## 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

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:

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:

Pyramid | Base in Meters | Height in Meters | Base/2 in Meters | Ratio of Height / (Base/2) | Angle Radians using (ATAN) | Convert Angle to Degrees | Variance 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 | 0.0367 | 0.025% |

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 | 0.1772 | 0.121% |

5.5 Seked | 230.4 | 146.61818 | 115.20 | 1.272727 | 0.90482709 | 51.84277 | 0.1182 | 0.081% |

## 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

SAMUEL LABOY says

MATHEMATICAL PATTERNS OF THE GREAT PYRAMID – eduardopiperet.wordpress.com

Samuel Laboy – Mathematical patterns of the Great Pyramid: A civil …

https://eduardopiperet.wordpress.com/2014/03/01/samuel-laboy-mathematical-patterns-of-the-great-pyramid-a-civil-engineer-looks-at-the-great-pyramid-of-giza/

Mar 1, 2014 … For many years, we have heard and read about the Great Pyramid of Egypt … can be created in any part of the world, with no reference to the pyramids. …. Video: Modern Evaluation of the Great Pyramid (2 hours 20 minutes).

Marti says

Why is it that, despite the fact that the “pyramides” are much older than Pythagoras, Westeners still want the mathematical calculations used to build them to be credited to an European who spent over 20 years learning in Africa? Is it a matter of an intellectual dishonesty? Or is it just ignorance?

SAMUEL LABOY says

READ MINISTRY OF ANTIQUITIES -SAMUEL LABOY

Ministry of Antiquities – Home | Facebook

https://www.facebook.com/Ministry-of-Antiquities-336764893195328/

THE DOCUMENTS PRESENTED BY SAMUEL LABOY IN RELATION TO THE …. Barcelona legend Carles Puyol visits the Grand Egyptian Museum and Giza …

SAMUEL LABOY says

SEE MY NEW DISCOVERIES ABOUT LEONARDO’S DRAWING

DAVINCI – YouTube

https://www.youtube.com/watch?v=P6KrJCz0Zaw

May 6, 2012 … SEE MY NEW DISCOVERIES ABOUT LEONARDO’S DRAWING: THE FIGURE … + Samuel Laboy oh about the vitruvian man.. i used the leonardo’s …

Samuel Laboy says

SEE MY NEW DISCOVERIES ABOUT LEONARDO’S DRAWING

DAVINCI – YouTube

https://www.youtube.com/watch?v=P6KrJCz0Zaw

May 6, 2012 … SEE MY NEW DISCOVERIES ABOUT LEONARDO’S DRAWING: THE FIGURE … + Samuel Laboy oh about the vitruvian man.. i used the leonardo’s …

Puddin'head says

Er…

Looking at the picture of the folded circle:

It seems to me that all you need is to determine the height that you want your pyramid to be and the thing builds itself.

1. Determine your desired height.

2. Cut a piece of string to that height.

3. Tie the string to a stick planted in the sand at the center of your future pyramid.

4. Walk the circle dictated by the string which will serve as its circumference.

5. Draw a square which touches your circle at 8 symmetrically distributed points as indicated in the picture above.

6. The square tells you exactly where the edges of your pyramid need to fall and you know the height – the rest is simple algebra.

The Egyptians wouldn’t need to know anything else that this post hoc analysis points out – all of the *interesting* geometrical attributes of the pyramid are the *necessary* geometrical attributes of a golden pyramid, which comes about almost unavoidably once you determine how tall you want it to be. All of the imagined complexity is only inferred after the fact, the method seems painfully simple.

Gary B Meisner says

Take a look again at your step 5, which says “Draw a square which touches your circle at 8 symmetrically distributed points as indicated in the picture above.” Therein lies the challenge. You could come up with a variety of sets of “8 symmetrically distributed points.” Only one set will produce a pyramid with the proportional characteristics of the Great Pyramid, so you need to have an exact plan for the position of those eight points BEFORE you can do anything else in your approach.

Puddin'head says

Ah, missed a step. However, the golden ratio is not all that remarkable a relationship – if you play around with geometry long enough, you are going to stumble upon it. Given any arbitrary height or base, calculating the other is a trivial matter.

The method of tracing out the circle with the same radius as the height just simplifies the surveying process and helps direct precisely where the edges need to fall while requiring minimal technology to get it done..

It also seems that you can get a pretty good approximation of true north just by following shadows across the ground.

I guess I’m just missing what is so remarkable about the planning of any of this.

Gary B Meisner says

I guess you’re just not seeing the golden ratio as others do. Kepler called it one of the two jewels of geometry, yet you see it is unremarkable. I would say that the fact that it can be derived from so many geometric constructions, limits and series that DOES make it remarkable. In addition to that it appears in an amazing number of surprising places in nature. What other number do you know that has that quality?

Puddin'head says

Here’s another approach that gets you in the same vicinity without needing to know anything about phi or pi.

Start off drawing a square with an edge length of 230.4 meters. Use an edge of this square as the hypotenuse of an isosceles triangle with its apex at the center of the square. Each leg of this triangle will be 162.9 meters. Take 90% of 162.9 and you get 146.6 meters.

This method gets a ratio of height to breadth of 0.6363, with the height being ~ 0.07% off of the quoted height of 146.5.

Given that we are using an assumed height from Wikipedia, and the distances quoted are rounded to half-meters, it seems more reasonable to assume that the Egyptians simply knocked 10% off of a number determined by their base edge length than it is to assume that they employed any geometric constants. In either case, we need to assume that they rounded to the nearest half-meter when they built it. The benefit to this method is that it requires no knowledge of the Pythagorean theorem nor of pi or phi.

Gary B Meisner says

I would have loved to have been at that committee meeting. The ancient Egyptians are planning the world’s largest structure to entomb their Pharoah for his journey into the afterlife. They’re aligning it to many important celestial markers. The Pharoah asks “how high will my pyramid be?” The lead architect says, “Hey, how about we just knock 10% off the base length? All in favor?” The Pharoah says, “Well, it is about time for lunch. Yeah, that sounds good enough. Motion passed.”

Steve Allcock says

Its easy to see why they chose a phi triangle.

If the triangle were steeper it would tend to collapse outwardly (like the bent pyramid), and if the triangle were shallower it would tend to collapse inwardly. So at phi its perfectly balanced and set to last millions of years.

Gary B Meisner says

That’s a very interesting theory. I’d like to see if any of the engineers in the audience can prove it to be true with some sort of structural engineering analysis. Any takers?