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This problem is giving me headaches for quite some time now I guess mainly because I am not a mathematician, but I would very much like to know if there is anyone here that has any deeper insight into this.

So, the Papyrus translation is given here. Additionally there is a discussion presented here on whether the translation of the object in question is correct. They say it could possibly refer to the area of a semi-cylinder.

That said, I am wondering if anyone knows how could these ancient Egyptians have derived this algorithm? There is a fraction omnipresent throughout this algorithm (1/9). What could that fraction represent? Is there any way we could explain this algorithm intuitively? I guess we can 'assume' that they didn't know calculus, so how?

P.S. I have asked this question already on Stack Exchange, but without satisfying results.

EDIT

So to be more specific. The algorithm obviously works ignoring the small error compared to our current formula. They were very practical and didn't have sophisticated mathematical knowledge. So this is why I essentially want that whoever answers this forgets about our modern mathematics and thinks exclusively in the context of this algorithm and its operations to somehow make sense of this.

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    $\begingroup$ Again, as I asked on your question on MathOverflow: what was unsatisfactory about the answers you received on StackExchange? $\endgroup$ Aug 9 at 14:18
  • $\begingroup$ @Carl-FredrikNybergBrodda Both the answers given were just explaining how the calculation compares to our modern form. I wish to know if there exists some sort of logic behind this. Like practically how could they have possibly arrived at this algorithm in the first place? It's in my question already basically, what does $1/9$ represent? Why take $1/9$ of remainders? What do all the little intermediary steps in this algorithm even represent in geometrical terms? $\endgroup$ Aug 9 at 15:48
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    $\begingroup$ A similar algorithm is used to compute a circular area in the Rhind papyrus. The $256/81$ is used as the approximation for $π$ there too, hence the division by $9$. A common speculation, based on a drawing in the papyrus, is that they got it by approximating a circle by a regular octagon, see e.g. Numberwarrior. The rest is their usual procedure for calculations with Egyptian fractions. $\endgroup$
    – Conifold
    Aug 9 at 19:50
  • $\begingroup$ @Conifold I saw that before, but that concerns a circle and not a spherical 3D object. How did they arrive from this circle algorithm to the hemisphere algorithm? $\endgroup$ Aug 10 at 13:01
  • $\begingroup$ @Conifold if you understand how that translates to a hemisphere algorithm then I would very much like to know how. $\endgroup$ Aug 10 at 13:29

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