The Great Pyramid

What does it look like? What are its Purposes?

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Ikonos satellite image.

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.

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What does it look like?

` 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.  WHAT STARS WERE AT THE NORTH POLE WHEN THE GREAT GOLDEN PYRAMID WAS BUILT? Since I think that the Great Golden Pyramid represents the Supernova Vela X of about 9000 BC, I think that it was built at that time.  Due to the precession of the equinoxes (a 26,000 year cycle stabilized by the moon and sun) `
`the north pole was in the constellation Hercules, near the head of Draco the Dragon.  The Egyptians identified Draco with a crocodile, the Typhon or card 15 of the Tarot Major Arcana.  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  - 34.2exterior width  - 38.698    interior width  - 26.7exterior 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 roof of the Upper Chamber are 5 Ceiling Chambers. `
` They are formed by 5 granite slabs and a 6th top roof of gabled limestone.    They were opened by excavations by Vyse in the 1800s.  The lowest of the 5 Ceiling Chambers was found to contain a black dust, possibly the remains of insects entombed in construction.  The limestone gable of the highest chamber is the only place (other than the Mid Chamber and some parts of the Grand Gallery) that contains salt deposits.   `

The Ceiling Chambers contain some markings that are shown in a photograph on page 51 of Mark Lehner's book The Complete Pyramids (Thames and Hudson 1997):

In my opinion, the "graffitti which included the name of the Pharaoh Khufu" is (like the other graffitti) of more recent origin than the construction of the Great Pyramid, perhaps being daubed on by workers during the 1837 expedition of Howard Vyse that blasted open a passage to that Ceiling Chamber.

` If the 6 slabs are considered to be lines of an I Ching hexagram, with granite slabs being unbroken lines and a limestone gable being a broken line, you get:  the I Ching hexagram Lake over Heaven, which can be interpreted as:  News comes from afar, deal with it by words, not war.   `

What are its Purposes?

It could serve two types of Purposes:

Symbolic -

to encode Information in its design and dimensions

You can see many Symbolic possibilities in these web pages about Pyramids.

Some examples are:

Information encoded in the thicknesses of its layers, or COURSES. (Jim Branson has been studying this in some detail.)

Correspondence between levels in the Great Golden Pyramid and Levels in Physics:

Correspondence between its dimensions, the timetable of History, and the Calendar:

and

Functional -

to act as a Machine to transmit and receive information (and perhaps material).

Transmission/Reception may be among many different forms of Life.

` What about the physical structure of the Machine? `

(This is conjecture based on e-mail conversation with J. C. Paul - however I am solely responsible for errors of fact that may appear here.)

`Information is received by the Pyramid as a receiver, and sent by the Pyramid as a transmitter.   Can the Great Golden Pyramid be a transmitter/receiver, or transceiver device for electromagnetic or other signals?   Consider the materials used to build the Pyramid: limestone and granite.  Limestone is made up of calcium and magnesium carbonates.  Calcium carbonate (calcite) is electromagnetically anisotropic (put a calcite crystal over a straight line, rotate the crystal, and you will see two images of the line, appearing to be in different places, because the effective velocity of light depends on the direction it goes through the calcite crystal).  Limestone can also be triboluminescent, meaning that if it is crushed or scratched or rubbed, it becomes luminescent.   Although I do not understand details, this means to me that limestone can be used to control electromagnetic radiation, and that physical stress can modulate the effect.  I CONJECTURE THAT LIMESTONE IS USED AS AN "ANTENNA" material for the Great Golden Pyramid transceiver.    Granite is largely made up of quartz.   Quartz is piezoelectric, meaning that a quartz crystal changes its electric charge distribution  when it vibrates (i.e. when it is subjected to change in pressure), and that a quartz crystal vibrates when it is subjected to a changing electromagnetic field.   Some examples of quartz conversion of sound vibrations to and from electromagnetism (beyond the old crystal radio) are:   ceramic sonar transducers;  ultrasonic electronics; and  piezoelectric/seismic location of quartz/gold mineral veins.  Again, I do not understand details, but I CONJECTURE THAT GRANITE IS USED AS A "CONVERTER" of vibration (sound) to and from electromagnetism for the Great Golden Pyramid transceiver.    If sound vibrations are the basis for the initial input of transmitted information and the final output of received information, then how does the Great Golden Pyramid act as a machine to produce the sound vibrationscontaining information to be transmitted?    Look at the overall plan of the Pyramid `
` and a closeup of the Upper Chamber area  `
` as shown by the Giza Plateau Computer Model of The Oriental Institute of The University of Chicago.    Since the walls, ceilings, and floor of the Mid Chamber are limestone, the antenna material, and since its niche looks like a place you would stand to receive a signal, I think that the MID CHAMBER IS THE RECEIVING CHAMBER.    Since the Upper Chamber has granite walls, floor, and ceiling, all of which can convert sound vibrations into electromagnetism, I think that the UPPER CHAMBER IS THE TRANSMITTING CHAMBER.    Its ceiling is granite-slab, the lowest of 5 granite-slabs separated by the Ceiling Chambers, with the ceiling of the top Ceiling Chamber being limestone gable.   If the Great Golden Pyramid is considered to be a Great Musical Instrument, the 5 Ceiling Chambers would act as resonating chambers to modulate the sound/electromagnetism conversion.   Another way to modulate the sound/electromagnetism is by the connection of the Upper Chamber to the Grand Gallery through the Antechamber. The choice of placement of granite leaves into the grooves of the Antechamber could modulate not only the connection between the Upper Chamber and the Grand Gallery, but also the conversion by the granite in the leaf.  Still another way to modulate the sound/electromagnetism would be by putting the source in the granite coffer (which is about the right size to contain the Ark of the Covenant including the poles on which it is carried) and then putting at least one granite leaf in a bottom-groove position in the Antechamber, so that the Upper Chamber could be flooded with liquid (water, mercury, or something else)up to the level of the top of the granite leaf, making a liquid lake above the Mid Chamber heaven.   Then, the source would be protected from the liquid by the granite coffer, and the liquid could act as a further modulator, or maybe even produce electromagnetic radiation through the mechanism of sonoluminescence.  For still further modulation, other materials might be placed between granite leaves in the Antechamber or in the granite coffer with the source.   For example, consider the research program of Russell at UBC:  "Our current research is based on reports, principally in the Soviet literature, that the conversion of acoustic to electromagnetic signals in quartz, sulphides, kimberlites [diamond or-bodies] and possibly other minerals, produces signals useful for geophysical exploration. Seismoelectric conversions are produced by:  linear responses due to electrokinetic conversions; piezoelectric conversions from quartz deposits; pulsed radio-frequency responses from sulphides."   After the transceiver has done its job, the granite leaf sections would be moved, allowing the spent liquid to drain away down the central groove of the floor of the Great Gallery to the Well Passage, and thence to the chambers underneath the pyramid, where they soak into the surrounding earth.   Some of the liquid would probably spill over into the passageway to the Mid Chamber, which might, as the liquid evaporates and interacts with its limestone walls, accumulate salt deposits. Other limestone-areas could also have accumulated salt deposits from liquid evaporation.      As you have probably noticed, I have (due to my ignorance) not given any details supporting my conjectures on acoustics or sound/electromagnetism conversion. An interesting reference is the book of Fletcher and Rossing, The Physics of Musical Instruments.   Also, I have not said how to produce the sound in the first place, except that the granite coffer might be used, and that it might have contained the Ark of the Covenant.   Some studies indicate that ultrasound may have been used in constructing the granite coffer and other pyramid structures.  A source for ideas about how such a sound machine might work is the Sphinx group WWW page about ACOUSTICS of MESOAMERICAN and ANDEAN STRUCTURES.  Particularly relevant are:  The Castillo at Chavin De Huantar in Peru was a ceremonial center with drains where water could be pushed through and the roar of the water could be heard through vents and chambers within the center itself. When this was done, the center literally roared. Ancient Peruvian Whistling pottery vessels made a sound when you poured water from them. They were tuned fairly precisely, if you blew into them. Two vessels blown simultaneously produced difference tones. Chichen Itza has a series of cones that produce musical tones when tapped with a wooden mallet.  At Tulum on the Yucatan coast there was a temple which gave a clear and long-range whistle or howl when the wind velocity and direction were correct.  At the Mayan Pyramid I at Tikal `
`a person standing on the top step speaking in a normal voice can be heard by those at ground level for some distance.  At the 3 pyramids of the Group of the Cross at Palenque `
`a three-way conversation can be held at the tops of the three pyramids.  The Tikal and Group of the Cross pyramids all have flat tops on which are placed temples with roof combs, as shown in this cross section of the Pyramid of the Cross:  `
`Note the similarity between the construction of Mesoamerican roof combs and the Grand Gallery of the Great Golden Pyramid of Giza.   Since the Great Golden Pyramid also has a flat top, a similar temple with roof comb could have been there originally.    If the Great Golden Pyramid is a sound/electromagnetism transceiver, with the FACES of the Pyramid acting as large antennae, then THE GEOMETRY OF THE FACES of the Great Golden Pyramid should be designed to make then effective antennae.  HERE is a page about the FACES and their geometry.    Sometimes the Earth's magnetic pole wanders near Giza.   HERE is a page about its layers, or COURSES.    More interesting math (and language) stuff is on an Egyptian Hieroglyphics page.   Some comments on human civilization at the time of the building of the Great Golden Pyramid are HERE.  Note that the boats found near the Pyramid (see the boat location shown on an overall plan of the Pyramid) indicate that it was built by a seafaring civilization. `

Images of such boats,

such as this one (scanned from Egypt, by Albert Siliotti (Thames and Hudson 1996)) from the tomb of Sheshonq II of the 22nd dynasty (about 945-712 BC, so it may be later than the time of the Exodus), of a sun barque which sails across the primordial ocean carrying the Sun disk, with two beings that may correspond to the two Cherubim of the Ark of the Covenant. The Sun disk is described in the Kbre Nege'st as a symbol for the Ark of the Covenant, which was taken by the son of Solomon and Sheba (with the help of angels and friends) from the Temple of Solomon to the Abyssinian Highlands of Africa).

Could the Ark of the Covenant have contained the Power of the Sun, in the sense that it contained a Cold Fusion Energy Source?

Could the Ark of the Covenant have been placed in the coffer of the Great Pyramid?

`  My favorite list of data about the Great Golden Pyramid is here, by Tim Hunkler.   `

The Golden Mean

can be found in natural structures such as sea shells and plants, and, since people like it, also in manmade structures and art.

The Golden Ratio, or Golden Mean, is PHI = (1 + sqrt(5))/2.

It is given by the continued fraction 1 + 1/ 1 + 1/ 1 + 1/ 1 + 1/ ...

It is also given by 1 + PHI = PHI^2.

Even though PHI is algebraic, and not transcendental, it is the most irrational number.

PHI is related to

( The spiral illustrations are from a Robin Hu web page. )

Marcus Chown, in an article in the New Scientist, 21/28 December 2002, pages 55-56, says:

"... The golden ratio ... is equal to ( 1 + sqrt(5) ) / 2 , or 1.6180339887... . ... you can square it simply by adding 1 and find its reciprocal simply by subtracting 1. ... if you take a golden rectangle ... and snip from it a square, you are left with another golden rectangle. ... The golden ratio ... crops up in
• ... the ... Fibonacci sequence ... 1, 1, 2, 3 ...
• ... phyllotaxis ... the arrangement of leaves on a vertical plant stem ... the most common offset angle turns out to be 137.5 degrees, which ... divide[s] 360 degrees into ... the golden ratio: 137.5 degrees an 222.5 degrees ...
• ... "quasicrystals" ... which have fivefold symmetry ...
• ... a rotating black hole flips from a negative to a positive specific heat when the square of its mass divided by the square of its spin parameter is equal to the golden ratio.

...".

Adam Maturana has a web page about the Golden Mean geometry of the Cydonia site on Mars.

`  REFERENCES: Many of the facts and images are taken from the following references - (however,  I am responsible for any erroneous speculations):  Bauval, R., The Orion Mystery (Crown 1994, 1995); Bauval, R. and Hancock, G., The Message of the Sphinx                             (Crown 1996); Berry, L., and Mason, B., Mineralogy (W. H. Freeman 1959); Cecil, T., Lie Sphere Geometry (Springer-Verlag 1992); Coe, M., The Maya, 4th ed (Thames and Hudson 1987); Ferguson, W., and Royce, J.,           Maya Ruins in Central America in Color           (Un. of New Mexico Press 1984); Fletcher, N., and Rossing, T., The Physics of Musical                      Instruments (Springer-Verlag 1991); Hancock, G., Fingerprints of the Gods (Crown 1995); Helgason, S., Groups and Geometric Analysis (Academic 1984); Huang, K., and Huang, R., I Ching (Workman 1987); Kappraff, J., Connections (McGraw-Hill 1991); Kaufmann, W., Universe (Freeman 1988, 1991, 1994); Lehner, M., The Complete Pyramids (Thames and Hudson 1997); Lemesurier, P., The Great Pyramid Decoded (Element 1977, 1993); Lemesurier, P., The Great Pyramid, Your Personal Guide                 (Element 1987); Levy, S., Automatic Generation of Hyperbolic Tilings,           in The Visual Mind, Emmer, ed. (Leonardo 1993); Montesinos, J., Classical Tessellations and Three-Manifolds                 (Springer-Verlag 1987);Schele, L., and Freidel, D., A Forest of Kings (William Morrow 1990); Tompkins, P., Secrets of the Great Pyramid (Allen Lane 1973); Weeks, J., The Shape of Space (Marcel Dekker 1985).   `

For doing calculations about resonant frequencies of the structures of the Great Golden Pyramid, Mike Mitton has pointed out that the CSG web site has a very useful Calculators and Converters web page.