# The Pyramids

## Introduction

The pyramids in Egypt developed in several stages. The first pyramid dates back to the third dynasty. King Djoser had a step pyramid constructed at Saqqara. Djoser's chief architect Imhotep is credited with developing the step pyramid from the older mastaba tomb. The mastaba is thought by some to represent a bed. Mastabas are large rectangular structures that form the monument for the deceased. The dead were buried in shafts that were sunk into the ground. Imhotep stacked successively smaller rectangular shaped structures to create what is now known as the step pyramid.

Miroslav Verner ^{[1]}. writes: "At the outset the structure had the form of a square mastaba (stage M1), which was gradually enlarged, first equally on all four sides (stage M2) and then only o the east side (stage M2). The mastaba <...> already had a step shape in the M3 stage. The step-shaped mastaba was finally built in two stages, first as a four step (P1) and then as a six-step (P2) pyramid, which no longer had - and this is a noteworthy point - a square base; it now had a rectangular base, oriented east-west."

Approximately fifty years after the reign of Djoser, Egypt was ruled by Pharaoh Sneferu. During the reign of this king the first true (smooth faced) pyramids were developed. The Egyptians seem to have used some trial and error methods to develop the appropriate angles of the faces. There are two rather interesting pyramids dating to the time of Sneferu that give us some insight into the developmental stages of the pyramid construction.

A pyramid in Meidum collapsed at some point. It has been suggested that it collapsed while under construction. But recent excavations seem to show that the structure slowly lost its cohesion.

In nearby Dashur Snefru's bent pyramid was started at a rather steep angle of incline measuring 60 degrees, but several adjustments had to be made. The subsurface at Dashur was likely to soft to support the downward forces of a pyramid built at a steep inclination. Originally the angle of inclination was adjusted down from 60 degrees to not quite 55. This required the Egyptian builders to enlarge the base. But halfway through the construction the angle was again changed. The bottom part of the structure was left as it was, but the rest of the pyramid had an angle of only 45 degrees. The latter adjustment gives the pyramid it's peculiar bent shape.^{[2]}

## Pyramids, the Pythagorean Theorem and Special Triangles

One theory about the construction of the pyramids involves the use of special right triangles. The Egyptians may have used (wooden?)right triangles during the construction process to help determine the slope of the pyramid.

The Pythagorean Theorem states that given a right triangle with sides of length a and b respectively and a hypothenuse of length c, the lengths satisfy the equation <math>a^2 + b^2 = c^2</math>. There are some special cases where the lengths are integers. One very famous example is the 3-4-5 triangle.

For the triangle depicted here for instance the seqed would be 3/4 = .75 The angle made between the base and the hypothenuse is <math>53.13^o</math>. This is extremely close to the angle of incline of the second pyramid at Giza - the pyramid of King Khafre - which has an angle of incline measuring <math>53.1^o</math>.

## Pyramids and Mathematical Papyri

Problems 56 - 59 of the Rhind Mathematical Papyrus compute the seqed of a triangle. The seqed is related to the slope of the side of the pyramid. If we look at a triangular cross section, then the seqed is defined to be (1/2 base of the pyramid) / height.