Hi All,
The purpose of this rather lengthy post is to demonstrate the Ancient Egyptians concepts and functions will not translate using the concepts from our current system of mathematics. Using Greek concepts and the meter to research the concepts of the Ancient Egyptians only propagates the "I can't see anything there, so there isn't anything there to see." Logic, and the pyramids continue to defy comprehension because the Greek concepts and the meter we currently employ are incapable of translating the concepts, principles and elements of Ancient Egyptian mathematics. Restating that while the Greek concepts will get close, but they miss by just enough to obscure the rational concepts employed by the Ancient Egyptians.

Wiki definition: The mathematical concept of a function expresses the intuitive idea that one quantity (the argument of the function, also known as the input) completely determines another quantity (the value, or the output). A function assigns a unique value to each input of a specified type.

If the mathematics on this page look different from what you are use to seeing it is because they are different, I have tried to avoid using the AE unit fractions as they would only confuse the observer unfamiliar with AE mathematics, so there are fractions and improper fractions coupled with rational concepts in lieu of the familiar irrational concepts and decimal system.

Since The Great Pyramid (aka G1) has been a troublemaker having spawned so much diverse mathematical conjecture, is it possible the structure (G1) is designed with nothing more than geometric and mathematical concepts? Concepts such as the pi, phi, and square root 2 etc.... in the simplest rational form would relate to the many diverse mathematical theories and hypotheses. Logically, G1 lends itself so many diverse theories that in order to relate to and accommodate these theories it would have to be constructed using the basic form of geometrical and mathematical concepts. To my amazement the structure seems to be designed by the concepts and these concepts determine the intended dimensions and length of unit measures normally illustrated by and associated with the pyramids.


G1d Cult Pyramid:
G1d is a satellite pyramid found in 1993 by Z. Hawass and was interpreted as being a cult pyramid originally thought to have aided the alignment of G1. Studying the geometry and numbers of G1d a small inconspicuous pyramid on the south side of G1 produced a stunning find. A sequence of geometric factors found to be far beyond remarkable and far beyond coincidence. G1d is a small pyramid approximately 23 meters or 44 cubits, a one-tenth-scale model of G1 built with the same 14/11 rise-run or 5.5 seked, which provides an 11:7 ratio of the base to height of the Pyramids.

Concepts and Values:
Subsequent research revealed the Ancient builders executed in rational form a series of concepts to form the pyramids dimensions. The dimensions of G1d are set by the geometric values of the structure itself, the values of 5.5 seked produce a 14/11 rise-run ratio, diagonal of a square 99/70 rational square root of 2, pi in the form of 22/7 and 22/21 ((22/7) /3) or (3/(22/7)) along with 7 and 11 values as the ratio for the height and base, all concepts which necessary to formulate the structure:

1. 14/11 run-rise of G1, ratio of base to height and height to apothem. The 14/11, also the ratio between the area of a square and circle with equal perimeter and circumference.
2. 99/70 or 140/99 Diagonal of the base, rational value that approximates our value for the square root 2, the diagonal of the base (140/99 * 99/70 = 2)
3. 22/21 equal to (22/7 / 3) or 21/22 equal to (3 /22/7) appearing as ratios in and between structures and components: G1 440 cubits / 22/21 Red 420 cubits
i.e. (G1’s 5 + 1/2 seked / 22/21 = 5 + 1/4 seked G2)
4. 22/7 rational value for pi
5. 7:11 height to base ratio of any 5 + 1/2 seked pyramidal structure

These factors define and determine the exterior measures of G1d and when multiplied times ten provide the exterior dimensions of G1, further defining the intended length of the cubit for these particular structures.

G1d
What is demonstrated by the pyramids is; while the derived dimensions are given in cubits they are expressed in feet and inches: whoever the builders, assuming the AE, were cognizant of and used units of measure akin to the current definitions of the British Imperial foot, inch, explaining why the measures of structures translate so easily from cubits to feet and inches and feet and inches to cubits.

The formula is: Run-rise squared * diagonal of a square / ((22/7) /3) * (22/7)

((14/11)^2) * (99/70) / (22/21) * (22/7) = 378/55 = 6 48/55 feet equal to 4 cubits expressed.

Applying the height to base ratio, multiply by 7 for the height and 11 for the base provides the dimensions of G1d as shown below:

(14/11)^2 * (99/70) / (22/21) * (22/7) * 7 = 48 6/55 feet = 28 cubits = G1d height
(14/11)^2 * (99/70) / (22/21) * (22/7) * 11 = 75 3/5 feet = 44 cubits = G1d base

Using the four-cubit unit value provides the following the parameters for G1d:
1. Circumference of circle or perimeter of square = 44 units (176 units)
2. Radius of circle and/or pyramid height = 7 units (28 units)
3. Diameter of a circle = 14 units (56 units)
4. Base length = 11 units (side of square) (44 units)

The conceptual formula produces the dimensions of G1d in four cubit units, and when multiplied by 10 as shown below providing the same 44/7 ratio for G1 as demonstrated in G1d.

Dimensions of G1:
(14/11)^2 * (99/70) / (22/21) * (22/7) * 7 * 10 = 481 1/11 feet 280 cubits G1 height
(14/11)^2 * (99/70) / (22/21) * (22/7) * 11 * 10 = 756 feet = 440 cubits = G1 base

Further dimensions:
The Kings Chamber:
The same formula multiplied by 5 yields the length of the Kings Chamber:
(14/11)^2 * (99/70) / (22/21) * (22/7) * 5 = 34 4/11 feet the length of the King’s Chamber 34 /4/11 * 12 = 412 4/11 (412.3636363…) inches falling well within the parameters established by Petrie’s survey of the Kings Chamber.

<[www.ronaldbirdsall.com];

Showing 412.14 + 412.78 + 411.88 + 412.53 = 1649.33
1649.33 / 4 = 412.3325 inches as the Kings Chamber average length

Continuing: 34 4/11 divided by 3 = 126/11 feet or 11 5/11 feet = (14/11)^2 /(140/99) * 10 = diameter of a 36 unit circumference circle.
Or 34 4/11 * 22/21 = 36

34 4/11 * 3 = 103 1/11
(diameter ^2 / 14/11 = area of a circle)
11 5/11 ^2 / 14/11 = 103 1/11

Kings Chamber continued in inches:
Working in concepts provides for the ability to assign any unit of measure to any part of the conceptual formula, taking the same formula that gives the base of G1, Red and Bent Pyramids in feet and assigning the value of inches we find:
((14/11)^2 /(140/99) * 10) ^2 / (14/11) = 103 1/11 inches
Further:
((14/11)^2 /(140/99) * 10) ^2 * 22/7 = 412 4/11 inches Kings Chamber length or the same as the number of inches in the surface area of a 36 inch circumference sphere, calculated as:

1. (diameter ^2) * (22/7) = surface area of a sphere)
2. (circumference ^2) / (22/7) ) = surface area of a sphere)

((14/11)^2 /(140/99) * 10) ^2 * (22/7 ^2) = 1296 inches = 1/7 G1 base.
Further; 1296 inches equals the perimeter of the North or South wall of the Kings’ Chamber, the length and height of the when the height is measured from the sub floor i.e. 412 4/11 + 412 4/11 + 235 7/11 + 235 7/11 = 1296 inches 1296 / 20 34/55 = 62 6/7 (20 * 22/7) cubits. 1296 * 7 = 9072 base G1

G1 and Other Pyramids:

((14/11)^2 /(140/99) * 10) ^2 / (14/11) = 103 1/11 feet
(diameter ^2 / 14/11 = area of a circle)
Or calculated from the area of a square:
((14/11)/(140/99) * 10)^2 * 14/11
9 * 9 = 81 area of square
81 * 14/11 = 103 1/11 feet area of circle

1. 103 1/11 * 7 + 1/3 = 756 feet base = G1 440 cubits.

2. 103 1/11 * 7 = 721 7/11 feet base Red Pyramid in unit fractions 721 + 1/2+1/8+1/88 feet = 420 cubits.

3. 103 1/11 * 6 = 618 6/11 feet base Bent Pyramid = 360 cubits
in unit fractions 618 + 1/2 + 1/22 = 360 cubits.
Graham Chase’s 360 cubit length explained here: [www.hallofmaat.com]
Where he states in a Footnote:
The distance between the corner sockets, or where they would have been, was measured by Petrie as an average of 361.6 cubits, (while Dorner claimed it was 362). There was a pavement around the pyramid of which traces remain. Petrie measure it as .63 cubits at one corner above the socket and 1.65 cubits above the socket at another. The pyramid had obviously settled unevenly before the pavement was added. Taking the average pavement height above the socket of 1.14 cubits and with an angle of 54 degrees for the pyramid we get a horizontal distance covered by the pavement of 0.8 cubits. So the pyramid base is 361.6 – 0.8 – 0.8 = 360.0

Which I feel is correct; however, if you feel Dorner’s measures are more accurate see 3a.
3a. 103 1/11 * 6 + 1/30 = 621 54/55 feet base Bent Pyramid in unit fractions 621 + 1/2 + 1/4 + 1/5 + 1/44 + 1/110 feet equaling 362 cubits.

Defining the cubit
(14/11)^2 * 10 (22/7/(140/99)) = 36 circumference. Using the method of circumference squared divided by 22/7 to get the surface area of the sphere (360^2 / (22/7)) / 100 = 4536/11 or 412 4/11 inches surface area of a 36 inch circumference sphere making one Royal Egyptian cubit equal to 1/20 the surface area of a 36-inch, 3 foot, 1 yard or 110/63 cubit circumference sphere.

So the cubit by concept is:
14/11 * 99/70 / 22/21 = 189/110 feet, equal to 1/20 th the surface area (34 4/11 feet) of a 3-foot circumference sphere.

(412 4/11 / 20 = 20 34/55 inches) (20.6181818…. inches)
(34 4/11 / 20 = 189/110 feet) (1.71818181…. feet)

412 4/11 * 22 (11*2) = 9072
9072 / 12 = 756 feet, base G1
412 4/11 * 14 = 5773 1/11
5773 1/11 / 12 = 481 1/11 feet, height G1

20 * 14/11 * 99/70 / 22/21 = surface area of a 36 inch circumference sphere, KC length.
10 * 14/11 * 99/70 / 22/21 = surface area of a 36 inch circumference hemisphere, KC width.
This formula provides the intended length of the Royal Egyptian cubit.

Other Factors:
The Sphinx
Base of the Sphinx of 140 cubits divided by the standard cubit division of 7 and 28, 140/7 = 20 cubits = length of the Kings Chamber 412 4/11 inches and equal to the surface area of a 36 inch circumference sphere. 140/28 = 5 cubits = 103 1/11 inches equal to the area of a 36 inch circumference circle.

Red Pyramid
Applying this same rule of cubit divisions to the base of Red Pyramid of 721 7/11 divided by 7 equals a number equal to the area of a 36-foot circle, 103 1/11 feet. The same 103 1/11 multiplied by 28 equals 2886 6/11 feet, which is the perimeter of Red Pyramid, and in inches the number 2886 6/11 equals the base of the Sphinx.

*There is a 22:21 (22/7/3) ratio between G1 and Red Pyramid (440:420). King’s Chamber length times 22 equals G1 base, times 21 equals Dashur North. Find additionally that the 14/11 rise-run ratio like the seked of G1 multiplied by 22/21 figure equals 4/3 run-rise ratio of G2. These are just a couple of examples of the many times the (22:21) ratio is employed within and around the pyramids of Giza and Dashur.

Squaring the Circle, cubing the sphere:
9-inch square:
9 * 9 = 81 area of square
81 * 14/11 = 103 1/11 inches area of circle
103 1/11 * 4 = 412 4/11 square inches = surface area of a 36-inch circumference sphere.

4*22/7r^2 = 9 ^2 * 4 * 14/11 = 88/(7 r^2) = 4536/11 = 412 4/11 inches

9 * 9 = 81 area of square
81 * 6 = 486 square inches = surface area of a 9 inch cube
(412 4/11 / 486 = 28/33) is always the ratio between the surface areas of a sphere and cube, i.e. Sphere (14^2) * 22/7 = 616…616 / 28/33 = 726…726 / 6 = 121 (11^2) cube.

14/11 * 2/3 = 28/33 equals the ratio between the surface area of a sphere and cube.

In conclusion
The Builders of Pyramids used a four-cubit measure expressed in the equivalent of British imperial foot and inch system of measures in the design of the pyramids.

(14/11)^2 * 99/70 / 22/21 * 22/7 * 11 = 75.6 feet = 44 cubits = G1d base
(14/11)^2 * 99/70 / 22/21 * 22/7 * 7 = 48 6/55 feet or 28 cubits = G1d height

The above equations for the dimensions of G1d demonstrate the AE’s extraordinary mathematical aptitude using conceptual functions to determine the size of G1d and G1 in foot/cubit units of measure. The really obvious and beyond remarkable thing is; it seems the builders of the pyramids were acquainted with what is now called the British Imperial system of measures including the inch, foot, yard, acre and statute mile. This clearly demonstrates the cubit is provided within the concepts of G1d in units of measure expressed by the foot and inch found within a 36-inch circumference sphere. Ancient Egyptian concepts demonstrating G1 also illustrate their perception of rules governing the entire universe.
The above information states these pyramids were designed by concepts of the structure, stated in foot and inch units of measure via the cubit produced by the geometric concepts of Meidum, G1 and G1d Pyramids. The results include many other unspoken implications generated by the concepts of the Ancient Egyptian structures.