Phi and the Great Pyramid of Khufu

Posted: November 19, 2015 by oldbrew in Maths, Measurement, Phi
Tags:
Great Pyramid of Giza from a 19th-century stereopticon card photo [credit: Wikipedia]

Great Pyramid of Giza from a 19th-century stereopticon card photo [credit: Wikipedia]

Let’s have a look at some numbers for the Great Pyramid.

Source: Building the Great Pyramid (aka Cheops)
Copyright 2006 Franz Löhner and Teresa Zuberbühler

Dimensions as designed (in Egyptian royal cubits):
Length: 440
Height: 280
Slope: 356

Original dimensions as built (a,h and c in the pyramid diagram below):
Length: 230.36m (half = 115.18m)
Height: 146.59m
Slope: 186.42m


Dimensions of the pyramid of Khufu [credit: Franz Löhner and Teresa Uberbühler]

Dimensions of the pyramid of Khufu
[credit: Franz Löhner and Teresa Uberbühler]


Consider the right-angled triangle formed by half the length, the height and the slope. In the diagram the right angle is in orange with a black dot in it.
In Egyptian cubits the triangle sides are: 220 (length = 440/2), 280 (height), 356 (slope).

Using the design numbers, the proportions of the sides of the triangle are:
Length:slope = 1:1.61818 (18 recurring)
Length:height = 1:1.2727 (27 recurring)

Phi = 1.6180339~
Square root of Phi = 1.2720196~

Using the dimensions as built, the proportions of the sides of the triangle are:
Length:slope = 1:1.61851
Length:height = 1:1.2727035

Wikipedia says: ‘A Kepler triangle is a right triangle with edge lengths in geometric progression. The ratio of the edges of a Kepler triangle is linked to the golden ratio…approximately 1 : 1.272 : 1.618. The squares of the edges of this triangle are in geometric progression according to the golden ratio.’ [link has a diagram].

The angle of 51.83° (or 51°50′),  has a cosine of 0.618 or phi [credit: goldennumbers.net]

The angle of 51.83° (or 51°50′), has a cosine of 0.618 or phi
[credit: goldennumbers.net]


Löhner and Zuberbühler say:
‘Pyramid angle α = 51° 50′ 40″ = inclination of the lateral surface.’

For an exact Kepler triangle, the angle should be about 51° 49′ 38.25″ (= 51.8273°). In this diagram we have 51.83°.

goldennumber.net says: ‘Although difficult to prove due to deterioration through the ages, this angle is believed by some to have been used by the Egyptians in the construction of the Great Pyramid of Cheops.’

Comments
  1. oldbrew says:

    There’s not much commentary in the post but hopefully the numbers speak for themselves.

    Kepler triangle:

    ‘Kepler triangles combine two key mathematical concepts — the Pythagorean theorem and the golden ratio.’

  2. oldbrew says:

    Re. the Dimensions as designed (in Egyptian royal cubits):
    Length: 440
    Height: 280
    Slope: 356

    Dividing by 8 gives: 55 L, 70 H, 89 S
    55 and 89 are Fibonacci numbers.

    The square root of (89² – 55²) = 69.9714~ (about 99.96% of 70).
    Those would be the proportions of the three sides of a true Kepler triangle (55, 69.9714 and 89).

  3. oldbrew says:

    Thanks, Michele – looks like you got there before me🙂

  4. Chaeremon says:

    @oldbrew, as you have seen with Saros + Inex + FMC + LNC + LAC, there are quite a bunch of relations possible with Phi😎

    My favorite here: take 356 (slope) as draconic then (sidereal orbits) – (anomalistic orbits) = 3.0 = 23½ FMC (therefore 26½ equinoctial years, an eclipse interval at opposite equinoces). And in addition, in contradistinction to the many fails, opposite apsides at both interval ends🙂

  5. JKrob says:

    Heh – and there are those who say; “see, it was Islam that developed all that!”…even though Islam did not come around till nearly 3000 years later (around 609-630 AD?)

  6. oldbrew says:

    The king’s chamber is also interesting. The triangle made by half the side wall (brown in the diagram) is in the proportions 2:3:square root 5 i.e. a Fibonacci triangle (3 is the hypotenuse or ‘long side’).

  7. oldbrew says:

    The pyramid website says:

    ‘A Pythagorean triple is a set of three positive whole numbers a, b, and c that are the side lengths of a right triangle.
    Remark: there is no Pythagorean triple that approximates the angle of 51.84° of the Khufu (Cheops) Pyramid (see value for angle β).’
    http://www.cheops-pyramide.ch/khufu-pyramid/pythagorean-triple.html

    This is true, but they overlooked the Kepler triangle solution based on Phi (see the goldennumbers.net graphic in the post). It ‘approximates the angle of 51.84°’ very well: 51.83°.

  8. p.g.sharrow says:

    Sorry guys , but the creation of the pyramids was an engineering problem, not a mathematical one.
    The object was the maximum stable height that loose stones could be stacked. A builders problem. Quite a number of the early structures failed due to too steep of an angle being used. The maximum angle used for a stable construction is set by gravity and physics. The maximum height was set by the amount of resources and time available to the builders. The Bureaucrat, Pharaoh, that ordered up the project, wanted a Monument! grave marker, that would last forever and be a testament to his greatness. Placement and internal structures was determined by the best Scientific thought, religion, of the day for the continuation of the godking’s soul. Building a stable mountain on the earth required that the builder observe and solve physical problems, because he HAD TO. The creation of these monuments bankrupted their society and the practice was,in time, abandoned. The physical constraints of site, material and technology that was solved is inspiring. The mathematical gymnastics is left to the later observers….pg

  9. dp says:

    So there are no vertical stone walls, then?

    Has anyone ever looked at the temperature history of this pyramid’s interior? As tombs go it is probably a pretty good indicator of climate change – or lack of it.

  10. oldbrew says:

    dp says: ‘Has anyone ever looked at the temperature history of this pyramid’s interior?’

    ‘Egypt pyramids scan finds mystery heat spots’ — 10 November 2015
    http://www.bbc.co.uk/news/world-middle-east-34773856

  11. rishrac says:

    I was looking at this. It seems that the royal cubit is 0.5236 meters. That 6 is a tenth of a meter. I can determine the 0.5236 actually 0.52358 with a compass and knowing the value of pi. What I can’t figure out is how they determined a meter. do you think they might have used a pendulum the distance traveled for 1 second? Scribing the circle with the radius will give me 6 10 minute arcs. One of those lengths on the arcs is the number, but how they determined a meter, and to that degree of accuracy is a puzzle. There is problems using a pendulum. That has to be more than a coincidence down to 1/10th of a millimeter
    I came across a term heterogeneous. I thought that with the modular form of current brick laying was more advanced. Turns out it isn’t. The different size shapes and angles makes them not prove to cracks or destruction by earthquakes. How they got them to fit so closely is also a mystery.
    I disagree with p.j. sharrow, its a mathematical wonder box. They had to know the math to build it. The golden numbers and pi and throughout it. You can’t build an object that big and so perfect without a detailed knowledge of math.
    If you have any ideas on how they ascertained the meter? And do it repeatable so that 500 miles away they would have the same measurement.

  12. oldbrew says:

    rishrac: Have a look at ‘Egyptian units of measurement’ here.
    http://www.cheops-pyramide.ch/khufu-pyramid/pyramid-alignment.html

  13. rishrac says:

    I saw documentary on it. The amount of math involved in building it is simply staggering. It also said it was an eight sided pyramid. It is aligned so that at equinox it splits the pyramid in two. There is a picture of it doing that. The glass pyramid in Paris is in the same proportions using the same math.

    It’s on you tube. Must see documentary. Somebody built that thing, but how? As they said, the only way they knew it was built within a 1/10 of a millimeter was when we had the ability recently to measure it.

  14. rishrac says:

    I looked at the instruments the Egyptians used. There can only be one conclusion, they didn’t build the pyramids with that technology.

  15. oldbrew says:

    Then there’s Karnak and Abu Simbel.

    ‘One famous aspect of Karnak is the Hypostyle Hall in the Precinct of Amun-Re, a hall area of 50,000 sq ft (5,000 m2) with 134 massive columns arranged in 16 rows. 122 of these columns are 10 meters tall, and the other 12 are 21 meters tall with a diameter of over three meters.
    The architraves on top of these columns are estimated to weigh 70 tons.’
    http://en.wikipedia.org/wiki/Karnak

    ‘Four colossal 20 meter statues of the pharaoh with the double Atef crown of Upper and Lower Egypt decorate the facade of the temple, which is 35 meters wide and is topped by a frieze with 22 baboons, worshippers of the sun and flank the entrance.’
    http://en.wikipedia.org/wiki/Abu_Simbel_temples