14. Phi in the Great Pyramid
by Tom Gilmore
All graphics by Tom Gilmore
See also The Mathematics of Phi
Part I – Phi Geometry in the Great Pyramid
The Phi Right-Triangle
Two exact Royal measures are given by the Nile Scholars for the Great Pyramid dimensions.
Pyramid = 440 Royal Cubits
Apothem of Pyramid = 356 Royal Cubits
The "apothem" of a pyramid is the line from the mid-base to the
The measures of the pyramid in Royal Cubits are diagrammed below left.
Applying the Pythagorean formula (Hypotenuse2 = Base2 + Height 2) to the right-triangle of height, apothem, and half-base, (height2 = 3562 - 2202), so height = ~279.8857 Royal Cubits. This is approximately 0.0004% short of the Royal approximate measure of 280 Royal Cubits.
The relative Royal dimensions of the right triangle consisting of the
apothem, height, and half-base, are the values Phi, root-of-Phi, and 1 (illustrated
The relative lengths are found by dividing all 3 sides by 220.
Half Base = 220/220 =1
Apothem =356/220 = 89/55 = 1.6181818 = Phi.
Height = 280/220 = 1.272727 = square-root of Phi (when Phi =1.6181818)
The measures of the apothem and half-base are 4 times the Phi ratio of 89/55 (89x4=356 and 55x4=220).
(The Fibonacci Series of the Phi ratio
1 (2) 2 (1.5) 3 (1.6) 5 (1.6) 8 (1.625) 13 (1.615384) 21 (1.619047) 34 (1.6176471) 55 (1.618 >) 89
The slope of the faces of the Great Pyramid is 51 degrees 52 minutes, the
same as the angle between the right triangle sides of 1 and Phi. The area of the triangular face is
½ Base x Height = ½ 2 x Phi = Phi
The ratio of base=440 and height=280 (divided by 40) is base=11, height=7.
Approximate Squaring of the Circle
It is mathematically impossible to exactly square the circle because the number Pi is transcendent.
The ratio of base to height in the Great Pyramid squares the
(Squaring the circle is essential to projecting global maps onto flat surfaces).
The circumference of the square base approximately equals the circumference
of the circle with a radius of the height (see diagram above).
The height of the pyramid is ~279.8857 Royal Cubits, and for this proof is rounded to 280.
The circumference of a circle is (Pi x 2r).
Taking the height of the pyramid (280) as the radius(r) of a circle, the circumference is (Pi x 560).
Using Pi = 22/7,
(Pi x 560) = (22/7 x 560) = (12320 / 7) = 1760
The circumference of the base is 440 x 4 = 1760.
Location of the Great Pyramid
The Great Pyramid is located is at the center of the combined land mass of Africa and Asia, at 290 58' 45" latitude, which is close to the 30th parallel. The geometry of the 30th parallel is that it is 1/3 of the distance on the surface from the equator to the north-pole, and that the center of the plane of the 30th parallel is 1/2 the distance from the center of the Earth to the north-pole. This is diagrammed below.
The 4 faces are also unique in having a slight (almost imperceptible) indentation bisecting the triangular surface, however this indentation is actually a curve at the base of the faces that diminishes as the stone masonry rises, and was only applied to the underlying stone blocks, while the finished limestone casing was flat.
Part II – Phi Measures in the Great Pyramid
Phi and Earth Measures
The table below records the various measures of distance by fractions of the circumference of the Earth at the equator. The fractions are cumulative, so for example the English foot is based on 24 hourly divisions of one full rotation, with 1000 miles per hour by definition, and 5280 feet per mile, for a circumference total of 126,720,000 feet. The geographic measure uses 360 degrees and the time measures use 24 hours. The advantage of using the time measure of the cubit is that it is commensurate with clocks, and thus convenient in calculating longitudinal position on the Earth’s surface using the positions of the stars in the night sky as compared to the clock time.
Degree and Time Measures (Base-12)
Geographic Royal Roman English
Foot Cubit Cubit Foot Statute Foot
360 degrees 24
hours 24 24
60 minutes 60 minutes 60 1000 miles 1000 miles
60 seconds 60 seconds 60 chains (60x100) chains (33x160)
100 feet 1000 cubits 880 6000 Roman feet 5280 Statute feet
129,600,000 86,400,000 76,032,000 144,000,000 126,720,000
English Statute Foot and Freemasonry
During the Dark Ages of Christianity, the populace was not allowed to travel, to keep them from relocating in search of less oppressive conditions. But the Church needed masons to build their ever more magnificent cathedrals (at the expense of the suffering poor), and so the masons were granted travel passes and allowed to build Masonic Lodges at construction sites and at intermediate intervals between construction sites to shelter the travelers. Being free to travel, they were called free-masons.
The masonic order has its origin in the guilds of Ptah in ancient Egypt, and preserved secret knowledge of ancient Egypt from eradication by the Church. As Protestant rebellion against the Church gained independence, the Masons became influential in the emerging democratic government of Great Britain.
The English could have retained the time-based mile-measure and their foot be exactly equal to the Geographic Foot by changing the Roman Mile of 6000 feet to being 5400 feet (24x1000x5400=129,600,000), but the Masons sought to incorporate the ratio of Phi into the time measure of feet, and the predominant Masonic presence in British government allowed them to influence the British Parliament in 1593 to declare there to be 5,280 feet to a mile by statute. The Masons chose this number because it preserves the Phi-Measure of the Royal Cubit, and also closely approximates the Geographic Foot.
Since (126,720/129,600 =
a Statute foot is 0.97… of a geographic foot.
Thus the Statute foot correlates closely with the Geographic foot.
Since (76,032/126,720 = 0.6),
a Statute foot is exactly 0.6 of a Royal cubit.
Thus, in the Statute foot, the measure of Phi is preserved.
Measures of the Base of the Great Pyramid
The base measures exactly 750 Geographic Feet
A Cubit is 1.5 Geographic Feet (129,600,000 / 86,400,000 = 1.5)
Thus, the base measures 500 Cubits (750/1.5)
1,000 cubits equals 1 second of time, so 500 cubits equals ½ second.
So the base measures 1/2 second of time (by Earth’s rotation at the equator).
A Cubit is 0.88 of a Royal Cubit (76,032,000 /
86,400,000 = 0.88)
500x.88=440 Royal Cubits (The base measure given by the Nile Scholars)
A Roman Foot is 0.9 Geographic Feet (129,600,000 /
144,000,000 = 0.9)
750/.9=833.33 Roman Feet (The base measure given by the Roman historian Pliny)
As shown previously a Statute foot
is .6 of a Royal cubit
440/0.6 =733.33 Statute Feet
The Remen (a Phi-Measure)
The Remen is defined as the approximate diagonal of a square Royal Cubit. Mathematically a Royal Cubit is 1.7045 Geographic feet, so using the diagonal of a square Royal Cubit, is approximately (sqrt (2 x 1.70452)) ~ (2.4106) feet in length. The Egyptian architects knew that the Royal Cubit was a Phi measure and thus already approximate, so defining the Remen as the approximate diagonal implies that it is not as precise a measure as the Royal Cubit. Some measures of the interior of the Great Pyramid reveal that the standard measure of the Remen was 2.3885 feet (and the preferred measure of 2 Remen was 4.7777 feet).
Following are examples of the 2-Remen measure of 4.7777 feet.
There are 3 stone plugs 2 Remen long lodged at the base of the ascending passage.
The 3 plugs lodged in the passage have been estimated to be just under 4.8 feet in length, which is 2 Remen (4.777 feet).
The Grand Gallery floor is of equal width as the ascending corridor, but there are raised shoulders on each side that are slotted at 27 intervals. These slots extend horizontally across the shelves and vertically up the walls. It is currently thought by many that these slots were intended for wooden beams suspending the granite plugs for release into the ascending passage, completely filling it. There are 27 paired slots, while the ascending passage is 129 feet, and (27x4.777=129).
All the passages, shafts, and chambers are estimated to be offset 23.9 feet east of the pyramid centerline.
This would be a 10 Remen offset, since 5x4.777 = 23.888 feet.
The height of the Grand Gallery over the height of the ascending passage is also 10 Remen.
One explanation of the eastern offset of the interior is that it was intentional and intended to help obscure the location of the sealed entrance. An alternate scenario is that initially there was a smaller pyramid (shown in green above) where the Great Pyramid now stands. The Queen's Chamber would have been centered, both north/south and east/west in this initial pyramid. When the truncated pyramid (truncated at the 50th layer) was built over the initial one, it was too close to the solid eastern end of the plateau, and this did not allow the larger pyramid to be exactly centered east to west over it.
It is likely that the Descending passage was cut at ground level from outside the Queen’s Chamber Pyramid (portion of passage in green above), and when the truncated pyramid was built over it, the in-line extension of the descending passage caused the entrance to be elevated. The descending angle of 260 31' 23" points exactly at the intersection of the axis of Earth's rotation with the celestial globe (for the 30th parallel where the pyramid is located). The passage was most certainly used to align the interior Grand Gallery (an astronomical observatory) exactly north/south, using the then North Star shining down the passage, and reflected up the equally inclined ascending passage at the junction.
The Queen’s Chamber “Sub-Pyramid”
Early explorers reported a half inch of a salty encrustation on the walls of the Queen's Chamber, and that this chamber originally had a putrid smell. The Queen’s Chamber shafts are about 8 x 5 inches. The shafts end 5 inches behind the walls, with a narrow slot connecting to the interior. These shafts might have stored chemicals to be combined in the chamber, producing Hydrogen gas. This would also account for the niche in the Queen’s Chamber that appears to have housed a tall cooling tower 15.3 feet high.
After the King’s Chamber was added to the pyramid, the combustion (Queen’s) chamber may have been used to generate Hydrogen to be piped to the King’s chamber, where it was excited to emit microwaves. The microwave wavelength from hydrogen emission is 8.309 inches and the south shaft in the King’s Chamber is slightly less than 8.4 inches wide (running straight to the pyramid surface). The north shaft doglegs around the antechamber and grand gallery. The north shaft is thought to have served to allow a flow of air through both shafts.
The Pyramid Inch
In the ante-chamber of the Kings Chamber is a granite ceiling slab with a curved embossment marking 25 equal divisions that measure a total of 25.0265 British inches, or 1.00106 British inches per division. The total length is not a phi measure, nor a time or Earth measure. The display of 25 pyramid inches has been left as a clue that at the time of building the Great Pyramid the exact polar diameter of the Earth was known.
The polar radius of the Earth is currently considered to be approximately (to 2 decimals) 3949.89 miles or (3949.89 x 5280 x 12) = 250,265,030.4 British inches. Dividing this by the pyramid inch of 1.00106, the radius is 250,000,030.37 pyramid inches.
Assuming a 4 decimal value of 3949.8896 miles, or 250,264,998.72 British inches (3949.8896 x 5280 x 12), and dividing the British inches by the pyramid inch of 1.00106 would be 250,000,005.05 pyramid inches.
In either case the polar radius is very nearly 250,000,000 pyramid inches (hence the 25 divisions).
It is significant that the polar diameter of the Earth was known, because the rotation of the Earth expands the equatorial diameter to a radius of 3963.2 miles, about 13.31 miles over the polar radius, and there is no physical way to measure the polar diameter except from an orbit, such as has been measured in modern times using satellites. Thus the pyramid inch is powerful evidence that the initial builders of the Great Pyramid had achieved space flight.
The French Meter is also a polar measure, but the meter is based on the surface distance from the equator to the North Pole (and it is less accurate than the pyramid inch). In addition, due to the discrepancy of the polar circumference to the equatorial circumference, the meter is incommensurate with all the equatorial measures, which is a monumental stupidity.
It was determined that a circle with a diameter equal to the length of the antechamber has a circumference equal to 365.242 pyramid inches, and this is the very close to the number of days to a solar year (the circumference is calculated by multiplying the length of the antechamber (116.26025 inches) by pi (3.14159)). This clearly indicates that the length of the antechamber was chosen for this ratio in order to further highlight their advanced understanding of the orbital dynamics in our solar system.
Unfortunately, the Pyramid Inch was usurped by religious fanatics (Pyramidology) to proselytize their spiritual desperation, by equating interior dimensions of the pyramid in inches to years of Biblical history (and the predicted future of “Revelation”). It is reminiscent of vainly equating the vaguely worded quatrains of Nostradamus to subsequent events. As a consequence, the actual significance of the Pyramid Inch has suffered from disrepute.
Five in the Pyramid
The number 5 is prevalent throughout the pyramid. The pyramid has 5 faces (the base is a face).
The sum of the length, width, and height of the King’s Chamber is 5 times the sum of the length, width, and height of the coffer in the chamber. The coffer is precisely carved from a single block of red Aswan granite, and which must have been within the chamber when it was constructed, since it could not fit through the opening into the chamber. The north/south centerline of the Kings Chamber coincides with the north/south centerline of the pyramid, in other words reverses the 10 Remen offset of the interior passages (refer to pyramid diagram above for the interior passages offset).
There are 5 flat granite ceilings above the King’s Chamber, the floor is 10 Royal cubits by 20 Royal cubits, and the floor is 10 times the 5 courses upward from the base (on the 50th layer). The Queen’s Chamber floor is 5 times the 5 courses of masonry upward from the base of the pyramid (on the 25th layer).
The Queen’s Chamber shafts are about 8 x 5 inches, and the shafts end 5 inches behind the walls, with a narrow slot connecting to the interior.
The Coptic word “pyrmet” means “division (or half) of 10”, which is 5.
The Pentagon/Decagon Connection
Five upright pentagons arranged in a circle
enclose a five-pointed star called a pentagram (shown at left above).
Five inverted pentagons arranged in a circle enclose an upright pentagon (shown left-middle).
Alternating 5 upright and 5 inverted pentagons by edges encloses a decagon (10-sided polygon, shown right-middle).
Alternating 5 upright and 5 inverted pentagons by corners encloses a decagram (10-pointed star, shown far right).