PBS NOVA has an Online Exploration of the Giza Pyramids and Sphinx with commentary by Mark Lehner and Zahi Hawass, who advocate an age of about 4,600 years for the Giza pyramids. Mark Lehner says "... we know ... the date of the pyramids ... mostly by context. The pyramids are surrounded by cemeteries of other tombs. ... we find organic materials ... that can be radiocarbon dated ... But primarily we date the pyramids by their position in the development of Egyptian architecture and material culture over the broad sweep of 3,000 years. ..."

I agree with the dates of the materials found in the area of the Giza pyramids, but I do NOT agree that those materials, tombs, etc., are the materials, tombs, etc., of the builders of the Giza Pyramids and Sphinx. In my opinion, the builders were either the Ice Age Civilization of 36,525 to 11,600 years ago, or, at the latest, a part of a global Early Human Civilization that broke up about 6,000 years ago, and that the Giza Sphinx and Pyramids were already built about 5,100 years ago when Menes reunited Lower Egypt, including Giza, with Southern Upper Egypt.

The largest of the Giza pyramids is usually called the Great Pyramid.

I prefer to call it the Great Golden Pyramid, because its geometry

is that of the Golden Mean.

The Golden Mean Ratio is given by PHI = (1 + sqrt(5))/2 = 1.618... .

The Great Golden Pyramid has height sqrt(PHI) = 1.272...

and base 1+1 = 2.

Each triangular face intersects the ground at an angle

of arcsin(sqrt(PHI)/PHI), a little less than 52 degrees.

Since 4/sqrt(PHI) = PI, approximately, so that the circumference

of a circle whose radius is the height of the pyramid is

2 x PI x radius = 2 x PI x sqrt(PHI) =

= 2 x (4/sqrt(PHI)) x sqrt(PHI) = 2 x 4 = 8 ,

and since the perimeter of the base of the pyramid is

2 + 2 + 2 + 2 = 8 ,

the Great Golden Pyramid approximately squares the circle.

Exterior measurements of the Great Golden Pyramid are

now very uncertain, because the casing of limestone blocks

has been removed over the years.

Some smaller blocks are in European museums,

but most of them are in Cairo,

where they are now part of dams, bridges,

and mosques (such as the Sultan Hasan Mosque).

Each side of the base of the Great Golden Pyramid

is about 440 Royal Cubits,

so that each semibase is 220 RC,

the total perimeter of its base is about 8 x 220 = 1760 RC.

Units of measurement often used for the Great Golden Pyramid include the Primitive Inch (PI = 2.54268 cm) and the Royal Cubit (RC = 20.6066 PI, or 52.396 cm).

Its height would be about 280 RC if

it had a capstone shaped like a Golden pyramid.

Therefore, the ratio of perimeter to height

would be approximately 2 PI: 1760/280 = 6.28... .

SteveR pointed out that an earlier version of this page had an error in which I wrote PI instead of 2 PI, a mental mistake that I probably made by thinking of PI as fundamental and neglecting the 2, and failing to proof-read carefully. Perhaps I should think of 2 PI as more fundamental than PI, as advocated by Bob Palais in his Mathematical Intelligencer (2001, v. 23, no. 3, pp. 7-8) article PI Is Wrong!.

Each face is an isosceles triangle of height PHI times the semibase.

Such an isosceles triangle has central angle

2 arccos(PHI/sqrt(1+PHI^2)) = 63.43... degrees.

If three such triangles are used to make

a pyramid (instead of the four at Giza),

and if they are mirrors to make a mirrorhouse,

the result is an icosahedral kaleidoscope.

In describing the Great Golden Pyramid,

I do not try to be exact about lengths, angles,

and such, but only approximate. One reason for

discussing it approximately is that the pyramid

itself has been disturbed,

perhaps by earthquakes over the last few thousand years,

or

perhaps by explosions inside the pyrmid,

as suggested by Chris Dunn because the

disturbance seems to be to internal structure of the pyramid

rather than to the base of pyramid.

Internal explosions were used by Vyse during

his explorations in the 1800s.

This gif image amplifies the magnitude of the disturbance

by a factor of 10 to show it more clearly.

Another reason is that my astronomical software

does not go back in time to 9000 BC, so I have

just estimated with a paper precession chart,

a 6-inch star globe, and protractor, ruler, and compass,

to see roughly what is where.

Still another reason for using rough estimates is

my general attitude that you can get a better

overall view of something by initially making

rough overall estimates than by trying to

achieve a lot of precision at first.

Rough first approximations can often show

you what is important and what is not,

and then later you can refine the important things.

The Great Golden Pyramid contains two interior chambers

and a number of interior shafts. The interior shafts

all generally lie in a north-south vertical plane

section of the pyramid, displaced a little east of

the center of the pyramid:

It (like many other Egyptian pyramids) has chambers underneath,

but the Great Golden Pyramid has far more elaborate interior chambers

than any other Egyptian pyramid.

The higher interior chamber, usually called the King's chamber,

is called here the Upper Chamber.

The lower interior chamber, usually called the Queen's chamber,

is called here the Mid Chamber.

The Entrance shaft starts at an entrance on the north face,

a little above ground level,

and descends at an angle of about 26 degrees.

Since the latitude of Giza is about 30 degrees North,

if you go into the entrance shaft and look back,

you will see a part of the sky centered about 4 degrees

from the North pole in the sky.

If you could see through the ground and the Earth as

you looked forward going down the Entrance shaft,

you would see an area near the South pole in the sky.

The Entrance shaft continues down to the conventional

chamber under (not inside) the pyramid.

However, part way there an Ascending shaft branches off,

and goes up to the two interior chambers.

That shaft goes up at a slope of about 26 degrees.

If you could see through the pyramid to the sky in its direction,

you would see (at a certain time of day) the Belt of Orion.

At another time of day, you would see the Southern Cross.

The height of the entire pyramid

(not including the capstone, which is and has been missing)

is 203 courses, or layers, of varying thicknesses,

totalling about 264.5 RC.

If a capstone went to the theoretical apex of the pyramid,

the total height would be about 282 RC, or,

roughly rounding, 280 RC.

At a little bit above the 25th course,

the Ascending shaft branches into 3 passages:

the Well Passage, descending vertically;

the horizontal branch to the Mid Chamber;

and the continuation into the Grand Gallery.

The Well Passage is a vertical descending branch,

leading through an irregular shaft down

to the chambers underneath the pyramid.

The horizontal branch leads to the Mid Chamber,

which is located on the East-West plane of the pyramid

just East of the North-South plane.

The Mid Chamber is made of limestone,

10 RC in the East-West direction,

11 RC in the North-South direction,

and the height of the North and South side walls

is (20/sqrt(5)) RC = about 8.94 RC.

The height of the top of the gable roof is about 11.8 RC.

A 5-step corbeled Niche, from 1 RC to 2 RC deep,

from 1 RC to 3 RC wide, and (20/sqrt(5)) RC = about 8.94 RC high,

has been carved in the East wall.

The top of the side walls is 50 RC above the base of the pyramid.

A crust of salt is on the walls of the Mid Chamber.

Two small shafts, with cross section

about 8 x 8 PI = 0.388 x 0.388 RC = 20.3 x 20.3 cm,

which is about the size of a waveguide set at the frequency of "... 21-centimeter Radiation ... "radio emission" from hydrogen when the electron flips its spin ... a critical technique for "seeing" the interstellar medium which is dominated by hydrogen ...",

go North and South from the Mid Chamber

at an angle of about 39 degrees.

Their openings were sealed until about 1872.

The top of the openings are about 3.3 RC from the floor,

the same level as the top of the bottom section of the Niche

and the roof of the entrance into the Mid Chamber.

The Southern shaft has been explored by a robot used

by Rudolf Gantenbrink of Munich. He has found it to be

blocked by a stone door with handles.

An new exploration was then undertaken to

see what is behind the stone door.

In 2002 Zahi Hawass and his team used a new robot to

drill through the door

and look behind it with a small camera

to find yet another door.

Since the 39 degree angles of the Mid Chamber shafts

are nearly the right-angle-complements of the 52 degree

angles of the faces of the pyramids with the ground,

the Mid Chamber shafts symbolically place the Mid Chamber

at the apex of an inverted intersecting and interpenetrating

Great Golden Pyramid.

In this way, the Mid Chamber (which is smaller than a

capstone that would complete the geometry of the pyramid)

might be regarded as being the interior chamber of a "virtual" capstone,

and the Niche might be a "door" to a "virtual gallery shaft"

leading to the "outside" of the "virtual" capstone.

Returning to the Ascending shaft, and continuing ascent,

you enter the Grand Gallery, an upward extension continuing

to ascend at the 26 degree slope toward the Antechamber

preceding the Upper Chamber.

The Grand Gallery has 7-step corbeled side walls,

2 RC wide at the top and 4 RC wide at the floor.

The floor has a central groove 2 RC wide,

flanked by 1 RC floor-ramps on each side.

The Grand Gallery is about 18 RC high vertically,

or about 16 RC high perpendicular to its floor.

It is about (200/sqrt(5)) = about 89 RC long.

There are 55 ramp-holes on the sides of the floor-ramps.

Some parts of the Grand Gallery walls contain salt deposits,

but not as much as in the Mid Chamber.

At about the level of the 50th course of the pyramid,

the Grand Gallery reaches the level of the Upper Chamber

and leads through a low passage way about 2 RC x 2 RC x 2.5 RC

to the Antechamber of the Upper Chamber.

The Antechamber is about 5 RC long, 2 RC wide and

a little over 5 RC high.

Its side walls have 4 pairs of vertical grooves,

each about 4 RC high.

3 go all the way to the floor.

The first pair of grooves stop 2 RC above the floor,

and contain a two-section granite leaf about 2 RC high,

each section being about 1 RC high.

You could slide the top section of the granite leaf

up and out of the first pair of grooves,

then move it over and drop it into

one of the other 3 pairs of grooves.

You could then do the same thing with the bottom section.

In this way,

you could make a 2 RC high wall in any one

of the other 3 pairs of grooves,

or you could make two 1 RC high walls in any two of them.

Such a 2 RC high wall in the last pair of grooves

would block the entrance to the Upper Chamber,

which is through a low passage way about 2 RC x 2 RC x 5 RC.

Other such configurations would not block the entrance,

but you could use them to act as "dams" to contain

water or other liquids in the Upper Chamber

and the Antechamber.

The Upper Chamber itself is made of granite,

about 10 RC wide (North-South),

about 20 RC long (East-West),

and about 5 x sqrt(5) RC = about 11.18 RC high.

The diagonal of each North-South wall is about 15 RC.

The diagonal of the whole chamber is about 25 RC.

The ratio of the North-South wall diagonal

to the East-West length to the whole chamber diagonal

is about 15:20:25 = 3:4:5 .

A granite coffer is inside.

It is about 2 RC high, about 1.9 RC wide, and about 4.36 RC long.

It is too wide to be taken in or out through the Entrance shaft.

In Primitive Inches, its dimensions are about:

exterior height - 41.213 interior depth - 34.2

exterior width - 38.698 interior width - 26.7

exterior length - 89.806 interior length - 77.8

The dimensions of the coffer make it possible that the Ark of the Covenant could fit inside it, including carrying poles, since according to Exodus (chapter 37) the Ark is (in terms of conventional cubits that are roughly 18 conventional incles) 2 cubits (36 conventional inches) long, 1.5 cubits (27 conventional inches) wide, and 1.5 cubits (27 conventional inches) high. The two Cherubim on top of the Ark would have about 34.2 - 27 = 7.2 conventional inches to fit inside the coffer.

Two small shafts, with cross section

about 8 x 8 PI = 0.388 x 0.388 RC = 20.3 x 20.3 cm,

which is about the size of a waveguide set at the frequency of "... 21-centimeter Radiation ... "radio emission" from hydrogen when the electron flips its spin ... a critical technique for "seeing" the interstellar medium which is dominated by hydrogen ...",

go North and South from the Upper Chamber.

The top of the Northern shaft is about 2 RC above the floor.

The shaft goes up at an angle of about 32 degrees to exit

the pyramid at the 101st course.

If you looked out the Northern shaft,

you would see the area of the North polar sky.

If you went outside the pyramid and looked down the shaft,

and could see through the pyramid and the Earth,

you would see the area of the South polar sky.

The top of the Southern shaft is also about 2 RC above the floor.

The shaft goes up at an angle of about 45 degrees to exit

the pyramid at the 102nd course.

If you went outside the pyramid and looked down the shaft,

and could see through the pyramid and the Earth,

then (at a certain time of day) you would see

the Center of the Milky Way galaxy in Saggitarius.

If you looked out the Southern shaft from within,

then (at the same certain time of day) you would see

the thinnest part of the Milky Way,

North of Betelgeuse, between Orion and Taurus.

Above the Grand Gallery is a Great Void leading to Ceiling Chambers

above the Upper Chamber.

(image from ScanPyramids web site) There is also an Entrance Void above the Entrance Shaft.

According to http://dx.doi.org/10.1038/nature24647 (2017)

"... The Great Pyramid ... was built on the Giza Plateau (Egypt) ...

Despite being one of the oldest and largest monuments on Earth,

there is no consensus about how it was built ... we imaged

the pyramid using muons, which are by-products of cosmic rays that

are only partially absorbed by stone ... The resulting cosmic-ray

muon radiography allows us to visualize the known and potentially

unknown voids in the pyramid in a non-invasive way. Here we report

the discovery of a large void (with a cross section similar to the

Grand Gallery and a length of 30 m minimum) above the Grand Gallery,

which constitutes the first major inner structure found

in the Great Pyramid since the 19th century ... This void, named

ScanPyramids Big Void, was first observed with nuclear emulsion

films ... installed in the Queen’s chamber (Nagoya University),

then confirmed with scintillator hodoscopes ... set up in the same

chamber (KEK) and re-confirmed with gas detectors ... outside of the

pyramid (CEA). This large void has therefore been detected with a

high confidence by three different muon detection technologies and

three independent analyses ...".

The Grand Gallery and Upper Chamber represent the 4-Real-dimensional Physical World Minkowski Spacetime

of Schwinger Sources and Human Tubulin / Microtubules. It is the RP1 x S3 Shilov Boundary Lie Sphere

of

the Lie Ball Complex Domain Type IV(4)(p=2) in which Planck Scale Cl(16) Cells mirror the information

content of the Human-scale Microtubule / Tubulin Quantum Consciousness structures.

The 4 Ceiling Chamber spaces ( 1-4 ) represent the 4 Complex Imaginary Dimensions of the Lie Ball Complex Domain.

Therefore, the Builders of the Great Pyramid represented the Real Shilov Boundary Physical world

by the Grand Gallery and Upper Chamber that are easily accessible by Humans with Microtubule Quantum Consciousness

and

they represented the Imaginary Complex World of Cl(16) Spacetime Cells mirroring the Human Microtubule World

as Ceiling Chamber spaces and the Great Void that are more accessible to Souls of the Spirit World than to Physical Humans.

HERE is some more about the Great Pyramid.

......