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THE GREAT PYRAMID MAY 2002
THE MATHEMATICS OF THE FOUR SHAFTS
By Jeremy Potter
Preliminary Information
It is well known that there are two shafts rising upwards from the King’s Chamber and a further two rising up from the Queen’s Chamber. The purpose of these shafts has always been a matter of conjecture, especially since it was discovered that the two lower shafts were built closed at both ends. The answer was that the shafts were actually diagonals on squares and rectangles seen on a pyramid section east to west and viewed from a distance. None of the slope angles at the Great Pyramid were measured in degrees, minutes, and seconds, but in tangents, and they were taken from a 90-degree triangle formed by half a pyramid on its base. The tangent of the pyramid base angle was found by dividing the height by half the base. The distances and angles on the pyramid, mainly in whole number cubits, could then be obtained by multiplying or dividing on the number given by the tangent. It was very simple but very profound. The tangential value of its base angles will then have been 280 / half-base 220 = Tan 1.272727 the original design slope angle of the Great Pyramid and given in number form. The equivalent value in degrees can then be found from the Standard Four Figure Mathematical tables issued by Cambridge University to give a slope of 51 degrees 50 minutes 30 seconds, or just less than 52 degrees, as commonly assumed.
Nothing more could be found at the Great Pyramid without an accurate survey given in architectural format so as to avoid ambiguity. The Italian engineers Rinaldi and Maragioglio had carried out such a survey in 1965 and they had given the existing distances and levels of the Great Pyramid on a set of large-scale drawings. Without recourse to those drawings it would be impossible to draw any reliable conclusions. They were used as a reference during the investigations that follow but the survey was done in metres and that made it necessary to convert metres to cubits, the unit used by the original builders.
The question was, how many cubits were there to the metre. It was an absurdity because the metre did not exist at the time of the pyramids whereas the unit foot might have done. The Cole Borchardt Survey of 1925 had found the original corner stones buried in the sand and they found that the distances between the four corners were very close to 756 feet. The builders had been trying to build a square of sides 756 feet and the differences were due to simple construction error. The pyramid might have been 280 cubits high and 440 cubits on its base, before its casing stones were taken, and if that were true then the correct conversion was 756 / 440 = 1.718181 feet per cubit. The British Museum says that it was 1.7183 feet per cubit. The difference is small indeed.
The geometrical principles that had been adopted for the planning of this unique pyramid were pyramids inside pyramid squares. The first of these was the Great Pyramid Square whose sides were 440 cubits inside which was the Great Pyramid on its base and 280 cubits high. The pyramid (or isosceles triangle) inside such a square will carry certain properties. One of these is that if the two slopes are extended, and reflected about the square until they exit the square, they will create 11 layers of 40 whole number cubits, and 14 divisions, making 154 sub-rectangles, all found by calculation using Tan 1.2727 recurring. The numbers were dedicated.
Piazzi Smythe, Astronomer Royal of Scotland had in 1868 published his Life and Work at the Great Pyramid in which he had advocated that the area of one sloping side was equal to the height squared, but he did not know then that the cubit was equal to 1.718181 feet because the corners stones had not then been found. Nor did he know about 280 and 440 cubits. He thought that the cubit was 25 inches long. He also claimed that the value for PIE could be found at the Great Pyramid and for that Sir Henry James, Director of the Ordnance Survey, derided him and he lost his credibility. In fact, he was right and he can now be vindicated.
280
356.0898
220 440
The value of П = 3.1416, the circumference of a circle divided by the diameter. If twice the pyramid height becomes the diameter of that circle and the pyramid perimeter is the circumference then Pyramid П is 1760 / 560 = 3.1428571. The number is close but not quite close enough to the correct value, and that was the problem. Very strangely, the identical number also resides in 440 cubits / 14 and to six decimal places but 10 times larger. This was not the true value for Pie at the Great Pyramid but it was there nonetheless. It was based Smythe’s other theory that the area of one of the sloping sides was equal to the height squared.
Height squared = 280 x 280 = 78400 sq cubits
Half base squared = 220 x 220 = 48400 sq cubits
78400 + 48400 = 126800 sq cubits
Square root 126800 = 356.0898 cubits
Area sloping side = base x half height
Area sloping side = 178.0449 x 440 sq cubits
Area sloping side = 78339.756 sq cubits
Square root 78339.769 = 279.8924 cubits
If the height of the square, derived from the area of one sloping side, is seen as a pyramid height, then it falls short of 280 cubits by a distance of 280 – 279.8924 = 0.1076 rather oddly non-recurring cubits. If then the distance of 0.1028 cubits is added to a pyramid 280 cubits high, that gives 280.1076 cubits for a new conceptual pyramid height. Now if the theory that the pyramid height was the radius of a circle, and the perimeter was the circumference, was correct, then the exact value for Pie will appear.
Pie = Perimeter / New pyramid height
Pie = (440 x 4) / (280.1076 x 2)
Pie = 1760 / 560.2152
Pie = 3.1416
Such an extraordinary result, on a pyramid that archaeologists claim was never designed at all, must prove that the pyramid was 280 cubits high and 440 cubits on its base when its original casing stones were in place. The long hidden meaning behind the four shafts can now be revealed as they had originally related with a pyramid 280 cubits high, and 440 cubits on its base.
THE SOUTH SHAFT, KING’S CHAMBER.
The South Shaft runs a short distance horizontally from the south wall of the King’s Chamber before rising out of the pyramid at an inclination of exactly 45 degrees until it passes out of the south slope, but because the original casing stones are missing at the point of exit, it is impossible to know the where exit level had occurred by physical measurement.
The Rinaldi and Maragioglio survey of 1965 had climbed the pyramid and taken levels off the existing pyramids stones around the shaft exit point and they found levels of 79.24 metres (150.30 cubits) on course number 102, and 79.88 metres (152.53 cubits) on course number 103, but they had not measured on course 104. They had then shown a drawn section through the shaft exit point and had extended the line of shaft at 153.31 cubits above base where they thought it had originally passed out of the pyramid when the casing stones were in place.
The South Shaft Exit Point
104 153.31
Level 154
103
102
Original slope
Existing Stones
The engineers had assumed that the line was on the shaft floor whereas it was actually on the upper surfaces. The reason for that was because the horizontal parts of the shafts align with the entrance corridor ceiling at the King’s Chamber below, and at a particular level that had formed a datum above base. If the upper surfaces are used then the level where the shaft had passed out of the original pyramid slope was very slightly higher, and it was almost certainly at 154 cubits above base, the same in cubits as the number of rectangles on the pyramid square. If the level of the exit point is 154 cubits above base then another pyramid will exist above that level and it will be 280 – 154 = 126 cubits high.
280
South
154
Shaft
Tan 1.2727
220 99 121
With the tangent on the pyramid slopes known, the length of the half-base of that pyramid can be calculated at 126 / Tan 1.2727 = 99 cubits to give a full base of 99 x 2= 198 cubits for a pyramid 126 cubits high. This will mean that the exit point of the South Shaft, King’s Chamber, is not only 154 cubits above base but also 99 cubits south of pyramid centre.
The relationship between the base of the upper pyramid and that of the Great Pyramid now produces another revelation. A pyramid square can now be added, so that its corners abut the pyramid slopes.
The First Pyramid Square at level 154 cubits
280
154
82
44
99 121
A notional square exists of sides 198 cubits and its upper corners abut the original slopes of the Great Pyramid at an assumed level of 154 cubits above base. The square will therefore extend below the base by 198 – 154 = 44 cubits and a second pyramid 126 cubits high can be constructed on the base of the square. Its tip will extend above the Great Pyramid base by 126 – 44 = 82 cubits, and on the pyramid centreline.
The Great Step at the Grand Gallery
Pyramid centre
Grand Gallery
To King’s Chamber 82 cubits above base
Great Step
Pyramid Tip
The level on the top of the Great Step at the Grand Gallery leading into the entrance corridor of the King’s Chamber is known because the RM Survey had taken the level at 42.96 metres above base, and that would convert to 42.96 x 1.0936 x 3 / 1.71818 = 82.03 cubits. But for the smallest of errors, that is 82.00 cubits above base, and it has been found according to the pyramid inside the pyramid square.
The pyramid geometry is proving the reason for the Step. In that case the builders were able to place a stone at a level high up inside the pyramid with incredible accuracy, and only tangents and pyramid squares would show where this was on the pyramid section.
The Second Pyramid Square at level 126 cubits.
280
154
126
116
121 99
Because there exists a level of 154 cubits on the shaft exit point there can also exist a pyramid 154 cubits high at that level with a half-base of 154 / Tan 1.2727 = 121 cubits. A second notional square therefore exists of sides 121 x 2 = 242 cubits and its topside will abut the pyramid slopes at a level of 99 x Tan 1.2727 = 126 cubits above base.
The base of the second pyramid square will extend below the base of the Great Pyramid by 242 – 126 = 116 cubits. The base dimensions are now reversed from 99 and 121 cubits for the first square to 121 and 99 cubits for the second square. The geometry is revolving about a centreline.
The Second Height on the Second Pyramid Square.
280
242
154
126
38
242
116
Because the square extends below base by 116 cubits and the pyramid is 154 cubits high, its tip must extend above the base by 154 – 116 = 38 cubits. The Great Pyramid is 280 cubits high and so the tip intrudes into the height by 280 – 38 = 242 cubits. But 242 cubits is also the length of side of the second pyramid square. The distance of 242 cubits now exists on the shaft exit point and the second pyramid tip.
The difference between the levels of 154 cubits, and 126 cubits, is 28 cubits, and that is 10 times smaller than 280 cubits, height of the Great Pyramid. The mean level between the levels of 154 cubits and 126 cubits will then be (28 / 2) + 126 = 140 cubits above base and that is half the height of 280 cubits. The pyramid has been bisected.
The Slope on the Diagonal
280
275
275
North
154
Line of Shaft
154
121 99 55 165
165
99 + 55 = 154 cubits
220 – 55 = 165 cubits
121 + 99 + 55 = 275 cubits
According to Petrie, the shaft slope angle was something slightly greater than 45 degrees, but the Gantenbrink survey of 1993 using a robotic crawling device, decided that the angle was at 45 degrees. It was actually the diagonal on a square at 45 degrees. In fact it was the diagonal of three consecutive squares at the pyramid.
If the shaft is a geometrical line on section it is also acting as a diagonal on squares of sides 154 cubits, 165 cubits, and 275 cubits crossing the pyramid base at 154 cubits due north of the exit point. Because the exit point is 99 cubits due south of pyramid centre, the point of crossing on the base must then be 154 – 99 = 55 cubits north of pyramid centre, and it divides the Great Pyramid base of 440 cubits into 8 sections exactly.
The Shafts at the King’s Chamber
South North
King’s Chamber
Datum 84
X
3 10
The Rinaldi and Maragioglio survey gives a vertical distance of 1.14 metres from the King’s Chamber floor to the upper reveal of the south shaft opening on the south wall, converting to 2.17 cubits. If a level of 82 cubits is taken on the floor from the Great Step then the level of point X at the change of direction of the shaft and on its upper surfaces is about 84.17 cubits above base giving a probable design level of 84 cubits above base for the datum at the King’s Chamber.
The distance between the north wall of the King’s Chamber and pyramid centre is 330.6 inches or (330.6 / 12) / 1.7183 = 16.03 cubits, as given by the Petrie Survey of 1880 and the RM survey had found that the chamber is 5.24 metres wide, or 10 cubits wide exactly. They also found that the distance between the point change of direction for the south shaft and the face of the south wall is 1.5 metres, or 2.86 cubits. The distance from point X to pyramid centre will then have been somewhere very close to 16.03 + 10 + 2.86 = 28.89 cubits. The design distance was probably 29 cubits to pyramid centre.
The Original Position of the South Shaft
154
70 Centre
South
Shaft Chamber
84
55
55
Tan 1.2727
220 55
The coordinates on point X on the shaft as seen on the east-west pyramid section will then be 84 cubits above base and 29 cubits south of pyramid centre. The true position of the South Shaft can now be shown relative to the pyramid geometry. The shaft rises at 45 degrees and is a diagonal on three squares. The lower square is of sides 55 cubits giving a level of 55 cubits on pyramid centre. The second square on the same diagonal is of sides 29 cubits, creating the next level of 55 + 29 = 84 cubits. The third square on the diagonal must then be of sides 154 – 84 = 70 cubits and its south side must be 70 + 29 = 99 cubits south of pyramid centre and that was the half base distance on the first pyramid square.
The square of sides 70 cubits will have a perimeter of 70 x 4 = 280 cubits and that is the Great Pyramid height. The length of the shaft from exit point to change of direction will be the hypotenuse of a square of sides 70 cubits, or the square root of 70 squared + 70 squared = 98.99 cubits, or as close to 99 cubits as it could get. The perimeter and the diagonal of a square of sides 70 cubits are mathematically meaningful. That must show that the line on the shaft as existing is the same line at that determined by the geometry of pyramids and pyramid squares.
THE NORTH SHAFT, KING’S CHAMBER
The first problem associated with this shaft is that it is not straight and has been found to divert laterally as it progresses upwards, during which time it also changes direction vertically as well. As a result there can be no specific diagonal on a rectangle or a square, and the tangent of the shaft slope is impossible to find, but that did not mean that the shaft was useless. The RM Survey had also taken the levels on the stone courses on the existing the North Slope where the shaft exits the pyramid.
The North Shaft Exit Point
151.31 cubits above base
151.25
103
102
Tan 1.2727
Tan 1.2500
Existing Stones
The level on course number 103 was given on the engineer’s drawing at 79.24 metres above base, converting to 151.31 cubits above base. They had extrapolated the line where they had thought the original shaft had passed out of the original pyramid slope when the casing stones were in place. They had naturally assumed a pyramid 280 cubits high, but they could have been wrong. No one knows if this shaft had passed out of the pyramid slope because the casing stones are missing. The level that they gave is suggestive that something quite different had occurred in this location, and it had involved another pyramid of another height.
The opposite shaft, when it was acting as a diagonal on a square, had produced a square of sides 275 cubits, falling short of the built pyramid height by 5 cubits. Perhaps that had been intended as a means of linking the north shaft with the south shaft. If that were so then the pyramid that had related with the north shaft had never been built in stone but was conceptual. It might have been 275 cubits high, and one way of finding out is to relate the two pyramid heights with shaft exit points where Z is the unknown level for the north shaft exit point.
154 x 275 = Z x 280
Z = (154 x 275) / 280 = 151.25 cubits
That looks just about right considering that 151.31 cubits had been found by survey on the floor of the shaft for a pyramid 280 cubits high, whereas if the pyramid was less high, giving a less steep base angle, and a lower shaft exit point, and on the upper surfaces, the exit point would be at 151.25 cubits above base, and must be correct.
The First Pyramid Square
275