One of the facts we are sure about is that the Egyptians were adept at comparing right-angled triangles. For example, in the Berlin Papyrus there are problems where a given square is to be written as the sum of two squares in a given ratio. The solutions involve the fact that 6² +8² = 10², and 12² + 16² = 20², which is familiar to us from our knowledge of the 3,4,5 right triangle. We are also certain that the ancient Egyptians used this knowledge of the 3,4,5 triangle in their constructions. Evidence suggests that pieces of knotted rope, with knots spaced at intervals of 3, 4 and 5 units were used to readily apply this knowledge.

There are two squares joined together. Each of the squares has sides of `1'. The diagonal AB is therefore

Now imagine you are an ancient Egyptian with a piece of rope, and the ability to tie knots. Cut a length of rope of length AB + BC + CA, and tie knots at B and at C.

Positions A and X represents both ends of the piece of rope. Now suppose you want to utilize the proportions of Phi in some manner. All you would have to do is to take hold of the knot at point C with one hand, then pull end A with the other, until they lie together as below.

You have now constructed the Golden Proportion.

You have a length of rope CA lying above a smaller length of rope CX, where X denotes the other end of the rope. X lies at the golden cut of line CA. If you doubt this remember that the positive value of Phi equates to

Which is exactly what you have constructed geometrically.

So the next time someone tries to tell you that the ancient Egyptians could not have known about Phi, just point out to them that any child with a piece of string and two knots can do it, so why not an Egyptian!