Hi Kanga,

You wrote: We also are unable to express irrational square roots either. Let's look at sqrt 2. How would we express it as a single fraction or ratio? We can't. We can only speak of rational approximations, and there are an infinite number of them.

I think this is where Ocham's razor might apply: when presented with competing hypothetical answers to a problem, one should select the answer that makes the fewest assumptions.

You wrote: In a discussion on sqrt 2, Legon pointed out that the Bent enclosure walls and the bases of the Red Pyramid and Userkaf's pyramid are simple multiples of 140 cubits long, and that their diagonals are multiples of 198 cubits. This points potentially to a sqrt 2 value of 99/70. Legon believed that the origin of this rational expression was the 10c x 10c square, whose diagonal is almost exactly 99 palms, giving rise to 99/70, as 1c = 70p. However, Legon wouldn't commit to the pyramid builders knowing sqrt 2! It was sufficient, he said, for them to know the diagonal-to-side ratio was 99:70.... I don't think Legon was being forthright enough.

There are others who believe the Egyptians knew sqrt 2. Take for example Robins and Shute. They wrote an article called "Irrational Numbers and Pyramids" (DE 18, 1990) in which they give two possible values for sqrt 2 of 10/7 and 17/12. The first, based on the "double remen," has a unit fraction form of 1 + 1/4 + 1/7 + 1/28, while second has a unit fraction form of 1 + 1/3 + 1/12. When squared, these produce 2 + 1/25 + 1/1225 and 2 + 1/144 respectively. Thus they prefer 17/12 over 10/7. But Robins and Shute never referred these values to the pyramids.

On the other hand, Legon's value of 99/70 - directly connected to the pyramids - when squared, produces 2 + 1/4900, a far superior result to 2 + 1/144, the square of 17/12.

The value 99/70 has the unit fraction form of 1 + 1/3 + 1/15 + 1/70. So as far as square roots are concerned, the number of unit fractions would not have been a deterrent to the Egyptians.

99/70 is not a bad figure, but only half of the equation from my perspective. There is a third option based on 10/9 the rise run of G1, which totals 140/99 = 1 + 1/3 + 1/18 + 1/66 + 1/99 derived as (1 + 1/4 + 1/44) x (1 + 1/9) which is G1.s rise run 14/11 x 10/9 = 140/99.
2 / 99/70 = 140/99, therefore 99/70 x 140/99 = 2.
i.e. 140/99 = 1.414141414…minus the sqrt 2 = 1.41213562 = 0.000072148.

The fact the Ancient Egyptians could not process irrational numbers is not proof they were not cognizant of these concepts in rational form. Nor of their importance in design calculations and constructions. The reality is there are fractional equivalents within 4 to 5 decimal places of accuracy for values listed above easily substituted for our current values. Let's even not mention the elephant in the room, to the Ancient Egyptians using unit fractions our irrational values simply did not exist.

Most who throw out these numbers don’t think about obtaining, by mathematical process, the square root 2, since it is only necessary is to press a button on a calculator. It is a process which seems, to me, to be more labor intensive, using our current guess & check method used for finding square roots centuries prior to the more advanced algorithm and the invention of the slide rule and calculator.

Regarding the Bent Pyramid gradients. Everyone is entitled to their opinion, but what would be the purpose for the Ancient Egyptians to choose to make the lower slope of Bent pyramid relative to the square root of 2 given they were unable to factor the square root of 2?

I have to maintain a more traditional approach and say the sqrt 2 does not apply. Since the lower and upper seked are related by a factor of 2/3. The lower seked of 4 + 1/2 + 1/3 + 1/15 palms run 7 and a rise 10, A seked of (4 9/10), 4 + 1/2 + 1/3 + 1/15 / 2/3 = 7 + 1/3 + 1/60, (i.e. 4 9/10 / 2/3 = 7 7/20) which when divided by the rise of 7 equals a run of 21 to a rise of 20 which like G2’s 3:4:5 forms another of the Pythagorean triple triangle with the values 20:21:29 for the upper section of Bent pyramid.

Bent Pyramid lower gradient Seked of 4 + 1/2 + 1/3 + 1/15
cotangent 7/10 = 34º 59’ 31.29”
tangent 10/7 = 55º 0’ 27.73”

divided by 2/3

Bent Pyramid upper gradient a Seked of 7 + 1/3 + 1/60 = 1 + 1/20 cubits
cotangent 21/20 = 46º 23’ 49.85”
tangent 20/21 = 43º 36’ 10.15”

This eliminates the averaging, mental and mathematical gymnastics required to fit Bent Pyramid into the square root 2.

The same 7 + 1/3 + 1/60 Seked, 1 + 1/20 cubits creating the 20:21:29 triangle is echoed in the gradient of the Red pyramid. Run rise 21 x 20 = 420 as in cubits base of Red pyramid. A 20:21:29 triangle has a perimeter of 70 and an area of 210.

If one were to ask why 20:21:29 if you can answer why they might also answer why G2 has a 3:4:5 is also a Pythagorean triple.

140/99 is the sum of multiplying the seked of G1's rise run by 1 + 1/9 or 1 + 1/9 divided by 11/14 (1/2 + 1/4 + 1/28) the run rise of the 5 1/2 seked. Giving the value of 140/99 or a decimal value of 1.4141414… a difference of 0.000072148 from the sqrt 2. Which like 22/7 is accurate enough for most calculations. i.e. The 1 + 1/2 +1/14 / 7 G1 base to height ratio as evidence.
i.e. 140/99 = 1.414141414…minus the sqrt 2 = 1.41213562 = 0.000072148

A considered opinion, anyone believing the Ancient Egyptians could process irrational numbers like the square root 2 and 3 are doing exactly what they were trained to do. Simultaneously they are ignoring all of the tenets of Ancient Egyptian mathematics of seked, unit fractions and Eye of Horace. With so many self imposed restrictions it is little wonder everyone fails to grasp their intent?

Regards,
Jacob