Phi, Pi and the Great Pyramid

August 18, 2012

Phi, Pi and the
  

Great Pyramid of Egypt at Giza

 


 

Great Pyramid of Giza, Egypt with golden ratio proportionsThe Great Pyramid of Egypt is based on Golden Ratio proportions

There is still some debate as to whether the Great Pyramid of Giza in Egypt, built around 2560 BC, was constructed with dimensions based on phi, the golden ratio.  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 compelling evidence, however, that the design of the pyramid embodied 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.

First, 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 146.53567, so the difference of only 0.3567 meters appears to be just a measurement or rounding difference.

The Great Pyramid has a surface ratio to base ratio of Phi, the Golden Ratio

A pyramid based on a 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.

The Great Pyramid also has a relationship to Pi

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 less than 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).

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

Some say that the relationships of the Great Pyramid’s dimensions to phi and pi either do not exist or happened 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 didn’t intend the precise 51.83 degree angle of a golden triangle, why would they have not used another simpler angle found in divisions of a circle such as 30, 45, 54 or 60 degrees?  If the dimensions of the pyramid were not based on both phi and pi, would it not be most reasonable to assume that phi was used since it is based on the visible base of the pyramid and not an invisible circle with the same circumference as that base?

Other possibilities for Phi and Pi relationships

Even if the Egyptians were using numbers that they understood to be the circumference of the circle to its diameter and the golden ratio that appeared in nature, it’s difficult to know if 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 approximatey 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 in the geometry shown about.  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 both pi and phi as we understand them today could have been the driving factors in the design of the pyramid.

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.”

 

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Leave a Comment

{ 28 comments… read them below or add one }

Nico January 5, 2013 at 4:26 pm

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)

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Panagiotis Stefanides February 6, 2013 at 1:34 pm
kerry February 11, 2013 at 2:44 am

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?

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Beto June 13, 2013 at 6:47 pm

Maybe it wasn’t the Egyptians who planned and built it.

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Johnny September 21, 2013 at 5:35 pm

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.

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kirazeek February 11, 2014 at 6:26 am

simply we are not alone

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Gage April 14, 2014 at 7:15 pm

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.

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D. April 8, 2014 at 7:03 pm

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.

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Elton June 13, 2013 at 9:45 am

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.

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greg August 17, 2013 at 6:41 am

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.

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Dave Lightbody September 3, 2013 at 4:08 pm

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.

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Gary Meisner September 3, 2013 at 10:31 pm

Actually, the article concludes with this: “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.” There’s still much we don’t know about the pyramids, and there are those who disagree with the traditional beliefs held about them. I respect your work and viewpoint, but think there is merit in presenting viable alternatives for the reader.

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mark worrell February 17, 2014 at 6:59 pm

Dr.lightbody…..I am sure your a busy man but I truely need to discuss some very important issues that I think I have answers that I just came across the last couple months…your article here is part of it and I think you may understand me on this one….I look forward to speaking with you if you choose so..take care Mark…..[email protected]

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Scott Tucker September 27, 2013 at 10:43 am

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….”

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Gary Meisner October 1, 2013 at 3:30 am

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.

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Paul Hansen November 1, 2013 at 6:36 am

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.

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Abdu November 23, 2013 at 3:05 pm

phi is mentioned as 256/121 the true figure is 196/121

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Sean December 22, 2013 at 6:43 pm

They used a wheel to measure the dimensions of the base. They unknowingly introduced pi.

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Gary Meisner December 22, 2013 at 11:09 pm

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?

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Rajwade Sunil Chintamani December 23, 2013 at 3:22 am

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.

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Gary Meisner December 27, 2013 at 11:01 am

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.

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hack_nug January 20, 2014 at 11:37 am

Didn’t they say the great pyramid had 8 sides?

Here’s a link with some facts about it (number 12 is the one I mention): http://www.ancient-code.com/25-facts-about-the-great-pyramid-of-giza/

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art c February 9, 2014 at 5:05 pm

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.

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lupsaman February 15, 2014 at 4:49 am

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.

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art February 16, 2014 at 10:36 am

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.

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Gary Meisner February 19, 2014 at 6:25 am

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?

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jacob March 10, 2014 at 7:02 pm

Do you guys make pyramids to these exact dimensions or know someone who does?

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Raymond March 11, 2014 at 9:42 am

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.

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