The strange system the pyramid builders and/or the ancient Egyptians used for measuring has been bothering me for a while. On the one hand, we have the logical divisions of the cubit, into nice 15cm chunks and smaller, while on the other hand we have the royal cubit as π/6 which is a horrible irrational number.
So on the face of it it seems odd that there should be some relationship between a cubit at 0.4500m and a royal cubit at 0.5236+ m. The difference is 0.07356+, and trying ratios does not get us very far.
But my intuition was insisting that there must be some easy connection between the two. So I played around a bit, and noticed that cubit/royal cubit was around 0.8594, which rang a bell, as 0.5236 x phi is 0.8472.
So I played around a bit with paper and pencil and eventually figured it out … the method below is not what I used, but is an easier way to get the same result. We start with one of the formulas for the cubit, taken from parts of the cubit.
But if we go back to what was probably the original values of the cubit, then the maths is much prettier. For this, we take the cubit as around 0.44 m, and then can do this: