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Topic: RMP # 48
Replies: 70   Last Post: Jul 23, 2010 8:57 AM

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 Franz Gnaedinger Posts: 330 Registered: 4/30/07
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 pseudo-octagon. Consider a
grid of 9 by 9 royal cubits, divived into 3+3+3 by 3+3+3
cubits. The pseudo-octagon 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 pseudo-octagon is 63 square
cubits. The pseudo-octagon 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

Date Subject Author
9/30/07 L. Cooper
10/1/07 Franz Gnaedinger
10/1/07 L. Cooper
10/1/07 Milo Gardner
10/1/07 L. Cooper
10/2/07 Milo Gardner
10/3/07 Milo Gardner
10/3/07 L. Cooper
10/3/07 Franz Gnaedinger
10/4/07 Milo Gardner
10/4/07 Milo Gardner
10/4/07 Milo Gardner
10/4/07 Milo Gardner
10/4/07 Milo Gardner
10/4/07 Milo Gardner
10/4/07 L. Cooper
10/5/07 L. Cooper
10/5/07 Milo Gardner
10/6/07 L. Cooper
10/6/07 Franz Gnaedinger
10/6/07 Milo Gardner
10/6/07 L. Cooper
10/6/07 Franz Gnaedinger
10/6/07 Milo Gardner
10/7/07 Franz Gnaedinger
10/7/07 Milo Gardner
10/7/07 Franz Gnaedinger
10/7/07 Milo Gardner
10/8/07 Franz Gnaedinger
10/8/07 Milo Gardner
10/8/07 Franz Gnaedinger
10/8/07 Milo Gardner
10/9/07 Franz Gnaedinger
10/9/07 L. Cooper
10/10/07 Franz Gnaedinger
10/11/07 Franz Gnaedinger
10/11/07 L. Cooper
10/12/07 Franz Gnaedinger
10/12/07 Franz Gnaedinger
10/12/07 L. Cooper
10/13/07 Franz Gnaedinger
10/13/07 L. Cooper
10/13/07 Franz Gnaedinger
10/15/07 Franz Gnaedinger
10/18/07 Ed Wall
10/19/07 Franz Gnaedinger
10/20/07 Milo Gardner
12/5/07 Milo Gardner
12/14/07 Franz Gnaedinger
12/14/07 Milo Gardner
12/14/07 Milo Gardner
10/11/07 Milo Gardner
10/12/07 L. Cooper
10/12/07 Milo Gardner
10/13/07 L. Cooper
7/11/10 Dioxippus
7/12/10 Milo Gardner
7/20/10 Dioxippus
7/21/10 Milo Gardner
7/21/10 Dioxippus
7/21/10 Milo Gardner
7/23/10 Dioxippus
7/23/10 Milo Gardner
10/6/07 Hossam Aboulfotouh
10/8/07 Milo Gardner
10/11/07 Hossam Aboulfotouh
10/12/07 Milo Gardner
10/13/07 Hossam Aboulfotouh
10/13/07 Hossam Aboulfotouh
10/19/07 Hossam Aboulfotouh
10/27/07 Matt Hugh