9. Sun Worship and Phi

Updated: Jul 22, 2020

Atum and Ra, First book of respirations of Usirur Collection Louvre Museum By Rama - Own work, CC BY-SA 2.0 fr, https://commons.wikimedia.org/w/index.php?curid=2991295

Phi, the Golden Ratio

What is Phi? It's a number close to 1.618. It's an irrational number, meaning that you can never fully write all the digits out, they go on forever. And it's also not a number.

At least, that's what John Anthony West says, in his Serpent in the Sky.

"Phi is not a number. It is a function." (p. 64)

What does he mean? Perhaps, simply, that its value is in its use to multiply things, to create sequences of numbers created by phi, or close to phi, in the case of Fibonacci numbers, or sequences of angles created by phi, in the case of some spirals. In fact, phi's main value seems to be geometric, it's application in form, either on paper, or in nature. And from that point of view, it doesn't matter that it is irrational, because if you are going to put pen to paper and draw something, or measure a face, a tree, a shell, you have to use an approximation. So whether phi is considered as a number or not, it is its use as a ratio, its applications that matter.

Phi in fact is central to geometry. It is found in certain geometric shapes, most obviously the pentagon and pentagram, but also many other shapes can be made 'golden' to conform to a phi ratio, for example, a golden rectangle or a golden triangle, or even a golden ellipse.

"Phi, pi, and the square roots of two, three and five are all that are required to form all Perfect geometric solids, and to define and describe all possible harmonic combinations. It is the web of interaction, this vast complex of harmonies, that we respond to as 'the world' - in this case, the physical world, but which is but one (the tangible, perceivable) aspect of he spiritual world, or world of consciousness. The key to this harmonic world is number, and the means by which number is to be understood is geometry."

John Anthony West, Serpent in the Sky, p 64.

Phi, as we've seen, is roughly 1.618.

Pi is roughly 3.1416, another irrational number, used in determining the areas and circumferences of circles.

And then the √2 is close to 1.4142.

√3 is close to 1.732, √4 is 2, √5 is close to 2.236, a number which can be used to calculate the value of phi.

(1 + square root of 5 ) / 2 ≈ 1.618033988749894848204586834...

Five irrational numbers, all crucial to geometry. All these irrational numbers have impressive properties and uses, for example, the aspect ratio of A4 paper is 1 / √2. Vitruvius made use of the square root of 2 in architecture. If the sides of a square are worth 1 unit, then the diagonal is worth √2, and this diagonal can be used to draw another square, with sides worth √2 units. Then a third square is drawn using the diagonal of the second square, and the resulting third square will be twice as big as the first, with sides worth 2 units. This is the quadratum technique, and it relies on the fact that the square root of 4 is 2, and the value of the diagonal of a square is the square root of the value of its sides squared and then added together.

√2 ≈ 1.41421356237309504880168872420969807856967187537694807317667973799...

The square root of 2 is sometimes called Pythagoras's number. Apparently, when one of the Pythagoreans, Hippasus, divulged the secret of the square root of 2 being irrational, he was murdered for it. It seems that numbers not only connect us to the spiritual world, of which everything is but a part, but also to a secret world in which various degrees of knowledge come at a personal cost - a secretive world where a person's discretion and loyalty are closely observed, a world of secrets about the nature of the universe and the way humans shape it.

√3 ≈ 1.73205080756887729352744634150587236694280525381038062805580…

The square root of 3 is also known as Theodorus' constant, named after Theodorus of Cyrene. The fraction 97/56 = (1.732142857…) for the square root of three can be used as an approximation. An equilateral triangle with a side length of 2 has a height of √3.

And pi, or π ≈ 3.1415926535897932384626433...

In the 3rd century BCE, Archimedes proved that ​223⁄71 < π < ​22⁄7 .

So what makes phi stand out among these irrational numbers?

A tiling with squares whose side lengths are successive Fibonacci numbers, By 克勞棣 - Own work, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=38708516

Phi does have some amazing properties.

Two quantities a and b are said to be in the golden ratio φ if a + b / a = a / b = φ .

These quantities may be lines, angles, areas of squares or circles. Or even quantities of time, as when you divide 24 hours into two parts, one the number of hours of daylight on a certain day, and the other part the number of hours of darkness.

So if you divide a whole by phi, you are left with two parts: the greater part, a, is bigger than b by exactly the same multiple as the whole (a+b) is bigger than a.

Actually, since phi is associated with life and growth, it may be more helpful to think of it as part of a multiplication rather than a division. If you take a quantity Q, and multiply it by φ, you get a new quantity S. The difference between Q and S is R, so Q + R = S. But also, curiously, this happens: R / φ = Q, S / φ = R, and Q x φ = S. And of course you can keep going. Let's take S and multiply it by φ : S x φ = U, U being a new amount, created by multiplying S by φ, and T the difference between S and U. The same thing happens: T / φ = S, U / φ = T, S + T = U.

Q is the smallest part, then R, then S, then T, then U.

Q + R = S, S + R = T, S + T = U

In this way you can create long sequences of numbers, or geometric shapes.

Phi is the only number of which the square of the number is equal to the number plus 1.

Φ² = Φ + 1

1.618² = 2.618

1.618 + 1 = 2.618

And this leads to the fact that for any n:

Phi n+2 = Phi n+1 + Phi n

Thus each two successive powers of phi add to the next one.

There are various number sequences formed by phi, the most famous of which is the Fibonacci number sequence.

Starting with 0 and 1, each new number in the sequence is simply the sum of the previous two.

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377 . . . (Fibonacci himself started with 1 not 0)

A page of Fibonacci's Liber Abaci from the Biblioteca Nazionale di Firenze showing (in box on right) the Fibonacci sequence with the position in the sequence labeled with Latin numbers an Public Domain, https://commons.wikimedia.org/w/index.php?curid=720501

Fibonacci was a 13th century Italian mathematician. His actual name was Leonardo of Pisa, or Leonardo Bonacci, the Fibonacci name was made up by a historian, 'Fi' simply standing for 'son of'. He's best known today for the number sequence that bears his name, but what we should really be remembering him for is introducing the Hindu - Arabic number system to Europe, with the digits 1, 2, 3, 4, 5, 6, 7, 8, 9 etc. It seems the Fibonacci number sequence was originally Indian in origin too, as had been described by Indian mathematicians as early as the sixth century.

The ratio of each successive pair of numbers in the sequence approximates Phi (1.618. . .) , as 5 divided by 3 is 1.666…, and 8 divided by 5 is 1.60.

Fibonacci used it to show how it can describe growth in nature, with the example of rabbits reproducing, but he didn't make the connection himself to the golden ratio (which he also used, but independently). This was made later by a German mathematician called Simon Jacob, in the 16th century.

3 / 2 = 1.5

5 / 3 = 1.66667

8 / 5 = 1.6

13 / 8 = 1.625

21 / 13 = 1.61538

34 / 21 = 1.61905

55 / 34 = 1.617647

The ratios of the successive numbers in the Fibonacci sequence progressively approach φ. After the 40th number in the sequence, the ratio is accurate to 15 decimal places.

1.618033988749895 . . .

Who knows when it was first discovered and applied by mankind, perhaps many many thousands of years ago. Humans have been fully modern human for at least several hundred thousand years, perhaps much more, so it might go way back not only to times of which we have no record but to times of which most people ( I think, mistakenly) think humans had very little brain power.

Both pi and phi feature in the design of the Great Pyramids, and phi is in their layout too, as we'll see below.

Phi appears both the patterns of life and growth and in the proportions living creatures and plants. Here are just some of the things that are phi-structured:

DNA, the human body, from head to toe, but also in the face, especially in what are thought of as beautiful faces, in the hand, even in the rhythm of the heartbeat, a fertilized egg divides and multiplies in way in which the ratio of the succeeding number of cells to the previous number of cells is phi, population growth in cities is said to be phi related, stock exchange patterns too. Phi can be seen in the structure of insects, animals, plants, and other creatures.

Below are just two examples, taken from Gary Meisner's fantastic website https://www.goldennumber.net/.


Here are three images from another great website: http://www.natures-word.com .The Spiral Movement of Leaf / Branch Distribution, "In an overwhelming number of plants, a given branch or leaf will grow out of the stem approximately 137.5 degrees around the stem relative to the prior branch. In other words, after a branch grows out of the plant, the plant grows up some amount and then sends out another branch rotated 137.5 degrees relative to the direction that the first branch grew out of." by Aidrian O'Connor.

The middle picture reminded me of my diagram below, to illustrate phi days.

Here is a link to a short fascinating video by Aidrian O'Connor of www.natures-word.com : https://www.youtube.com/watch?v=fZT0g6F-Hrg

From a purely mathematical point of view, phi has lots of inbuilt fun to offer. For example, the 216th number is the Fibonacci sequence is 619220451666590135228675387863297874269396512. The sum of all the digits in that number add up to 216, as well. This was observed by Lucien Khan, see https://www.goldennumber.net.

There's always something more to discover: for example, you can look for the Fibonacci numbers in Pascal’s Triangle, developed by the French Mathematician Blaise Pascal, and formed by starting with an apex of 1. Hours of bliss.


Phi in the Ancient World

The golden ratio is endlessly fascinating. Perhaps even more fascinating, if that's at all possible, is the way it has been used over the centuries.

I have come to think of the Michael - Apollo - Artemis axis as a phi line, or at least as a series of phi lines, so I really want to know about instances of phi that have survived since the Bronze Age or Neolithic..

The Pyramids and Cairo, By Сашка Денисов, CC BY 3.0, https://commons.wikimedia.org/w/index.php?curid=54784693

In order to make sense of it, Egypt beckons: it is one of the places where phi controlled artefacts and structures have endured the best.

Many researchers have looked into the phi ratios in the Great pyramids at Giza, and the best account online that I know of is by Gary Meisner, at https://www.goldennumber.net/phi-pi-great-pyramid-egypt.

But let's start with Flinders Petrie.

William Flinders Petrie

William Flinders Petrie was a British archaeologist and egyptologist. Born on June 3rd, 1853 in Charlton, Kent. At thirteen, he read Piazzi Smyth's Our Inheritance in the Great Pyramids, the author was apparently an acquaintance of his dad's. Petrie was primarily self-taught and had no formal schooling. At the age of eight, he was tutored in French, Latin, and Greek, until he had some kind of collapse, after which he was taught at home.

At the age of 19, he produced the most accurate survey of Stonehenge.

In 1880, at the age of 24, Flinders Petrie published his first book called Stonehenge: Plans, Description, and Theories; this book would become the basis for future discoveries at that site. That same year, he began his more than forty years of exploration and examination of Egypt and the Middle East.

For two years, while he worked at Giza, he lived in an old tomb in the rock.

"I had a doorway in the middle into my living room, a window on one side for my bedroom, and another window opposite for a store-room. I resided here for a great part of two years; and often when in draughty houses, or chilly tents, I have wished myself back in my tomb. No place is so equable in heat and cold, as a room cut out in solid rock; it seems as good as a fire is in cold weather, and deliciously cool in the heat."

" I settled at Gizeh in December, 1880, and lived there till the end of May, 1881; I returned thither in the middle of October that year, and (excepting two months up the Nile, and a fortnight elsewhere), lived on there till the end of April, 1882; thus spending nine months at Gizeh. Excellent accommodation was to be had in a rock-hewn tomb, or rather three tombs joined together, formerly used by Mr. Waynman Dixon, C.E.; his door and shutters I strengthened; and fitting up shelves and a hammock bedstead, I found the place as convenient as anything that could be wished. The tombs were sheltered from the strong and hot south-west winds, and preserved an admirably uniform temperature; not varying beyond 58º to 64º F. during the winter, and only reaching 80º during three days of hot wind, which was at 96º to 100º outside."

The Pyramids and Temples of Gizeh, Flinders Petrie

Photo taken by Flinders Petrie from the tomb where he lived in 1880, By William Matthew Flinders Petrie (1853-1942) - http://framingarchaeologist.blogspot.co.uk/, Public Domain, https://commons.wikimedia.org/w/index.php?curid=31260505

He had travelled to the Pyramids at Giza in 1880 to survey them, understand their geometry and properly investigate how they were constructed, and he was the first person in modern times to do so, using first hand observation.

After Giza, Flinders Petrie explored and excavated over thirty sites in the Middle East. One of his most famous finds was a Stele of Mernepath at Thebes which contains the earliest known Egyptian references to Israel. One of his trainees, Howard Carter, discovered the tomb of Tutankhamun in 1922.

In 1923 he was knighted for services to British archaeology and Egyptology. In 1927, Flinders Petrie returned to Palestine uncovering ruins and remained there until his death at the age of eighty-nine, in Jerusalem on July 28, 1942.

How to measure a pyramid

Map of Giza pyramid complex, work created in Inkscape. Windrose made by Brosen. Author MesserWoland, Wikimedia Commons

The orientation of the pyramids is a good place to start, if only because it gives an indication of the precision of the architect's and builders' work.

It is worth quoting Flinders Petrie at length here:

"93. The orientation of the Great Pyramid is about 4' West of North; a difference very perceptible, and so much larger than the errors of setting out the form (which average 12"), that such a divergence might be wondered at. When, however, it is seen that the passage, which was probably set out by a different observation, nearly agrees in this divergence, it seems unlikely to be a mere mistake. And when, further, the Second Pyramid sides, and also its passages, all diverge similarly to the W. of North, the presumption of some change in the position of the North point itself, seems strongly indicated. The Third and lesser Pyramids are so inferior in work, that they ought not to interfere with the determination from the accurate remains; they would, however, scarcely affect the mean deviation if included with the better data. The azimuths of the two large Pyramids are thus:—

Great Pyramid, casing sides Great Pyramid, core sides – 5' 16" ± 10" Second Pyramid, casing sides – 5' 26" ± 16" Second Pyramid, passage (Smyth) – 5' 37" ± 10" ? Great Pyramid, passage (Smyth) – 5' 49" ± 7" – 3' 43" ± 6"

"[p. 126] In considering these results, the difference of the casing and core azimuths of the Great Pyramid shows that probably a re-determination of the N. was made after the core was finished; and it must be remembered that the orientation would be far more difficult to fix after, than during, the construction; as a high face of masonry, for a plumb-line, would not be available. The passages of the Great and Second Pyramids are the most valuable elements; as, being so nearly at the polar altitude; a very short plumb-line would transfer the observations to the fixed plane. Considering, then, that the Great Pyramid core agrees with the passages far closer than does the casing, the inference seems to be that the casing was fixed by a re-determination of N., by the men who finished the building. These men had not the facilities of the earlier workers; and are shown, by the inferiority of the later work in the Pyramid, to have been far less careful. Hence the casing may probably be left out of consideration, in view of the close agreement of the four other determinations, one of which — the passage — was laid out by the most skilful workmen of the Great Pyramid, with their utmost regularity, the mean variation of the built part being but 1/50 inch.

The simple mean of the last four data is – 5' 32" ± 6"; their divergences being just what would be expected from their intrinsic probable errors. The passages are, however, probably far the most accurate lines in their execution and as the Second Pyramid is inferior in its workmanship, – 5' 45" ± 5" might be well taken as the result from them alone. On the whole, considering the various values of the data, – 5' 40" ± 10" may be taken as a safe statement of the suggested place of the pole, at the epoch of the Pyramid builders."

William Flinders Petrie, The Pyramids and Temples of Gizeh

In other words the workmanship and planning of the Great Pyramid is so precise that Flinders Petrie wonders at a divergence of just a few seconds of degree from north. This same divergence is found both inside and outside the structure. It might in fact help to date the construction, if the orientation of the pole at a certain date can be matched to the orientation of the Great Pyramid, and with the assumption that the workmanship was initially so perfect that the builders achieved true north to the second. But the earth's axis does not leave any trace of its past inclinations, as far as I know - unlike the magnetic pole which can apparently explain the microscopic structure of various volcanic rocks.

Flinders Petrie briefly discusses the possibility of a change in the axis of the earth, and reckons that, going by the pattern of change in recent centuries, the difference in the orientation of North 4,000 to 6,000 years ago should only be about one minute. However, the orientation of the Great Pyramid is about 5 minutes West of North. Either something happened to change the axis abruptly, or if the change in the axis was gradual, the pyramids are much older than 6,000 years. There are various interesting theories about meteorite and comet impacts, polar ice cap imbalance, and earth crust displacement to back up sudden catastrophic changes. Flinders Petrie ponders on the possibility of ocean currents tugging at the polar axis.

Section of the Great Pyramid Passages, from The Pyramids and Temples of Gizeh, by William Flinders Petrie, taken from http://www.ronaldbirdsall.com/gizeh/petrie/photo/plate9.html

It's hard to imagine exactly how difficult it must have been to start measuring the pyramids. When Flinders Petrie got there, they were surrounded by sand, broken casing stones, and some precarious large blocks weighing a ton or more. He had to figure out where the pyramid began and where it ended, which parts were still buried, what might once have been.

From these extracts you can get an idea of the challenges he faced:

"[p. 45] The limestone pavement was found on the N. side first by Howard Vyse, having a maximum remaining width of 402 inches; but the edge of this part is broken and irregular, and there is mortar on the rock beyond it, showing that it has extended further. On examination I found the edge of the rock-cut bed in which it was laid, and was able to trace it in many parts. At no part has the paving been found complete up to the edge of its bed or socket, and it is not certain, therefore, how closely it fitted into it; perhaps there was a margin, as around the casing stones in the corner sockets"

"28. The basalt pavement is a magnificent work, which covered more than a third of an acre. The blocks of basalt are all sawn and fitted together; they are laid upon a bed of limestone, which is of such a fine quality that the Arabs lately destroyed a large part of the work to extract the limestone for burning. I was assured that the limestone invariably occurs under every block, even though in only a thin layer. Only about a quarter of this pavement remains in situ, and none of it around the edges; the position of it can therefore only be settled by the edge of the rock-cut bed of it. This bed was traced by excavating around its N., E., and S. sides; but on the inner side, next to the Pyramid, no edge could be found; and considering how near it approached to the normal edge of the limestone pavement, and that it is within two inches of the same level as that, it seems most probable that it joined it, and hence the lack of any termination of its bed."

Flinders Petrie was able to determine the length of the sides, and then measure the angle of the sides against the ground, to determine the true original height. You can see from the photo below, from the mid to late 19th century, that there was a lot of sand around the pyramid complex in the time Flinders Petrie was there.

A. D. White Architectural Photographs, Cornell University Library Accession Number: 15/5/3090.01504 Title: Giza. Pyramid of Khufu and Sphinx Pyramid. Photograph date: ca. 1865-ca. 1889 Location: Africa: Egypt; Giza URI: http://hdl.handle.net/1813.001 Cornell University Library Wikimedia Commons

Incidentally, the fact that a desert surrounds the pyramid complex, and the trouble the authorities have to go to, to remove enormous amounts of sand to prevent build ups such as you can see in the photo above, is one good argument in favour of the pyramids and sphinx having been built before the Sahara had reached the outskirts of the Nile Delta - a point made by Graham Hancock in his Fingerprints of the Gods, and by John Anthony West in Serpent in the Sky .

Here are some of the results Flinders Petrie came up with, all in inches:

"On the whole, we probably cannot do better than take 51º 52' ± 2' as the nearest approximation to the mean angle of the Pyramid, allowing some weight to the South side.

The mean base being 9068.8 ± .5 inches, this yields a height of 5776.0 ± 7.0 inches."

"This square, of the original base of the Great Pyramid casing on the platform, is of these dimensions:—

N 9069.4 E 9067.7 S 9069.5 W 9068.6"

"Level of the King's Chamber is where the Pyramid diagonal equals the base side. Diagonal of passage section (vertical) rises parallel to the Pyramid face 51º 52' Diagonal of inside end of coffer rises parallel to the Pyramid face 51º 52'. Diagonal of inside bottom of coffer is double its height, or 4 cubits. Diagonal of floor of King's Chamber is double its height."

"Angle of casing as measured By theory of 34 slope to 21 base 51º 52' ± 2'"

"On the whole, we probably cannot do better than take 51º 52' ± 2' as the nearest approximation to the mean angle of the Pyramid, allowing some weight to the South side.

The mean base being 9068.8 ± .5 inches, this yields a height of 5776.0 ± 7.0 inches."

"41. In the Queen's Chamber it seems, from the foregoing statement, that the ridge of the roof is exactly in the mid-place of the Pyramid, equidistant from N. and S. sides; it only varies from this plane by a less amount than the probable error of the determination.

The size of the chamber (after allowing suitably in each part for the incrustation of salt) is on an average 205.85 wide, and 226.47 long, 184.47 high on N. and S. walls, and 245.1 high to the top of the roof ridge on E. and W. walls. "

" Wide High

King's Chamber, mean dimensions 206.13 235.2 or 5 x 41.22 and 47.04 Gallery, lower part, vertically 82.42 92.4 to 94.6 2 x 41.21 and 46.2 to 47.3 Passages 40.6 to 42.6 46.2 to 48.6 40.6 to 42.6 and 46.2 to 48.6 Ramps of gallery, vertically 19.3 to 20.4 22.65 to 23.76 ½ x 38.6 to 40.8 and 45.3 to 47.5 Width 206.12 ± .12 squared, is 100 cubits of 20.612 ± .012 Length 412.24 ± .12 squared, is 400 cubits of 20.612 ± .006 Height 230.09 ± .15 squared, is 125 cubits of 20.580 ± .014 "

"Level of the King's Chamber is where the Pyramid diagonal equals the base side. Diagonal of passage section (vertical) rises parallel to the Pyramid face 51º 52' Diagonal of inside end of coffer rises parallel to the Pyramid face 51º 52'. Diagonal of inside bottom of coffer is double its height, or 4 cubits. Diagonal of floor of King's Chamber is double its height."

His measurements are still considered perfectly accurate, as far as I know - despite all that sand and rubble he had to deal with. So, because they are available freely online, are very precise, and extensive, I have decided to use them. I also use the most recent survey of the Great Pyramid by Glen Dash but there is much less data and it's much less precise, measured in metres with only three decimal points.

The layout of the Pyramids on the ground is interesting. Perhaps it is the most interesting part, especially since Robert Bauval's brilliant insights - that the layout of the pyramids matches the layout of the three stars of the belt of the constellation Orion, a constellation associated with one of the main figure heads of the ancient Egyptian religion: Osiris. Until then, I think Flinders Petrie's view prevailed, that there was no particular method to the planning of the ground layout of the pyramids. Here is another extract from Flinders Petrie's book The Pyramids and Temples of Gizeh, on the ground layout:

"92. [p. 125] The relative positions of the three larger Pyramids to one another were completely fixed in the triangulation, which included them all. The following are their distances apart, as measured on parallels inclined – 5' to true N.— i.e., at the mean azimuth of the First and Second Pyramids; and also the distances, and the angles from these parallels, of the direct lines from one Pyramid to another:—

N E Direct

Centre of First to centre of Second Pyramid 13931.6 and 13165.8 = 19168.4 at 43º 22' 52" Centre of First to centre of Third Pyramid 29102.0 and 22616.0 = 36857.7 at 37º 51' 6" Centre of Second to centre of Third Pyramid 15170.4 and 9450.2 = 17873.2 at 34º 10' 11"

There does not appear to be any exact relation between their centres, or between the corners; and from the nature and appearance of the ground, and the irregularity of the peribolus walls, it would not seem likely that any connection had been planned."

(Italics my own)

The Great Pyramid is a golden pyramid

Here is a link to the recent survey of the Great Pyramid

by Glen Dash: http://dashfoundation.com/downloads/archaeology/as-published/AERAGRAM16_2_GDash.pdf

This gives a mean average side of 230.363 metres, which works out at 9069.4 inches.

According to Wikipedia, the Great Pyramid has a base of 230.4 meters (755.9 feet) and an estimated original height of 146.5 meters (480.6 feet).

Though Flinders Petrie's figures, are slightly different, they are much more precise. He gives a mean value of 9,068.8 inches ± 0.5 for each base side, or 755.7333 feet, which works out at 230.34751 metres ± 0.0127.

If the cubit is 20.62 inches (0.5237 meters) according to Flinders Petrie, the average base length is 439.806 cubits according to Flinders Petrie's measurements or 439.8351 cubits according to the most recent survey. Flinders Petrie varies the value of the cubit though, sometimes 20.632 inches, sometimes 20.612 inches, sometimes 20.580, depending on the area he is measuring, and as he tries to make sense of the figures in cubits. If you use the cubit value of 20.612 inches, the base side is 440.0058 cubits long.

For the height, Flinders Petrie has 5776.0 ± 7.0 inches, which is 481.3333 feet, or 146.71039 metres, or 280.1163 cubits of 20.62 inches, or 280.2251 cubits of 20.612 inches. This he works out with the value of the mean angle of the slope of the pyramid of 51º 52' ± 2'. (or 51.8667º ± 0.0333º).

What's the ratio between base and height?

Working in inches:

5,776 / 9,068.8 = 0.6369

9,068.8 / 5,776 = 1.5701

1.5701 is close to half of pi, and 0.6369 is close to what you would expect the ratio of height and base in a golden pyramid to be.

Height, slant height and mean base values of the Great Pyramid, by Melissa Campbell

Curiously, no mention is made of phi by Flinders Petrie in this book by the way. I suppose he wasn't looking for it, or perhaps he thought it best to stick to his findings from his measuring tape and theodolite for the purposes of the book. In order to look for phi in a pyramid, to see if it is golden, you need to look for phi in the ratio of the height to the base, or half the base. Also, the surface area of the four sides would be linked by phi to the surface area of the base. Using Flinders Petrie's figures in inches:

The base length 9068.8 divided by 2 is 4,534.4‬. So half the base length is 4,534.4 inches.

The slant height can be calculated as 7,343.2 inches.

This divided by half the base side is 1.61944, a figure close to φ.

The slant height divided by Phi would be 4,538.358 inches. That's a difference of just under 4 inches to the actual half base length.

The square root of 1.61803 is 1.272018. The ratio of pyramid height to half the base length should match this.

The height 5776.0 divided by 1.272018 is 4,540.816. That's a difference of 6.416 inches from the actual half base.

If we allow for these differences of 4 and 6 inches approximately between the actual half base length and the ideal phi engineered half base length, then we can safely say that the Great Pyramid is a golden pyramid. Half the base length being in fact 4,534.4 inches long, this amounts to a difference of between 0.0882% and 0.1415%.

What about areas? You would expect them to be in Phi ratio too.

What is the area of each face of the pyramid? It is found by multiplying the base side by the slant height and dividing by 2. The average base length is 9,068.8 inches, which gives a base area of 82,243,133.44 in². This times the slant height of 7,343.23 in and divided by 2 gives a value of 33,297,459.52 in² per side, or face of the pyramid, and a total of 133,189,838.08‬ for all 4 sides. 133,189,838.08 divided by 82,243,133.44 is 1.61946, very close to φ. 133,189,838.08 divided by φ is 82,316,049.814, a difference of 72,916.373 in² between this and the actual base area.

All this means that the Great Pyramid has a base surface area in φ ratio to its sides surface area, and that the slant height is in φ ratio to half the base length.

A Piece of π

Taking the Glen Dash figure for the base side of the pyramid of 230.363 m, and a height of 146.71039 m, this link between the base and height of the ratio becomes apparent in metres:

2 x base side - height = 314.01561

This is close to 10 x pi. Was this intentional?

In metres, (base / 2) + height = (230.363 / 2) + 146.71039 = 262.02539, which is very close to φ² x 100 = 261.8021. Phi is, to an error of 2.518 cm, linked to the dimension in metres of height and half base, added together. These two observations are from Patrice Pooyard's film The Revelation of the Pyramids.

I think I first read about Phi and pi being integral to the Great Pyramid in Fingerprints of the Gods by Graham Hancock.

Perhaps I had known about it before, but reading the book for the first time had such an impact that I can't remember much about what I knew before it, as it's so full of fascinating facts and questions. Perhaps you could say that the whole was bigger than the sum of its parts.

I remember feeling fascinated by the mysterious dimensions of the Pyramid, its impossible precise workmanship, the huge size of the stone blocks, and the links to Phi and pi.

In fact, Graham Hancock doesn't mention phi till he gets to the King's Chamber, well inside the pyramid. Pi is the main draw, at first, and this is because the Great Pyramid embodies spherical properties: it's base sides combine to form a measure that is (almost) in pi ratio with its height, just as a circumference of a circle is in pi ratio to its radius.

base side x 4 ≈ height x π x 2

or: base side x 2 ≈ height x π

The average base length is 9,068.8 inches, which times four makes 36,275.2 inches, or 3,022.933 feet (according to Flinders Petrie's figures). For the height, Flinders Petrie has 5,776.0 ± 7.0 inches, which is 481.3333 feet.

Taking π as 3.1416, we get: 5,776 x 3.1416 x 2 = 36,291.7632, which converts to 3,024.3136 feet.

This is 16.5632 inches greater than the four average base lengths added together - the difference between an exact π height to base perimeter ratio and the actual base perimeter is just over 16 inches.

9,068.8 x 2 / 3.1416 = 5773.36389

This is 2.63611 inches less than the actual height.

If we take Flinders Petrie's value for the south face base side, the greatest of them, at 9,069.5 inches, this difference between pi projected value and actual value of the height in inches is even smaller.

9,069.5 x 2 / 3.1416 = 5,773.809

(only 2.1905 inches difference with 5,776).

Using the most recent base side figure by Glen Dash of 9,069.4 inches, this gives 5,773.74586, a difference of 2.2541 inches with 5,776.

The slant height of 7,323.2 inches can also be included here.

5,776 x 4 / 7343.2 = 3.14631

5,776 x 4 / 3.1416 = 7,354.21441

With an error of 11 inches, height and slant height are related to each other by 4π.

Graham Hancock draws the reader's attention to the similarities between the dimensions of the Earth and the Great Pyramid, as well as the Pyramid of the Sun at Teotihuacan in Mexico:

"(...) the original height of the monument (481.3949 feet), and the perimeter of its base (3023.16 feet), stood in the same ratio to each other as did the radius of a sphere to its circumference. This ratio was 2pi (2 x 3.14) and to express it the builders had been obliged to specify the tricky and idiosyncratic angle of 52 for the pyramid's sides (since any greater or lesser slope would have meant a different height-to-perimeter ratio)."

"(...) the so-called Pyramid of the Sun at Teotihuacan in Mexico also expressed knowledge and deliberate use of the transcendental number pi; in its case the height (233.5 feet) stood in a relationship of 4pi to the perimeter of its base (2932.76 feet)"

Fingerprints of the Gods, by Graham Hancock, Kindle Loc 5957 45%

Graham Hancock quotes Livio Stecchini from Secrets of the Great Pyramid:

"The Great Pyramid was a projection on four triangular surfaces. The apex represented the pole and the perimeter represented the equator. This is the reason why the perimeter is in relation 2pi to the height"

Fingerprints of the Gods, by Graham Hancock, Kindle Loc 7990 60%

And what is the ratio between the size of the Great Pyramid and the size of the Northern Hemisphere?

Graham Hancock uses the best modern figure of 24,902.45 miles (131,484,936 feet) for the equatorial circumference, and the polar radius of 3,949.921 miles (20,855,582.88 feet).

The pyramid base perimeter is, with Flinders Petrie's measurements, 9,068.8 x 4 = 36,275.2 inches (3,022.93212 feet), or with Glen Dash's measurements 9069.4094 x 4 = 36,277.6378 inches, (3,023.1352 feet).

131,484,936 / 3,022.933 = 43,495.8155

131,484,936 / 3,023.1352 = 43,492.9063

Average: 43,494.3609

The scale used for the base perimeter is: 1 : 43,494.3609

The pyramid height is 5,776.0 feet. If the height to base ratio were exactly π in the Great Pyramid, then taking the height as 5,776, the perimeter base would be 3,024.3136 feet, as we saw above. If we divide the circumference of the Earth in feet by this figure, we get: 131,484,936 / 3,024.3136 = 43,475.9596.

Alternatively, you could use a value of 24,883 miles for the equatorial circumference, used by Robin Heath in his book The Lost Science of Measuring the Earth, which appears to be an ancient value. This then gives a scale of 1:43,461.843

Why is the π ratio not exact in the pyramid? Is it because the Earth is not in fact a precise sphere, but bulges at the equator and is flatter at the poles? There is, it's true, a difference of 16 inches between what would be the base perimeter of the Great Pyramid if it were exactly in pi ratio to its height, and the actual base perimeter. However, the actual pyramid measurements are 16 and a half inches smaller than a perfect π height to base ratio, to bigger, and so cannot reflect an equatorial bulge.

What about the scale used then to represent the Earth in this Great Pyramid? It seems to be 1 : 43,494.3609 or 1:43,461.843 .

Graham Hancock shaves a bit off this 43,495.8155 number to approach 43,200.

What's the significance of 43,200?

It's probably best to go back to an earlier section of Fingerprints, where Graham Hancock quotes Giorgio de Santillana from Hamlet's Mill, and mention the concept of precession.

"During the course of each year the earth's movement along its orbit causes the stellar background against which the sun is seen to rise to change from month to month: (...) At present, on the vernal equinox, the sun rises due east between Pisces and Aquarius. The effect of precession is to cause the 'vernal point' to be reached fractionally earlier in the orbit each year with the result that it very gradually shifts through all 12 houses of the zodiac, spending 2160 years 'in' each sign and making a complete circuit in 25,920 years"

Quoted in Fingerprints of the Gods, Kindle Loc 4597, 34%

The key here is 25,920 years. This is one Great Year, one big circle travelled by the sun. And 10 / 6 of 25,920 is 43,200.

The scale used to determine the height and base of the Great Pyramid is linked to a cycle of the sun, or at least, to 10 / 6 of one: precession - if you are happy to accept that 43,495.8155 is close enough to 43,200. It's not a perfect fit.

But this is weird. The equatorial circumference of the earth in miles divided by 108, a precessional number, gives a value close to the base of one side of the Great Pyramid in metres (230.34751 metres). The ancient value for the equatorial circumference is 24,883.2 miles, according to Robin Heath and John Michell in The Lost Science of Measuring the Earth, which is a little short of the current estimate of 24,901.461 miles, according to Wikipedia. The equatorial circumference divided by 27 gives a value close to the base circumference of the Great Pyramid, 921.39004 metres, according to Flinders Petrie's measurements, converted from inches.

24,883.2 / 108 = 230.4

24,883.2 / 27 = 921.6

24,901.461 / 108 = 230.569

24,901.461 / 27 = 922.2763

This doesn't work quite as well with the ratio between pyramid height and Earth polar radius. The height of the pyramid is 146.5 metres according to Wikipedia and 146.71039 metres according to Flinders Petrie, the polar radius is 3,950 miles, according to Wikipedia, and the ancient value is 3,949 5/7 miles.

3950 / 27 = 146.2963

146.28571 / 27 = 146.2857

3,949.7142857 / 146.71039 = 26.9218

With this in mind, you could begin to think of pi as linked to the sun too, together with Phi.

Pi in the Great Pyramid. If the height were 2.19 inches smaller, well within Flinders Petrie's margin of error, the circumference of the circle would match 4 x mean base perfectly.

Pi and Phi as Easy Bedfellows

How can a geometric structure embody both φ and π? How do they get along?

The Great Pyramid's outer structure has a height to base perimeter ratio close to π.

base side x 2 ≈ height x π

It is also a golden pyramid in that it has a base surface area in φ ratio to its sides surface area, and that the slant height is in φ ratio to half the base perimeter.

The slant height of the pyramid divided by the height is: 7,323.2 / 5,776 = 1.27133, which multiplied by 3.1416 gives 3.99401.

The slant height of the pyramid divided by the square root of phi is:

7,323.2 / √ 1.618 = 5,757.15087

5,776 - 5,757.150