Hello all,

It occurs to me that if I wanted to convey through geometry that I knew pi (be it as the irrational number 3.1415926xxx or the approximation 22/7), then the least likely thing I would do is to draw a circle and or construct a circular building and leave it at that.

A circle is one of the easiest geometric figures to draw – given that the person drawing it is equipped with a drafting compass or two pegs and a piece of string.
However, I could go through life drawing circles and building circular structures here there and everywhere and at the same time be blissfully unaware of the existence of pi.

If, for whatever reason, I wanted to let somebody know I knew pi, then I would simply write it down somewhere.
But what if, for whatever reason, I was not allowed to record this knowledge in any form of writing?
What would I do then?
How could I convey my knowledge to somebody else (whoever that somebody may be) without breaking the rules?
Well, I think I would do what I now suggest Khufu’s architect did and that is to work the pi ratio into rectangles.*

The classic and most self-evident example of this method at work is, I suggest, the dimensions of the side (north and south) walls of the King’s Chamber.
Here we have rectangular walls with a perimeter (twice the length plus twice the height) equal to the length of the walls multiplied by pi (again leaving aside whether it is the irrational number 3.1415926... or the approximation 22/7).
Hence, perimeter of wall is to length of wall as circumference is to diameter.

I further suggest that Khufu’s architect then developed his idea further and displayed his knowledge of pi also as: linear dimension A is to linear dimension B as circumference is to diameter or diameter is to circumference (and variations thereof in order to take in half circle and radius).
For example:
The width (east and west) of the Antechamber above the wainscots is 3.143 royal cubits and the width of the Antechamber between the faces of the wainscots is 2 royal cubits (please note I am using our decimal notation purely for convenience).
I suggest that the architect initially saw this as:
The width of the Antechamber above the wainscots is to the width of the Antechamber between the faces of the wainscots as half the circumference of a circle is to the diameter of that circle.
Thus: diameter @ 2 royal cubits multiplied by 3.143 = circumference 6.286 royal cubits.
Half the circumference = 3.143 royal cubits.

For a second example (and staying with the Antechamber):
The height of the Antechamber above the top of the west wainscot is 1.818 royal cubits and the height of the Antechamber from the level of the base of the King’s Chamber walls to the top of the west wainscot is 5.714 royal cubits.
I suggest that the architect initially saw this as:
The height of the Antechamber above the top of the west wainscot is to the height of the Antechamber from the level of the base of the King’s Chamber walls to the top of the west wainscot as the diameter of a circle is to the circumference of that circle.
Thus: circumference @ 5.714 royal cubits divided by 3.143 = diameter 1.818 royal cubits.

I suggest that this technique came about through the architect realising that pi (as the equivalent 3 1/7 or 22/7) is inherent in the seked 5½ (1 royal cubit rise to 5½ palms run) and then conceiving the idea of recording his knowledge of pi as 3/ 1/7 or 22/7 in the form of squares, rectangles and linear measurements not only in the superstructure of his king’s pyramid, but also in its passages and chambers.

MJ

Despite a lot of folks thinking otherwise, this is not the same as ‘Squaring the Circle’,