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Yehuda Lave, Spiritual Advisor and Counselor

Concentration: Focusing on Now

Totally focusing on the present moment is the key to concentration.

Thoughts about the past, be they positive or negative, prevent concentration. Similarly, worrying about the future keeps your mind away from the present moment. Thinking about anything but the immediate task at hand is distracting. Integrating the message that "life is now" will make it easier to keep your mind focused on the present. This attitude will help you concentrate on what is right before you.

To study effectively, we need to focus our total attention on what we are reading. To listen wholeheartedly, we need to focus our total attention on what someone is saying. To pray with all our heart and all our soul, we need to focus our total attention on the words we are saying to the Almighty.

I'm having some minor surgery this morning (umbilical Hernia) so I would appreciate everyone's prayers for a speedy recovery. You know what the definition of minor surgery is. When it happens to someone else!

Love Yehuda Lave

Ancient History 15 Facts that prove the Great Pyramid of Giza was built by an extremely advanced ancient civilization

Ivan

While many questions remain unanswered about the Great Pyramid of Giza, we still remain perplexed by the complexity and precision of this incredible ancient structure.

Researchers have yet to answer numerous enigmas about the Great Pyramid, enigmas that prevent us from solving the puzzle surrounding this mysterious antique construction that has baffled researchers, historians, and tourists alike. Interestingly, the great pyramid is the only remaining structure of the 7 wonders of the ancient world.

It remains a profound mystery the fact that the Great Pyramid of Giza was built with such precision, how people managed to quarry and transport huge blocks of stone and incorporate them creating the pyramid is a mystery that no one has been able to properly explain. The fact that The Great Pyramid is the most accurately aligned structure in existence and faces true north with only 3/60th of a degree of error is something mind-boggling.

Not only is the Great Pyramid one of the most accurately aligned structures on Earth, there are numerous other details about it that are even more incredible.

In this article, we go through twenty points about the Great Pyramid which are the ultimate evidence that this incredible ancient structure was built by an ancient civilization that was far more advanced than what mainstream scholars are willing to admit.

The Great Pyramid of Giza, a monument like no other The numeric value of 144,000: A key role in the Building process of the Pyramid

It's fascinating to read about the numerous details and studies on the Great Pyramid of Giza, but there are many 'unknown' details about the Pyramid that are not mentioned in history books and schools, these points are indicative of a far more advanced civilization which participated in the planning and construction of the great Pyramid, evidence of that are the numerous complex mathematical formulas incorporated and used in the construction. Interestingly, the outer mantle was composed of 144,000 casing stones, all of them highly polished and flat to an accuracy of 1/100th of an inch, about 100 inches thick and weighing approx. 15 tons each. It is believed that the numeric value of 144,000 plays a key role in the harmonic connection that eventually determined the exact size of the structure. (source) (source)

The Great Pyramid shined like a star. It was covered with casing stones of highly polished limestone

The Great Pyramid of Giza was originally covered with casing stones (made of highly polished limestone). These casing stones reflected the sun's light and made the pyramid shine like a jewel. They are no longer present being used by Arabs to build mosques after an earthquake in the 14th century loosened many of them. It has been calculated that the original pyramid with its casing stones would act like gigantic mirrors and reflect light so powerful that it would be visible from the moon as a shining star on earth. Appropriately, the ancient Egyptians called the Great Pyramid "Ikhet", meaning the "Glorious Light". How these blocks were transported and assembled into the pyramid is still a mystery. (source)

The Great Pyramid is the only Pyramid in Egypt with both descending and ascending inner passages

The fact that the Great Pyramid of Giza is the only one in Egypt with descending and ascending inner passages is a fact that cannot be overlooked when comparing it to other similar structures in Egypt. While the reason behind it still remains a mystery, it is evident that the Great Pyramid was the most unique structure built in ancient Egypt.

Aligned true North

The Great Pyramid of Giza is the most accurately aligned structure in existence and faces true north with only 3/60th of a degree of error. The position of the North Pole moves over time and the pyramid was exactly aligned at one time. Furthermore, the Great Pyramid is located at the center of the land mass of the earth. The east/west parallel that crosses the most land and the north/south meridian that crosses the most land intersect in two places on the earth, one in the ocean and the other at the Great Pyramid.

The only 8-sided Pyramid in Egypt

This is a fact unknown to many people. The Great Pyramid of Giza is the only Pyramid discovered to date which in fact has eight sides. The four faces of the pyramid are slightly concave, the only pyramid to have been built this way.

The centers of the four sides are indented with an extraordinary degree of precision forming the only 8 sided pyramid, this effect is not visible from the ground or from a distance but only from the air, and then only under the proper lighting conditions. This phenomenon is only detectable from the air at dawn and sunset on the spring and autumn equinoxes, when the sun casts shadows on the pyramid. (Check out the above image)

The Value of Pi represented in the Great Pyramid

The relationship between Pi (p) and Phi (F) is expressed in the fundamental proportions of the Great Pyramid. Even though textbooks and mainstream scholars suggest that the ancient Greeks were those who discovered the relationship of Pi, it seems that the builder of the Great Pyramid predated the ancient Greeks by quite some time. Pi is the relationship between the radius of a circle and its circumference. The mathematical formula is:

Circumference = 2 * pi * radius (C = 2 * pi * r)

According to reports, the vertical height of the pyramid holds the same relationship to the perimeter of its base (distance around the pyramid) as the radius of a circle bears to its circumference. If we equate the height of the pyramid to the radius of a circle than the distance around the pyramid is equal to the circumference of that circle.

The celestial connection

While many believe there is a direct correlation between the constellation of Orion and the Pyramids at the Giza plateau, many people are unaware of the fact that the Descending Passage of the Great Pyramid pointed to the pole star Alpha Draconis, circa 2170-2144 BCE. This was the North Star at that point in time. No other star has aligned with the passage since then.

Orion and The Great Pyramid

The southern shaft in the King's Chamber pointed to the star Al Nitak (Zeta Orionis) in the constellation Orion, circa 2450 BCE. The Orion constellation was associated with the Egyptian god Osiris. No other star aligned with this shaft during that time in history.

The Sun, math and the Great Pyramid

Twice the perimeter of the bottom of the granite coffer times 10^8 is the sun's mean radius. [270.45378502 Pyramid Inches* 10^8 = 427,316 miles]. The height of the pyramid times 10**9 = Avg. distance to the sun. {5813.2355653 * 10**9 * (1 mi / 63291.58 PI) = 91,848,500 mi} Mean Distance to the Sun: Half of the length of the diagonal of the base times 10**6 = average distance to the sun Mean Distance to Sun: The height of the pyramid times 10**9 represents the mean radius of the earth's orbit around the sun or Astronomical Unit. { 5813.235565376 pyramid inches x 10**9 = 91,848,816.9 miles} Mean Distance to Moon: ] The length of the Jubilee passage times 7 times 10**7 is the mean distance to the moon. {215.973053 PI * 7 * 10**7 =1.5118e10 PI = 238,865 miles } (source)

The Great Pyramid and planet Earth

The weight of the pyramid is estimated at 5,955,000 tons. Multiplied by 10^8 gives a reasonable estimate of the earth's mass. With the mantle in place, the Great Pyramid could be seen from the mountains in Israel and probably the moon as well (citation needed). The sacred cubit times 10**7 = polar radius of the earth (distance from North Pole to Earth's center) {25 PI * 10**7 * (1.001081 in / 1 PI) * (1 ft / 12 in) * (1 mi/ 5280 ft) = 3950 miles }

The curvature designed into the faces of the pyramid exactly matches the radius of the earth. (source) (source)

Not for mummies

The Great Pyramid of Giza was erected, according to mainstream scholars, to serve as the eternal resting place for a Pharaoh. Contrary to the mainstream theories, no mummy has ever been discovered in the Great Pyramid of Giza. This important fact provides much needed space to theorize about the possible use of the Great Pyramid of Giza which, as we can see, was not meant to serve as a tomb.

When it was first entered by the Arabs in 820 AD, the only thing found in the pyramid was an empty granite box in the King's chamber called the "coffer". (source)

Built in harmony with the galaxy

According to reports, on midnight of the autumnal equinox in the year when the builder of the Great Pyramid finished its construction process, a line extending from the apex pointed to the star Alcyone.

Alcyone is the brightest star in the Pleiades open cluster, which is a young cluster, aged at less than 50 million years. It is located approximately 400 light years from Earth. (source)

It is believed that our solar system revolves around this star accompanied with other solar systems much like the planets in our solar system revolve around the sun. How the ancient builders of the Pyramid have such advanced astronomical knowledge still remains a mystery.

The Ark of the Covenant and the Great Pyramid of Giza

A detail that was unknown to me until not long ago is that the volume or cubic capacity of the Coffer in the King's chamber is exactly the same volume to the Ark of the Covenant as described in the Bible. Interestingly, The granite coffer in the "King's Chamber" is too big to fit through the passages and so it must have been put in place during construction.

The mysterious coffin in the Great Pyramid

If the great coffin wasn't meant to house the remains of a Pharaoh, then what was its real purpose? The coffer was made out of a block of solid granite. This would have required bronze saws 8-9 ft. long set with teeth of sapphires. Hollowing out of the interior would require tubular drills of the same material applied with a tremendous vertical force. Microscopic analysis of the coffer reveals that it was made with a fixed point drill that used hard jewel bits and a drilling force of 2 tons.

8 BIGGEST Mysteries of the Pyramids You've Never Heard Of

The Great Pyramid of Giza Latitude Positioning & Proof of an Ultra Advanced Ancient Civilization

As remarkable as it may seem, the precise global location of the Great Pyramid of Giza in Egypt proves beyond all doubt the existence of a highly advanced civilisation in ancient times. Carefully considered, just the singular fact that is the placement of this mighty megalithic structure forces one to conclude:

1) The builders possessed highly sophisticated knowledge of mathematics and geometry

2) Had knowledge of the true dimensions of the earth to extreme precision

3) Possessed exceptionally advanced technical instrumentation to site The Great Pyramid

As astounding as it may seem, in all three areas, the civilisation responsible for the monument was at least on par with the present age. Indeed, the mathematics employed to position the pyramid in ancient times would seem to have necessitated certain advanced breakthroughs in the field of geometry; such as were only recently discovered (or re-discovered) in the past few centuries.

Moreover, the latitude placement of The Great Pyramid of Giza would appear to have been determined by a key ratio linked to the size of the earth; the exceptional accuracy achieved in realising this, implying that the builders knew the earth's true dimensions to a level in accordance with the most advanced earth model developed in the modern era: The WGS84 earth ellipsoid model (World Geodetic System, established in 1984) [1].

Furthermore, in actually pin-pointing the location of the site at which to build the megalithic structure, highly sophisticated equipment would have to have been used. Simple stone and wooden tools as assumed by conventional historians could never have sufficed. Though difficult for some to accept, electronic instrumentation is not outside the bounds of possibility.

High Precision Placement

Upon careful evaluation, there is simply no way that the precise positioning of the Great Pyramid was due to chance. And it is doubtful also that it was built as a local project purely for the local populous. Indeed, located exactly where it is upon the earth reveals the remarkable intent of the builders; their aim being to achieve a very elegant mathematical connection between the pyramid itself, and the earth form.

Where is The Great Pyramid of Giza located?

The co-ordinates of the structure under the WGS84 earth model are as follows:

29 degrees, 58 minutes, 51 seconds of arc - North (Latitude)
31 degrees, 09 minutes, 0 seconds of arc - East (Longitude)

Of the two noted values, the latitude is of greatest significance, as the placement of the pyramid north of the equator is the very thing that forces one to conclude the builders knew the true and exacting dimensions of the earth.

The Intent of the Pyramid Builders

Due to its axial spin the earth is an ellipsoid form, being a slightly squashed sphere with a marked equatorial bulge [2]. In consequence, all arc length measures from any point on the equator directly connecting up to either pole are complex elliptical curves. All arc length measures swept out over the surface of the earth on or parallel to the equator, are basic circular curves.

The critical point is: elliptical geometry is infinitely more complex to model than simple circular geometry. To place The Great Pyramid where they did the builders had to have possessed advanced mathematical knowledge in both areas; for this seems to have enabled them to locate the structure where they did in order to deliberately capture a certain special mathematical ratio. This being between two key measures swept out over the surface of the earth:

The 'small circle circumference' of the earth (parallel to the equator) at the latitude of the structure.

The elliptical arc length up from the earth equator to the centre of the structure.

Using the modern WGS84 earth ellipsoid model combined with the latitude position of The Great Pyramid, a highly accurate set of values can be had for the two noted measures, including their ratio value [3]:

Circular 'Small Circle' Circumference at Pyramid Latitude = 20917.36136 miles
Elliptical Arc - Earth Equator to centre of Pyramid = 2061.702612 miles

Ratio = 20917.36136 / 2061.702612 = 10.14567340

Now the key value of interest here is the ratio, which indeed at first one might be tempted to dismiss as being of no importance. This would be a great mistake however, as the value appears to have immense significance. Specifically, the sequence of numbers that comprise it closely match those that make up none other than the ratio between the current earth tropical year, and an ideal value of 360 days:

365.2421897 / 360 = 1.014561638

Essentially, the noted ratio associated with The Great Pyramid is practically dead on 10 times the value for a change in the earth year from an ideal of 360 days to its present value.

Now it has previously been discussed at length in other articles that dealt with Egyptian astronomy, that the ancient Egyptians themselves were adamant in their belief that the earth did once possess a 360 day orbit. But that a fundamental shift or disruption to the cosmos in its early stages, altered the orbit of our planet, giving it its current tropical year of 365.2421897 days.

A Chance Relationship? I think Not

Now before anyone dismisses the above numerical match as due to coincidence, consider the margin of error. Let us judge the placement of The Great Pyramid as evaluated above, against what its latitude would have to be in order to capture a precise 10.14561638 ratio, as based upon the currently observed tropical year divided by 360 (x10).

To attain the exact ratio, the structure would have to be located at the precise latitude of 29:58:51.4477 North, with the two key earth measures as follows:

Circular 'Small Circle' Circumference at Pyramid Latitude = 20917.33070 miles
Elliptical Arc - Earth Equator to centre of Pyramid = 2061.711178 miles

Ratio = 20917.33070 / 2061.711178 = 10.14561638

The result reveals that to attain the value of 10.14561638 exactly, the monument would need to be moved slightly further north from its present position. Specifically:

2061.711178 – 2061.702612 = 0.008566 miles, or 45.22 feet.

Now one should consider this difference in light of the fact that the full base side-length of The Great Pyramid itself is some 756 feet. The difference is thus 'contained' within the structure as placed. Moreover, in terms of achieving a location based upon global considerations, one should also note that the true elliptical circumference of the earth (pole to pole) under the WGS84 model is 131259392.7 feet (24859.73347 miles).

Conclusions

In light of the above analysis one must conclude that the ratio of 10.14561638 (10 times the tropical year increase from 360 to 365.2421897 days) was intentional. The builders actively sought to realise this ratio between the noted small circle circumference, and the elliptical arc up to the pyramid from the equator. The very latitude placement was thus determined by the ratio. It was the governing principle that led to it being built precisely where it was, north of the equator.

Knowledge of the Earth's True Dimensions

Now it is critical to note that in accurately determining the elliptical arc distance from the equator to the centre of the pyramid, it is undeniable that the builders knew the dimensions of the earth form to our present day standard (WGS84).

This is inescapable due to the mathematical proof that the smallest section of an ellipse instantly determines the curvature and size of the complete form, including all its associated dimensions. Therefore, in determining so accurately just a small arc section of the earth's full elliptical (polar) circumference, they had to know and be capable of deriving to extreme precision, both the equatorial and polar radius values of the earth, to exceptional accuracy. And this obviously gives the pyramid builders the complete earth form and all its associated elements.

Advanced Instrumentation in Siting the Structure

Now one last point to note is the challenge of going from an 'on paper calculation', to physically isolating the site at which to build The Great Pyramid. And here one may consider a modern project: the artificial islands built off the coast of Dubai that form the project known as 'The World'. The islands themselves are composed mainly of sand from the shallow waters off Dubai, roughly shaped and arranged into a map of the world.

It is interesting to note that the precision placement of the sand to construct the individual islands and shape their 'coastlines', was in line with a margin of error of 10 metres (32.8 feet). And to achieve this, the builders of 'The World' Project made use of the Global Positioning Satellite (GPS) system.

Bearing this in mind, it is difficult to accept that those who built The Great Pyramid of Giza used primitive instruments for isolating the spot upon the earth at which to build the structure. High technology is strongly hinted at. A further article will explore this in greater detail.

Notes [1] In conventional history as taught, at the high point of ancient Greek culture, the so called 'father of geometry' Eratosthenes (Circa 3rd Century BC), was able to derive a value for the diameter of our planet only some 50 miles in error. And this from a precise value of 7899 miles as determined by modern instrumentation. Now as one might expect, prior to Eratosthenes, other determinations are simply assumed by scholars to be more inaccurate (an unfortunate bias in historical analysis). The WGS84 earth model as developed in 1984 is based upon exceptionally accurate measurements of the earth form combined with elliptical geometry.

[2] Viewed 'side-on' the polar circumference of the planet is an ellipse. Viewed 'top-down' the equatorial circumference is circular. All 'sliced cuts' through the earth parallel to the axis from a side view will generate ellipse shapes of ever smaller size. By contrast, all cuts through the earth parallel to the equator (at 90 degrees to the earth axis) produce a series of perfect circles, such as get ever smaller in size as one approaches either pole.

[3] The small circle circumference value and the elliptical arc were both calculated using Ed Williams':
Great Circle Calculator

The Great Pyramid and the Speed of Light

The Great Pyramid encodes enormous amount of numerical coincidences ( pi, Phi, dimensions and movement of our planet, axial tilt, procession, speed of light, and more…)
We can only wonder if the ancient architects were fully aware of these special numbers encoded in their design — or are these numbers simply the result of selecting 2 numbers (7 and 11) for proportions for the Great Pyramid???

The design of the Great Pyramid is based on the ratio 11:7. This ratio (equal 1.571) is perfect approximation of the "squaring the circle" principle.

For the Great Pyramid, Base to Height Ratio 440/280 is exactly 11/7

Most "pyramidologists" appear to be "chasing their tails" uncovering huge amount of "numerical coincidences" embedded in the Great Pyramid…

It is simply unbelievable, however ALL of these numerical coincidences are result of selecting just 4 numbers for the pyramid design: 7, 11 (height to base ratio), 40 (the scale factor), and the 4th key number is the value of the measuring unit: Royal Cubit = 20.62 " = 0.524 m.

This single, fundamental design principle: 11 : 7 Base to Height Ratio generates ALL amazing mathematical properties of the Great Pyramid:

the Golden Ratio Phi=1.618 (the Great Pyramid is a Golden Pyramid: length of the slope side (356) divided by half of the side (440/2 = 220) height is equal to 1.6181818… which is the Golden Ratio Phi
squaring the circle ratio 1.571 (base/height = 44/28 = 1.571)
pi=3.14159… (2 x base/height = 2 x 44/28 = 3.14286 which is very close approximation of "pi" = 3.14159…)
Perimeter of the square base, 4×440=1760, is the same as circumference of the circle with radius = height: 2x "pi" x height (2x 22/7 x 280=1760)
The ratio of the perimeter to height of 1760/280 cubits equates to 2x pi
to an accuracy of better than 0.05%
Side of the base (440) plus double height (2x 280=560) = 1,000
Perimeter of the square base is equal 4×440=1760 RC = 0.5 nautical mile = 1/7,200th of the radius length of the earth
the slop angle 51°.843
The Pyramid exhibits in the design both pi and by Phi, given the similarity
of 2/ sqrt(phi) (2 divided by the square root of Phi) with pi/2 :

11/ 7 equal 1.5714
2/ sqrt(89/55) equal 1.5722
2/ sqrt(Phi) equal 1.5723
pi/ 2 equal 1.5708

Royal Cubit = 0.5236 m, pi – Phi2 = 0.5231
and more…

Does Great Pyramid encode "fractal" value of the speed of light?

The speed of light in a vacuum is 299, 792, 458 meters per second or 983,571,056.43045 feet per second or 186,282.397 miles per second.

Base of the Great Pyramid is a square with side B = 44o Royal Egyptian Cubits. Let's draw two circles: one inscribed and one superscribed on the square of the base of the Great Pyramid.

Circumference of superscribed circle: 2x pi x R
Circumference of the inscribed circle: 2x pi x r

The difference of the circumference of both circles (lets call it C) is:

C = 2 x pi x (R – r) = 2 x pi x [ B/sqrt(2) – B/2 ] =
2x pi x B x [ 1/sqrt(2) – 1/2 ] = 1.301290285 x B

The length of the Egyptian Royal Cubit

Based on "The Pyramids and Temples of Gizeh" by W.M. Flinders Petrie. 1883.

The unit of measuring length used by the ancient architects in the construction of the Great Pyramid was the Royal Cubit.

Petrie estimated the value of the Royal Cubit using some key dimensions of the Great Pyramid:
By the base length of the Pyramid, if 440 cubits (section '43): 20.611 ± .002
By the base of King's Chamber, corrected for opening of joints: 20.632 ± .004 inches
By the Queen's Chamber, if dimensions squared are in square cubits: 20.61   ± .020
By the antechamber: 20.58   ± .020
By the ascending and Queen's Chamber passage lengths (section 149): 20.622 ± .002
By the gallery width:   20.605 ± .032

The Average value of the RC (based on above numbers) is 20.61 inches.

It's almost universally accepted that archaeologist Flinders Petrie's determination of the royal cubit length at 20.632 inches, from his measurements of the King's Chamber in the Great Pyramid of Giza, was the likely measure to survey the dimensions of that pyramid, 440 royal cubits per base side, but the experts stop there, not then letting you know that those 1,760 royal cubits which total the Great Pyramid's base perimeter length, when multiplied by 20.632 inches, equals half a modern nautical mile, or 1/7,200th of the radius length of the earth, so there certainly is a connection.

By most accurate series of measurements Petrie concluded that the royal cubit standard for the Great Pyramid was 20.620 ± .005 inches ( 523.7 mm).

Let's use the lower value 20.615 " for the Royal Cubit (allowed by the 0.005 " accuracy):

the Royal Egyptian Cubit (RC) is 20.615 " = 0.523621 m

B = 440RC = 9070.6″ = 755.883 feet = 230.393 m

Therefore C = 1.301290285 x B = 983.6 feet = 299.8 m

The value of C x 106 is surprisingly very close to
the speed of light in m/s

The speed of light in a vacuum is
983, 571, 056.43 feet per second =
299, 792, 458 meters per second.

Another very strange Great Pyramid "coincidence" related to the speed of light was discovered by John Charles Webb Jr. :
Precise latitude of the centre of the Grand Gallery (inside GP) is
29° 58′ 45.28″ N = 29.9792458° N
The speed of light in vacuum, usually denoted by c, is a universal physical constant important in many areas of physics ( 299,792,458 metres /s ).

– – –

Another interesting coincidence encoded in the Royal Cubit:

Royal Egyptian Cubit = 0.524 m
pi = 3.1416
3.1416 feet = 0.957 m = 1.8263 RC
2x (0.957 / 0.524 ) = 2x 1.8263 = 3.6526 = 365.26 x 10-2

Number of days in a year: 365.25

Another interpretation of the above:
circumference of a a circle with diameter equal 100 feet is
2 x pi x 100 feet = 2 x pi x 1200″ =7539.8″ = 365.74 RC
(or 356.25 RC measured in King's Chamber)

– – –

The Missing Capstone Theory

The pyramidion (capstone) found near the Red Pyramid is nearly "96 cm from the top, measured along the edge". This (and other capstones found in Egypt) suggests that Egyptians were not making huge capstones for their pyramids. Therefore assumption that missing capstone of the Great Pyramid was huge (30 feet at the base) is wrong. It is very likely that in addition to the missing pyramidion there is a layer of stones missing as well. Therefore the capstone was much smaller than the missing part of the pyramid.

To maintain perfect slope angle such a capstone would have to have identical proportions as the whole pyramid: 7:11. If we use the "Red Pyramid" capstone to estimate possible size of the capstone from the Great Pyramid, it seems reasonable to select 2 Royal Cubits (140th of the pyramid's height) for its height. The capstone's base in this case would be 3.14286 RC (as the result of keeping the same scale and proportions as the GP) . This is very close approximation of "pi".

The missing capstone of the Great Pyramid was very likely 2 Royal Cubits in height (3.4267 feet = 1.0475 m ) with square base of 3.1428 Royal Cubits (5.4 feet = 1.646 m).

The Great Pyramid of Giza based on J.H Cole survey

Phi, Pi and the Great Pyramid of Egypt at Giza

August 18, 2012 by Gary Meisner

The Great Pyramid of Egypt closely embodies Golden Ratio proportions.

There is debate as to the geometry used in the design of the Great Pyramid of Giza in Egypt. Built around 2560 BC, its once flat, smooth outer shell is gone and all that remains is the roughly-shaped inner core, so it is difficult to know with absolute certainty. The outer shell remains though at the cone, so this does help to establish the original dimensions.

There is evidence, however, that the design of the pyramid may embody these foundations of mathematics and geometry:

Phi, the Golden Ratio that appears throughout nature.
Pi, the circumference of a circle in relation to its diameter.
The Pythagorean Theorem – Credited by tradition to mathematician Pythagoras (about 570 – 495 BC), which can be expressed as a² + b² = c².

So how might the Great Pyramid have embodied these concepts? There are a number of theories to explore.

Update: See also the article on golden ratios in the site design of the Giza Pyramid complex.

A pyramid based on Phi varies by only 0.025% from the Great Pyramid's estimated dimensions

Phi is the only number which has the mathematical property of its square being one more than itself:

Φ + 1 = Φ²

1.618… + 1 = 2.618…

By applying the above Pythagorean equation to this, we can construct a right triangle, of sides a, b and c, or in this case a Golden Triangle of sides √Φ, 1 and Φ, which looks like this:

This creates a pyramid with a base width of 2 (i.e., two triangles above placed back-to-back) and a height of the square root of Phi, 1.272. The ratio of the height to the base is 0.636.

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). This also creates a height to base ratio of 0.636, which indicates it is indeed a Golden Triangles, at least to within three significant decimal places of accuracy. If the base is indeed exactly 230.4 meters then a perfect golden ratio would have a height of 146.5367. This varies from the estimated actual dimensions of the Great Pyramid by only 0.0367 meters (1.4 inches) or 0.025%, which could be just a measurement or rounding difference.

A pyramid based on golden triangle would have other interesting properties. The surface area of the four sides would be a golden ratio of the surface area of the base. The area of each trianglular side is the base x height / 2, or 2 x Φ/2 or Φ. The surface area of the base is 2 x 2, or 4. So four sides is 4 x Φ / 4, or Φ for the ratio of sides to base.

A pyramid based on Pi varies by only 0.1% from the Great Pyramid's estimated dimensions

There is another interesting aspect of this pyramid. Construct a circle with a circumference of 8, the same as the perimeter of this pyramid with its base width of 2. Then fold the arc of the semi-circle at a right angle, as illustrated below in "Revelation of the Pyramids". The height of the semi-circle will be the radius of the circle, which is 8/pi/2 or 1.273.

This is only 1/10th of a percent different than the height of 1.272 computed above using the Golden Triangle. Applying this to the 146.5 meter height of the pyramid would result in a difference in height between the two methods of only 0.14 meters (5.5 inches).

A pyramid based on areas is identical in geometry to one based on Phi

In addition to the relationships of the pyramid's geometry to phi and pi, it's also possible that the pyramid was constructed using a completely different approach that simply produced the phi relationship. The writings of Herodotus make a vague and debated reference to a relationship between the area of the surface of the face of the pyramid to that of the area of a square formed by its height. If that's the case, this is expressed as follows:

Area of the Face = Area of the Square formed by the Height (h)

(2r × s) / 2 = h²

We also know by the Pythagorean Theorem that r² + h² = s² , which is equal to s² – r² = h², so

r × s = s² – r²

Let the base r equal 1 to express the other dimensions in relation to it:

s = s² – 1

Solve for zero:

s² – s – 1 = 0

Using the quadratic formula, the only positive solution is where s = Phi, 1.618…..

This same relationship is shown on the Mathematics of Phi article, where we how Phi is calculated based on dividing a line so that the ratio of the line to the larger section is the same as the ratio of the larger section to the smaller section. If the height area to side area was the basis for the dimensions of the Great Pyramid, it would be in a perfect Phi relationship, whether or not that was intended by its designers. If so, it would demonstrate another of the many geometric constructions which embody Phi.

A pyramid based on a constant gradient varies by 0.8% from the Great Pyramid's estimated dimensions

Yet another possibility is that the Great Pyramid is based on another method, known as the seked. The seked is a measure of slope or gradient. It is based on the Egyptian system of measure in which 1 cubit = 7 palms and 1 palm = 4 digits. The theory is that the Great Pyramid is based on the application of a gradient of 5.5 sekeds. This measure means that for a pyramid height of 1 cubit, which is 7 palms, its base would be 5.5 palms. The ratio of height to base then is 7 divided by 5.5, which is 1.2727. This is very close to the square root of Phi, which is 1.27202. The slope of a pyramid created with sekeds would be 51.84°, while that of a pyramid based on phi is 51.83°. The seked method was known to be used for the construction of some pyramids, but not all. If used on the Great Pyramid it should have resulted in a height of 146.618 meters on a base of 230.4 meters. This is 0.118 meters (4.7 inches) greater than the actual estimated height of the Great Pyramid. This variance of 0.8% thus does not match the geometry of the Great Pyramid as closely as the geometries based on phi or pi. This result is very close to the dimensions of the Great Pyramid. The question remains though as to why 5.5 would be chosen over some other number for the gradient. What was more appealing about 5.5 rather than simply using a gradient based on 5 or 6? Even without a mathematical knowledge of Phi, a simple awareness of the golden ratio observed in nature might have led choosing this proportion.

Illustration of the Seked method (Image credit to David Furlong):

Its near perfect alignment to due north shows that little was left to chance

One thing that is clear is that the dimensions and geometries were did not happen by chance. Would a civilization with the technological skill and knowledge to align the pyramid to within 1/15th of a degree to true north leave the dimensions of the pyramid to chance? If they did not intend the geometry that resulted in a rather precise angle like 51.83 degrees, why would they have not used another simpler angle found in divisions of a circle such as 30, 45, 54 or 60 degrees? Only one other Egyptian pyramid used this geometry or angle of incline, the Meidum pyramid, and it's a step pyramid with three tiers. Given that there are several ways based in simple geometry by which the Great Pyramid could have ended up with this precise angle, it seems unreasonable to suggest that none of them apply, until another equally plausible and accurate theory can be presented.

Other possibilities for Phi and Pi relationships

If the Egyptians were using numbers that they understood to be the circumference of the circle to its diameter or the golden ratio that appeared in nature, it's difficult to assume that they truly understood the actual decimal representations of pi and phi as we understand them now. Since references to phi don't appear in the historical record until the time of the Greeks hundreds of years later, some contend that the Egyptians did not have this knowledge and instead used integer approximations that achieved the same relationships and results in the design.

A rather amazing mathematical fact is that pi and the square root of phi can be approximated with a high degree of accuracy using simple integers. Pi can be approximated as 22/7, resulting in a repeating decimal number 3.142857142857… which is different from Pi by only 4/100's of a percent. The square root of Phi can be approximately by 14/11, resulting in a repeating decimal number 1.2727…, which is different from Phi by less than 6/100's of a percent. That means that Phi can be approximated as 196/121.

The Great Pyramid could thus have been based on 22/7 or 14/11, which is the same as 7/5.5, in the geometries shown above. Even if the Egyptians only understood pi and/or phi through their integer approximations, the fact that the pyramid uses them shows that there was likely some understanding and intent of their mathematical importance in their application. It's possible though that the pyramid dimensions could have been intended to represent only one of these numbers, either pi or phi, and the mathematics would have included the other automatically.

We really don't know with certainty how the pyramid was designed as this knowledge could have existed and then been lost. The builders of such incredible architecture may have had far greater knowledge and sophistication than we may know, and it's possible that pi , phi or both, as we understand them today, could have been the factors in the design of the pyramid. It could be that they chose other approaches that resulted in almost identical geometries.

A detail of the geomatries and calculations is below:

PyramidBase in MetersHeight in MetersBase/2 in MetersRatio of Height / (Base/2)Angle Radians using (ATAN)Convert Angle to DegreesVariance from Actual in Meters% Variance from ActualGreat Pyramid of Giza 230.4 146.50000 115.20 1.271701 0.90443531 51.82033Phi Geometry 2.0 1.27202 1.00 1.272020 0.90455689 51.82729Phi to Scale 230.4 146.53666 115.20 1.272020 0.90455689 51.82729 0.03670.025%Pi Geometry (8/pi/2) 2.0 1.27324 1.00 1.273240 0.90502258 51.85397Pi to Scale 230.4 146.67720 115.20 1.273240 0.90502258 51.85397 0.17720.121%5.5 Seked 230.4 146.61818 115.20 1.272727 0.90482709 51.84277 0.11820.081%One fact and one interesting question remains

The fact is that whatever method was used in its design, the end result represents the geometry of a phi-based triangle with a high degree of accuracy.

The interesting question is "why did they choose this specific shape geometry and configuration of three pyramids for the Great Pyramid?" It's different than the rest and was clearly done with intent. Was it because it appeared more beautiful, more aligned with nature? If not that, what other reasons did they have that captured this one number associated with nature and beauty?

Construct your own pyramid to the same proportions as the Great Pyramid

Use the template below in gif or pdf format:

Thanks go to Jacques Grimault for these insights, and for other fascinating facts and speculations presented about ancient pyramids in the movie on "The Revelations of the Pyramids."

References:

http://en.wikipedia.org/wiki/Great_Pyramid_of_Giza

http://en.wikipedia.org/wiki/Ancient_Egyptian_units_of_measurement

http://www.kch42.dial.pipex.com/sekes0.htm

http://earthmatrix.com/great/pyramid.htm

Great Pyramid Giza Site Complex Golden Ratios