SOMETHING NEW - IN EGYPTOLOGY THE GEOMETRIC sOLUTION OF THE GREAT PYRAMID By: samuel laboy THE PERFECT symbol solution  based on Phi UNITS (= f) As known, the original geometrical plan of the Great Pyramid’s structure never has been found since it was built 4,500 years ago. Many theories have been exposed to try to explain its geometry, but none of them has been considered acceptable by mainstream Egyptologists, they continue to be only theories. However, my studies and research on these matters allowed me to design and be able to present to all a new geometrical designed plan which finally exactly fits the geometry of the Great Pyramid. My designed plan proves to be simple, accurate, and that corresponds section by section to the entire structure of the Great Pyramid. My geometrical design method allows designing other pyramids as well. Although, mainstream Egyptologists are still resistant to look into my findings, I know they will. The reason is that this is not another theory about the pyramids, this is a fact, something real and that eventually, they will concur with my findings. I exposed my findings, now the time involved in getting into them depends on the scholars.  In my four decades personal research about the Egyptian pyramids, I cover the following items, which I discussed in my report books. In was my idea that for the design of the Egyptian pyramids, and to provide uniformity, it was required the use of an architectural template. I spent several years looking to create this type of template. Finally, I discovered or created, the geometrical template I was looking for. This Symbol, which consists of a circle, a triangle and a square, was created using only a straightedge and a compass as tools, and  independently from the pyramid's structures dimensions, or shape. However, in the contrary, It can be said and evidenced, that it is  the real origin of the pyramid's designs and their constructions. Let’s take a look to this architectural template, and its geometric arrangement. As it will be seen, the unchanging arrangement of the Perfect Symbol figure can be proportionally increased or decreased by changing its radius. The radius can be expressed in any unit of measurements, be in Egyptian cubits, feet, inches, meters, or any unit of our interest. Therefore, all their sections in the figure are proportional  to the radius of the circle, and will be expressed in the same measuring units. PERFECT SYMBOL Copyright © 2010 Samuel Laboy  All Rights Reserved Observe iin the figure that the triangle between points Q, H and T, with its base at the horizontal diameter AB, looks like the vertical cross section of a pyramid, where the pyramid’s height would be the vertical distance OQ. The face-angle of the pyramid would be determined by the ratio of its diameter (D) divided by (b), which represents its side lengths. This angle in the template is constant, no matter the size of the designed pyramid. It will be also observed that the distance HT represents the side length of the pyramid’s base, and also the sides of the symbol’s square. These settings are just perfect for the design of a pyramid. Therefore, using this template arrangement, the size, characteristics and properties of the conceived pyramid’s cross section, in the figure  can be calculated. In the figure, the angles are constant, and all dimensions are proportional. In this case, each time the size of the radius is changed, we will have a different size for the entire pyramid. Additionally, the units of measurements will always correspond to those units used for the radius. In my last two books, and my videos presentations, I illustrate and demonstrate the geometric process I used to design the pyramid’s model, without the use of numbers, calculations, formulas, or any reference to the Great Pyramid’s dimensions. My geometric designed and dimensionless plan of the model pyramid included all the exterior and interior structural elements locations, as observed in the Great pyramid’s structure. The following figure shows how the dimensionless plan of the Great Pyramid looks. DIMENSIONLESS PLAN OF THE GREAT PYRAMID By: Samuel Laboy In order for the cross sectional view of the conceived pyramid in the symbol  to represent the cross sectional view of the Great Pyramid; both cross sections will have to be exactly equal. It is important to note that only one number for the radius, from an infinite numbers available, will create the entire  dimensional scale of the conceived pyramid. Each time the radius is changed, a new pyramid will emerged  with new dimensions. Therefore, we have to be lucky enough to select a number and unit of measurement for the radius of the designed model, so that it coincides with the height of the Great pyramids. However, besides been of equal heights, all angles, exterior dimensions, and all interior dimensions and elevation of chambers, will have to be equal.  In other words, if the Great Pyramid’s structure is really framed by the Perfect Symbol arrangement, as stated, the dimensions; calculated in the designed symbol, and those measured in the Great Pyramid, will correspond section by section and will be exactly equal. In my books and the DVD I show how to design the entire dimensionless pyramid's model based only on he Perfect Symbol. I also select a number and unit of measurement to convert my dimensionless plan in a real construction plan. Finally, I presented a comprehensive and detailed comparison, to demonstrate that both plans, my designed one and that of the Great Pyramid are incredibly exactly equal. Any person, in any country of the world, with basic knowledge in geometry and mathematics, can verify or rebuild my entire exercise and will obtained equal results and conclusions as presented here. Therefore, it is clear that this is not another theory about the pyramids; it is a fact, something real. Is it possible that I can guess thousands of measurements, angles, elevations, properties and characteristics of the largest man-made structure of the world?  My designed plan, besides giving the exact dimensions, angles, properties and characteristics of the Great Pyramid’s structure, reveals the solution to many unanswered questions about this famous structure. As a matter of fact, there are many so-called mysteries of the great Pyramid, which are clearly exposed by this designed plan.   The following geometric plan of a pyramid corresponds to the calculated dimensions of the author’s dimensionless and independently designed plan model for the Great Pyramid’s structure. It exactly corresponds, section by section with the dimensions, characteristics and properties that have been measured and observed in the Great Pyramid In my books, and DVD-Video, as well in my website http://www.samuellaboy.com, is exposed “free” all this important material for reading and uploading of those interested, both in my English book  and in my Spanish book, where you can verify my claims and the importance of this findings and discoveries. It can be concluded that this is an important discovery for the world and for the Egyptian History. It can be said that if the Egyptian Engineers did not use the Perfect Symbol for the design of the Great Pyramid, they will have paid any amount of gold to buy and obtain this system for their designers. DIMENSIONS CALCULATED FOR THE DIMENSIONLESS PLAN By: Samuel Laboy THE DIMENSIONS IN THIS PLAN WERE CALCULATED AND SET INDEPENDENTLY FROM THOSE DIMENSIONS MEASURED AT THE GREAT PYRAMID’S STRUCTURE, NEVERTHELESS THEY WERE FOUND TO BE EXACTLY EQUAL. THEREFORE, THIS PLAN REPRESENTS THE GREAT PYRAMID’S “BLUE PRINT” PLAN FOR ITS CONSTRUCTION. KNOWN GREAT PYRAMID'S SKETCH    NEW FORMULAS FROM SAMUEL LABOY TO DETERMINE GREAT PYRAMID’S DIMENSIONS   THE FOLLOWING FIGURE REPRESENTS THE GEOMETRIC SOLUTION OF THE PERFECT SYMBOL EXPRESSED IN PHI UNITS. THIS CONSTITUTES EVIDENCE THAT THE PERFECT SYMBOL IS THE ORIGIN OF THE GREAT PYRAMID’ STRUCTURE AND REPRESENTS ITS CONSTRUCTION PLAN. Since my Perfect Symbol represents the frame of the Great Pyramid’s structure, I developed a group of formulas which are applicable to its structure. By means of these formulas, the dimensions for any pyramid, proportional in shape to the Great Pyramid, can be easily calculated. You can use any unit of measurements for the radius in the formulas, be in Egyptian cubits, in meters, feet, inches, or any other measurement unit of your preference. The results will be obtained in the same units you select for the diameter. "PERFECT SYMBOL FRAME" AND "GEOMETRIC SOLUTION" FOR THE GREAT PYRAMID'S STRUCTURE PLAN, BASE ON THE PHI (= f) FUNCTION The following formulas are to calculate the dimensions of the Perfect Symbol, based on a radius equal to R = 1. When R is set up equal to any other desire number for the pyramid’s height, the results of the formulas will represents the dimensions for that pyramid. When the radius is set up as 480.6637 ft., the formulas will apply to the Great Pyramid’s structure and the Symbol's drawing will be transformed in the entire "frame plan" for the Great Pyramid. Symbolisms used in the Perfect Symbol in reference to the Great Pyramid’s structure, to determine the dimensions between the structural sections using the indicated formulas. The following symbolisms between the Great Pyramid’s segments and the Perfect Symbol are adopted to work with my geometrical method: The radius of the circle will correspond to the height of the pyramid. For example, if you want to build a replica of the Great Pyramid 60 inches height, substitute the 60 inches for the radius in the formulas. The result will be equal to the require dimension. If you are interested in the Great Pyramid’s dimensions, use 480.6637 ft. for the radius. In this special case, you can corroborate the results with those lengths which have been measured at the pyramid’s structure. As stated, the radius is the only numerical value to be input to the formulas to determine the particular dimensions. 1. The radius of the circle (R) represents the height of the pyramid. 2. The horizontal diameter (AB = D = 2R) represents the leveled terrain use for the pyramid’s design. 3. The vertical diameter (QK = D = 2R) represents the pyramid's vertical axis. 4. The triangle QHT represents the Pyramid's vertical cross-sectional view, as seen in a plane cut through the center of the pyramid’s sides. 5. The area enclosed between points P, H, T, and N, is located below the ground level of the pyramid’s base use for the design of the pyramid’s underground work. 6. The Perfect Symbol’s square, represents the square base of the pyramid, as seen in a vertical plane. The distance HT represents the pyramid's side length. Remember, to change from one unit of measurement to another, you have to use their corresponding equivalencies. In this case, since we are more acquainted with the English System of Measurements than with the Egyptian cubits, the calculations will be shown in feet and decimal units. However, you can used any unit of measurement as the radius.   FORMULAS DEVELOPED BY: SAMUEL LABOY   ●          To calculate the face angle of the Great Pyramid,  using a unitary circle (R = 1): The slope determination of the pyramid’s inclined face,  = (D / b) = (2)  / (2 / √ f) = √ f = 1.27201965. The corresponding angle for this ratio is 51.82729237° = 51° 49’ 38.25”. ●          To determine the side lengths (b) of the pyramid’s square base: (between points I, II, III, and IV)  = D / √f = 961.327352 / 1.27201965 = 755.7488 ft. In 1881 a British scholar, Sir Flinders Petrie, who was called the “Father of modern Egyptian Archaeology,” undertook the most accurate survey, up to that time, of the Great Pyramid and other monuments on the Giza Plateau. Sir Flinders Petrie measured the base of the pyramid very accurately. He said: “the mean base value was 9,068.8 ±.5 inches (= that is, from 755.70 ft. to 755.77 ft.). He said that this is the only measurement, from all reported in the outside of the pyramid, which can be considered certain”. My calculation for the model’s squared base sides is 755.7488 ft. which represents the average of Petrie’s measurement. So, it can be considered in complete agreement with his measurements. If Petrie had calculated the face angle from his own numbers, he would have determined a more precise angle for the pyramid and its height. The face angle is given by the tangent function of (D / b), or 480.75 (2) / 755.75 = 1.272246113. Using his numbers, the angle with this tangent function is 51.83224799° = 51° 49’ 56”. However, Petrie finally reported his average measured angle as 51° 52’. The correct angle calculated from the Perfect Symbol is 51° 49’ 38.25” Petrie also measured the pyramid’s height as 5,776.00 inches (481.333 ft.), plus or minus 7 inches. This means that any height from 5769 inches (= 480.75 ft.) to 5,783 inches (= 481.916) could be acceptable. However, his calculations were based on an assumed face slope angle of 51° 52’, more or less 2’. Therefore, our calculations of 480.6637 ft. for the pyramid’s height and 755.7488 ft. for the base length are correct. So is the face angle of the pyramid as 51° 49’ 38.25”. OTHER FORMULAS TO CALCULATE IMPORTANT DIMENSIONS FOR THE GREAT PYRAMID: ●          Vertical distance from point Q to point X: = D / f = 961.327352 / f = 594.1330 ft. Vertical distance from point Q to point X’: = D / f2 = 961.327352 / f2 = 367.1944 ft. ●         Vertical distance from point X’ to point X:             = D / f3 = 961.327352 / f3 = 226.9386 ft. ●         Vertical distance from point O to point X:             = R / f3 = 480.6637 / f3 = 113.4693 ft. ●          Inclined distance from point Q to point P, or from point Q to point N: D / √f = 961.327352 / 1.27201965 = 755.7488 ft. Curiously, this distance coincides with the pyramid’s side length. ●          Inclined distance from point K to point P   = Distance from point K to P = Distance from point K to point N = (√5 -1) (R) = (√5 -1) (480.6637) = 594.1330 ft.   ●          Inclined distance from point H to point Q = or from Q T = the apothem of the pyramid. = (√f) (R) = (√f) (480.6637) = 611.41 ft.   ●          Vertical distance from center point O to point X: = (480.6637 / f3) = 113.4693 ft.   ●          Vertical distance from point O to point X’: = (480.6637 / f3) = 113.4693 ft.   ●          Vertical distance from point X to K: = (3 - √5) (R) = (3 - √5) (480.6637) = 367.1944 ft.   ●         Horizontal distance from point P to point N: = (4 R / √f3) = (4)  (480.6637) / (√f3) = 934.1569 ft.   ●          The ratio of the area of the square that circumscribe the Great Pyramid’s design circle, (961.327352)2,  and the area of the pyramid square base (755.7488)2 is equal to = f. The results of all these calculations, and others presented in my books, revealed without any doubt that all dimensions calculated for the Perfect Symbol using a radius equivalent to 480.6637 ft., exactly correspond to the dimensions measured at the Great Pyramid’s structure. All additional supporting data and formulas for the calculated dimensions are explained in my Report Book “A Civil Engineer looks at the Great Pyramid” and its Supplement”. I have proven using direct calculations, and by the use of formulas, that the Perfect Symbol is the real origin of the design plan of the Great Pyramid. This presentation definitely and conclusively proves that the radius of 480.6637 ft. is the only number that correctly fits and frames the Great Pyramid’s dimensions. If you are interested in using other units of measurements, just set the number of units from the other system equivalent to 480.6637 ft.   A. THE FUNCTIONS OF PI = π B. THE FUNCTION OF PHI = f C. THE NUMBER 153 IN THE GREAT PYRAMID π It is being said that the functions of π and f, were created and used as parameters by our Creator for the proportioning of the entire universe. The π function is equivalent to 3.14159… where the end of its decimal point had been calculated up to billions of decimal places and never shows its end, it is generally shortened to 3.1416. This function is used in Math’s and Geometry, especially to determine the circumference of a circle, which is a common geometric figure in our existence. The figure of the Sun, represented by a circle, symbolizes the Sun God in many cultures. The circumference of the circle is equivalent to the product of its diameter by π, or C = D π.  f From the other side, the Golden Proportion, Golden Mean, Divine proportion, or the f function and other names, is currently identified in Mathematics and Geometry, with the Greek letter f, and is equivalent to 1.6180339... As the function of π, its decimal point never ends. The f function is a fundamental part of the Perfect Symbol, and takes care of the design’s beauty and symmetry of the figures. Apparently, it has been used by architects, engineers, painters, etc. to compose their artwork. It is created by means of a series of numbers, by a square or from a triangle with the ratio 1:2 in its sides, and from sections of the figure of a circle. The Perfect Symbols offers a very good example of the properties and characteristics of this function. The symbol constitutes and serves as the conversion “key” between the circle, the triangle and the square. It appears that f was used by our Creator to set the organization of the planets around the Sun and to set the circular and outward appearance of the shapes of the galaxies. Let’s make clear, the fact that the Egyptians used cubits as their measuring units, did not deprive them from using the Perfect Symbol, or the f function. Both can be used with any units of measurement. However, and almost for sure, the Egyptians could have used other names, or symbols to identify the f function, and the Perfect Symbol. The Egyptian’s knowledge of the functions of f and π is not confirmed yet by the surviving written papyri, or mathematical tablets about ancient mathematics. However, the two functions had been clearly exposed in their construction works.  The two constants and their functions, appears constantly in the structural properties and proportions of their structures and designed objects. Both the π and f functions are in-built parts of the circle and other geometric figures, like squares, and triangles. To find them in as-built structures is natural. The perfect Symbol, provided by Geometry, constitutes a “bridge” to go from one geometrical figure to the other. For example, when the diameter of the circle is multiplied by Pi, we get the circumference of the circle, but when the same diameter is multiplied by f3, give us the perimeter of the square base. These are unique conditions which relate the circumference of the circle set up to design a pyramid, with the perimeter of its square base. These unique properties were applied to the Red Pyramid and the top section of the Bent Pyramid.  153 This number when multiplied by π and expressed in feet gives us the vertical height of the Great Pyramid. The number 153 generates many curiosities. It is mentioned in the Bible, Verse John 21:11, to indicate the number of fishes (153) brought in by his disciples, the fishing-net, to Jesus. This number also represents the sum of numbers from 1 to 17. That is, 1 + 2 +3 + 4+ … 17 = 153. It also represents the sum of the cubit of each number (1)3 + (5) 3 + (3)3 = 153, and many other curiosities. I will explain my findings about the manifestations of this number in the Great Pyramid’s dimensions. It is known that the Egyptians use cubits as their units of measurements. The smallest unit was one cubit. This is something that really could have happened and which explains some fundamentals events in the ancient Egyptian life. My suggestion fulfills the empty space to arrive to this extraordinary number, repeatedly found in measurements in ancient Egyptian constructions. As known by the Egyptian constructions, the Egyptian engineers knew about the properties of the f function and wanted to introduce it in their cubits system. With this purpose, the Egyptian engineers divided their basic unit of measurement, (1) cubit, by f  = (1.6180339), and obtained (1 / f) cu. = 0.6180339 cu. This Egyptian unit is equivalent to the inverse function of f,  or  1 / f. For example, they needed to construct a measuring wooden rod of 12 unit’s length, it would be equivalent to 12 (1 / f) = 7.416407865 cu. So, their first scaled rod length will measure 7.416407865 cu., consisting of 12 units of (1 / f) = 7.416407865 cu. According to the famous scientist Isaac Newton and Sir W. Petrie’s calculations, one (1) Egyptian cubit is equivalent to 20.63 inches in the English System. Therefore, 7.416407865 cu is equivalent to 7.416407865 (20.63) = 153.000 inches… Eureka!!! Consequently, the first Egyptian measuring wooden rod was 153 inches, which represents 7.416407865 cubits. So, their first scaled measuring rod length measured 7.416407865 cu and was equivalent to 153 inches. It means that when they measure 1 Egyptian rod length, they were really measuring 153 inches in the British System. Since 153 inches is equal to (12 / f) cu.), if they wanted a longer measuring rod, for example an exact 12 cubits length rod, would be equivalent to 153 (f) = 247.5592 inches. Consequently, the Egyptian engineers had available a longer rod length of (12 (f) / (f) cu.), or 12 cubits In a simple way, since 7.416407865 (f) = 12 Egyptian cubits, 153 inches multiplied by (f), is also equivalent to 12 Egyptian cubits. For example, if they used the 7.416407865 cubits rod (= 153 in) = to measure the ceiling of the Gran Gallery which have been measured as 153 ft., the total length would be equivalent to 153 (12) in / 153 in = they measured 12 rods of (153 inches) = 1,836 inches = 153 ft. Petrie’s measurements in the Great Pyramid can be considered a heavy support for this idea. He indicates that the vertical distance from the pavement outside the Great Pyramid to the level of the ceiling of the Antechamber was equal to 153 ft., which is equivalent to say, 12 units of 153 inches, or 12 units of 7.416407865 cubits. It can also be observed that 153 ft. represents the inclined length of the Grand Gallery’s ceiling. From the other part, according to Petrie, the height of the Antechamber from its real base floor to the ceiling is 153 inches, or 7.416407865 cubits. We can see here the use of the two suggested measuring rods applied to some important measures in the Great Pyramid’s structure. However, these are not the only ones. Note that the length of the circumference of a circle having the length of the ceiling of the Grand Gallery, represents the height of the Great Pyramid (153)(3.14159) = 480.6637. As a short summary: 153 (π) ft. = 480.6637 ft. - Great Pyramid’s height 153 inches = (12) (1 / f) cu = (12) (0.6180339 cubits) = 7.416407865 cubits 1 cu = 153 (f) / 12 = 20.63 inches 1 cu = (7.416407865) (f) / 12 12 cu = [(12) (7.416407865) (f)] / 12 153 ft. = 153 (12) in = 1,836 in 1,836 in / 20.63 = 88.996 cu C = 88.996 (π) = 279.59 cu 1 cu = 1.719161113 ft. These numeric equivalencies represent great findings, especially for the Egyptologists, if they desire to explain the manifestation of the number 153 in the Great Pyramid. For example, to use Egyptian cubits: R = 153 (Pi) = 480.6637 ft. R in ft. changed to cubits:     (480.6637 ft.)  (12 in) = 5,767.9641 inches 5,767.9641 inches / 20.63 = 279.5911 Egyptian cubits. To solve the formulas and obtain the results in Egyptian cubits, use R = 279.5911 cu. and calculate the entire dimensionless plan, or use Laboy's formulas.   Formulas developed by Engineer Samuel Laboy for the most important pyramids of Egypt (based on the f function).    R= radius    D = diameter    C= circumference      f = 1.6180339...   Face slope = (D / b)       Face angle = Tan inv.  (D / b)     PYRAMID        SLOPE           SIDE LENGTH         HEIGHT                   ANGLE                          D / b                (b)                               (R)                             ° /'  /"  ------------------------------------------------------------------------------------------------------ Khufu                   Öf               (D / Öf) = 2R / Öf      (b Öf) / 2                  51.82729337°                                                                                                                             51° 49' 38.3"      Chephren          (4Ö2) / f³        (f³R Ö2)/ 4             2Ö2 b / (f³)                 53.17273225°                                                                                                                             53° 10' 21.8"   Mycerinus           2 / f                   (R)(f)                      (b) / f                      51.02655266°                                                                                                                             51° 01' 35.5"   Red Pyramid         4 / f³             (R) f³ / 2                  (2 b) / f³                     43.35819755°                                                                                                                             43° 21' 29.51"   BENT PYRAMID   PYRAMID           SLOPE            SIDE LENGTH        HEIGHT                      ANGLE   Lower angle          = Ö2                    = (R) Ö2               = (b) / Ö2                    54.73561032°                                                                                                                               54° 44' 8.2"   Upper angle         = 4 / f³              = (R) f³ / 2           = (2 b) / f³                    43.35819755°                                                                                                                               43°21' 29.51" --------------------------------------------------------------------------------------------------------------- Known b = 620.00 ft.                                                                                 Lower Angle -  Extended - height   (b) / Ö2           = 620.00 (Ö2)             = 876.8124 ft.   Known b = 722.25 ft. Upper Angle Extended - Height      2 (b) / f³         = 2 (722.25) / f³        = 341.00 ft. ------------------------------------------------------------------------------------------------------------- Examples:  Calculations of the pyramid's heights                                                                                                                               HEIGHT   PYRAMID OF KHUFU = (b Öf) / 2 = (755.7488) (1.27201965) / 2       = 480.66366 ft.             Known b = 755.7488 ft                                                                                         PYRAMID OF CHEPHREN= 2 Ö2 b / (f³) = 2 Ö2 (706.24) / f³            = 471.5572 ft. Known b = 706.24 ft.   PYRAMID OF MYCERINUS = (b) / f = 335 / f = 207.04 ft., (345) / f = 213.22 ft.             Known b = 335.00 ft.                                                                                                    Known b = 345.00 ft.                                                                                         RED PYRAMID = (2 b) / f³ =                     (2) (722.25) / f³                       = 341.00 ft.     Known b = 722.25 ft.                                                                                            Bent Pyramid: Lower Face Ratio               = Ö2 = 1.414213562   Bent Pyramid: Lower Face Angle               = 54.73561032° = 54° 44’ 8.197”   Bent Pyramid: Upper Face Ratio                = 4 / f³ = 0.94427191   Bent Pyramid: Upper Face Angle               = 43.35819755° = 43° 21’ 29.51”   Note: The pyramid of Chephren’s angle corresponds to the product of the two superimposed upper and lower pyramid models, which composed the Bent pyramid.   Bent Pyramid’s upper section ratio = (4 / f³)   Bent Pyramid’s lower section ratio = Ö2   Chephren’s Pyramid: Face Ratio                = (Ö2)(4 / f³) = 1.335402142   Chephren’s Pyramid: Face Angle               = 53   Red Pyramid: Face ratio                              = 4 / f³ = 0.94427191   Red Pyramid: Face Angle                            = 43° 21’ 29.51”