Roger Herz-Fischler, The Shape of the Great Pyramid. Waterloo, Ontario: Wilfrid Laurier University Press, 2000. To order this book from Amazon.com, click here.

Reviewed by Mark A. Reynolds

Just when we think there could be nothing more written to help us with our studies of the Great Pyramid at Giza, a new book arrives. Some are worthy efforts, as Mark Lehner has shown with his masterful The Complete Pyramids [1997], while others are exercises in futility and folly. It is a risky enterprise to attempt further work on the subject, for the writer may find that anything less than a life's work devoted to the subject can raise more questions and riddles than it will resolve. Fortunately, this is not the case with Roger Herz-Fischler's new book, The Shape of the Great Pyramid. Herz-Fischler is perhaps best known for his comprehensive chronicling of the historical and mathematical aspects of the golden section in A Mathematical History of Division in Extreme and Mean Ratio [1987; reprinted 1998]. In The Shape of the Great Pyramid, Herz-Fischler removes much of the mathematical complexity and controversy associated with the golden section in favor of a discussion of additional mathematical possibilities that might be present in the structure. The book centers on one of the key questions concerning the Great Pyramid: is there a mathematical plan that was utilized in the construction of the building, or was the structure built by sheer force of will and muscle, with no working plan as a guide?

As a researcher in theoretical probability, Herz-Fischler opts for a straightforward compendium of the various historical, philosophical and sociological theories on the geometry and mathematical relationships believed to be contained in the shape of the Khufu Pyramid. He provides us with a list of those theories that have been presented over the centuries, from Herodotus to W. F. Petrie and Piazzi Smyth. The book offers an overview and detailed summary of eleven major hypotheses regarding the mathematical systems that could have possibly been used for the geometric structure of the Great Pyramid. The introductory chapter presents the problems at issue regarding the Pyramid's shape: 1) was there a design plan prior to the building of the pyramid?; 2) what system of numbers and geometry were selected for that plan?; 3) was that plan carried out, and carried out faithfully or changed by the builders?

The body of the book includes eleven chapters that present eleven different theories on the pyramid's measures as developed by scholars throughout history, all of them feasible due to the present condition of the pyramid. Of particular interest is the juxtaposition of theories involving simple whole number relationships with those containing the irrational golden section systems (which the author has chosen to label by the letter "G", a problem for P. D. Ouspensky, no doubt!). In a fascinating and curious way, the simple whole number theories and the more complex numbers are exceedingly close to one another when ratios, angles, and lengths are calculated, and Herz-Fischler presents a variety of charts on these relationships for the observed and measured quantities found within each theory. Fortunately for the reader, decimal places are only taken out to the thousandths place.

The book is completed with chapters of summary and conclusion, covering the philosophical considerations for why one theory may supersede or outlive another, and a case study on the sociology of these theories with the example of the "pi theory". This discussion centers on the popularity of the issue of quadrature between the base and height of the pyramid in 19th-century Europe. Several helpful appendices and a good range of footnotes, along with a good bibliography, conclude the work. Unfortunately, there is no index, but one may not be essential given the format of chapter headings and layout of subject matter.

There is a reasonable amount of mathematical content for the interested mathematician and geometer, but nothing approaching the calculus or set theory, and most of it should be easily digestible for most readers interested in pyramid studies. The math presented is logically based on the math that was available to the original builders or on the mathematics employed in a particular theory; especially noted is the seked theory in the Rhind Papyrus as well as natural fractions for the slopes of the rise and run of triangular shapes. Herz-Fischler does not go beyond arithmetic, geometry, and trigonometry, so most academicians should be able to follow the math presented.

There are always a few problems with any text. In this one, unfortunately, the publisher did not provide many illustrations in the text, and the reader is required to go to their website (http.//www.wlu.ca/~wwwpress/) to view the documents, photographs and drawings referred to in the book. If one has no computer, one forfeits the opportunity to see them. (This trend in books to include computer images may work out in the future, but there are critical factors that need to be considered before this medium becomes the only source for seeing the visual aspects of a literary work. Until a better time comes, perhaps it is best to have both hard copy and computer images available to the reader.) Fortunately, there are only 15 such illustrations, and the reader isn't missing much. The photographs and drawings provided do far less than the line art constructions in the text to provide us with any necessary, additional understanding to the thoughts presented in the book. They are all black and white images, so printing costs could not have been near those presented with four color work. The selection of illustrations is interesting for those who study pyramidology and history, but they will be relatively meaningless for those with a less passionate interest in the subject. Some of the terms used by the author may be initially confusing. In addition to "G" for the golden section, some readers may not know that the "Kepler Triangle" is the "Triangle of Price", which is the triangle that is the right, scalene triangle whose sides are in the geometric progression, 1, Öf, f. Also, I do wish that the author would have footnoted the obvious errors made by other authors. Especially so is the quoted paragraph by R. Ballard (p. 243) regarding the "Star Cheops". Referring to the folded down plan of the Great Pyramid, Ballard states, "… Thus do we close the geometric flower Pentalpha, and close it into a pyramid." Food for fodder for certain of the mathematical community for sure, and juicy prime for some of the sacred geometry crowd, no doubt, and who, should note, that this was written in 1882. The "Triangle of Pentalpha" is one of the isosceles triangles — the 72°/36°/72°— in the pentagram, so called for its primary importance in its ability to generate the pentagon in only three turns, not five. Although it is fairly commonly held that the Great Pyramid contains both the golden section and its square root (both which develop during the course of generating the pentagonal system from the golden section rectangle), the splayed plan bears no resemblance, congruency, or content to the Triangle of Pentalpha. The triangular faces of the pyramid are very close approximations to the diagonals of two golden section rectangles, which are tangent to each other's long sides, and the base of which is the base of each of that rectangle's short sides added together. This is not to say that Ballard is completely wrong, because a different kind of pyramid could be generated having a square base and four faces that are Triangles of Pentalpha. The result would be vaguely similar to the Transamerica building in San Francisco, and not even remotely similar to the Great Pyramid, the subject of Ballard's comment and the line drawing shown in fig. 70 in Ballard's book. These items, however, should in no way discourage someone interested in the subject of Great Pyramid geometry from reading the book, for it has much to offer under one cover.

Herz-Fischler is perhaps at his best in the book when he summarizes the various hypotheses presented, and in the conclusions he provides for the reader. There are no surprises in his conclusions, but this is not the point of the work. Rather, the volume provides an academic source for the major systems of measure connected to the geometry of the Great Pyramid, and a ready reference for the numbers and formulae associated with that geometry. He incorporates social and historical factors into his conclusions, and, as I prefer the growing trend towards an interdisciplinary approach to scholarship, I believe that Mr. Herz-Fischler has provided us with a path in that direction.

Although the book is in paperback, yet the stock is higher quality, the binding is good, the price is fairly reasonable, and the font and point size are easy on the eye. The publisher has provided a good colophon on the last page for graphic designers interested in the specific Adobe specifications regarding production. There is an 18-page bibliography for those wishing to further investigate the rich and exhaustive work done on this last Ancient Wonder of the World. The Shape of the Great Pyramid is a welcome addition to that bibliography for an overview of all the geometric theories generated by this object of awe.

REFERENCES
Herz-Fischler, Roger. 1987. A Mathematical History of Division in Extreme and Mean Ratio. Waterloo, Ontario: Wilfrid Laurier University Press. Reprinted as A Mathematical History of the Golden Number (New York: Dover, 1998). To order this book from Amazon.com, click here.

Lehner, Mark. 1997. The Complete Pyramids. London: Thames and Hudson. To order this book from Amazon.com, click here.

ABOUT THE REVIEWER
Mark A. Reynolds is the editor of the Geometer's Angle for the NNJ. He is the author of "A Comparative Geometric Analysis of the Heights and Bases of the Great Pyramid of Khufu and the Pyramid of the Sun at Teotihuacan" published in the NNJ vol. 1, no. 4 (October 1999).

Copyright ©2001 Kim Williams