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PLANNING THE PYRAMIDS

A New Theory Shows these Architects Were Way Ahead of Their Time

BY MARTIN RUGGLES

For thousands of years, scientists from around the world have tried to understand how the ancient Egyptians

erected their giant pyramids, especially the Great Pyramid at Giza. Now, an architect and researcher at the

Norwegian University of Science and Technology (NTNU) in Trondheim says he has discovered a previously

unappreciated dimension to the problem—the extraordinarily precise planning and direction that such a massive

structure would have required. Is it possible that millennia ago the monument’s architects were the first to use a

technique now considered essential to the construction of modern skyscrapers?

According to Ole J. Bryn, in a new scientif ic paper, prior researchers have been so preoccupied, if not overwhelmed,

by the logistics of dealing with the massive weight and sheer numbers of stone blocks involved that they have

virtually overlooked other major problems that would have faced the ancient builders : How, for instance, did the

Egyptians know exactly where to put the enormously heavy building blocks? And how was the master

architect able to communicate detailed, highly precise plans to a work force of 10,000 illiterate men?

These were among the questions that confronted Bryn when he began his examination of the Great Pyramid. The

so-called Khufu pyramid, better known as the Pyramid of Cheops, consists of 2.3 million limestone blocks weighing

roughly 7 million tons. At 146.6 meters high, it held the record as the tallest structure ever built for at least 4000

years.

What Bryn discovered was actually quite simple. The Egyptians, he believes, must have invented the modern

building grid by separating the structure’s measuring system from the physical building itself, thus introducing “ tol-

erance,” as it is called in today’s engineering and architectural professions.

According to Bryn, “ the Egyptian pyramids are the only true pyramids in the world culminating in an Apex point [which

implies] the need for extreme precision on a grand scale. To aim for a point 146.6 meters (28 0 Royal Cubits) up in the

sky with only a plumb line and a string implies a crucial need for numerous points of measure. These points, says

Bryn, have to be evenly distributed over the face of the pyramid in order for the geometry to be carefully controlled.”

In his scientif ic article published in May 2010 (Retracing Khufu’s Great Pyramid in the Nordic Journal of Archi-

tectural Research, vol 22, no. 1/2, 2010), Bryn discusses aspects that can explain the construction of a multitude of

the Egyptian pyramids by taking the building grid, and not the physical building itself, as the starting point for the

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analysis.

Ea rly T heories

Khufu’s Great Pyramid (purportedly built between 2606-2573 BC) on the Giza plateau is arguably the most studied

structure in the history of mankind. Its immense size, sophistication, and endurance has nurtured a great number of

theories, some more plausible than others, on how it was built and why its dimensions are what they are.

In one of the earliest theories, the historian Herodotus, claimed Egyptian priests told him the Great Pyramid was

designed so that the area of each face was equal to the square of its height.

During the nineteenth century, as the Great Pyramid was measured and re-measured, a number of explorers,

Egyptologists, and early scientists suggested explanations concerning its shape and size. For example in 18 59

John Taylor, an English publisher and Egyptologist, introduced a theory that included π and square roots. In his model,

the perimeter was twice π times the height. He went on to claim that the Great Pyramid was intended to, “ make a

record of the measure of the Earth.”

In 18 63 French Architect Viollet-le-Duc introduced the use of the 3-4-5 triangle to explain the structure’s geometry.

The mathematical theory related to this triangle had been developed 2000 years earlier by the Greek philosopher

Pythagoras, and bears his name—the much celebrated Pythagorean Theorem.

Other theories claimed that the exterior slope angle could be formed with the vesica pisces; that the Great Pyra-

mid’s geometry is an accurate representation of the northern hemisphere; that it incorporates the Greek “ Golden

section,” etc. And while some of these ideas have fallen by the wayside, to this day, none solve the crucial

engineering challenge, which is how to build a massive four-sided building whose four sides meet at a predetermined

apex point. “ The great pyramid, being 146.6 m, (28 0 royal cubits) high, ” says Bryn, “ would need a finely tuned, highly

accurate, but simple, system to enable [its builders] to reach the apex point.”

As Bryn explains in his paper, “ Most theories on the construction of the Great Pyramid deal with the logistics of

moving roughly seven million tons of stone and the physical erection of the monument.” The natural starting point for

the pyramid’s architect, however, argues Bryn, would be development of a system which would make it possible to

communicate the required design and accuracy to presumably illiterate workers.

Building Grid

Building grids are crucial and indispensable tools in any modern project. They “ provide points of measurement that are

not part of the built structure,” Bryn explains, “ thereby introducing the tolerance needed to achieve the requisite

precision.” Today the use of building grids and tolerance is taken for granted when erecting large buildings, yet the

separation of the measuring system from the building’s constituent parts is not recognized as an invention in its own

right. F ive factors are essential for producing a practical building grid:

The grid must provide enough points of measure to be practical.

The numbers in the grid should be divisible, s imilar, and whole.

The grid should provide a practical way of labeling geographical positions in the building under construction.

A true building grid must have a physical structure ris ing with it; i.e., a core.

The grid for a true pyramid must be three-dimensional in order for the apex point to be reached.

A functional building grid would have to be developed from the units of measurement that the Egyptians used; we

know about a large number of units in their architectural designs in the third (2727 – 2655 BC) and fourth (2655 – 248 4

BC) dynasties. Bryn has used the basic set of units used in religious and royal structures of Egypt’s first dynasty:

The Royal cubit (Rc).

T he Roya l Cubit

In 18 64 Astronomer Royal of Scotland Charles Piazzi Smyth (18 19-1900) went to Giza and made accurate meas-

urements on the Great Pyramid. He then published the first work on the units of measurements used on the struc-

ture. His theory attempted to link “ the pyramid inch” to the Brit ish inch. He was inspired by, and collaborated with,

John Taylor to block the introduction of the metric system in Britain.

The ideas of Smyth and Taylor were ultimately discarded because, it was argued, Piazzi Smyth did not take into

consideration the well-known historical evidence that the ancient Egyptians used the royal cubit, and most scholars

felt it was highly unlikely that the great pyramid builders would have used other measures. The Egyptian Royal cubit

represented the forearm of the pharaoh and was divided into 7 palms. The latter was divided into 4 fingers or digits.

In the early 18 8 0s, when W.H. Flinders Petrie traveled to Giza, he was the first truly modern archaeologist to study

the Great Pyramid. His measurements on the Giza plateau in general, and on the Great Pyramid in particular, are so

accurate, that they remain, to this day, the main source of measurements used. Based on Petrie’s measure-

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ments, it is now established that the exact length of an Egyptian Royal cubit, as used for the Great Pyramid,

was 52.355 cm.

To measure angles, the Egyptians used the ‘rise and run’ method, where a seked is the horizontal distance run for

every 1 Rc rise (7 palms). The seked of the slope of Khufu’s pyramid is 5 palms, 2 f ingers run, for every one Rc

rise. Likewise, the seked of the slightly steeper Khafre pyramid is 5 palms, 1 finger. The seked was used to define

most pyramid slopes in the Old Kingdom.

Researchers have looked in vain for the tools used at the building site to implement this angle of slope. In his new

paper, Bryn argues that the seked was most probably used on the drawing board and not at the building site. This

specific rise and run had to be translated into other means of precision when transferred to a building site. “ The only

precise ‘tool’ available to the Egyptians on this scale,” says Bryn, “ was gravity. I t has always worked.” And in his

paper, Bryn shows how it was possible for the pyramids to reach the elusive apex point using nothing more than a

plumb line and a string.

Apex P oint Key

Bryn has studied the plans from the 30 oldest Egyptian pyramids, and uncovered a sys tem of precision that could

have made it possible for the Egyptians to reach the pyramid’s last and highest point, the apex, with an impressive

degree of accuracy. By exploring and making a plan of the pyramid, it is possible to prepare modern project documenta-

tion of not just one, but all, pyramids from any given period.

“ The building grid,” says Bryn, “ was probably necessary in order for the apex point of the pyramid to be established,

with the apex point itself being a true grid point.” The pyramidion is the last piece of masonry placed on site, but that

point must have been built into the project documentation during the entire time span of the monument’s erection.

One may even claim that the entire building’s purpose was to reach that s ingular point. By focusing on the grid rather

than the building itself, it is possible to retrace the architectural drawings showing the internal structure and thereby

explain the pyramid’s advanced geometry.

“ As long as the architect knows the main dimensions of a pyramid, he can project the building as he would have done

it with a modern building but with building methods and measurements known from ancient Egypt,” says Bryn.

T he Core

“ The need for a structure to rise with the grid is obvious,” he adds. When putting in place the curtain wall on a

skyscraper, the precision is already (hopefully) in place within the concrete and steel core.

A step pyramid is ideal to raise a grid. We know from the earliest third dynasty step pyramids in Egypt that they were

built with accreditation walls or layers. In those early pyramids the accreditation layers were tilted inwards at an angle.

The tilt ing was probably done to support temporary ramps from step to step and/or to provide structural stability.

Each part of the pyramid between two steps is referred to as a ‘mastaba.’ Each accreditation wall or layer

formed one step of the step pyramid and defined the base of the next mastaba by reducing the height of the wall

layer by layer.

To this day, the interior or core of Khufu’s pyramid is hidden from view by the preserved masonry. Most archaeol-

ogists, however, believe that the pyramid is comprised of horizontally laid masonry with the same height of the ma-

sonry visible on the surface today. The core of Khufu’s great pyramid, Bryn believes , is most probably a step pyramid.

This step pyramid was used to transfer the pyramid’s precision system, the building grid, vertically; and the blocks

seen today are backing blocks that do not necessarily align with those of the underlying mastaba. Bryn argues that

the accreditation walls forming the stepped core of a true pyramid are made with horizontally laid masonry. (One ex-

ception is the Meidum pyramid which started out as a step pyramid with tilted masonry.)

The Egyptians most probably used the base of each mastaba for measuring the face of the finished pyramid. “ It is

quite possible,” says Bryn, “ that it was the mastabas which were first carefully geographically aligned North-South,

and not the baseline we see today.” Note that the grid points used to measure out to the face of the pyramid would

have been placed at the base, and not the top, of each mastaba.

By erecting walls with horizontally laid masonry, they could introduce tolerance between the accreditation layers,

thereby transferring the building grid vertically from mastaba to mastaba with a plumb line. The outlines of each ac-

creditation wall were carefully measured out on top of each mastaba. Those lines and point could now be adjusted us-

ing the 3-4-5 triangle and the diagonal. Those lines represented the building grid.

T he ‘Dia m ond Ma trix’

Aerial photographs taken in the 1920s reveal that Khufu’s pyramid was not set on a square base. The center line on

each face is moved inwards by a short distance at the base, yet the deviation is so small that it cannot be detected

at ground level given the present-day, uneven surfaces. Most researchers seem to have overlooked or ignored this

fact, possibly because it is diff icult to explain the level of precision involved. Bryn confesses that he, also, has not

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found any serious explanation for this fact.

He does suggest, however, that, though the motivation behind the diamond matrix is unknown, it could have been

religious or aesthetic. It would undoubtedly have been easier to make a straight true pyramid, he points out. So,

Khufu’s pyramid’s diamond matrix, Bryn believes, testif ies clearly to the possible invention and use of a versatile

building grid, with a very high degree of f lexibility, without requiring π, the golden section, or any other form of advanced

mathematics.

A New Ma p

“ The Apex point of a True pyramid,” concludes Bryn, “ could not have been reached without the precision and tol-

erance embedded in a simple and redundant 3-dimensional grid with the precision system separated from the build-

ing’s constituent parts. Since this idea is taken for granted in the architectural profession, and among every brick layer

and mason, what is most surpris ing is that it has never been applied to the most widely discussed building in the

history of mankind.”

If the principles behind Bryn’s drawings are correct, then archaeologists will now have a new “ map” that demon-

strates that the pyramids are not s imply a “ bunch of heavy rocks with unknown structures” but, rather, are incredibly

precise structures.

Ole J. Bryn’s findings were presented and explained at the exhibit ion The Apex Point in Trondheim September of

2010.The exhibit ion was an official part of the program to celebrate the centenary (1910-2010) of the Norwegian Uni-

versity of Science and Technology.

The 45-year-old Bryn is a former practicing architect and currently holds a position as Associate Professor at the

Faculty of Architecture and Fine Art, at NTNU.

According to the NTNU press release, the development of Bryn’s theories on the building grids used in Egyptian

pyramids benefited from cooperation with Dr. Michel Barsoum, Grosvenor and Distinguished Professor at the

Department of Materials Science and Engineering, Drexel University, Philadelphia.

You can download an English language PDF of Bryn’s paper at: http://www.ntnu.edu/c/document_library/

get_file?uuid=d1d69d25-b3cc-4b17-9c8b-c16dbbd2d199&groupId=139701

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