the ancient cubit because accurate terrestrial dimensions mattered for his natural philosophy. Jomard proposed that the pyramid embodied a geodetic relationship. Taylor argued that its geometry recorded the Earth's circumference. Although they differed in method and conclusion, they shared a common belief: that systems of measure were ultimately geodetic. A unit of length was not an arbitrary convention but a reflection of the Earth itself.

This historical perspective is important because it places ancient metrology within a much broader intellectual tradition. Rather than treating measures as isolated human inventions, many scholars regarded them as expressions of the dimensions of the world in which humanity lives. Whether or not every proposed relationship proves convincing, the underlying principle, that measurement should ultimately be grounded in the Earth, has a long and distinguished history.

To measure, in this tradition, was not merely to quantify. It was to participate in the ordering of the cosmos. Seen in this light, the architecture of Giza is not static but dynamic. Its transformations are not merely mathematical operations but expressions of a broader cosmological vision, in which the order observed in the heavens is translated into the built landscape. The monument becomes a point of mediation between earth and sky, between the measurable and the intelligible, between temporal existence and the enduring patterns from which it is thought to derive.

Notes

Edme-François Jomard, Description de l'Égypte, ou Recueil des observations et des recherches qui ont été faites en Égypte pendant l'expédition de l'armée française, Antiquités, vol. III, part 1 (Paris: Imprimerie Impériale, 1809), 520. Heidelberg University Library Digital Collections. https://doi.org/10.11588/diglit.5428#0527
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.
Ibid.
Hancock, Graham (1995). Fingerprints of the Gods: The Evidence of Earth's Lost Civilization. Crown Publishers.
Peter Tompkins, Secrets of the Great Pyramid. New York: Harper & Row, 1971
Stecchini, Livio Catullo. "Notes on the Relation of Ancient Measures to the Great Pyramid." In Peter Tompkins, Secrets of the Great Pyramid. New York: Harper & Row, 1971, Appendix, especially pp. 371–378.
Taylor, John. The Great Pyramid: Why Was It Built? And Who Built It? London: Longman, Brown, Green, Longmans & Roberts, 1859.