The north and south walls of the King's Chamber are made up of five courses of 2 royal cubits and 2 palms, so the full height to the footing of the floor is precisely 11 royal cubits and 3 palms

The length of these walls is 20 royal cubits, and can be envisaged as the diameter of a circle, as proposed by Smyth, and confirmed by Petrie.

The perimeter of these walls, to the footing of the first course is:

20 royal cubits + 20 royal cubits + 11 royal cubits & 3 palms + 11 royal cubits & 3 palms

Perimeter = 62 royal cubits and 6 palms = 62 6/7 royal cubits = 1,760 digits

In this model the length of the chamber is the diameter of a circle, and the width of the chamber corresponds to the radius of that circle.

The scale model is one digit to one royal cubit, so the radius of 10 royal cubits (280 digits) becomes 280 royal cubits as the height of the pyramid, and the perimeter of 1,760 digits becomes 1,760 royal cubits as the perimeter of the base square (4 x 440 royal cubits).

The division of the royal cubit into 28 digits allows a simple expression of the pi shape, but may have been fortuitous:
The cubit of 7 palms, with four digits (width of finger) per palm allowed very rapid measurement along a wall or on a bench, as quick as measuring in English feet, because the natural cubit of the forearm is close to 6 palms, and one can use say the left forearm and the right palm to measure off lengths. Try it!

The square of the cotangent to square the circle is exactly equal to 360/28pi when evaluated for the pi approximation 22/7. The division of the royal cubit into 28 digits can be viewed as an innovation to represent pi using the pi approximation 22/7 (which may have been regarded as exact).

The triangle latent in design of the Great Step of the Grand Gallery has a hypotenuse of 112 digits. This triangle can be inverted so that 112 digits becomes the base. A vertical line drawn down from the right angle splits 112 digits into 90 digits and 22 digits.

The square of the cotangent to square the circle also equals 90/22

Both Professor Smyth and Professor Petrie were certain about the pi shape of the pyramid from the pi design of the King's Chamber.

Smyth also argued that precise slope of the the floor of the Grand Gallery 'squared the circle'. Petrie accepted that the measured slope was very close to the theoretical slope, but saw this as a coincidence.

When I saw Smyth's diagram I knew that a sloping floor with a ceiling parallel to the floor can be expressed, symbolically, in the ratio of the perpendicular to vertical height.

I calculated a perpendicular height of 15 royal cubits and a vertical height equal to the square root of 280 in royal cubits, which are indeed the actual heights near the north end wall.

It is not necessary to actually evaluate the square root, because the triangle can be drawn, and a template made. The slope can be constructed from the following right angle triangle:

Base 15, short vertical side sr55 (which can be measured from a triangle with a base of 3 and a hypotenuse of 8). The hypotenuse is then equal to the square root of 280.

I am sure that Petrie would have accepted my proof, as set out in full in my monograph on the Grand Gallery (2006)

The Queen's Chamber is a complementary design, as set out at www.ancientcalendar.co.uk

This website also includes details of the pi geometry of the King's Chamber and Grand Gallery.

All three chambers have a pi design based on the division of the royal cubit into 28 digits, which is the natural biological division of this unit of length.

Mark