
Re: RMP # 48
Posted:
Oct 2, 2007 8:54 AM


Lee,
One of the MMP's most noted problems does not include the idea of slicing a semisphere, as you have interjected. I had not mentioned the point (though others have). The most noted problem in the MMP reports slicing a truncated pyramid into exact pieces. Adding up the pieces finds the area, using Egyptian fractions, no more, no less.
The antiBlack Athena review of MMP 10 stressed a proposed link to Eudoxus and Archimedes without stating the 1/4th geometric (infinite) series Egyptian fraction details,
4A/3 = A + A/4 + A/16 + A/64 + ... + A/4n + ...
which Archimedes converted to a finite Egyptian fraction series,
4A/3 = A + A/4 + A/12
as one slice needed to exactly sum the area, much as the MMP did 1,500 years earlier.
Returning to MMP 10, Peet's views of Egyptian fraction arithmetic have been halfcorrect, hence misleading in certain respects. Peet threw out the baby out with the bath water when he discussed the correct geometrical cylinder view of the problem. However, read Peet's review of the Akhmim Wooden Tablet, and his slam against the 1906 paper written by Daressy.He did not understand the basis of the Egyptian fractions being used (remainder arithmetic, with Egyptian fractions being scaled to 1/320 hekat remainders), though he did agree that a ro meant 1/320 of a hekat. It took 96 years, until 2002, for the typographical errors introduced by Daressy to be corrected by Vymazalova, a Charles U. graduate student. Peet did the same thing (not grasping the nonadditive aspects of Egyptian fraction series) when discussing MMP 10. Peet did not understand the finite basis of Egyptian Egyptian fraction series, since he only saw the additive aspects of the EMLR, RMP and all the Egyptian fraction documents.
Beauty is in the eye of beholder  and Peet saw no nonadditive beauty in Egyptian fractions, as Ahmes and the Middle Kingdom scribes saw when they solved the HorusEye problem exactly writing
1 = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + ...
as an exact Egyptian fraction series,
by using any rational number as an exact remainder, such as, the remainder 1/64 as 5/320 into ro units, per
1 = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + (4 + 1)/320
1 = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + 4 ro + 1ro.
Best Regards,
Milo Gardner

