Phi, Pi and the Great Pyramid of Egypt at Giza

The Great Pyramid of Egypt closely embodies Golden Ratio proportions.

Great Pyramid of Giza, Egypt with 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 certainty.

There is evidence, however, that the design of the pyramid embodies 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.

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

or

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:

Golden Ratio Triangle of Great Pyramid of Egypt

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.

Great Pyramid of Giza showing Phi and Pi relationships

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:

pyramid-dimensions

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):

Great-Pyramid-Seked-Angle

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 256/121.

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 Actual
Great Pyramid of Giza        230.4   146.50000       115.20   1.271701   0.90443531      51.82033
Phi Geometry            2.0       1.27202            1.00   1.272020   0.90455689      51.82729
Phi 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      51.85397
Pi 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:

Great Pyramid golden triangle model template

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

 

 

 

 

Comments

  1. Nico says

    The base of the great pyramid could be equal to b=c/π/(sqrt(2)-1) which is in my calculations somewhere close to 230.380924 meters using:
    b=base, c=lightspeed (current best estimate) = 299792458 m/s, π=3.141592653589793238462643383279 (some “signigficant” number of decimals:-),
    sqrt(2)-1=0.414213562 which also happens to be a nice number since 1/(sqrt(2)-1)=2.414213562 (which is exactly 2 higher)

    also,
    If you do some calculations on the height it turns out to be b=1/2*π*h to come exactly to the golden ratios for the surfaces. (b=base,h=height)

  2. kerry says

    I find it quite interesting that there have been no plans or blueprints found of the great pyramid as the Egyptians were meticulous when it come to writing. surely they had plans in which to work from?

      • Johnny says

        I think you are correct.I don’t think the ancient Egyptians build the Great Pyramid or the Sphinx.I think they were build by some advanced civilization either living on Earth or from another place in the Universe.

        • Gage says

          They probably just got rid of them so robbers wouldn’t be able to get into the tomb that held the pharaoh and his treasure.

      • D. says

        There are Egyptian figures all through the region…of course they built it. Romans and Greeks learned mathematics, religion, “government” agriculture, and self-defense from them. The tradition of learning was mostly oral and done by doing, not necessary by writing everything down.

        The Egyptian culture was well advanced hundreds of years before anyone else “discovered” the Pyramids and Egyptian culture.

  3. Elton says

    There are three possible culprits for the design and construction of the Great Pyramid. They are:

    Father Abraham who came out of Ur and brought with him the secrets of Astronomy.

    Joseph, who was sold into Egypt who was trained by Jacob/Israel in how to read Hieroglyphs.

    And someone so bizarre that it’s unthinkable to the Archaeologist’s mind. A far out, far away, lost Civilization that had mapped the Earth’s longitudes with the same precision that we have. We used time, they used the Precession of the Equinox.

  4. greg says

    I also notice the ratio in courses to width in the grand gallery. 22 steps, at 7 ft of width.
    I discussed this in respect to resonance with Christopher Dunn in Glenview Il in 1999.
    I never heard back on that.

  5. says

    Interesting to see this article repeat the same (and understandable) erroneous conclusion drawn by many previous works regarding the Great Pyramid and the supposed deliberate occurrence of Phi in its dimensions. In fact, the occurrence of Phi was an inadvertent by-product of the deliberate use of circular proportions in the structure, by its Old Kingdom Egyptian architects. The circular proportions (which explain why Pi is observable) were thought to bestow encircling ritual protection on the monuments, and this was included deliberately there and on other pyramids and tombs, and in the proportions of the Great Pyramid’s tomb chamber. Egyptology Professor Petrie, Verner, Edwards and myself (background in archaeology and engineering and studied this for 10 years) all concur on this point. It was circular proportions that determined the shape of the pyramid at Giza for Khufu, not Phi or any Golden Ratio, which were culturally unknown and even irrelevant for the Ancient Egyptians whose life was based on practical, functional geometry and basic symbolic systems.
    SEE HERE FOR a website that discusses some research I carried out on the Great Pyramid with the National Museum of Scotland recently:
    http://arkysite.wordpress.com/2013/05/07/the-edinburgh-casing-stone-a-piece-of-giza-at-the-national-museum-of-scotland/

    Dr Dave Lightbody.

  6. Scott Tucker says

    You state that “phi is the only number which has the mathematical property of its square being one more than itself.” This is not true. If you solve the equation x^2 – x – 1 = 0, you get 2 solutions, one of which is phi. The other number is negative and would be irrelevant in constructing something. But to be accurate, you should probably change your assertion to “phi is the only *positive* number….”

    • says

      That is partially correct. There are indeed two solutions. One is 1.618…, which is known as Phi with an upper case. The other is the -0.618, and 0.618 is the reciprocal of Phi, also often known as phi with a lower case. So both of these numbers are known as “phi”, with one of the solutions being its negative version.

  7. Paul Hansen says

    Isaiah 19:19+20 In ancient Hebrew letters also had numerical value. If you use the measurements that were used at the time, the pyramid inch then the numerical value of the words in these 2 verses in Isaiah add up to be the height of the Giza Pyramid. The pyramid inch was derived by taking distance from north pole to south pole thru the core of the earth and dividing that distance by 500,000,000. The pyramid inch was also used in Solomon’s Temple and the arch of the covenant which is also the same exact measurements of the stone arc found inside the King’s Chamber in the Great Pyramid.

    • says

      And perhaps they unknowingly aligned the pyramid to within 1/15th of a degree to true north too? Doesn’t the evidence more readily suggest a high degree of sophistication in their knowledge?

  8. Rajwade Sunil Chintamani says

    I am doing search work on music theorapy. I would like to construct a pyramid based on golden ratio. Size of the pyramid around 4 or 6 feet base. What should be height. Kindly guide me. also I would like to small pyramid about 4 inch what should be height.

    • says

      Use the Golden Triangle above to make such calculations. It’s base is shown as 1, so the full base for a pyramid is 2. If you want a 4 foot base, you multiply the dimensions by 2. The base is then 4. The height is 2 times the square root of Phi, which is 2.544. The hypotenuse is 2 times Phi, which is 3.236.

  9. art c says

    Simple experimenting will show that for a rectangle with given perimeter, maximum area is obtained if it is a square, and for a triangle the maximum area is obtained if it is equilateral. For a pyramid maximum volume is obtained if base is square and sides are equilateral triangles with sides same base side.

  10. lupsaman says

    it’s almost funny how people in charge who have this information can hide it from the controlable masses and rewrite hystory, and supress knowlwdge…. but we are about to wake up.

  11. art says

    A rectangle of given perimeter will have maximum area if it is a square.
    A triangle of given perimeter will have maximum area if it is equilateral.
    A pyramid will have maximum volume if it base is a square and its sides are
    equilateral triangles.
    This requires only simple math to arrive at.
    No pi or phi needed.

    • says

      That may be, but then the sides of the Great Pyramid are not equilateral triangles. The triangles on each side of the pyramid have a base with a relative length of 2, and the two other sides have a length of 1.618 (Phi). If they could have maximized the volume or surface area by using equilateral triangles, especially given the high cost of construction, then why didn’t they do so?

  12. Raymond says

    I’ve heard that the Great Pyramid is also 8 sided to signify the equinoxes. I saw a documentary that included some history, but mostly about the Pyramid its’ self. In it, there was a photo of the Pyramid taken from a plane a few years after the second World War, during the equinox, and it clearly showed that half of each of the two of the sides facing away from the sun shadowed on one half and (sort of) light up on the other, and that this can not be seen during the rest of the year. Do you have more information about this? I can not seem to find that documentary again.

  13. says

    Primary concerns of mankind was,is and always will be are Ownership and Orientation that leads to counting,calculation, navigation and astronomy to start with mankind knowledge.Signs for counting(numbers)lead phonetic signs(alphabet)through out the History.
    Early use of pen and paper as well as tools goes back to civilizations in Asia and Africa because of easy access to natural facilities like minerals,texture of soil,open skies and so on.
    Historical documents are not quiet referable.
    We should not be surprised if Chinese,Egyptians,Indians,Babylonians or Persians knew about Square roots or Divine proportions as leading nations, since when a curious person starts measuring shadows as natural time peace in different times of days and seasons and makes breaks for constructions and studies it’s diagonal proportion he will come up with,not so miraculous.

  14. shogun says

    The fact that the pyramid faces true north and during the spring and fall equinoxes the pyramid shows 8 sides from the sky is enough to baffle anyone with any sense of wonder. forget the fact of the impossible feat of its construction, its mathematical and symbolic properties is enough for one to realize this is the work of something divine.

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