
Re: RMP # 48
Posted:
Oct 3, 2007 11:51 PM


RMP 48 is not only concerned with the square and circle but also with the inscribed pseudooctagon. Consider a grid of 9 by 9 royal cubits, divived into 3+3+3 by 3+3+3 cubits. The pseudooctagon in this grid has four sides of length 3 rc each, and four longer sides (diagonals of the small squares in the corners of the grid). As you can easily find, the area of the pseudooctagon is 63 square cubits. The pseudooctagon is close to the circle we can inscribe into the square. A close number to 63 is 64, so let us assume that the area of the circle measures 64 square cubits, or 8 by 8 cubits. The diameter of the circle measures 9 royal cubits, the area about 8 x 8 royal cubits, and from this you get the famous rule of the Rhind Mathematical Papyrus relating square and circle: a circle of diameter 9 and a square of side 8 have practically the same area.
Now for the algebraic aspect of this practical formula.
The number of the circle is smaller than 4 but a little more than 3. Begin with 4/1 and add repeatedly 3/1 in the way that was forbidden in school:
4/1 (plus 3/1) 7/2 10/3 13/4 16/5 19/6 22/7 25/8 28/9
Pi is more than 3 but slightly smaller than 22/7. Begin another sequence with 3/1 and add repeatedly 22/7, again in the forbidden way:
3/1 (plus 22/7) 25/8 47/15 ... 333/106 355/113 377/120
Pi is more than 9/3 but smaller than 19/6 :
9/3 (plus 19/6) 28/7 ... 256/81
256/81 is the implicit value of pi in the above formula.
Regards, Franz Gnaedinger

