If one takes the figure of 940 feet (that is, 286.512 meters) as the side length factorizing 940 gives 20 x 47 and 47 (a prime number) times 5 gives 235 which is the number of lunar months in 19 solar years: the Metonic period. image by Google Earth
This is the larger of three bounding periods for the sun, moon, and earth. The lower boundary is exactly 19 eclipse years, called the Saros eclipse period of 18.03 solar years. . Within that range of 18-19 years lies the moon’s nodal period of 18.618 years, this being the time taken for the two lunar nodes, of the lunar orbit, to travel once backwards around the ecliptic. It is only at these nodal points that eclipses of sun and moon can occur, when both bodies are sitting on the nodes.
The first article on Ushtogai showed how, by daily counting all the tumuli in a special way, the 6800 days of the nodal period would keep a tally in days, to quantify where the nodes were on the ecliptic as well as predicting the lunar maximum and minimum standstills.
It now seems that, if the absolute size of the monument’s perimeter was able to count the 19-year Metonic, not by counting days but rather, counting the 235 lunar months of the Metonic period. The lunar month would then be 16 feet long. And, within that counting, one could also have counted the 223 lunar months between eclipses having the same appearance. The diameter of a circle drawn within the square would then have a diameter of 235 (lunar months) divided by 4 = 58.75 lunar months which, times the 16 feet per month, is the 940 feet of the square’s side length.
Figure 1. The size of Ushtogai Square, side length 940 feet, is 235 x 4 feet, making its perimeter able to count 235 lunar months of 16 feet.
In Cappadocia, present-day Turkey, this type of geometrical usage can be seen within a rock-cut church called Ayvali Kelise, only then in miniature to form a circular apse, just over 100 times smaller! The church was built in the early Christian period (see figure 2).
The Ushtagai Square has the basic form for the equal perimeter geometry. If so, that would form a tradition at least 10,000 years old. As a counting framework for the 18-19 solar year recurrences of aspects between the the Sun, Moon, Earth, eclipses and nodes the Square appears to be both a tour-de-force in a form of astronomy now largely forgotten.
Figure 3 Showing the circle equal in perimeter to the Ushtagai Square, the size of the Earth (in-circle of diameter 11) and Moon (four circles of diameter 3.)
As an earthwork where tumuli punctuate geometrical lines, it is a highly portable symbol of great time and a highly specific astronomical construction. It was an observatory and also a snapshot within celestial time, built just after the Ice Age had ended.
The Taj Mahal is perhaps the most recognizable building on earth. It was built by a Moghul king as a memorial for his dead queen and for love itself. The Mughals became famous for their architecture and the Persian notion of the sacred garden though their roots were in Central Asia.
I had been working on Angkor Wat, for my soon to be released book: Sacred Geometry in Ancient Goddess Cultures, where the dominant form of its three inner walls surrounding the inner sanctum were in the rectangular ratio of outer walls of six to five. A little later I came across a BBC program about the Mughals and construction of the Taj by a late Moghul ruler, indicating how this style almost certainly arose due the Central Asian influences and amongst these the Samanids and the Kwajaghan (meaning “Masters of Wisdom”). I had also been working on the facades of two major Gothic Cathedrals (see post), and when the dimensions of the façade of the Taj Mahal was established, it too had dimensions six to five. An online document decoding the Taj Mahal, established the likely unit of measure as the Gaz of 8/3 feet (a step of 2.5 feet of 16/15 English feet; the Persepolitan root foot *(see below: John Neal. 2017. 81-82 ). Here, the façade is 84 by 70 gaz.
A very significant feature of the six by five rectangle is that its perimeter is 2 times 11 (or 22) and in the Taj Mahal, 14 gaz times 22 equals 308 gaz as perimeter of the façade. Since π (or “pi”) was often taken to be 22/7, an equal perimeter of circle would require a radius 7 to be 22 in circumference or, in gaz, a radius of 49 gaz, again giving 308 gaz (821 + 1/3rd feet).
With Angkor Wat, this feature of a circle could be exploited to count time around the walkways of the rectangular walls, just as if time was flowing around a circle. But, in this case, the architecture is taking us on a symbolic journey. One can see how the central façade can “explain” the three outer rectangles to the towers left and right, and to the pinnacle boss of the onion dome, synonymous with Mughal architecture. But the basic form of equal perimeter involves a circle diameter 11 whose out-square equals 44, if π equals 22/7. The circle of equal perimeter is then a diameter of 14 since 14 x 22/7 = 44. The circle shown here is obviously the outer circle of equal perimeter, for which there must be a circle of 11 whose out-square is 44. Seven gaz times 11 equals 77 and this indeed gives the out-square (of EP) as 308. The inner circle (see below) can then be seen to be the size of the onion dome.
The equal perimeter geometry is, when magnified by 720, a model of the Earth and Moon in miles and, in Sacred Geometry: Language of the Angels, it was shown time and again that domed monuments were pictures of the Earth within a strong but hidden tradition. There also seems to be a correlation between the moon circle’s size (of 42 gaz) and the windows of the octagonal outer building surrounding the tomb itself, and also the two visible cupolas. In order to draw the two circles, it was necessary to draw the two diagonals of the rectangle and this gives a central point in the façade as being the upper central window, a phenomenon quite clear in Chartres (post # 3) where the diagonals of its 3 by 4 façade locate the center of the circular rose window.
The elevation of the Earth and Moon above the equal perimeter façade is surely of sublime design to celebrate love, resurrection. The octagon design takes its inspiration from the cosmology of Islam and beliefs concerning the afterlife.
Bibliography of Ancient Metrology
Berriman, A. E. Historical Metrology. London: J. M. Dent and Sons, 1953.
Heath, Robin, and John Michell. Lost Science of Measuring the Earth: Discovering the Sacred Geometry of the Ancients. Kempton, Ill.: Adventures Unlimited Press, 2006. Reprint edition of The Measure of Albion.
Heath, Richard. Sacred Geometry: Language of the Angels. Vermont: Inner Traditions 2022.
Michell, John. Ancient Metrology. Bristol, England: Pentacle Press, 1981.
Neal, John. All Done with Mirrors. London: Secret Academy, 2000.
—-. Ancient Metrology. Vol. 1, A Numerical Code—Metrological Continuity in Neolithic, Bronze, and Iron Age Europe. Glastonbury, England: Squeeze, 2016 – read 1.6 Pi and the World.
—-. Ancient Metrology. Vol. 2, The Geographic Correlation—Arabian, Egyptian, and Chinese Metrology. Glastonbury, England: Squeeze, 2017.
—-. Ancient Metrology, Vol. 3, The Worldwide Diffusion – Ancient Egyptian, and American Metrology. The Squeeze Press: 2024.
Petri, W. M. Flinders. Inductive Metrology. 1877. Reprint, Cambridge: Cambridge University Press, 2013.
An essay by Richard Heath on the numbers found in the book
“Plato and the Quran”, by Noor Bosra.
SUMMARY
Multiplication of the consecutive numbers 1 to 6 equals 720 and this, applied, to the equal perimeter model, numerically scaled it up to the cosmic size of the Earth and Moon in miles. The Quranic formula, to instead multiply the numbers 3 to 8 and generate 20160, doubled the cosmic model’s 10080 and hence the whole geometry was doubled in scale. The further addition of the consecutive numbers 3 to 8, further signaled 33 years, pointing to the equal area model of the Solar Hero and to the Moon’s nodal period because, 33 divided by 18.618 equals √𝝅 = 1.778 and the area function squares that. This equality of a circle’s perimeter and a circle’s area to the same square can then be drawn to unite the cosmic model of Earth and Moon’s size, to the relationship of the Solar Hero period of 33 years, as the square, to a circle of radius 18.618 years, as appears quite clearly in the plan of Islam’s Dome of the Rock[1].
Chapter 23: The Seven Sleepers of the Cave
One of Noor Bosra’s key themes is that the Quran was composed out of the cultural context of the eastern Mediterranean, Arabia, the ancient Near East, and beyond. One example is the story, from different sources, of people who sleep in a cave to escape religious persecution. Waking after what appears to be a single night, they find many years have passed yet their bodies have not aged and the danger they hid from has now passed.
Noor Bosra provides comprehensive literary references, in a good order starting with the Quran’s chapter 18, the story running for 18 Suras. The situation for these verses was ironic, since they were demanded by Mohammad’s religious persecutors to demand proof that his source of knowledge was competent, or his revelations should be discredited. A further irony is that Mohammad often used to ascend to his favourite cave on Mount Hira, where many key recitations were uttered.
One previous story of the sleepers is about early Christians from Ephesus in Asia Minor. In 250 BCE the persecution was then from Rome. But a roughly similar story is found in an apocryphal Jewish text, written shortly after 110 BCE, about the prophet Jeremiah who slept in the desert alone for 66 years to escape a king. In contrast, the Christians slept for 372 years and the Quranic sleepers, 309 years.
The numbers 300 and 9:
“they remained in the cave three hundred years plus nine more”
On page 440 Noor Bosra finds, in 309, the number of lunar months in 25 solar years of 365.25 days, which was the Egyptian Sothic year (of the helical rising of Sirius) from which the Nile flood was predicted. Four Sothic years is 1461 days and the accounting for the extra quarter day per solar year might explain that, in the Quranic version, the sleepers are found with a Dog to protect their cave, just as the Jackel god Anubis protected the dead in Egyptian tombs, Sirius being the dog star.
In the ancient world, coincidences of astronomical cycles within a whole number of time periods made these “natural” numbers sacred and to be associated with archetypes such as a dog (the constellation Canis Major). 309 lunar months is just 6 days less than 25 whole solar years but what about 365 days?
The Egyptians had two other years of 360 and 365 days that, when combined, allowed them to establish historical dates for over 1000 years. And if we divide 309 lunar months (in days) by 365 days, the result is 24.99986 years, which is within 1.3 days of 25 years of 365 days. This then has a further resonance with the Venus synod of 584 days, since the latter is 8/5[2] (1.6) of 365 days, and their common factor of 73.
Noor Bosra explains how the Apis bull, one of the deepest Egyptian traditions, was connected to this golden mean, a proportion that enables a unique division of three points as two identical proportions: of the least to the middle as of the middle to the whole – a sort of holism. We should note that Noor provides excellent resources of background information regarding astronomical numbers, numerical traditions and historical facts.
The varied accounts of the number of sleepers
Noor tells us that in no previous versions of the sleeper story were there any strong references to the number of sleepers, while in the Quran, all the numbers are playfully doubted as between 3 to 8, as if none of the locals, who remembered them emerging from the cave, could quite remember. These varied accounts were,
3 [sleepers] and the 4th was their Dog and,
5 [sleepers] and the 6thwas their Dog and,
7 [sleepers] and the 8th was their Dog,
taking a form congruent with a rhyme-based puzzle on successive numbers between 3 and 8. Noor Bosra suggests these three couplets form a code when multiplied, to give 20160, or when added, to give 33.
The first clue for me was that, in multiplying successive numbers, a factorial process was indicated, similar to the factorial[3] for 6 (notated as 6!) that equals 720: an important harmonic prime which, when multiplied by the three numbers 3, 11, and 14, becomes the familiar geometry of equal perimeter of square and circle based upon π =22/7[4] , but here expanded by 720 in an ancient “cosmic” model of the relative size of the Earth and Moon (ratio 11/3), in units of miles containing 5280 feet. But first, what is the equal perimeter geometry of circle and square? Figure 1 shows that the approximation to π of 22/7 causes a radius r of 7 to lead to a circumference of 44 (2 πr) and that the square of equal perimeter then has a side length of 11 whose in circle is diameter 11.
Figure 1 The equal perimeter model of circle and square
When 11 is multiplied by 720, the diameter of the Earth is interpreted as being 7960 miles.
The circle equal in perimeter to the out square of the Earth is then to be interpreted as 14 multiplied by 720, or 10080 miles in diameter, the limits of the sub-lunary world and of sacred spaces.
The diameter of the Moon is then interpreted as 3 multiplied by 720 = 2160 miles.
This cosmic geometrical model[5] [John Michell 2008] is accurately the size of the mean (spherical) Earth and of the Moon, long before the scientific revolutions of the last millennium, both Islamic and European. It suggests our planet and moon conform to a numerical plan, as an accurate overall approximation.
But the Quran appears to have used the alternative way, provided by Noor Bosra on page 443, of specifying the cosmic perimeter model, using 3 x 4 x 5 x 6 x 7 x 8 = 20160 to replace 720 x 14 = 10080. This 20160 is twice the normal 10080 of the cosmic model so that, the diameter of the Earth must also be 11/14 of that: that is, 7920 doubled to 15840. The equal perimeter length must also then double from 31680 to 63360. The results (now in half miles of 2640 feet) enabled the rhyme-form puzzle to replace the factorial expansion of the simple geometry and create numbers exactly twice that of the cosmic geometry, as shown in figure 2. By doubling the numbers of the cosmic model, the traditional numbers associated with the cosmic model of the earth were hidden, while upgrading the previous versions to become the subject for the Quranic version, the seven sleepers of the cave, all sleeping through history to avoid persecution.
Figure 2 The simplest, cosmic and Quranic geometries, the Quranic differentiating itself from the cosmic using an alternative factorial form to double the cosmic numbers.
Adding the numbers of the Sleepers
Noor then adds the numbers 3 to 8 to get 33, the number associated with the Solar Hero who are, as with Jesus and Mithras, said to die after 33 years. [6] This number has a strange relationship to the Moon’s nodal period of 18.618 years in that, the ratio between these numbers is the square root of pi as 3.141679 that is 1.77248. For this reason a square of side 33 is equal in area to the circle of radius 18.618[7] – a unique situation for a whole number as small as 33. The creation a circle and square of equal area has in fact been proven impossible to achieve by algebraic geometrical methods but 33 and 18.618 provide a practical synthesis[8].
It should be no surprise that the Quran might have referred to the nodal cycle in this way, since Mount Hira provides an approximate span, to the east, of the moon’s maximum (extreme) moonrises every 18.618 years, to the north and south, when viewed from the Kaaba, which is right angles to the lunar maximum to the north[9]. If the square side, equal to the diameter of the Earth, is reused then the required unit is one 33rd of the diameter. 15840, divided by 33, equals a unit of 480 and 18.618 x 480 equals 8936.64 as a radius, and a diameter of twice that. Figure 3 shows the radius of the circle of equal area to the square if that is 33 years and the radius 18.618 years.
Figure 3 The Quranic geometry with the nodal period added, given Noor’s clue of adding the numbers between 3 and 8 to give 33 years
This equality of a circle’s perimeter and a circle’s area, to the same square, can then be drawn to unite the cosmic model of Earth and Moon’s size, to the relationship of the Solar Hero period of 33 years, as the square, to a circle of radius 18.618 years, as appears quite clearly in the plan of Islam’s Dome of the Rock in Jerusalem.
Figure 4 left: The plan of the Dome of the Rock. Right: The Dome itself with Dome of the Chain before it, from Heath, 2021. This bedrock is the scene of Mohammad’s Journey called Miraj[10].
There appears to be some continuity between the built heritage of the subsequent Islamic empire, and the Quran’s subliminal reference to the cosmic model doubled using the story of the seven sleepers to provide the numbers 20160 and 33 through the multiplication and addition of the numbers [3, 4, 5, 6, 7, 8], respectively, centuries before. Plato and the Quran therefore appears to be both an excellent introduction to the influences acting upon the Quran when composed but also to esoteric number traditions to which the Prophet was connected, deserving of further attention.
Bibliography
Bosra, Noor. Plato And The Quran: Number and Allegory from Ancient Mesopotamia and Greece to Islam. London: Noor Bosra Publishing. 2023.
Heath, Richard. Sacred Geometry: Language of the Angels. Rochester, Vt.: Inner Traditions, 2021
Michell, John. Dimensions of Paradise: Sacred Geometry, Ancient Science, and the Heavenly Order on Earth. Rochester, Vt.: Inner Traditions, 2008.
[2] The numbers 5 and 8 are early members of the Fibonacci series in which successive terms approximate the golden mean of 1.618034, in this case to 1.6.
[3] A factorial number is all the numbers up to its limit, so that factorial 3 (or 3!) equals 1 + 2 + 3 equalling 6
[4] Pronounced pi, π is the relationship between a circle’s perimeter and its diameter which if the diameter is 7 then the perimeter is accurately 22, leading to the ratio 22/7 widely adopted by early numerate cultures.
[6] This periodicity denotes the exact location of the sun east on rising, or west on setting, since the extra quarter day is closely the fraction 32/132 which equals 0.24 days while 132 = 4 x 33.
[7] First identified by Robin Heath, personal communication.
[8] then found within astronomical time as the Solar Hero of 33 years and Nodal period of 18.618, reach into the structure of the Metonic period (19 × 19.618 × 18.618) and Saros periods (19 × 18.6182), solar year (19.618 × 18.618), and eclipse year (18.6182), all in solar days.
[9] See chapter 8: The Focal Buildings of Islam of Sacred Number: Language of the Angels, page 185
[10] The Israʾand Miʿraj( Arabic: الإسراء والمعراج, al-’Isrā’ wal-Miʿrāj) are the two parts of a Night Journey that Muslims believe the Islamic prophet Muhammad (AD 570–632) took during a single night around the year AD 621 (1 BH– 0 BH) … miʿraj means rising, or going up to a high place. – Wikipedia. To paraphrase, in the Miraj, Muhammad “ascends to heaven” whilst still alive.
above: The Basilica plan at some stage gained a front extension using a golden rectangle. below: Later Plan for St. Peter’s 16th–17th century. Anonymous. Metropolitan Museum.
The question is whether the extension from a square was related the previous square design. The original square seems quite reworked but similar still to the original square. The four gates were transformed into three ambulatories defining four circles left, above, right and centre, see below.
Equal Perimeter models at the center of St Peter’s Basilica
Equal Perimeter Models
The central circle can be considered as 11 units in diameter so that its out-square is then 44 units. The circle of equal perimeter to the square will then be 14 units in diameter and the difference of 3 defines a circle diameter 3 units. The 11-circle represents the Earth while the 3-circle represents the Moon, to very high precision – hence making this model a representative of the Mysteries inherited from deep antiquity; at least the megalithic age and/or early dynastic Egypt, when the earth’s size can be seen in Stonehenge and Great Pyramid. This inner EP model, is diagonal so that the pillars represent four moons.
An outer Equal Perimeter model is in the cardinal directions (this alternation also found in the Cosmati pavement at Westminster Abbey, and inner models are related to the microcosm of the human being relative to the slightly larger model of Moons). The two sizes of Moon define the circles at the center, around St Peter’s monument. The mandala-like character of the Equal Perimeter model give here the impressions of a flower’s petals and leaves.
Golden Rectangles
You may remember a recent post about double squares and golden rectangles, where a half-circle that fits a Square has root 5 diagonal radius which, arced down, generates a golden triangle. It is therefore possible to fit the square part of the original design and draw the circle that fits the half-diagonal of the square as shown below.
The golden extension of the Basilica’s Square Plan
By eye, the square’s side is one {1} and the new side length below is 1/φ and the two together are 1 + 1/φ = φ (D’B’ below) which is the magic of the Golden Mean. This insight can be quantified to grasp this design as a useful generality:
Quantifying how the golden mean rectangles are generating phi (φ)
Establishing the lengths from the unit square and point O, the center of the right hand side. OA’ is then √5/2. When this is arced, the square is placed inside a half circle A’C, BC is √5/2 + 1/2 = 1/φ.
The rectangle sides ACD’B’ are the golden mean relative to the width A’B = 1, the unit square’s side, but that unit side length A’B is the golden mean relative to the side of the golden rectangle BC. In addition the length B’D’ is the golden mean squared relative to BC, the side of the golden rectangle.
Commentary
It seems that the equal perimeter models within the square design of Bramante were adjusted. The golden mean was used to extend the Basilica (originally an Orthodox square building named after St Basil) into a golden rectangle. This could be done by adding the equivalent lesser golden rectangle, relative to the unit square through the properties of the out half-circle from O.
The series of golden rectangles can travel out in four directions, each coming naturally from a single unitary square. The likely threefold symbolic message, added by the extension seems to be the primacy of the unitary square, of St Peter (on whom the Church was to be founded) and of the Pope (as a living symbol of St Peter).
In Malcolm Stewart’s book on Sacred Geometry, his starcut diagram was applied to Raphael’s painting The School of Athens to create radiants to the people standing around the Athenium Lyceum. “If the starcut was the central geometrical determinant for Raphael’s formal depiction of classical philosophy” it was a “known authoritative device” or framework for geometrical understanding. Stewart found a potential antecedent for such a technique Donato Brahmante’s plan for St Peter’s (see above) which was square like a starcut diagram.
left: Stewarts book cover right: The simplest version of the starcut square where the sides are divided by two and the outer square is four squares of nine, which is 62 = 36 squares and there an octagon within the crossing lines. If there were 72 squares, then the octagon’s vertices would all be on crossings.
A starcut diagram works as a linear interpolator of lines drawn between its sides which are then divided by a number of points that radiate out to other points. The inner lines in this one are eight in number, three per side. Malcolm Stewart shows (see below) the number of coincidences between the plan and a starcut, as if the design was partly arrived at by establishing this pattern. The cardinal cross between its four entrances could have be arrived at, as could the corner octagons with their entrance and side circles lying on starcut radiants. And the central square has corners defining the central space and pillars for supporting the dome.
There seems to be other signs of starcutting such as Honnecourt’s Man, that masons were using such frameworks to build all manner of buildings, sculptures and designs. To investigate further, I made a diagram of my own, over Bramante’s plan and used the method of modular analysis, based on the fact that the central cross of walk ways is one fifth of the square’s side length so that 5 by 5 squares (in red) will define that feature. But there also seems to be a 3 by 3 grid of squares at work (shown in blue) to define the central space in the standard style of the Basilica from the Orthodox (Eastern Church) tradition, this then accounting for most of Stewart’s dotted lines.
Reconstructing most of Malcolm Stewart’s fig. 8.18 using grids of five and three, and applying modular analysis to the Basilica, to quantify it in relative units 1/120th of its side length.
The plan has no scale from which metrology can be deduced, but the smallest number able to hold these two grids together is 60. But to resolve the width of the corner octagons (as 15) I have used a side length of 120. The squares of 24 divided by the octagon width is 24/15 = 8/5 = 1.6. On can see that the starcut diagram was probably part of modular analysis, a technique popular in modern studies of cathedrals which, of necessity, can’t have been designed except as a meaningful whole. But this design would go through many hands including Michelangelo, Carlo Maderno and Gian Lorenzo Bernini to become a transcept cathedral design (see below).
Later Plan for St. Peter’s 16th–17th century. Anonymous. Metropolitan Museum.
My own book on sacred geometry found a different framework was often present in such capital buildings, a model called Equal Perimeter which is a model of pi as 22/7 but is also the basis for a cosmological model of the Earth and the Moon, as 3/11ths of the Earth in size. This model is principally a circle the same perimeter size as a given circle’s circumference, the square being symbolic of the earth in its side length, as a scaled down mean diameter for the Earth. The basilica square limits could then the Earth and the circle of equal perimeter and size of the Moon, as shown overlaid below. Just as the presence of starcut or modular frameworks were linked to a medieval tradition, perhaps parts of that tradition were conscious of this long lost knowledge of the size of the Earth and Moon.
The Equal Perimeter model seems quite clear within the Basilica as originally conceived by Bramante.
It would seem that the equal perimeter design was in use in medieval times because the Cosmati pavement of Westminster Abbey holds it very clearly, and it was the Pope who sent Cosmati guildsmen for its construction. If the basilica was completed on 18 November 1626, the Westminster pavement was completed by 1268 for king Henry III. Its mosaic is depicted in Hans Holbein’s The Ambassadors. The interpretation I gave to it is in my Sacred Geometry book was first published here.
In summary, sacred geometry became a repository for esoteric information and techniques useful for laying out the capital buildings and other religious artifacts in which the exoteric aspects of religion are performed. Rituals often have a deeper meaning, only accessible when one seeks to understand rather than merely know them. It may be that this was a necessary compromise between the outer and inner meaning of life in those times.
Cosmati Great Pavement at Westminster Abbey as a model of the Earth and Moon. [Copyright: Dean and Chapter of Westminster]
There are plans to walk again on the moon (above is a NASA visualization), but there is a sense in which the surface of the moon belongs to the surface of the earth, since the earth’s circumference is 4 times the mean diameter of the earth, minus the moon’s circumference.
The Earth and Moon were formed out of an early collision which left the two bodies in an unusual relationship to one another, in more ways than one. Here we discuss the diameter (and circumference) of each body as a sphere as being in the ratio 11 to 3. The diameter of the Moon is 2160 miles so that the common unit is 720 miles (the harmonic constant) and the diameter of the spherical mean earth would be 7920 miles.
