The amount of days in a solar year plus the proportions for the Equatorial circumference of the Earth and the proportions of the Great Pyramid of Giza according to Golden Pi = 4/√φ = 3.144605511029693144: The Great Pyramid of Giza is a geodetic model of Planet Earth. The measurements mentioned below are the ideal measurements. A meter is equal to 100 centimeters 1 Solon cubit = 40 times √φ = 50.88078598056276 centimeters. If 1 Solon cubit is divided into 20 equal units of measure then 20 inches can be derived because 1 Solon cubit is equal to 20 inches. 1 Saylen cubit = 50 times √φ = 63.60098247570345 centimeters. If 1 Saylen cubit is divided into 25 equal units of measure then 25 inches can be derived because 1 Saylen cubit is equal to 25 inches. 1 inch = 2 times √φ = 2.544039299028138 centimeters. 1 foot = 12 inches. 1 foot = 24 times √φ = 30.528471588337656 centimeters. If the shortest edge length of a Kepler right triangle is equal to 1 then the hypotenuse of the Kepelr right triangle is equal to The Golden ratio = (√(5) plus 1)/2 = φ = 1.618033988749895 according to the Pythagorean theorem. The Golden ratio in Trigonometry = (cosine (36 degrees) times 2) = 1.618033988749895. If the shortest edge length of a Kepler right triangle is equal to 1 then second longest edge length of the Kepelr right triangle is the square root of the Golden ratio = √φ = 1.272019649514069 according to the Pythagorean theorem. The width for the square base of the Great Pyramid of Giza is equal to 756 feet. • If the shortest edge length of a Kepler right triangle is equal to 1 foot and then the second edge length of the Kepler right triangle is multiplied 378 equal times the result will be the height of the Great Pyramid of Giza = 378 times √φ = 480.823427516318082 feet according to Golden Pi = 4/√φ = 3.144605511029693144. • If the shortest edge length of a Kepler right triangle is equal to 1 foot and then the hypotenuse of the Kepler right triangle is multiplied 378 equal times the result will be the slant height of the Great Pyramid of Giza = 378 times φ = 611.61684774746031 feet according to Golden Pi = 4/√φ = 3.144605511029693144. • If the shorter edge length of a Golden ratio = (√(5) plus 1)/2 = φ = 1.618033988749895 rectangle is equal to 1 foot and then the diagonal of the Golden ratio = (√(5) plus 1)/2 = φ = 1.618033988749895 rectangle is multiplied 378 equal times the result will be the edge height of the Great Pyramid of Giza = 378 times = Cosine (18) degrees times 2 = 1.902113032590307 = 718.998726319136046 feet according to Golden Pi = 4/√φ = 3.144605511029693144.Cosine (18) degrees times 2 = 1.902113032590307. Cosine (18) degrees times 2 = 1.902113032590307 squared = φ plus 2 = 3.618033988749895. • If the shortest edge length of a Kepler right triangle is equal to 1 foot = 24 times √φ = 30.528471588337656 centimeters then the second longest edge length of that Kepler right triangle is equal to 38.832815729997479 centimeters.24 times √φ = 30.528471588337656 centimeters times the square root of the Golden ratio = √φ = 1.272019649514069 = 38.832815729997479 centimeters. 1 foot = 24 times √φ = 30.528471588337656 centimeters times the square root of the Golden ratio = √φ = 1.272019649514069 = 38.832815729997479 centimeters times 378 = the height of the Great Pyramid of Giza of Giza = 14678.804345939047062 centimeters. The height of the Great Pyramid of Giza = 14678.804345939047062 centimeters divided by 24 times √φ = 30.528471588337656 centimeters = the height of the Great Pyramid of Giza = 480.823427516318088 feet. 378 times √φ = 1.272019649514069 = 480.823427516318088. • If the shortest edge length of a Kepler right triangle is equal to 1 foot = 24 times √φ = 30.528471588337656 centimeters then the hypotenuse of that Kepler right triangle is equal to 49.39610465451582 centimeters. 24 times √φ = 30.528471588337656 centimeters times the Golden ratio = (√(5) plus 1)/2 = φ = 1.618033988749895 = 49.39610465451582 centimeters. 1 foot = 24 times √φ = 30.528471588337656 centimeters times the Golden ratio = (√(5) plus 1)/2 = φ = 1.618033988749895 378 = 49.39610465451582 centimeters times 378 = slant the height of the Great Pyramid of Giza = 18671.727559406979971centimeters. The Slant height of the Great Pyramid = 18671.727559406979971 centimeters divided by 24 times √φ = 30.528471588337656 centimeters = the slant height of the Great Pyramid of Giza = 611.61684774746031 feet. 378 times √φ = 1.272019649514069 = 611.61684774746031. • If the shortest edge length of a Golden ratio = (√(5) plus 1)/2 = φ = 1.618033988749895 rectangle is equal to 1 foot = 24 times √φ = 30.528471588337656 centimeters then the length of the diagonal of that Golden ratio = (√(5) plus 1)/2 = φ = 1.618033988749895 rectangle is equal to 58.068603673239965 centimeters. 24 times √φ = 30.528471588337656 centimeters times Cosine (18) degrees times 2 = 1.902113032590307 = 58.068603673239965 centimeters. Cosine (18) degrees times 2 = 1.902113032590307 squared = φ plus 2 = 3.618033988749895. 1 foot = 24 times √φ = 30.528471588337656 centimeters times = Cosine (18) degrees times 2 = 1.902113032590307 = 58.068603673239965 centimeters times 378 = the edge height of the Great Pyramid of Giza of Giza = 21949.932188484706835 centimeters. The edge height of the Great Pyramid = 21949.932188484706835 centimeters divided by 24 times √φ = 30.528471588337656 centimeters = the edge height of the Great Pyramid of Giza = 718.998726319136046 feet. 378 times Cosine (18) degrees times 2 = 1.902113032590307 = 718.998726319136046. Cosine (18) degrees times 2 = 1.902113032590307. Cosine (18) degrees times 2 = 1.902113032590307 squared = φ plus 2 = 3.618033988749895. A Kepler right triangle can be created from the construction of a Golden ratio = (√(5) plus 1)/2 = φ = 1.618033988749895 rectangle by using Compass and straight edge and obviously a marker for the drawing surface. The amount of days in a Solar year = 4/√φ times 7920 times 5280/(10 ^ 3 times 360) = 365.277376161209156. The amount of days in a Solar year = 4/√φ = 3.144605511029693144 times 7920 times 5280/(10 ^ 3 times 360) = 365.277376161209156. The equatorial circumference of planet Earth = 10 ^ 3 times 360 times 365.277376161209156 = 131499855.41803529616 feet. The equatorial circumference of planet Earth = 4/√φ = 3.144605511029693144 times 7920 = 24905.275647355169727 statute miles. 131499855.41803529616 feet divided by 86400 = half the perimeter of the socle of the Great Pyramid of Giza = 1521.989067338371483 feet. Half the perimeter of the socle of the Great Pyramid of Giza times 86400 is also equal to the equatorial circumference of planet Earth = 131499855.41803529616 feet. 484 divided by √φ times 2 = the width of the socle of the Great Pyramid of Giza = 760.99453366918572 feet. Half the width of the socle of the Great Pyramid of Giza = 380.49726683459286 feet times √φ = 484 feet. 484/√φ times 2 times 2 times 86400 = 131499855.41803529616 feet. 131499855.41803529616 feet divided by 5280 = The perimeter of the socle of the Great Pyramid of Giza = 484/√φ times 8 = 3043.978134676742966 feet. 484 feet /√φ times 8 = 3043.978134676742966 feet times 12 = 24905.275647355169727 statute miles. 24905.275647355169727 statute miles divided by 7920 statute miles = Golden Pi = 4/√φ = 3.144605511029693144. There are 929.28 meters in the square perimeter of the socle of the Great Pyramid of Giza according to Golden Pi = 4/√φ = 3.144605511029693144. 484/√φ times 8 = 3043.978134676742966 times 12 = 36527.737616120915592 divided by 100 = the exact amount of days in a solar year = 365.277376161209156. The equatorial diameter of our planet Earth = 41817600 feet. 41817600 feet divided 86400 = the height of the Great Pyramid of Giza = 378 times √φ = 480.823427516318082 feet plus the height of the socle of the Great Pyramid of Giza = 3.176572483681918 feet. 10 ^ 3 times 360 times 484/(√φ) times 8 times 12/(100) = 131499855.41803529616 feet. The height of the Great Pyramid if Giza is 378 times √φ = 480.823427516318082 feet. The width of the square base of the Great Pyramid of Giza is 756 feet. The perimeter of the square base of the Great Pyramid of Giza = 3024 feet. 9 factorial = 362880. At 10 degrees latitude the length of a degree is 9 factorial =362880 feet. The amount of inches in the perimeter of the square of the Great Pyramid of Giza = 36288. 36288 times 10 = 362880. The width for the square base of the Great Pyramid of Giza = 756 feet. The perimeter of the square base of the Great Pyramid of Giza = 3024 feet. There are 36288 inches in the perimeter of the square base of the Great Pyramid of Giza. 3024 times 12 = 36288. 756 times 4 times 12 = 36288. (9!)/10 = 36288. The equatorial circumference of planet Earth: https://joedubs.com/four-earthly-elements/equatorial-circumference-of-earth/ Kepler right triangle diagram with squares upon the edges of the Kepler right triangle: https://drive.google.com/file/d/1iBtXYy06yv9UWtGP5mwMXyt80vaysFvR/view?usp=sharing Kepler right triangle construction method: https://drive.google.com/file/d/15DNXB_xNP2f2jCroC0FyNUyBYGhVAA2J/view?usp=sharing PYTHAGOREAN THEOREM: https://en.wikipedia.org/wiki/Pythagorean_theorem Golden ratio: https://en.wikipedia.org/wiki/Golden_ratio The history of the meter: The history of the meter: https://www.factinate.com/editorial/meter-history/ The meter: https://en.wikipedia.org/wiki/Metre The meter is based now on the speed of light: https://www.youtube.com/watch?v=vgqUyFaUDcI

Pyramids and the meter

Belgian television showed the film The Revelation of the Pyramids. It contains an intriguing suggestion for a mathematical relationship. Let us debunk it, though keep the intrigue.

I have three reasons to look into this. The first reason is the earlier weblog on the use of ‘archi’ Θ = 2 π = 6.283185307… rather than π as the key mathematical concept for the measurement of the circle. Other people suggest ‘tau’ τ but that looks too much like the radius r and thus will cause much confusion in the classroom. The second reason is the earlier weblog on the mathematics of Jesus. Since the holy family fled to Egypt there is ample reason to look what was happening there. The third reason is that the film suggests that there was an ancient advanced civilisation. Since we may all be disappointed about how we ourselves are doing as a civilisation, it would be great when we could discover that others in the past have been doing much better.

We will also use ‘phi’ φ = 1.618033989… or the golden ratio. This has the property that φ2 = 1 + φ, or alternatively that φ = 1 / φ + 1. It allows a particular interesting application of the Pythagorean Theorem. A right angled triangle with base a = 1 and height b = √φ generates a hypothenusa of c = √ (a2 + b2) = √(1 + φ) = √ φ2 = φ. The associated square has the surface φ2, and by using a circle of radius φ we can find that same value in the length of the interval 1 + φ. It appears that these dimensions have been used in the pyramid of Cheops. To measure length the Egyptians used the ell or the (royal) cubit of approximately 0.5236 meters (wikipedia: between 52.3 and 52.9 cm). The pyramid of Cheops has a height of 280 cubits and a full base of 440 cubits. That shape however consists of two right angled triangles. The proper triangle has a base of 220 cubits. The ratio is 280 / 220 = 14 / 11. It so happens that 11 * √φ = 13.99221614… ≈ 14. Thus the Egyptians chose a ratio in integer numbers that closely matches the real value of the golden ratio.

The film The Revelation of the Pyramids now presents the startling equation:

π = 0.5236 + φ2 or π = cubit + φ2

Startling about this is that π and φ are pure numbers while the length of the cubit only makes sense when everything is expressed by using the meter as the standard length. The pure numbers π and φ come about as ratio’s and thus by dividing lengths so that they do not depend upon any choice of measurement standard. But the value of the cubit changes if we switch from meters to feet and inches.

A first step is to check for accuracy. We find that π – φ2 = 0.5235586648… Thus the relation only holds by approximation, though the accuracy is eery.

A second step is to divide both sides by the cubit, or rather by the pure value π – φ2. Then we find:

π / (π – φ2) = 1 + φ2 / (π – φ2)

6.000459671… = 1 + 5.000459671…

There we are.

Do you see it ? Well, it took me some moments to find the proper sequence of explaning, so let us follow these steps.

A major point is that the use of π has been playing a misleading role in this analysis. It gives only a half circle and it is much better to use Θ and the whole circle.

The first point is the surprise that φ2 / (π – φ2) = 5.000459671… Reworked, we get:

φ2 / Θ ≈ 5 / 12

What is to say about that ? Well, it apparently is a mathematical property, like 11 * √φ ≈ 14. Sometimes mathematical numbers with complex properties and long decimal expansions can get close to ratio’s of specific integer values. This may be surprising, but it is a mathematical surprise. It cannot be a base for concluding that the ancient Egyptians knew about the decimal expansions of these numbers and their particular ratio. Once you decide to build a pyramid using the ratio of 14 / 11 since it is pleasing to the eye and with structural stability, then you are stuck with the implied mathematics, but that does not imply that you know more about the implied mathematics.

Secondly, let us assume that the Egyptians had their ell or cubit as an arbitrary length (based upon the human body). They also divided the year in 12 months and day and night in 12 hours each. Thus for them it makes sense to measure the circumference of a circle by 12 cubits, like we still do in our clocks. Of these 12 pieces of a pie, six can be allocated to π, five to φ2, and then one remains (all with a proportionality factor).

A small problem in this discussion is that the Egyptians might use either flexible ropes (circle) or rigid yardsticks (polygon). Let us assume flexible ropes (circle) first, as they have been nicknamed ‘rope-stretchers’. (See the appendix for approximation by a polygon.)

The radius r of that circle follows from Θ r ≈ 12 cubit, giving r ≈ 1.909859317… cubit ≈ 1.91 cubit. For the Egyptians there was nothing special about that number for that radius. The film shows that the capstone of the pyramid would have this side. That is not inconceivable given this geometry. (If the Egyptians had Θ ≈ 44 / 7 from π ≈ 22 / 7 then r ≈ 12 / Θ cubit = 12 * 7 / 44 cubit = 21 / 11 cubit = 1.90909 cubit. For them still no special value.)

It is only for us, who have adopted the meter (rather than feet and inches), that a sense of wonder arises. For r ≈ 1.909859317… cubit = 1.909859317.. * 0.5236 = 1.000002338… meters ! Alternatively put, if we take a circle with radius 1 meter then the division of the circumference by 12 gives us the Egyptian (royal) unit of measurement, namely via Θ r = 12 cubit or one cubit = Θ / 12 = 0.5235987756…. This uses the arcs rather than the sides of the polygon, and presumes that the flexible rope subsequently is transferred to a yardstick. (For the polygon, see the appendix.)

To understand what is happening here requires us to look into the history about the selection of the meter as the European standard of measurement. Officially, the French Academy decided in 1791 that a meter was to be one ten-millionth of the distance from the Earth’s equator to the North Pole (at sea level) (wikipedia). The expedition by Napoleon to Egypt took place in 1798-1801, thus later, and the results of the new Egyptology will not have been available immediately. From this we may tend to infer that the ancient Egyptians knew about the size of the Earth and reasoned like the French. It seems more reasonable to think differently. To start with, it is already curious to take something that is difficult to measure, such as the distance from the Earth’s equator to the North Pole, to define a standard. It seems more reasonable to assume that there were already circulating measures and that the story about the equator was only an embellishment. Apparently the circle with a circumference of 12 ells had been surviving over the ages and still made it into the discussion.

But the film then should be about what happened in France and not about mysteries in ancient Egypt.

NB. There is ample discussion about the measurements of the pyramid. The top is missing so we can only guess what the Egyptians intended. See the original Petrie measurements (base 9068 and height 5776 +/- 7 inches) and this discussion with drawings. Indeed, if the base is 220 cubits and the Egyptians had a precise estimate of √φ then the height would be 279.8443229 cubits, which is only a 0.06% of the whole height or one finger of a cubit short of 280. Because of this uncertainty, we cannot infer on these grounds that the Egyptians didn’t have a precise estimate of φ. It are other documents that show us that there were severe limits to their number system. We can neither infer that they were aware of the implication that φ2 = 1 + φ, We can observe however that they used geometry and architecture that closely matches these results. See the website by Gary Meisner for how you can create your own golden ratio paper pyramid.

PM. Sir Flinders Petrie (1853-1942) suggests that the basic inspiration lies in the circle rather than in the golden ratio. A circle with radius 7 has a circumference of 7 Θ ≈ 7 * 44 / 7 = 44, using the approximation π ≈ 22 / 7. This 44 gives a square with sides 11. A circle with radius 7 has the circumference of a square with sides 11. Thus we find the numbers 14 and 11 again.

The argument then is that the Great Pyramid expresses Θ ≈ 4 * 440 / 280 = 44 / 7, and that the golden ratio is only a by-product. If this is the case then this knowledge about Θ has been kept secret or has been lost since later documents apparently don’t mention it. It is a bit curious how that knowledge can get lost when that very same pyramid is standing in front of you. Mankind however has achieved greater mysteries. Note that there is no quick transformation into φ2 / Θ ≈ 5 / 12. Via Pythagoras φ2 ≈ 1 + (14 / 11)2 = 317 / 121 and now φ2 ≈ 5 / 12 * 44 / 7 = 55 / 21. For us these are approximations only but for the Egyptians it sufficed that the construction worked. The Petrie approach to start with the circle and 14 / 11 ratio seems simplest indeed. Still, the builders will not have been insensitive to the lure of the golden ratio, and it is remarkable that they have hit upon this very shape.

Appendix

We can also assume that they did not use flexible ropes but rigid yardsticks to lay out a polygon with circumference of 12 cubits, and imagined it enclosed by a circle. We can calculate the sine of a half slice, with Sin[angle] = h / r. A half slice has h = 0.5236 / 2 and the associated angle = Θ / 12 / 2 rad or 15 degrees. We find r = 1.01152. The enclosing circle has a radius that is 1% or one centimeter longer than the meter.

A slice of Θ / 12 of the polygon: The inner circle has r = 1 and h = Sin[Θ / 24] = 0.2588, the outer circle has r = 1.01152 and h = 0.5236 / 2 = 0.2618.

https://boycottholland.wordpress.com/2013/01/27/pyramids-and-the-meter/

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Pyramids and the meter