[mdp2021]

According to Bruno Jarrosson, the Egyptians had a marked preference for some classes of fractions, e.g. those with unitary numerator (so that, for example, representing a quantity as a composition (e.g. sum) of more fractions of the form 1/n was preferred to using single fractions of the form m/n). So, that fraction may have been privileged over others in this framework.

(One immediately notes that 256 and 81 are simple powers. As just a possibility, it could have been found intriguing that (2^8)/(3^4) could "reveal some underlying structure".)

[082349872349872]

Note that fractions of the form 1/n + 1/m + ... make equity obvious for non-arithmetically sophisticated populations.

Given 3 loaves of bread, and 4 workers eating lunch, we'd probably be fine if 3 of them got 3/4 loaf each and the last got the 3 1/4 slices. To make the fairness of the distribution obvious, the Egyptians might have given all 4 workers the same pair of slices: 1/2 + 1/4.

(Note that for less fungible items, there still might be some practicality in the ancient Egyptian system: if we have 3 5 meter ladders to divide between 4 people, giving everyone one 2,5 meter ladder and one 1,25m would be much fairer than giving 3 people a 3,75m ladder but the 4th three 1,25m ladders.)

[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

[eru]

Yes, I was thinking about the simple powers, too.

And I don't know whether they had enough math (or cared enough to do all the hard work with the math they had available to them), to figure out what's the simplest good approximation for the real value of Pi.