Some interesting facts about the Golden Ratio and the pyrimids of giza... A lot of math involved but I think you'll get the gist of the amazing mathmatics involved in the building and knowledge of math 3000 years ago when they built these things...

In architecture, art, plants, painings and even book design throughout history, the use of the golden ratio has made the aesthetics look more pleasing to the eye and through this ratio man has copied nature to produce some of the most wonderful works of art and architecture on earth.


Pyramids

A regular square pyramid is determined by its medial right triangle, whose edges are the pyramid's apothem (a), semi-base (b), and height (h); the face inclination angle is also marked. Mathematical proportions b:h:a are of particular interest in relation to Egyptian pyramids.Both Egyptian pyramids and those mathematical regular square pyramids that resemble them can be analyzed with respect to the golden ratio and other ratios.


Mathematical pyramids and triangles
A pyramid in which the apothem (slant height along the bisector of a face) is equal to φ times the semi-base (half the base width) is sometimes called a golden pyramid. The isosceles triangle that is the face of such a pyramid can be constructed from the two halves of a diagonally split golden rectangle (of size semi-base by apothem), joining the medium-length edges to make the apothem. The height of this pyramid is times the semi-base (that is, the slope of the face is ); the square of the height is equal to the area of a face, φ times the square of the semi-base.

The medial right triangle of this "golden" pyramid, with sides is interesting in its own right, demonstrating via the Pythagorean theorem the relationship or . This "Kepler triangle" is the only right triangle proportion with edge lengths in geometric progression, just as the 3–4–5 triangle is the only right triangle proportion with edge lengths in arithmetic progression. The angle with tangent corresponds to the angle that the side of the pyramid makes with respect to the ground, 51.827… degrees (51° 49' 38").

A nearly similar pyramid shape, but with rational proportions, is described in the Rhind Mathematical Papyrus (the source of a large part of modern knowledge of ancient Egyptian mathematics), based on the 3:4:5 triangle; the face slope corresponding to the angle with tangent 4/3 is 53.13 degrees (53 degrees and 8 minutes). The slant height or apothem is 5/3 or 1.666… times the semi-base. The Rhind papyrus has another pyramid problem as well, again with rational slope (expressed as run over rise). Egyptian mathematics did not include the notion of irrational numbers, and the rational inverse slope (run/rise, multiplied by a factor of 7 to convert to their conventional units of palms per cubit) was used in the building of pyramids.

Another mathematical pyramid with proportions almost identical to the "golden" one is the one with perimeter equal to 2π times the height, or h:b = 4:π. This triangle has a face angle of 51.854° (51°51'), very close to the 51.827° of the Kepler triangle. This pyramid relationship corresponds to the coincidental relationship .

Egyptian pyramids very close in proportion to these mathematical pyramids are known.


Egyptian pyramids
In the mid nineteenth century, Röber studied various Egyptian pyramids including Khafre, Menkaure and some of the Gizeh, Sakkara and Abusir groups, and was interpreted as saying that half the base of the side of the pyramid is the middle mean of the side, forming what other authors identified as the Kepler triangle; many other mathematical theories of the shape of the pyramids have also been explored.

One Egyptian pyramid is remarkably close to a "golden pyramid" – the Great Pyramid of Giza (also known as the Pyramid of Cheops or Khufu). Its slope of 51° 52' is extremely close to the "golden" pyramid inclination of 51° 50' and the π-based pyramid inclination of 51° 51'; other pyramids at Giza (Chephren, 52° 20', and Mycerinus, 50° 47') are also quite close. Whether the relationship to the golden ratio in these pyramids is by design or by accident remains controversial. Several other Egyptian pyramids are very close to the rational 3:4:5 shape.

Adding fuel to controversy over the architectural authorship of the Great Pyramid, Eric Temple Bell, mathematician and historian, claimed in 1950 that Egyptian mathematics would not have supported the ability to calculate the slant height of the pyramids, or the ratio to the height, except in the case of the 3:4:5 pyramid, since the 3:4:5 triangle was the only right triangle known to the Egyptians and they did not know the Pythagorean theorem nor any way to reason about irrationals such as π or φ.

Michael Rice asserts that principal authorities on the history of Egyptian architecture have argued that the Egyptians were well acquainted with the golden ratio and that it is part of mathematics of the Pyramids, citing Giedon (1957). Historians of science have always debated whether the Egyptians had any such knowledge or not, contending rather that its appearance in an Egyptian building is the result of chance.

In 1859, the pyramidologist John Taylor claimed that, in the Great Pyramid of Giza, the golden ratio is represented by the ratio of the length of the face (the slope height), inclined at an angle θ to the ground, to half the length of the side of the square base, equivalent to the secant of the angle θ. The above two lengths were about 186.4 and 115.2 meters respectively. The ratio of these lengths is the golden ratio, accurate to more digits than either of the original measurements. Similarly, Howard Vyse, according to Matila Ghyka, reported the great pyramid height 148.2 m, and half-base 116.4 m, yielding 1.6189 for the ratio of slant height to half-base, again more accurate than the data variability.

In my real life encounters that mkae sense to the average guy, look at it this way...

When I do construction, to find the beginning square corner of the building site, I take a stake in the ground and run a line off of it to 30 foot and place a stake. Then I run a string from the first stake out about 40 foot and another string from the second stake out to 50 foot and where the 40 and 50 foot strings cross is where I stake the last stake and I'll always have a perfect 90 degree corner starting at the first stake... This is the 3,4,5 triangle or 30,40,50 or 300,400,500... It always makes a perfect corner with a 90 degree angle at my first stake! Remember this and you'll build perfect pyrimids every time.

The slope angle of the Great Pyramid relates to Universal gravitational harmonics, sacred geometry, in fact, ...meaning not Limited to Particular circumstances, (such as the relative mass of a celestial object)........

High gravity can produce certain effects that tend more towards flattening, yet a pile of sand (isolated from influences like high atmospheric pressure, friction-tension between sand particles, etc.) under high gravity will behave much like a pile of sand under lower gravity...... under low gravity there may be fewer peripheral influences, allowing friction-variables to play at keeping the pile somewhat steeper.