Below is a diagram of how the ancient 3, 4, 5 right-angle triangle was so useful to the ancient Egyptian civilization, and even underlies modern biological sciences and mathematics (See Figure V-5 below). You will note that two 3, 4, 5 triangles, with their 90-degree right angles opposed, placed together on their 5-unit side, form a rectangle and that this 5-unit side then forms the diagonal of this rectangle. This is one of the basic building blocks of ancient Egyptian architecture and art.
Inherent in the above triangle is the progression of the Greek "Golden Section" and the "Golden Rectangle," the modern Fibonacci Series and the Summation Series used in modern sciences. For the Golden Section: line CP divided by line PQ equals 1.618.
For the Fibonacci Series and the Summation Series to be found within the above diagram, BC divided by BQ (the dotted line) equals 2/1; B'C divided by AB' equals 2/3; AD divided by OB equals 5/3; and AB divided by BO equals 8/5 and this can be continued on to infinity. This reads, in arithmetic, as 2/1 , 2/3 , 3/5 , 5/8 , 8/13 ...(1+1=2; 2+1= 3; 3+2= 5; 5+3= 8; 8+5= 13 ...) also read as 1, 2, 3, 5, 8, 13 ... always adding the last two numbers for the next number. This is why these series are called progressive and progressions.
Since there were no blueprints when temple and tomb building began, it became necessary to establish a right-angle triangle and then lay out full-sized plans on the ground. This system evolved into a religious practice involving the Egyptian God-Pharaoh when a new temple was built, and then termed "cording the temple" and using a golden peg for the first marker (See Figure V-6 below).
First, to establish the north-south axis, the rope-stretchers marked the rise and setting of the constellation of the Great Bear on a false horizon. The rope-stretchers then marked the center of this arc and "pegged" the knot at B at this point on the ground.
They then stretched the rope in a line away from the constellation and pegged down the third knot, C, thus establishing the north-south direction. They then moved the knots A and D so that they created a 90-degree angle at knot B, thus forming a right-angle triangle and the east-west directional line by way of the rope at knots A and D. This is how they measured the Great Temple of Amun Re at Karnak and is also the base for the Great Pyramid at Giza (See Figure V-7 below).
W. A. Price and Howard Vyse both reported on the proportions of the Great Pyramid at Giza, Egypt (Also called the Cheops Pyramid). Its height at the center is 146.2 meters. Its width at the base side is 232.8 meters. By dividing the base side by 2 gives us 116.4 meters (OM above) and then dividing the height (SO above) of 148.2 meters by half of the width of the base we obtain the proportional ratio of 1.273 which is the square root of 1.618, a number which is central to the importance of dynamic symmetry. The formula is: OM/2 = SO/OM = 1.273 = root 1.618 .