Evidence of the Golden Ratio in the Great Pyramid complex.
There are many pyramid theories and questions as to who built the pyramids in ancient Egypt. It’s commonly known though in Egyptology that the proportions of the Great Pyramid of Egypt are within inches of a golden ratio-based pyramid. This is discussed in detail in my article Phi, Pi and the Great Pyramid.
Some say, however, that this single piece of evidence is just a simple coincidence. The primary rationale given to deny this claim is that there is no written historical evidence that the ancient Egyptians had any knowledge of the golden ratio.
That’s a reasonable objection, but what if the evidence showed that the application of the golden ratio at Giza was not limited to the Great Pyramid? That evidence is presented in this article.
The golden ratio connection to the Great Pyramid of Khufu
Let’s quickly review of the evidence on this wonder of ancient architecture. A golden ratio pyramid is based on a triangle whose three sides represent the mathematical relationship that defines the golden ratio. This triangle, known as a Kepler triangle, has base of length 1 and a hypotenuse whose length is Phi. The height of this right triangle is the square root of Phi. Using the Pythagorean theorem, (a²+b²=c²), this triangle represents one of the golden ratio’s unique properties:
1 + Phi = Phi ²
The geometry of the Great Pyramid of Giza varies by less than 0.025% from that of a perfect golden triangle pyramid. This is a difference of just inches in its height of 480.6′. Could the pyramid be this close to the golden ratio in its design, but with no intent of this by its builders? That seems rather amazing, if not unlikely. If the builders thought the golden ratio was important enough to use for this critical measure, would they use it only in this one instance?
What if the golden ratio appears in other aspects in the pyramid complex?
I began to investigate this question after receiving a comment from Pat Miller, a visitor to this site. Pat observed,
The first time I saw the 6 smaller pyramids of Giza, the image of the scale rulers placed next to artifacts came to mind. Three points denote a circle, so 9 points could denote 3 circles. Has anyone, or could someone, calculate their sizes and spacial relationships?”
I have yet to find a relationship based on those circles. The question though inspired me to look at the relationships of the pyramids and the topography of the site.
The Golden Ratio in the three small pyramids with the Tomb of Queen Hetepheres
I used PhiMatrix, with its pixel-level accuracy, to see what ratios appeared in the site’s topography. Every line in the PhiMatrix grid is in a golden ratio proportion to the ones on either side of it.
I first placed the PhiMatrix grid over the “scale rulers,” the three small pyramids next to the Great Pyramid. Here I found a golden ratio in the placement and dimension of the middle pyramid in relation to the perimeters of the two outside pyramids:
The Pyramids of Queens reveals yet another Golden Ratio
There are three other small pyramids, which include the Pyramids of Queens. These are next to the Pyramid of Menakaure, the smallest of the three large pyramids. I measured these next. Although irregularly shaped, the distance between the corner points of their bases revealed the very same golden ratio relationship that appeared in the three small pyramids next to Khufu’s pyramid. Click for larger image:
The Golden Ratio in the pyramids of Khufu and Khafre
I then used PhiMatrix to investigate the relative positions of the two largest pyramids of the complex. Here I found yet another very surprising relationship. Create a rectangle with a perimeter that outlines the bases of the two larger pyramids, and the relationships are found:
- The base of the middle pyramid is a golden ratio of the width of that perimeter rectangle.
- The base of the middle pyramid in relation to the height of this perimeter rectangle is also very close to the golden ratio. It measures at 1.608, a variance of only 0.06%.
- Taken together, this results in a simple, yet elegant, golden ratio geometry. This geometry defines a relationship between the pyramids of Khufu and Khafre.
Click on images below for the high resolution images:
The Golden Ratio in the pyramids of Khafre and Menkaure
I next investigated the topography of the smaller pyramid in relation to the middle pyramid. Here too, I found a relationship. Create another rectangle with a perimeter that outlines the bases of the middle and smaller pyramids. The base of the smaller pyramid is Phi² (a golden ratio of a golden ratio) of the width of that perimeter rectangle.
The red line closest to the center represents the primary golden ratio of the width of the perimeter rectangle. The red line bordering the right side of the small pyramid is the golden ratio of that primary golden ratio.
Analysis of the Giza site plan by others
After first publishing this article, John Gury alerted me to some similar analysis of the Giza complex site. One of the plans expressed a concept similar to the Khufu-Khafre relationship I show above. This was by Jiri Mruzek from his site at http://www.vejprty.com/gizaplan.htm. Note that design is based on a square. The center of the square defines the middle pyramid. Each arm of the cross is a golden rectangle:
The apex points of the Khafre and Menkaure pyramids define the corners a Golden Rectangle!
Another site layout, proposed by Chris Tedder, provided a far simpler and more elegant approach yet. It’s based on two golden rectangles (one in portrait orientation, one in landscape orientation) whose corners align with the apex of each pyramid. I’ve included the original illustration below on the left. I then recreated it with the highest resolution Giza site map that fit on a 1920×1080 screen, and rotated it 90 degrees. The overlay of the PhiMatrix grid reveals how simple and precise this relationship is. I also applied this approach to a satellite image from Google Maps. I used PhiMatrix to create a crosshair (in green) on each pyramid to eliminate the perspective distortion and show the true position of its apex. I set the grid transparency low so as to not obscure the actual lines of the pyramid bases. The bases of the pyramids are not as affected by perspective distortion as are their apex points. Click on the images for full resolution.
I then combined the concepts of both Mruzek and Tedder to create this view of the site plan. The interesting addition is that the apex of Menakaure (the small pyrmid) is fairly closely aligned to the third golden ratio line of the square, as indicated by φ³. Click HERE on on image below for full resolution:
Validation of the results based on surveys of the Giza site
Seeking to validate these results with other measurement, my investigation took me to an analysis by Glen Dash based on very precise site surveys of the Giza site. Glen combined the results of Flinders Petrie, who surveyed the Pyramids in 1880 and 1881, and those of David Goodman and Mark Lehner, resurveyed the Giza Plateau in 1984 and 1985. Glen’s analysis provides the locations of the apex and base of each of the three pyramids to within ±10 centimeters, or about 4 inches. I used these measures to calculate the ratios presented above. Click on images below for full resolution:
This analysis shows a very close alignment of the pyramids to these golden ratios:
- The golden rectangle formed by the apex points of Khafre and Menakure has sides with a ratio of 1.613.
- The golden rectangle formed by the apex points of Khufu and Menakure has sides with a ratio of 1.618.
- The ratio of the base of Khufu to that of Khafre averages to a ratio of 1.610.
A progression of Golden Ratios
The analysis shows that the Giza site thus shows the applications of several basic golden ratio concepts:
- The bases of Khufu and Khafre express the concept of dividing a line at its golden ratio point.
- The apexes of Khafre and Menakure express the concept of the golden rectangle.
- The pyramid triangle of Khufu expresses the concept of the golden rectangle, aka the Kepler triangle.
Golden ratios beget golden ratios
One of the unique properties of the golden ratio is that when you DO use it in design, it creates multiple golden ratio relationships. This is illustrated by the expanded PhiMatrix grid below. Each line is in golden ratio proportion to the ones on either side of it, so imagine how many golden ratios there are to be found. It should be no surprise that the Giza site, if in fact based on the golden ratio, would produce multiple site layout variations that all contain golden ratios.
Multiple golden ratios in the design of the Great Pyramid complex
Could these golden ratio relationships be the basis for the sizes and positions of the pyramids at Giza? The ancient Egyptians did not reveal that to us in the writings they left behind. Perhaps, however, the evidence is beginning to speak for itself.
This analysis now provides us with at least eight instances of golden ratio proportions in the topography of the Giza site. This is in addition to the geometry of the Great Pyramid itself, which is within 0.25% of the proportions of a perfect golden ratio pyramid. The chance that golden ratios would appear nine or more times in the positions and sizes of these pyramids by coincidence alone seems quite unlikely.
If the SETI program received a message from space of these mathematical relationships, I suspect they’d be proclaiming the discovery of extra-terrestrial intelligence. When found in the site plan of Giza, should we chalk it up to chance and coincidence, or conclude that its design was based on these mathematical concepts?
This combined look at the evidences for golden ratios in the site plan of the Pyramids of Giza gives far more reason yet to believe that it does in fact exist, and was included in the design plan of the builders.
Which specific site design was used by the Egyptians in reality is anybody’s guess. The fact is that reasonably accurate plans can revealing golden ratios can be developed with reasonably simple approaches. This, in turn, should provide compelling evidence that the original site plans were based on a similar or related methodology.
Golden ratios are a simple and common outcome of many geometric constructions
Appearances of the golden ratio in nature and design should not be a surprise. The ratio appears in a variety of places in nature, and in features of the human face. This can lead to it being perceived as natural or aesthetically pleasing. This in turn can lead to its use in man’s own creative endeavors. It’s also very commonly found in a variety of simple geometric constructions, such as those shown below (AC/AB=1.618):
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There is thus no need to assume that any use of the golden ratio requires a knowledge of irrational numbers or advanced mathematics. It’s really quite easy to construct with just a length of rope and a straight edge. It sometimes shows up even when you don’t even plan it.
Methodology of the analysis
The topographies shown above are based on maps of the Great Pyramid Complex site provided by ZeeMaps.com. In my first pass at this analysis, I used satellite images from Google Earth and Google Maps. The same relationships were evident, but there are shadows and parallax distortion in these images. This made it clear that a topographical map of the site would provide more accurate results yet. Click on images for higher resolution.
ZeeMaps uses Google Maps to display its maps. My analysis of course assumes that the topography provided in these maps is accurate. If so, this provides compelling evidence that the ancient Egyptians may have used the golden ratio in their construction of the pyramids at Giza. This could have been with intent. It also could have resulted from the use of some simple geometries in laying out the site.
One thing that seems certain: With all the technology, precision and resources that went into the construction of these pyramids, the builders must have had something very specific in mind in determining their sizes, positions and geometries. The sides of the square base of the Great Pyramid align to within 0.7 degrees of the four compass points based on true north. Its base was squared to within 0.2 degrees. It’s unreasonable to assume that the relative sizes and positions of the three large pyramids and six small pyramids were just an afterthought left to chance.
Other nearby ratios do not fit the topographical evidence as well
To minimize any doubt that the proportions found in the topography represent ratios other than the golden ratio, consider the images below. The grid lines on these images show the positions of the base of the pyramids if ratios of 1.600 and 1.625 had been used rather than 1.618. The alignment of the pyramids is clearly closer to the golden ratio. Click on images for higher resolution.
Other theories on the placement of the pyramids – The Orion Connection
It has been proposed by Graham Hancock and Robert Bauval that the positions of the three large pyramids correspond to the positions of the three stars in the belt of the constellation Orion. This is a rather brilliant observation, and is supported by the alignment of the air shaft of the Great Pyramid to the belt of the constellation Orion, as it appeared around 2500 BC. I tested this alignment of the stars by overlaying an image of the constellation to a topography of the pyramid complex. Click on the images for higher resolution.
The relative distances between the pyramids matched quite perfectly to that of the stars. The angle of the lines formed by their positions was slightly off by my analysis. I aligned two stars of the belt to the two smaller pyramids. When this was done, the third star was positioned slightly to the left of the apex of the large pyramid. By my measurement, the angle formed by the three stars is about 172.6°, while that of the three pyramids is about 175.2°. The alignment of the apex of each pyramids to the stars is thus very close, but could have been even closer yet if this were the intention of the builders.
The relative brightness of the stars in Orion do not correlate to the relative sizes of the pyramids
I also noted that while the stars are of roughly equal brightness in the sky, the relative sizes of the pyramids are very different. The center star in the belt is the dimmest of the three, but the southernmost pyramid is by far the smallest of the three. If the pyramids were meant to be a terrestrial representation of the night sky, what was the basis for their very different sizes?
The Orion Connection explains the positions of the pyramids, but not their sizes
Even if the connection to Orion is true (Hancock and Bauval offer a compelling body of other evidence that it is), this defines the only the position of the apex of each pyramid. It does not explain their relative sizes. Could the golden ratio be the key factor yet that led the builders to the positions and dimensions of the pyramids? Based on my measurements of the pyramids presented above, the application of the golden ratio to the topography of the site provides a even higher degree of accuracy than the stars of Orion’s belt. It appears to better explain both the placement and the dimensions of the pyramids.
The Great Sphinx
There’s one more monument at the Giza site that I’d be remiss to exclude from this analysis: The Great Sphinx. The satellite and aerial images of the Sphinx on Google Maps and Google Earth are too blurry to use with any reliability. Look, however, at the topographical map of the Sphinx on ZeeMaps. I think you’ll be find the images rather interesting as well. Click on any of them for higher resolution.
One more step towards a deeper understanding of a great mystery
I find these results intriguing and compelling. There are admittedly many questions that remain. If the golden ratio were applied to the pyramids with intent, there are many alternative approaches to their design that could have been included in the site plan and the sizes and proportions of the pyramids. Still, it seems that the relationships shown are fundamental enough to not be deemed arbitary. They are also quite precise in describing the dimensions and proportions that exist. This should make them unlikely to be the result of coincidence or chance.
I hope these discoveries and analyses provides others with an insight to build upon in doing more research yet. There is still much to understand about the history, mathematics, design and purpose of the building of the pyramids. I hope too that those with access to the site of the Great Pyramid complex will take more accurate measures yet to confirm or deny the proportions that the maps appear to reveal.
You, as visitors to this site, are my peer review group. You can use PhiMatrix for free to examine this evidence for yourself. For those of you who use Autocad, there are site details available on the Internet that can be used for more detailed analysis yet. I invite your analysis and comments.
Gary Meisner
www.goldennumber.net
References:
ZeeMaps.com maps of the Great Pyramid of Giza Complex, with editable version.
Google Maps Satellite imagery of the Great Pyramid of Giza Complex
PhiMatrix Design and Analysis Software
Graham Hancock and Robert Bauval on the Great Pyramid and the Orion Connection
Giza Site Plan analysis by Jiri Mruzek
Giza Site Plan analysis by Glen Dash
Wikipedia on the Great Pyramid of Giza
Wikipedia on the Pyramid of Khafre