The Great Pyramid & Squaring the Circle
by Andras Goczey
| The modern world has every right to stand amazed at the building accuracy of the Cheops pyramid of theoretically 146.7 m height. It is a well known fact that, according to the intention of builders, the perimeter of the base square of pyramid is equal to the perimeter of the great circle of a hemisphere the radius of which is equal to the height (reference to the Earth). According to our recent knowledge, the building difference compared to the calculated dimension of the base edge lies just between 20 to 30 cm! The basic problem, however, consists of that, when building that Wonder of the World, the above tolerance can only be achieved by using Pi = 3.142 “that was unknown to them”. The use of Pi = 3.16 value attributed to their knowledge according to the Rhind papyrus predestinates an error in the base edge far exceeding one meter! Furthermore, they could not perform divisions and multiplications but only to a limited extent and they could adapt these operations by reducing them to additions! They must have known some more accurate Pi value that could be used without some particular calculations and drawing even by means of additions to solve their task! Might it be as follows? The most natural human measuring unit is the span (it is not this that popularized with them). In this particular case, suppose this span equal to around 14 to 15 cm. In the „Rhind papyrus”, the one is a mark of stick form. |
Neither places of value nor fractions were used, therefore:
if the height of Cheops Pyramid is supposed to be equal to 1000 stick unit, i.e. one lotus flower, the use of decimal system allows the bravura to be performed as follows:
1. Reference to the Sun:
It is well known that the height supposed to be equal to a lotus flower i.e. thousand sticks = 146.7 meter multiplied by thousand by thousand by thousand, i.e. by billion gives the distance to the Sun (it is well known that the minimal distance between the Sun and the Earth is equal to approx. 147 million kilometer).
2. Reference to the Earth:
If the height of 1000 units, i.e. the distance of one lotus flower is assumed to be equal to the radius of a semisphere (reference to the Earth), then the perimeter of the great circle of this semisphere shall be equal to the perimeter of the basic square of the Cheops Pyramid according to the message of the builders.
Measure a section of thousand stick length (lotus flower) on the half diagonal of this square first, then one of hundred sticks length (helix) from its end point, then one of ten sticks length (hairpin) from the end point of the previous two sections, then one of one stick length from the end point of the previous three sections; then, a simple addition of these four part sections gives the length of the half diagonal of the requested square requested to be 1111 units! — (András Goczey)
Performing the verification for a circle with radius of 1000 units i.e. r = 1000, the above drawing procedure for the diagonal of the requested basic square gives 1111
The half edge of the basic square is given by dividing 1111 by square root of two.
Multiplying this ratio by four results in the half perimeter which gives the thousands of Pi i.e. 3142!
Half of the perimeter of a circle with radius of 1000 units is equal to the Pi-fold of its radius (Pi = 3.142) i.e. 3142.
It follows that, by using the above method of drawing, the perimeter of the basic square is equal to that of the great circle. Thus, the task can be solved!
Copyright by András Goczey
Did you Know?
“According to the decision adopted by the Hungarian Academy of Sciences in the mid of 19th century: „dissertations dealing with quadrature of the circle, section of circle into three sections, the invention of perpetuum mobile will be refused without consideration”
From World Mysteries @ http://blog.world-mysteries.com/science/cheops-pyramid-squaring-the-circle-pi-and-the-decimal-system/
Geometry of the Great Pyramid
DIMENSIONS of the Great Pyramid
by morphvs
The following article is Copyright © 2001-2003 aiwaz.net_institute.
All rights reserved.
All rights reserved.
The base of 9069 inches is approximately 440 royal cubits (the difference is 9 inches which is not a remarkable difference if we consider the whole dimension and consider that the employed data represent only an estimation of the real values) whereas the calculated height, 5776 inches, is precisely 280 royal cubits. The relation 440:280 can be reduced to 11:7, which gives an approximation of the half value of Pi.
|
The engagement of Pi value in the main dimensions suggests also a very accurate angle of 51° 52' ± 2' of the slopes which expresses the value of Pi. Another coincidence is the relation between the height of the pyramid's triangle in relation to a half of the side of the pyramid, since it appears to be the Golden Section, or the specific ratio ruling this set of proportions, F = (sqr(5)+1)/2 = 1.618 = 356:220. This ratio, 356:220 = 89:55 is also contained in the first of Fibonacci Series:
1 2 3 5 8 13 21 34 55 89 144 ...
But analysis of other architectural and artistic forms suggested that the greatest masters skillfully juggled the proportional canons without losing the coherent system, for they knew that these systems can be interconnected if the path that links them is found.
That is obvious In the case of the Great Pyramid where two different principles are interweaved without interference ruling different angles of the composition, which is most importantly a most simple one, namely 11:7, a most simple ratio obviously signifying such infinite mysteries as the value of P and most 'natural' value of F. In spite of common miss-understanding of architectural composition, the most mysterious and praised compositions are very simple but not devoid of anthropomorphic appeal, since everything is made out of human proportions, just like Vitruvius describing the rations of the human body, very simple and very clean. The numbers 7 and in 11 are successive factors in the second of Fibonacci progressions that approximate geometry of the pentagram:
1 3 4 7 11 18 29 47 76 123 ...
The summary of the selected main mean dimensions is:
dimension | b. inch | m | royal cub. | palm | digit |
base | 9068.8 | 230.35 | 440 | 3,080 | 12,320 |
height | 5776 | 146.71 | 280 | 1,960 | 7,840 |
sum | 720 | 5,040 | 20,160 | ||
slope | 7343.2 | 186.52 | 356 | 2,492 | 9,968 |
edge | 8630.4 | 219.21 | 418 | 2,926 | 11,704 |
The main source of all kinds of delusions and speculations about our mythical past for the western man comes of course from Plato. With the myth of Atlantis he planted the necessary seed of mythical Eden, a culture of high intelligence that lived before the known history. If Plato received any wisdom from the ancient Egypt it could perhaps be traced in the canon of numbers that is so latently present throughout his work, but never on the surface. This canon seems to appear in the descriptions of his fantastic cities where everything is most carefully calculated and proportioned. The topic of Plato's Laws is the description of the ideal state called Magnesia which is entirely composed out of the mysterious number 5,040.
The distance* when Earth is closest to Sun (perihelion) is 147x106 km, which is translated into royal cubits 280x109, hinting at the height of the Great pyramid,
280 royal cubits.
The above article comes from aiwaz.net_institute - Great Pyramid and Giza plateau and is Copyright © 2001-2003 aiwaz.net_institute. All rights reserved.
* Related links: Astronomic & Cosmographic Data, Nasa site with planetary data
The Golden Ratio & Squaring the Circle in the Great Pyramid
A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less. [Euclid]
The extreme and mean ratio is also known as the golden ratio.
If the smaller part = 1, and larger part = G, the golden ratio requires that
G is equal approximately 1.6180
Does the Great Pyramid contain the Golden Ratio?
Assuming that the height of the GP = 146.515 m, and base = 230.363 m, and using simple math we find that half of the base is 115.182 m and the "slant height" is 186.369 m
Dividing the "slant height" (186.369m) by "half base" (115.182m) gives = 1.6180, which is practically equal to the golden ration!
Although the problem of squaring the circle was proven mathematically impossible in the 19th century (as pi, being irrational, cannot be exactly measured), the Earth, the moon, and the Great Pyramid, are all coming about as close as you can get to the solution!
If the base of the Great Pyramid is equated with the diameter of the earth, then the radius of the moon can be generated by subtracting the radius of the earth from the height of the pyramid (see the picture below).
Click here to view larger picture.
Also the square (in orange), with the side equal to the radius of the Earth, and the circle (in blue), with radius equal to the radius of the Earth plus the radius of the moon, are very nearly equal in perimeters:
Orange Square Perimeter = 2+2+2+2=8
Blue Circle Circumference = 2*pi*1.273=8
Blue Circle Circumference = 2*pi*1.273=8
Note:
Earth, Radius, Mean = 6,370,973.27862 m *
Moon, Radius, Mean = 1,738,000 m.*
Moon Radius divided by Earth Radius = 0.2728 *
Earth, Radius, Mean = 6,370,973.27862 m *
Moon, Radius, Mean = 1,738,000 m.*
Moon Radius divided by Earth Radius = 0.2728 *
* Source: Astronomic and Cosmographic Data
Let's re-phrase the above arguments **
This new circle will actually pass exactly through the apex of the pyramid. And now the “wow”: A circle drawn with its centre at the apex of the pyramid and its radius just long enough to touch the earth circle, will have the circumference of the moon! Neat, huh! And the small triangle formed by the moon and the earth square will be a perfect 345 triangle (which doesn’t seem to mean much.)
Was the golden ratio intentionally built into the Great Pyramid of Cheops?
Why would anyone intentionally build the golden ratio into a pyramid, or other structure? What was the significance of to the Egyptians? And did the ancient Egyptians intentionally design the Great Pyramid to square the circle?
The answer to these questions is uncertain since designing the Great Pyramid according to the simple rules explained by the graphic below would give the pyramid automatically (by coincidence? ) all its "magic" qualities.
The height of the Great Pyramid times 2π exactly equals the perimeter of the pyramid. This proportions result from elegant design of the pyramid with the height equal two diameters of a circle and the base equal to the circumference of the circle. Click here or on the image below to see larger picture.
| Angle of casing measured By theory of 34 slope to 21 base Height : circumference :: radius to circle 9 height on 10 base diagonally 7 height to 22 circumference area of face = area of height squared (or sine) = cotangent, and many other relations) 2 height vertical to 3 height diagonal 5 height on 4 base | 51º 52' ± 2' (51.867) 51º 51' 20" 51º 51' 14.3" 51º 50' 39.1" 51º 50' 34.0" 51º 49' 38.3" 51º 40' 16.2" 51º 20' 25" |
** Page 184, The Pyramids and Temples of Gizeh
by Sir W.M.Flinders Petrie 1883
Comparing the Great Pyramid with the Pyramid of the Sun in Teotihuacan
The Pyramid of the Sun and the Great Pyramid of Egypt are almost or very nearly equal to one another in base perimeter. The Pyramid of the Sun is "almost" half the height of the Great Pyramid. There is a slight difference. The Great Pyramid is 1.03 - times larger than the base of the Pyramid of the Sun. Conversely, the base of the Pyramid of the Sun is 97% of the Great Pyramid's base.
The ratio of the base perimeter to the height:
Great Pyramid | Pyramid of the Sun |
6.2800001... : 1 (deviates by 0.05 % from the 6.2831853 value for 2 x pi) | 12.560171... : 1 (deviates by 0.05 % from the 12.566371 value for 4 x pi) |
Comparing the Great Pyramid with the Pyramid of the Sun in Teotihuacan
The Pyramid of the Sun and the Great Pyramid of Egypt are almost or very nearly equal to one another in base perimeter. The Pyramid of the Sun is "almost" half the height of the Great Pyramid. There is a slight difference. The Great Pyramid is 1.03 - times larger than the base of the Pyramid of the Sun. Conversely, the base of the Pyramid of the Sun is 97% of the Great Pyramid's base.
The Great Pyramid - Metrological Standard
The Great Pyramid is generally regarded as a tomb and as grandiose memorial to the pharaoh who commissioned it. The opposing view is that of the pyramid being the culminating achievement of those who practised an advanced science in prehistory.
The Great Pyramid is a repository of universal standards, it is a model of the earth against which any standard could be confirmed and corrected if necessary.
It is exactly the imperishable standard, which the French had sought to create by the devising of the metre, but infinitely more practical and intelligent.
From classical times, the Great pyramid has always been acknowledged as having mathematical, metrological and geodetic functions. But ancient Greek and Roman writers were further removed in time from the designers of the Great Pyramid than they are from us. They had merely inherited fragments of a much older cosmology; the science in which it was founded having long since disappeared.
The Following articles are © 2000 by Larry Orcutt, Catchpenny Mysteries
The Concave Faces of the Great Pyramid
Aerial photo by Groves, 1940 (detail).
In his book The Egyptian Pyramids: A Comprehensive, Illustrated Reference, J.P. Lepre wrote:
One very unusual feature of the Great Pyramid is a concavity of the core that makes the monument an eight-sided figure, rather than four-sided like every other Egyptian pyramid. That is to say, that its four sides are hollowed in or indented along their central lines, from base to peak. This concavity divides each of the apparent four sides in half, creating a very special and unusual eight-sided pyramid; and it is executed to such an extraordinary degree of precision as to enter the realm of the uncanny. For, viewed from any ground position or distance, this concavity is quite invisible to the naked eye. The hollowing-in can be noticed only from the air, and only at certain times of the day. This explains why virtually every available photograph of the Great Pyramid does not show the hollowing-in phenomenon, and why the concavity was never discovered until the age of aviation. It was discovered quite by accident in 1940, when a British Air Force pilot, P. Groves, was flying over the pyramid. He happened to notice the concavity and captured it in the now-famous photograph. [p. 65]
I.E.S. Edwards wrote, "In the Great Pyramid the packing-blocks were laid in such a way that they sloped slightly inwards towards the centre of each course, with a result that a noticeable depression runs down the middle of each face -- a peculiarity shared, as far as is known, by no other pyramid" (The Pyramids of Egypt, 1975, p. 207). Maragioglio and Rinaldi described a similar concavity on the pyramid of Menkaure, the third pyramid at Giza. Miroslav Verner wrote that the faces of the Red Pyramid at Dahshur are also "slightly concave."
Diagram of the concavity (not to scale).
What was the purpose for concave Great Pyramid sides? Maragioglio and Rinaldi felt this feature would help bond the casing to the core. Verner agreed: "As in the case of the earlier Red Pyramid, the slightly concave walls were intended to increase the stability of the pyramid's mantle [i.e. casing stones]" (The Pyramids, 2001, p. 195). Martin Isler outlined the various theories in his article "Concerning the Concave Faces on the Great Pyramid" (Journal of the American Research Center in Egypt, 20:1983, pp. 27-32):
1. To give a curved form to the nucleus in order to prevent the faces from sliding.
2. The casing block in the center would be larger and would serve more suitably as a guide for other blocks in the same course.
3. To better bond the nucleus to the casing.
4. For aesthetic reasons, as concave faces would make the structure more pleasing to the eye.
5. When the casing stones were later removed, they were tumbled down the faces, and thereby wore down the center of the pyramids more than the edges.
6. Natural erosion of wind-swept sand had a greater effect on the center.
The concavity has prompted more improbable theories, usually in support of some larger agenda. David Davidson (cited by Peter Tompkins in Secrets of the Great Pyramid, pp. 108-114) defended the discredited Piazzi Smyth by attempting to demonstrate that if measurements included the hollowing, they would provide three base measurements that describe the three lengths of the year: solar, sidereal, and "anomalistic." (These lines, on the diagram below, would be AB, AEFB, and AMB.) What Davidson is assuming is that the concavity, present today in the core structure of the pyramid, would extend to the finished cased surface. There is no evidence for this; indeed the extant casing is perfectly flat. Maragioglio and Rinaldi observed that the granite casing of Menkaure's pyramid was flat, but above the granite the packing-blocks formed a concavity in the center of each face. The evidence indicates that the concavity is a functional feature of the core structure that was hidden from sight when the casing stones were applied.
Three proposed "baselines" of the Great Pyramid (not to scale).
John Williams, author of Williams' Hydraulic Theory to Cheops' Pyramid wrote that "the only advantage that I can see - and it is a great one - for having a concave face on a structure is to contain extremely high internal pressures - the type of pressures that would result from using a hydraulic method of my description. Think of this in terms of an egg shell, arch or gabling." This explanation is also voiced by other purveyors of the "pump-theory" such as Edward J. Kunkel (author of The Pharaoh's Pump, 1962) and Richard Noone (author of 5/5/2000: Ice: The Ultimate Disaster, 1982). Unfortunately, they fail to understand how an arch or load-bearing gable works. A supporting arch is designed to convert the downward force, or weight, of a structure to an outward force, which in turn is transferred to a buttress, a pier, or an abutment. An arch simply redirects the force; it does not make it vanish. If the sides of the Great Pyramid were designed as arches, then those arches would serve to direct the load into thin air. It doesn't make sense. The eggshell analogy is yet less applicable because the pyramid is not egg-shaped. Like the arch, the egg is strong because it transfers load pressure, in this case into vertical as well as horizontal forces that are distributed more evenly along the structure of the egg due to its shape.
Kunkel likened each pyramid face to a dam. He claimed that each side bends inward against the pressure of the water inside the pyramid just as a dam (Hoover Dam for example) bends towards the force of the water it holds back. An arch dam employs the same structural principles as the arch (described above). The dam curves towards the hydrostatic pressure from the water behind it, which in turn is distributed horizontally to abutments on the side walls against which the dam is built. Again, the pyramid lacks such abutments.
In Ancient Egyptian Construction and Architecture, Clarke and Englebach wrote:
Most pyramids have individual peculiarities which are as yet difficult to explain. For instance, in the Great Pyramid, as possibly in certain others, a large depression in the packing-blocks runs down the middle of each face, implying a line of extra-thick facing there. Though there is no special difficulty in arranging the blocks of a course in such a manner that they increase in size at the middle, there is no satisfactory explanation of the feature any more than there is of the 'girdle-blocks' [in the Great Pyramid's ascending passage] already discussed. [p. 128]
© 2000 by Larry Orcutt, Catchpenny Mysteries
The Great Pyramid's "Air Shafts"
While shafts in the King's Chamber had been described as early as 1610, the shafts in the Queen's Chamber were not discovered until 1872. In that year, Waynman Dixon and his friend Dr. Grant found a crack in the south wall of the Queen's Chamber. After pushing a long wire into the crack, indicating that a void was behind it, Dixon hired a carpenter named Bill Grundy to cut through the wall. A rectangular channel, 8.6 inches wide and 8 inches high, was found leading 7 feet into the pyramid before turning upward at about a 32º angle. With the two similar shafts of the King's Chamber in mind, Dixon measured a like position on the north wall, and Grundy chiseled away and, as expected, found the opening of a similar channel.
The men lit fires inside the shafts in an attempt to find where they led. The smoke stagnated in the northern shaft but disappeared into the southern shaft. No smoke was seen to exit the pyramid on the outside. Three artifacts were discovered inside the shafts: a small bronze grapnel hook, a bit of cedar-like wood, and a "grey-granite, or green-stone" ball weighing 8.325 grains thought to be an Egyptian "mina" weight ball.
Shafts and passages of the Great Pyramid at Giza.
The Shafts of the Queen's Chamber Described
The openings of both shafts are located at the same level in the chamber, at the joint at the top of the second course of granite wall-stone; the ceilings of the shafts are level with the joint.
The northern shaft runs horizontally for just over six feet (76"), then turns upward at a mean angle of 37º 28'. The shaft terminates about 20 feet short of the outside of the pyramid. The total length of the northern shaft is about 240 feet and rises at an angle of 38º for the majority of its length.
The southern shaft also runs horizontally for just over six feet (80"), then turns upward at a mean angle of 38º 28'. The total length of the southern shaft is about 250 feet and, as its northern counterpart, ascends at an angle of 38º for the majority of its length and comes to an end about 20 feet short of the outside of the pyramid.
The Shafts of the King's Chamber Described
The openings of both shafts are located at roughly the same level in the chamber, at the joint at the top of the first course of granite wall-stone. The northern opening is slightly lower, its ceiling being level with the joint, while the floor of the southern opening is roughly level with the joint.
The northern shaft is rectangular, about 7 inches wide by 5 inches high, a shape it maintains throughout its length. The shaft begins on the horizontal for about 6 feet then takes a series of four bends. While maintaining its general upward angle, it shifts first to the north-northwest then back to north, then to north-northeast, and finally back to true north. It has been speculated by some that this unexplained semicircular diversion might have been necessary to avoid some heretofore undiscovered feature of the pyramid. The total length of the northern shaft is about 235 feet and rises at an angle of 31º (with a variation of between 30º 43' and 32º 4') for the majority of its length.
Though the first eight feet of the northern shaft is intact, the next thirty or so feet have been excavated by treasure seekers, presumably following the direction of the shaft in search of treasure. The breach to the shaft was made in the west wall of the short passage leading from the antechamber to the King's Chamber. A modern iron grate today guards the mouth of this breach.
The southern shaft is different in appearance. Its mouth is larger, about 18" wide by 24" high. The dimensions are reduced to about 12" by 18" within a few feet, and then narrows yet more to about 8" by 12". The shape is not rectangular, as is the northern shaft, but has a dome shape where it enters the chamber, with a narrow floor, the angle of the walls being slightly obtuse, and a dome-shaped ceiling. The shaft is horizontal and true south for about 6 feet. At the first bend, its shape changes to an oval, and continues thusly for about 8 feet. Its orientation also changes slightly from true south to south-southwest.
At the second bend its shape changes yet again to a rectangle, with a height greater than its width. It retains this shape for the 160 feet to the outside of the pyramid where it emerges at the 101st course of stone. It also changes directions once again at the second bend to a more severe south-southwest diversion. The total length of the southern shaft is about 175 feet and ascends at an angle of 45º (with a variation of between 44º 26' and 45º 30') for the majority of its length.
The Function of the Shafts
When Sandys described the Great Pyramid in 1610, he wrote of the shafts:
In the walls, on each side of the upper room, there are two holes, one opposite to another, their ends not discernible, nor big enough to be crept into -- sooty within, and made, as they say, by a flame of fire which darted through it.
Greaves also wrote of the King's Chamber shafts in 1638. Considering the presence of the lampblack inside, he concluded that the shafts had been intended as receptacles for an "eternal lamp." In 1692, M. Maillet wrote that the shafts served as means of communication for those who were buried alive with the dead king. Not only did the shafts provide air, he reasoned, but they also provides a passage for food which was placed in boxes and pulled through by rope.
A symbolic function should also be attributed to the so-called "air-shafts," which had nothing to do with conducting air. No other pyramid contains chambers and passages so high in the body of masonry as Khufu's and so the builders provided the King's Chamber with small model passages to allow the king's spirit to ascend to the stars. (The Complete Pyramids, 1997, p. 114)
There are also, however, reasons why it is not likely that the shafts were meant to serve as "launching ramps" for the king's ka. When, in 1964, Alexander Badawy and Virginia Trimble determined that the shafts are "aimed" at certain "imperishable" circumpolar stars and at the constellation of Orion, the function of the shafts as cultic features seemed certain. But the ka did not require a physical means of egress from a tomb -- false doors served this purpose quite nicely both before and after Khufu's reign. The passage that ascends to the entrance of the pyramid is also directed at the circumpolar stars in the manner of previous pyramids. The northern shafts for such a use would have been a needless and bothersome redundancy, although admittedly the Egyptians were not adverse to redundancies.
That fact that no other pyramid in Egypt is known to posses similar shafts as those of the Great Pyramid is problematic. If the shafts were so important for either ventilation or as passages for the king's ka, then why were they omitted in other funerary structures? It is obvious that the builders of Khufu's pyramid went to a jolly lot of trouble to incorporate the shafts into the design of the pyramid, but the true reason why still remains a mystery.
© 2000 by Larry Orcutt, Catchpenny Mysteries
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Sorry! It is in hungarian, but very interesting!
ReplyDelete"Giza helyszínrajz Nap-Hold-youtube"
best wishes, Gőczey András from Budapest