Read This! The MAA Online book review column The Shape of the Great Pyramid by Roger Herz-Fischler Reviewed by Underwood Dudley

The rulers of ancient Egypt evidently wished their tombs to be as splendid as they imagined their reigns were. The earliest kings were buried in large structures (150' by 90' is big for a coffin) and as time went on they grew bigger still. Later kings added a second story, and then stories on top of that, so we have mastabas and stepped pyramids. A straight line is more appealing than a jagged one, so the step pyramids evolved into true pyramids, larger and larger, climaxing with the Great Pyramid, built sometime around 2600 BC. The Egyptians continued to build pyramids after that but eventually gave the practice up in favor of putting royal tombs underground.

The Great Pyramid and the two others occupying the Giza plateau (not to mention the Sphinx) are sights indeed and it is not surprising that they have excited comment from writers from Herodotus to the present. A question that naturally arises is, how was it built? How did that multitude of stone blocks get to where they are? Get there they did, and it clearly took a lot of work. The accepted opinion is that they were floated down the Nile, up a canal to near the work site, and then manhandled into place by one means or another. Some pyramid theorists argue that an earth ramp was constructed and the blocks dragged up it; the ramp was built higher and higher until, when the capstone was put into place, the entire pyramid was encased in dirt that was then carried away. It's also possible that some kind of hoists were used. One writer has maintained that the pyramid was built from the top down, difficult as that is to imagine. However it was constructed it must have been a grand sight when it was new, with its marble casing shining in the sun.

Another question that naturally arises is, how was it designed? The blocks were not piled up without plan. There was an architect admiring his work and accepting well-deserved congratulations at the dedication ceremony. How did he design the pyramid? That question is what the book is about.

Dr. Herz-Fischler quite properly pays no attention to almost all of the nonsense that the pyramid has inspired. Those who assert that the pyramid was built by marooned space aliens as a beacon are not mentioned, nor those who say that it was refugees from the sunken continent of Atlantis who were responsible. The theory that the pyramid was supernatural in origin is ignored. The pyramidologists, who find in measurements of the pyramid the history of the race, past and future, get only passing notice. The pyramid was built by people and designed by people, probably rather ordinary people. But why, exactly, does it have the shape that it does?

Even this question has inspired some nonsense. The common-sense explanation would be that the architect decided on what slope the sides should have and instructed the engineers to decrease the run by the appropriate quantity for every unit of rise. This, however, is only one of the many explanations of why the pyramid has the shape it does. Some writers assert that the pyramid exhibits the value of pi, sometimes to a surprising number of decimal places, or the golden number, the orbits of the planets (including Neptune, Uranus, and Pluto), or other information not known to the ancient Egyptians.

I think that common sense is right. The observed value of the slope is consistent with a rise of 14 over a run of 11. Those numbers may appear unusual, but the ancient Egyptian cubit was divided into seven palms, and each palm into four fingers, so the slope of the pyramid is 5 palms 2 fingers per cubit, not surprising numbers. The Rhind papyrus (c. 1650 BC) has in it some pyramid problems—given the height and side, find the slope and the like—that were based on pyramids that looked like the Great Pyramid, and their presence suggests that the slope of the side was the first thing that came to ancient Egyptian minds when pyramids were considered.

Dr. Herz-Fischler gives many other explanations for the shape of the pyramid. He made up none of them; he is only reporting what some writers, in some cases schools of writers, have said. Here are some of them. The pyramid was designed so that:

1. The distance up the line where two sides intersect (the arris) is the length of a side of the base. (This is way off, leading to an inclination of 54.7 degrees instead of the actual 51.8 degrees.)

2. The slant height is to the side as 5 is to 4.

3. (side)/(height) = 8/5.

4. The perimeter of the base is the circumference of a circle whose radius is the height. (So that the pyramid encodes the value of pi. But the ancient Egyptians didn't know the value of pi. The Rhind papyrus says that the area of a circle with diameter d is the square of 8d/9, which amounts to pi = (16/9)² = 3.16..., not very close.)

5. The angle of inclination is 360/7 degrees.

6. The ratio of the slant height to the base is the golden number phi = (1+Sqrt(5))/2.

7. The ratio of the side to the height is the golden number.

8. The square of the height is the area of a face.

9. (arris)/(half-diagonal of base) = 10/9.

10. (arris)/(height) = 3/2.

Given an object like the Great Pyramid from which many numbers can be taken, it is no surprise that there are combinations of them that can be expressed as ratios of small integers. From there it is but a step for a pyramid theorist to assume that whatever ratio he has fixed upon must have been intended by the pyramid's designers because such a thing could not happen by chance. Well, yes it could have happened by chance. Not all of the ratios in explanations 1-10 occurred by design. In fact, since they are mutually exclusive, at least nine of them are either false or coincidental. The golden number, pi, and those beguiling ratios have been projected onto the pyramid and were not designed into it.

Dr. Herz-Fischler presents all the theories in detail with thorough scholarship. He also gives some interesting details about their authors, quite a few of whom were nineteenth-century Englishmen. He does not, however, come to any conclusion about which is right. The closest he comes is on page 168, where he says

Builders decided that slopes in what we would refer to the 51° to 54° range were both aesthetically pleasing and feasible from a construction viewpoint. Individual pyramids may then have been designed with the builder deciding on the length of the base (presumably in whole cubits) and picking the inclination "by eye".

Unfortunately with the present state of our knowledge it is not possible to arrive at a definite conclusion as to the shape of the Great Pyramid.

It is traditional for reviewers to close by pointing out a few misprints. I couldn't find any! But the book is not perfect: the author made the unfortunate decision to write R(a) when he meant

Publication Data: The Shape of the Great Pyramid, by Roger Herz-Fischler. Wilfrid Laurier University Press, Waterloo, Ontario, 2000. Paperbound, xii + 293 pp., $29.95. ISBN 0-88920-324-5.

Woody Dudley (dudley@depauw.edu) teaches at DePauw University, Greencastle, Indiana. His Numerology (an MAA publication) contains a couple of chapters on pyramids.

Go to... The main Read This! page The index of all reviews The MAA Online front page

Find out... About the Mathematical Association of America Why you should be a member of the MAA How you can help support the activities of the MAA

Read This! is the MAA Online book review column. Contributions are welcome; contact the editor if you'd like to be one of our reviewers. Books for review should be sent to the editor: Fernando Gouvêa, Dept. of Math&CS, Colby College, Waterville, ME 04901. Publishers, please check our reviews information page.