The Mathematics of the Great Pyramid
The Queen's Chamber
Petrie’s measurements for the Queen’s Chamber are slightly less than given here, and his measurements for the King’s Chamber are slightly greater. The rationale for basing the measurements of both chambers on the use of the Golden Proportion in the design becomes obvious when one sees what it does in the King’s Chamber and Queen’s Chamber. It could be no other measurement. Both chambers use the Golden cubit as the basic unit of design.
The Queen’s Chamber then is 10 cubits wide (206.0113296 inches) and 11 cubits long(226.6124626 inches) making a floor area of 46684.73472 square inches.
6.18033988 inches before the chamber’s North-South center (the pyramid’s north-south axis), which is defined by the roof ridge, is the north edge of the off-center Niche. The Niche is 61.8033988 inches wide, 41.20226592 inches deep and 183.965057 inches high. The measurement of the three edges of its base line (arris) total 144.2079307 inches, which is the same as the remainder of the side, each 7 cubits. Since both chambers are designed with the Golden Cubit as the principle unit of measurement, ratios between the Niche and other multiples of the cubit are essentially x divided by 7, making ratios with the 6 repeating digits, 142857. The ratio between the Niche’s arris and the width of the chamber is 1.42857142 inches, with the length, 1.57142857, and with the sum of twice the length, (452.5743495 inches) 3.142857142, pi in the world of 7.
From the center of the chamber’s entrance to the north end of the east wall, and then following the floor line south along the wall, including the Niche, to the south wall is 329.6181274 inches. Its square root is 18.15538839, which, divided by 10 is half the Third Line to First Line ratio of 3.631077677, and times 100 is the length of the Grand Gallery’s passage.
The ratio between the Queen’s Chamber’s arris of 865.2475843 inches and the King’s Chamber’s arris of 1236.06797 inches is 1.42857142, the inverse of 7, the same as the Niche’s ratio with the Queen’s Chamber’s width. Adding a fourth side of 61.8033988 inches to make the Niche a rectangle, its measurement is 206.0113296 inches, the same as the side itself.
Going counter clockwise from the center of the chamber’s entrance to a point exactly opposite it, or the sum of the North, West, and South sides, minus the 41.20226592 inch wide entrance, measures 618.033988 inches. Going clockwise to the same point measures 247.2135963 inches, the square root of 200000. minus 200. A tenth of that, 24.72135963 inches, is how far the center of the niche is southward from the chamber’s north-south center.
The pyramid’s north-south axis is marked by both the face of the Great Step and the roof ridge of the Queen’s Chamber. Since the Great Step’s face is directly above the chamber’s center (roof ridge), and the step is 61.8033988 inches to the south wall of the Grand Gallery, the Niche’s north edge is 6.18033988 inches north from the vertical plane of the Great Step’s face. The Subterranean Chamber is 44.721359 inches south of the pyramid’s centerline.
The walls of the chamber are made up of 6 courses. The bottom two courses are at the level of the roof of the entrance and form the bottom section of the Niche. The four courses above that are very irregular in their height all around the chamber and unlike the width of the five corbelled sections, which have consistent progression of their dimensions, there seems to be no intrinsic value to the heights. The top width of the Niche is one cubit of 20.60113296 inches and each section below it is one half a cubit, (10.30056648 inches), wider, at 30.9016994 inches, 41.20226592 inches and 51.5028324 inches, until the bottom is 3 cubits, 61.8033988 inches.
The irregularity of the heights makes computing the volume of the Niche difficult, but using the measurements Petrie gives and all possible variations fail to deliver any interesting volume either as a number or in relation to anything else. This is the one place besides the coffer in the King’s Chamber (which has great relevance), one would expect an important number, since the niche represents us. The same applies to the volume of the chamber itself.
However, other important things do occur within the Niche. The first is the 4472.1359 inches representing the ratio between Fourth Line (one side of a square), and First Line, (the circle). This ratio is the distance from the pyramid entrance to the base of the Great Step, and is the distance from the pyramid entrance to a specific point in the space between the Subterranean Chamber and its anti-chamber. On a different route it also terminates within the Queen’s Chamber.
There are 4180.16817 inches to the entrance of the Queen’s Chamber from the Pyramid entrance, using the Ascending Passage’s measurement. There are a number of directions one could take within the chamber in order to reach 4472.1359 inches, but the only one that makes sense and winds up in an important place, is the route that stays in the center of the passage, proceeds directly south to a point 20.60113296 inches directly in front of the Niche’s north-south center line. From there proceeding 41.20226592 inches into the Niche to its east-west, north-south center lines and then going vertical for 112.7801699 inches. This totals 4472.1359 inches. The square root of the square root of 1.618033988 is 1.1278384, which, times a hundred, is only .003 of an inch greater.
Then, the remaining space to the top of the Niche measures 60.429109 inches. Minus .00160106 of an inch and then squared becomes 3651.483716, 100 times Third Line (one side triangle) and First Line. So the ratio that creates the square which leads to the three dimensional cube, terminates at a junction of the Golden Proportion and the symbol for action, in a 5 sectioned three dimensional space, representing the human being. Its termination is not exact for either ratio, which might be the point in this off center volume. The sum of the differences is .00527966. The termination point is too high for one and too low for the other.
Taking the long and only possible route to the Queen’s Chamber via the Well Shaft, due to the blocking of the Ascending Passage, is 3445.04497 inches longer than the direct Ascending Passage route. The 4472.135955 inches from the pyramid entrance, terminates 620.7039739 inches at some unknown point in the lower part of the passage. The long route winds up in front of the center of the Niche after 7738.807772 inches. This number has no intrinsic value, nor does extending it into the Niche or any aspect of the chamber enhance it.
However,, the total inches to the East-West center of the pyramid in the King’s Chamber, minus the Queen’s Chamber and passage, is 8660.254038 inches. The difference between the two numbers is 921.4462658, which is significant. It is but .00769012 of an inch from 921.4539559, which is the metric equivalent of the 36277.73439 inches of the pyramid’s perimeter, using 39.3701 as the conversion rate. Whether or not this is evidence of the knowledge and use of the contemporary meter is debatable, because there is another close approximation of the meter in the King’s Chamber, which is different. The height of the chamber from the floor level up is 230.3276684 inches, which, times 4 is 921.3106736, and converts to 36272.09335, not a pyramid number.
If, however, one divides the 36277.73439 inch perimeter by 921.4462658, the conversion rate is 39.37042857. The interesting aspect of this are the last digits, 7’s repeating trademark which dominates the ratios in the Queen’s Chamber. Its relationship to 7 is that it is the 7th part of 275.593. In any case, the long route to the Niche is the pyramid perimeter (and also the Sixth Line to Third Line ratio in the world according to 3) short of the complete route of 8660.254038.
There are other important dimensions that terminate within the Niche. The first is one side of the pentagon (Fifth Line) in the virtual circle on whose diameter the Great Pyramid sits. Measuring from the bottom of the circle, a vertical 6787.159471 inches terminates 1013.656779 inches above the pyramid’s base. This is in the top section of the Niche. 1013.993502, .336723 above this is the ratio between 35.77708764 (one eighth of the 286.2167011 inch displacement of the passage system) and 36277.73439 (pyramid’s perimeter).
678.7159471 is used often in the elevation of the passages. Subtracting that from the 1693.76875 elevation of the King’s Chamber passage floor the remainder is 1015.052083 inches above the base. .135743 of an inch below this is 1014.917064, the 35.7708764 ratio with 36310.77677, which is the Third line ratio in the Golden Proportioned world view. Half of the .135743 difference between the two is .06787159471, illustrating the many involvements of ratio between Fifth Line and Third Line.
Each end of the two cantilevered roof sections differ in width from 118.53 and 120 inches. This is too much leeway to ascertain what the intended measurement was, as there are so many possibilities, each with its own set of attributes. 118.9207115 is an example. Its square, 14142.13562, is the square root of 200000000.
One beautiful one on the low side of the possibilities is 118.9406966 inches. This would be the radius of a circle whose center is the roof peak and whose 206.0113296 inch width of the top of the wall, is one side of the circle’s inscribed triangle. making the perimeter of the triangle, 618.033988 inches. The altitude of the triangle would be 178.4110449, making the inverse of its square, the Golden Proportion pi, 3.141640789. One side of the circle’s inscribed pentagon is 139.8231747, whose square, times 10, is the volume of the King’s Chamber, 19550520.19 cubic inches. This, divided by the square root of 5 is 874.3258420, whose cube root is 20.60113296, the golden cubit. 356.822089 inches (times 10), half of the circle’s hexagon’s measurement, 713.6441796 inches, is the golden proportion of the Great Pyramid’s height of 5773.502692 inches. The radius, 118.9406966 inches, divided by 20.60113296, (Golden Cubit) is 5.773502692.
So, the number 7 ushers one into a seven sided chamber having a width of 10 cubits, a length of 11 cubits, a roof of 6 pairs of butting roof stones (12 altogether) whose ridge defines the north-south axis of the pyramid, a niche off center by the square root of 20 minus 2, having 5 sections with 4 corbels, that is 2 cubits deep, 3 cubits wide, with a height of the square root of 3 and an arris of 7 cubits, and in which the ratio for Fourth Line to First Line times a thousand (4472.135955 inches), terminates at a specific point in the center of the niche, dividing it vertically to within .005 of an inch into the Golden Proportion and the ratio between Third Line and First line.
The Queen’s Chamber then is 10 cubits wide (206.0113296 inches) and 11 cubits long(226.6124626 inches) making a floor area of 46684.73472 square inches.
6.18033988 inches before the chamber’s North-South center (the pyramid’s north-south axis), which is defined by the roof ridge, is the north edge of the off-center Niche. The Niche is 61.8033988 inches wide, 41.20226592 inches deep and 183.965057 inches high. The measurement of the three edges of its base line (arris) total 144.2079307 inches, which is the same as the remainder of the side, each 7 cubits. Since both chambers are designed with the Golden Cubit as the principle unit of measurement, ratios between the Niche and other multiples of the cubit are essentially x divided by 7, making ratios with the 6 repeating digits, 142857. The ratio between the Niche’s arris and the width of the chamber is 1.42857142 inches, with the length, 1.57142857, and with the sum of twice the length, (452.5743495 inches) 3.142857142, pi in the world of 7.
From the center of the chamber’s entrance to the north end of the east wall, and then following the floor line south along the wall, including the Niche, to the south wall is 329.6181274 inches. Its square root is 18.15538839, which, divided by 10 is half the Third Line to First Line ratio of 3.631077677, and times 100 is the length of the Grand Gallery’s passage.
The ratio between the Queen’s Chamber’s arris of 865.2475843 inches and the King’s Chamber’s arris of 1236.06797 inches is 1.42857142, the inverse of 7, the same as the Niche’s ratio with the Queen’s Chamber’s width. Adding a fourth side of 61.8033988 inches to make the Niche a rectangle, its measurement is 206.0113296 inches, the same as the side itself.
Going counter clockwise from the center of the chamber’s entrance to a point exactly opposite it, or the sum of the North, West, and South sides, minus the 41.20226592 inch wide entrance, measures 618.033988 inches. Going clockwise to the same point measures 247.2135963 inches, the square root of 200000. minus 200. A tenth of that, 24.72135963 inches, is how far the center of the niche is southward from the chamber’s north-south center.
The pyramid’s north-south axis is marked by both the face of the Great Step and the roof ridge of the Queen’s Chamber. Since the Great Step’s face is directly above the chamber’s center (roof ridge), and the step is 61.8033988 inches to the south wall of the Grand Gallery, the Niche’s north edge is 6.18033988 inches north from the vertical plane of the Great Step’s face. The Subterranean Chamber is 44.721359 inches south of the pyramid’s centerline.
The walls of the chamber are made up of 6 courses. The bottom two courses are at the level of the roof of the entrance and form the bottom section of the Niche. The four courses above that are very irregular in their height all around the chamber and unlike the width of the five corbelled sections, which have consistent progression of their dimensions, there seems to be no intrinsic value to the heights. The top width of the Niche is one cubit of 20.60113296 inches and each section below it is one half a cubit, (10.30056648 inches), wider, at 30.9016994 inches, 41.20226592 inches and 51.5028324 inches, until the bottom is 3 cubits, 61.8033988 inches.
The irregularity of the heights makes computing the volume of the Niche difficult, but using the measurements Petrie gives and all possible variations fail to deliver any interesting volume either as a number or in relation to anything else. This is the one place besides the coffer in the King’s Chamber (which has great relevance), one would expect an important number, since the niche represents us. The same applies to the volume of the chamber itself.
However, other important things do occur within the Niche. The first is the 4472.1359 inches representing the ratio between Fourth Line (one side of a square), and First Line, (the circle). This ratio is the distance from the pyramid entrance to the base of the Great Step, and is the distance from the pyramid entrance to a specific point in the space between the Subterranean Chamber and its anti-chamber. On a different route it also terminates within the Queen’s Chamber.
There are 4180.16817 inches to the entrance of the Queen’s Chamber from the Pyramid entrance, using the Ascending Passage’s measurement. There are a number of directions one could take within the chamber in order to reach 4472.1359 inches, but the only one that makes sense and winds up in an important place, is the route that stays in the center of the passage, proceeds directly south to a point 20.60113296 inches directly in front of the Niche’s north-south center line. From there proceeding 41.20226592 inches into the Niche to its east-west, north-south center lines and then going vertical for 112.7801699 inches. This totals 4472.1359 inches. The square root of the square root of 1.618033988 is 1.1278384, which, times a hundred, is only .003 of an inch greater.
Then, the remaining space to the top of the Niche measures 60.429109 inches. Minus .00160106 of an inch and then squared becomes 3651.483716, 100 times Third Line (one side triangle) and First Line. So the ratio that creates the square which leads to the three dimensional cube, terminates at a junction of the Golden Proportion and the symbol for action, in a 5 sectioned three dimensional space, representing the human being. Its termination is not exact for either ratio, which might be the point in this off center volume. The sum of the differences is .00527966. The termination point is too high for one and too low for the other.
Taking the long and only possible route to the Queen’s Chamber via the Well Shaft, due to the blocking of the Ascending Passage, is 3445.04497 inches longer than the direct Ascending Passage route. The 4472.135955 inches from the pyramid entrance, terminates 620.7039739 inches at some unknown point in the lower part of the passage. The long route winds up in front of the center of the Niche after 7738.807772 inches. This number has no intrinsic value, nor does extending it into the Niche or any aspect of the chamber enhance it.
However,, the total inches to the East-West center of the pyramid in the King’s Chamber, minus the Queen’s Chamber and passage, is 8660.254038 inches. The difference between the two numbers is 921.4462658, which is significant. It is but .00769012 of an inch from 921.4539559, which is the metric equivalent of the 36277.73439 inches of the pyramid’s perimeter, using 39.3701 as the conversion rate. Whether or not this is evidence of the knowledge and use of the contemporary meter is debatable, because there is another close approximation of the meter in the King’s Chamber, which is different. The height of the chamber from the floor level up is 230.3276684 inches, which, times 4 is 921.3106736, and converts to 36272.09335, not a pyramid number.
If, however, one divides the 36277.73439 inch perimeter by 921.4462658, the conversion rate is 39.37042857. The interesting aspect of this are the last digits, 7’s repeating trademark which dominates the ratios in the Queen’s Chamber. Its relationship to 7 is that it is the 7th part of 275.593. In any case, the long route to the Niche is the pyramid perimeter (and also the Sixth Line to Third Line ratio in the world according to 3) short of the complete route of 8660.254038.
There are other important dimensions that terminate within the Niche. The first is one side of the pentagon (Fifth Line) in the virtual circle on whose diameter the Great Pyramid sits. Measuring from the bottom of the circle, a vertical 6787.159471 inches terminates 1013.656779 inches above the pyramid’s base. This is in the top section of the Niche. 1013.993502, .336723 above this is the ratio between 35.77708764 (one eighth of the 286.2167011 inch displacement of the passage system) and 36277.73439 (pyramid’s perimeter).
678.7159471 is used often in the elevation of the passages. Subtracting that from the 1693.76875 elevation of the King’s Chamber passage floor the remainder is 1015.052083 inches above the base. .135743 of an inch below this is 1014.917064, the 35.7708764 ratio with 36310.77677, which is the Third line ratio in the Golden Proportioned world view. Half of the .135743 difference between the two is .06787159471, illustrating the many involvements of ratio between Fifth Line and Third Line.
Each end of the two cantilevered roof sections differ in width from 118.53 and 120 inches. This is too much leeway to ascertain what the intended measurement was, as there are so many possibilities, each with its own set of attributes. 118.9207115 is an example. Its square, 14142.13562, is the square root of 200000000.
One beautiful one on the low side of the possibilities is 118.9406966 inches. This would be the radius of a circle whose center is the roof peak and whose 206.0113296 inch width of the top of the wall, is one side of the circle’s inscribed triangle. making the perimeter of the triangle, 618.033988 inches. The altitude of the triangle would be 178.4110449, making the inverse of its square, the Golden Proportion pi, 3.141640789. One side of the circle’s inscribed pentagon is 139.8231747, whose square, times 10, is the volume of the King’s Chamber, 19550520.19 cubic inches. This, divided by the square root of 5 is 874.3258420, whose cube root is 20.60113296, the golden cubit. 356.822089 inches (times 10), half of the circle’s hexagon’s measurement, 713.6441796 inches, is the golden proportion of the Great Pyramid’s height of 5773.502692 inches. The radius, 118.9406966 inches, divided by 20.60113296, (Golden Cubit) is 5.773502692.
So, the number 7 ushers one into a seven sided chamber having a width of 10 cubits, a length of 11 cubits, a roof of 6 pairs of butting roof stones (12 altogether) whose ridge defines the north-south axis of the pyramid, a niche off center by the square root of 20 minus 2, having 5 sections with 4 corbels, that is 2 cubits deep, 3 cubits wide, with a height of the square root of 3 and an arris of 7 cubits, and in which the ratio for Fourth Line to First Line times a thousand (4472.135955 inches), terminates at a specific point in the center of the niche, dividing it vertically to within .005 of an inch into the Golden Proportion and the ratio between Third Line and First line.
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