[ben_w]

Well spotted that they're simple powers; riffing on that idea, if that is the aesthetic behind it, I'd guess that they're supposed to be the same base or the same power, so either (4^4)/(3^4) or (16^2)/(9^2)?

Edit:

Or perhaps (((4^2)/(3^3))^2)^1 and then it feels like an ancient aesthetic-precursor to Euler's identity?

[mdp2021]

> the aesthetic behind it

It could also have been practical: "Radius to circumference? Easy: double many times, then take thirds a few times".

[eru]

Some more speculation. Have a look at https://en.wikipedia.org/wiki/Ancient_Egyptian_multiplicatio...

Egyption multiplication was (implicitly) based on powers of two. In some sense weirdly similar to modern bit-twiddling.

Note that 4/3 in binary is 1.0101 0101 0101 0101 0101 0101 0101 0101...

So that suggests the following algorithm, expressed in modern day Python:

    def mul4_3(x):
      table = []
      while x > 0:
        table.append(x)
        x //= 4
      return sum(table)

I deliberately used a 'table', because that's what a scribe would do.

Repeat this function four times, and you will have multiplied by 256/81.

I have no clue whether they would have done anything resembling this procedure; this is just to show that it's plausible given how their multiplication worked.

I don't know how this relates to 'Egyption fractions'.

P.S. Just for fun the same thing for 22/7:

    def f22_7(x):
      y = (x << 1) + x
      while x > 0:
        x >>= 3
        y += x
      return y

Not actually harder to execute by hand, I'd say, but perhaps harder to come up with?

[mdp2021]

> Note that 4/3 in binary is

This is consistent with the notion from Jarrosson, because that means that 4/3 can be represented as

  1 + 1/4 + 1/16 + 1/64 + 1/256 + 1/1024 ... 

which is a sum of fractions with unitary numerator - the representation ancient Egyptians are said to absolutely prefer.

The approximated ratio of the circumference to its diameter can be represented by four iterations ( (4/3)^4 ) of infinite series of sums of fractions with unitary numerator.

[eru]

> [...] the representation ancient Egyptians are said to absolutely prefer.

Well, they can also represent it as 1 + 1/3.. It's just that I assumed they have an easier time dividing by two than dividing by three.

If you leave it as a fraction, instead of dividing your integers, 1/3 is fine.

[ben_w]

Certainly feels plausible, that possibly makes a lot of sense.

[ben_w]

And now I spotted the mistake:

> (((4^2)/(3^3))^2)^1

should be (((4^2)/(3^2))^2)^1