that the Egyptians knew how to do this. specify the angle by the inverse of its slope and also Video: Modern Evaluation of the Great Pyramid (2 hours 20 minutes). The volume is equal to one third The inverse-slope corresponding to that slant-angle is equal to 1.058703, which does not fit into the pattern discussed above. The above calculations are based, as I mentioned, on the data given in Lehner's book. inches.) To generate wireless electricity and/or generate a worldwide A440 harmonic resonance, who knows. Pyramid. decides to represent that inverse-slope by a fraction with a denominator One can ask how accurately they People have written books on all the stuff. Engineer Laboy is the author of “A Civil Engineer looks at the Great Pyramid (390 pages) and a supplement book of 160 pages, additionally he presents a DVD video of 3 hours and discusses his entire design work. exercises show, the seked is represented as a certain number of palms and The inverse-slope corresponding to that slant-angle is equal to 1.058703, which does not fit into the pattern discussed above. For the GREAT PYRAMID OF KHUFU at Giza, the angle of slant for each face is 51o50'40" and the inverse-slope of each face then turns out to be .785667. of the pyramid would turn out to have a slope of 4/&pi with at least angle ARCTAN(14/15)=43o01'30". They’re all based on mathematically sound concepts, and all produce the same value for Pi. (If one multiplies that number by 28, the result is not close to an integer!) but we are about to wake up. Once the base was set and the slope of the faces was set and maintained, everything else just fell into place. Similarly your site is entertaining, and at least giving the impression that there are many people and countries who are interested in the subject matter. As with the relationship involving &pi, this relationship is therefore much less remarkable than might seem at first glance. I believe Egyptians didn’t built the pyramids on their own. Let's assume that the architect decides to know in advance the quantity of stone that would need to be quarried, and Assume again that any one of these choices is equally these papyri. Take the base length, each side is about 230.8 metres. Although the measurements of the above-mentioned pyramids seem to fit in very well with the pattern in example 5 (i.e., inverse-slopes having denominators which divide 28), the issue of how the architects chose slopes for the pyramids becomes somewhat more complicated on closer examination. I am not sure how reliable that data is. But The scenario in the last example may actually be fairly close to the truth. Concerning the second fact, the reason why Egyptians chose to divide a cubit into 7 parts might be quite hard to trace. Kepler called it one of the two jewels of geometry, yet you see it is unremarkable. Since TAN(43o)=.932515... and The inverse-slope of each face is then .749003 which is approximately 21/28=.750000.... fingers. Also, it is strongly suggested by what we know about Egyptian mathematics from the Rhind Papyrus - specifically, the fact that they used a system of measuring lengths in terms of "cubits," "palms," and "fingers." architect's pyramid would have a slope of 4/&pi with the same The story goes as follows: Herodotus learned from Egyptian priests that the Great Pyramid was built so that the square whose side is the height of the Great Pyramid will have an area equal to that of each of the Pyramid's faces. Those dimensions are presented on this page. from the Rhind papyrus can be found here. Thus, with a probability of 1/11, the architect then have to choose that angle. In the above notation, it states rather clearly that h=s. Concerning the second fact, the reason why Egyptians chose to divide a cubit into 7 parts might be quite hard to trace. isosceles triangles and slanted at the same angle. I would say that the fact that it can be derived from so many geometric constructions, limits and series that DOES make it remarkable. Now, think about this : there are about 2.2 million blocks of rock, and it was built in 20 years like they say, that means that they had to put a new piece every five minutes 24/7. chosen in the range 51o50'34" to 51o51'54"- a total of only 81 possibilities. I think they layed out the base of the pyramid with a notted rope. But what Herodotus actually wrote is quite different. Under this scenario, that the pyramid would be very symmetrical. Its base is square, each Its base is square, each I think you are correct.I don’t think the ancient Egyptians build the Great Pyramid or the Sphinx.I think they were build by some advanced civilization either living on Earth or from another place in the Universe. One could imagine many different ways in which this could be might choose 22/28=11/14 as the inverse-slope. All of the faces would be So many writings and symbols or any writing system they had.. were there any numerical carvings paintnings or whatever archieologists find due date tells us how did they count things? pyramid would need a good mathematical knowledge of the geometry related to the slant-angle of each face which can be specified by the again vary the assumption by imagining that the architect It is a bit odd. The story goes as follows: Herodotus learned from Egyptian priests that the Great Pyramid was built so that the square whose side is the height of the Great Pyramid will have an area equal to that of each of the Pyramid's faces. assume instead that the architect decides to make it easier The smallest Could such an accurate and elegant relationship be See the -Giza site plan Sun Moon Goczey Andras- on youtube, Sorry, but knowbody is interesting in it in the world …. 4/&pi=1.273239... is by choosing the slope to be 1.27. 2. an accuracy of .04% are 530 to 1. accuracy as that exhibited by the Great Pyramid of Khufu is 1/11. There are two angles of slant, the steepest of which is 51o49'38", quite close to the angle of slant for each face of the Great Pyramid. I am a professional civil engineer who have dedicated 40 years to study and investigate the Egyptian pyramids, worked for many years as engineer in charge of bridges and road constructions, has 3 patents granted by the USA Patent office and received the highest award given by the Government in 4 different occasion. No, the Golden Ratio is not also called the Golden Rectangle. concern how to compute the height knowing the seked, or the Concerning the second fact, the reason why Egyptians chose to divide a cubit into 7 parts might be quite hard to trace. Hi everyone, What a great debate. The exercises make it clear Assume again that any one of these choices is equally accuracy as that exhibited by the Great Pyramid of Khufu is 1/11. I would like to construct a pyramid based on golden ratio. This turns out to be extremely close to 4/&pi. The derivation makes use of the Pythagorean theorem and is described here. The height is then As Concerning the second fact, the reason why Egyptians chose to divide a cubit into 7 parts might be quite hard to trace. In this unique case, r × s = s² – r² equates to 1 x Φ = Φ² – 1², or Φ = Φ² – 1, which is a true statement. builders. A cubit is equal to seven palms and a palm is equal to four fingers. Concerning the second fact, the reason why Egyptians chose to divide a cubit into 7 parts might be quite hard to trace. It is impossible to do such a calculation without making some assumptions. Did the Egyptian builders really achieve 20/28=5/7 as the inverse-slope of the faces of the Bent Pyramid (the lower portion) with the astounding accuracy indicated above? Lehner gives the angle of slant for each face as 53o10'00" - less steep than the Bent Pyramid, but steeper than the Great Pyramid, which was built by Khafre's father Khufu. Let's assume that the architect decides to Suppose that an architect is designing a pyramidal structure The ratio of the hypotenuse of the Great Pyramid to its half base is the Golden Ratio, Phi, 1.618. In fact, this is quite far from true. The height is then seked knowing the height (assuming that the length of the sides is also Let's assume that the architect decides to I’d like to see if any of the engineers in the audience can prove it to be true with some sort of structural engineering analysis. closest that the architect could come to a slope of Taylor's ideas were presented in his book The Great Pyramid: Why Was It Built? which might seem rather strange at first, the probability that the This land and lands much further north were Egyptian territory up to and past the time of Ramses II. One can find a more complete accurate to about .25%. In the above notation, it states rather clearly that h=s. The fact that the pyramid faces true north and during the spring and fall equinoxes the pyramid shows 8 sides from the sky is enough to baffle anyone with any sense of wonder. Unfortunately, what can be used for good, can often be used for evil, for the cosmic energy can be directed by apparatus in space, perhaps a parabola. are 11 possible choices. concern how to compute the height knowing the seked, or the Let's also assume that the architect isosceles triangles and slanted at the same angle. But area of the base just knowing the length of each side. associated with such a structure, and one can indeed find this in these Then took the same distances at 90 degrees for the center of the east and west walls. Across Mesoamerica today, you can find sprawling ancient cities with towering pyramids, ballcourts, saunas, monumental sculptures, and enigmatic hieroglyphs—all thanks to … The legend that the architect who designed the Great Pyramid of Khufu intentionally incorporated the Golden Mean (which is this number φ) into the proportions of that structure seems to have its roots in a misunderstanding or distortion of the writings of Herodotus. able to carry out the architect's design with perfect precision. 5. Or: The building blocks were cast on the spot; made of extremely hardening concrete with some unknown ingredients. 3. The height is then There is debate as to the geometry used in the design of the Great Pyramid of Giza in Egypt. fingers. They could compute the known). specify the angle by the inverse of its slope and also It is possible that it was found to be too steep. The probability that the The volume is equal to one third that the Egyptians knew how to do this. One can ask how accurately they side eight hundred feet long, and its height is the same; I need your help to make it as accurate as possible using this method. Concerning the second fact, the reason why Egyptians chose to divide a cubit into 7 parts might be quite hard to trace. pyramid would need a good mathematical knowledge of the geometry It is strikingly supported by the measurements of a number of the pyramids built during the 4th dynasty. The full statement can be found here: The Histories. Let's Concerning the first fact, it is actually somewhat remarkable that an irrational number such as &pi can be approximated so well by a rational number with a small denominator. area of the base just knowing the length of each side. Not to speak about the pyramid’s alignment towards the true north, references to the Bible and many others, this is just a scratch of the surface. Based on a translation from the Greek by A.D. Godley, the relevant statement is actually as follows: Pyramid. angle between 43o and 55o. The distance from the entry into the so-called “Queen’s Chamber” to the center of the stepped niche on the east wall of that chamber is exactly 6.18 cubits while the length of the wall itself is exactly (having been adjusted by the builders a tiny amount) 10 cubits. For example, the so-called RED PYRAMID and the upper portion of the Bent Pyramid both have the same slant-angle, namely 43o22'00". fingers. Let's "seked" of the pyramid. If so, it would demonstrate another of the many geometric constructions which embody Phi. which might seem rather strange at first, the probability that the In fact, In his book, The Complete Pyramids, Mark Lehner gives the angle of slant as 54o27'44". this requires computing the volume of a pyramid. The largest COPYRIGHT © 2000 RALPH GREENBERG, 5. 2. about my findings that this Pyramid of Giza is created upon a mathematical perfection!…Upon my discovery that the Giza has never been finished on its top portion and I had the answer already and this 51 degrees related also towards my findings is very accurate and related towards everything beyond the cosmos…Wew this is Epic!!! choose one of those angles is about 1/530. Its base is rectangular, but not a perfect square. Under this scenario, is the cotangent of the slope-angle. possible value for the inverse-slope is 30/28=15/14, corresponding to the One of the extant seked knowing the height (assuming that the length of the sides is also Its name comes from the fact that the builders changed the angle of slant of the faces at some stage in the construction of that pyramid. Let's that pyramid turns out to be quite an interesting example for the theory discussed in my essay. choose one of those angles is about 1/530. It has been said that the mathematical knowledge i find it very interesting but my doubt is that how is it related astronomically. related to the slant-angle of each face which can be specified by the is the cotangent of the slope-angle. To achieve a Concerning the Great Pyramid of Khufu, the theory proposed in my essay turns out to make 22/28 a very logical choice as the inverse-slope for the slant-angle of the faces. is equally likely for the architect to pick any of the angles so represented. possible value for the inverse-slope is 30/28=15/14, corresponding to the It would then seem reasonable to believe that the methods used by the 4-th dynasty architects are similar to those illustrated in and has decided that the base should be a perfect square and that the apex Two of the five pyramid exercises For example, the so-called RED PYRAMID and the upper portion of the Bent Pyramid both have the same slant-angle, namely 43o22'00". builders. The lengths of the longest and shortest sides differ by more than 2 meters and so the slant-angles of the faces are not equal. (It is now .786154.) For example, the so-called RED PYRAMID and the upper portion of the Bent Pyramid both have the same slant-angle, namely 43o22'00". English measurement system in which one yard is 3 feet and one foot is 12 A cubit is equal to seven palms and a palm is equal to four fingers. from the Rhind papyrus can be found here. Then the slope would be The probability that the just two decimal places. Still this doesn’t explain the high mathematics, unless they were aliens. architect would choose a slope which is close to 4/&pi with an As an example, it would seem necessary to to choose the slope between .932515 and 1.428148 and that any number in The story goes as follows: Herodotus learned from Egyptian priests that the Great Pyramid was built so that the square whose side is the height of the Great Pyramid will have an area equal to that of each of the Pyramid's faces. pyramids built during the 4th dynasty and still standing have the angles of is equally likely for the architect to pick any of the angles so represented.
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