While playing around with the calculator, looking at relationships between the numbers at Giza, I took another look at the diagonal of the great pyramid.
The great pyramid has a base of 440 ₢. This means the diagonal is 440√2, which is 622.25 ₢. This is about 4 ₢ more than 1000/φ, which is 618 ₢. The error percentage is about 0.6%, which is annoyingly close.
So I got to wondering what base size would produce a diagonal more or less exactly 1000/φ.
The answer turns out to be 437, which is 3 ₢ less the existing size.
437√2 = 618.011 which is as close as you are going to get using whole-cubit dimensions.
But the oddness does not stop there.
437² is 190969, a pattern which I immediately recognised due to familiarity with ₢ of ≈0.5236, and more importantly in this case, the inverse, which is 1.909859317…
So if we divide 100000/437², we get 0.523645199 which is a good approximation of ₢.
So now one simple number, 437, links directly to not only the golden ratio φ, but also the royal cubit ₢.
If you rearrange these identities you eventually end up with the familiar approximation
₢ ≈ φ²/5
Neither Wikipedia’s list of integers nor the Notable Properties of Specific Numbers note anything specific about 437. But there clearly is.
I can’t find 437 popping up anywhere on Giza (yet).
We can also run things in the other direction…. 437φ = 707.0808531 ≈ 1000/√2.
In similar vein, though not as good, we can repeat that with the plastic ratio:
437ρ = 578.9017473 ≈ 1000/√3
(1000/437ρ)² comes to 2.98394 rather than 3, so not as close as I would like.