**KING'S CHAMBER SHAFTS GEOMETRY**

**The south wall of the King's chamber is 26 cubits south
of the midline of the pyramid. The ceiling of the southern shaft is 84 cubits
above ground level at the south wall of the King's chamber. The southern shaft
begins it's angle of ascent three cubits south of the southern wall of the
King's chamber. In 1993, Rudolph Gantenbrink used a specially designed robot
to measure the angle of incline along the entire length of the shaft and found
the mean angle to be 45°, or a slope of 1/1. Given 280 cubits for the completed
height and 440 cubits for the baselength of the pyramid, and the measurements
of the shaft given above, the length of the ascending portion of the shaft
to the outer casing of the pyramid is 99 cubits. The horizontal and vertical
distance of the ascending portion of the shaft is 70 cubits, and the ceiling
of the shaft exits the outer casing of the pyramid 154 cubits above ground
level.**

**The north wall of the King's chamber is 16 cubits south
of the midline of the pyramid. The ceiling of the northern shaft is 84 cubits
above ground level at the north wall of the King's chamber. The northern shaft
begins its angle of ascent five cubits north of the northern wall of the King's
chamber. Gantenbrink measured the angle of ascent along the entire length
of the shaft. The lower portion of the northern shaft is constructed around
the grand gallery and Gantenbrink found variations in the angle of incline
in this short portion of the shaft. Gantenbrink numbered the ceiling blocks
in the shaft. The first block is the five cubit horizontal section. Ceiling
blocks 2-6 are the part of the shaft that goes around the grand gallery. Gantenbrink
found that the angle of ascent from block six to the point the shaft exits
the pyramid is 32.6°. Gantenbrink found the mean angle of the entire inclined
portion of the shaft to be a slope of 7/11 or an angle of 32.47°. Given the
exterior dimensions of the pyramid and the measurements of the shaft given
above, the vertical distance of the ascending portion of the shaft is 70 cubits,
the horizontal distance is 110 cubits, and the ceiling of the shaft exits
the outer casing of the pyramid 154 cubits above ground level.**

**Extended downward, the two King's chamber shafts intersect
seven cubits below the point they begin their ascent. The extended portions
of the shafts show the slopes of both shafts in the lowest common denomination
of whole numbers of cubits: 7/7 for the southern shaft and 7/11 for the northern
shaft. **

**The rise and run of the inclined parts of the shafts
are 10x multiples of the slopes of 7/7 (70/70) and 7/11 (70/110). The rise
and run of the shafts from the intersection point to the point the shafts
exit the pyramid are 11x multiples of the slopes (77/77 and 77/121). The slope
of the pyramid is 14/11 (280/220). The 7/11 slope of the southern shaft is
one-half of the slope of the pyramid. The intersection point is 77 cubits
above ground level and 77 cubits below the height the shafts exit the pyramid.
The intersection point is 22 cubits south of the midline of the pyramid and
seven cubits below the height of the shafts in the king's chamber, producing
a right triangle with sides in the ratio of 22/7. The sidelength of the pyramid
154 cubits above ground level is 198 cubits (77 + 121), equal to the vertical
distance from the apex of the pyarmid to the floor of the King's chamber (280-82).
198 cubits is also equal to the height of the pyramid divided by the square
root of 2 (280/1.4142 = 198). Thus, the horizontal area of the pyramid 154
cubits above ground level is equal to one-half of the square area of the height
of the pyramid. Because the area of the faces of the pyramid are equal to
the area of the square height of the pyramid, the horizontal area of the pyramid
154 cubits above ground level is also equal to one-half of the area of the
faces of the pyramid.**

**The height of the pyramid up from the exit points of
the shafts is 126 cubits (280-154). Extended upward, the northern shaft intersects
the height of the pyramid 198 cubits horizontally from the point the northern
shaft exits the pyramid (198 x 7/11 = 126).The 99 cubit length of the ascending
part of the southern shaft is one-half of the 198 cubit sidelength of the
pyramid 154 cubits above ground level. The horizontal distance from the shafts
intersection point to the point the southern shaft exits the pyramid is 121
cubits. The horizontal distance from the exit points to the outer edges of
the pyramid at ground level are also 121 cubits (121 x 14/11 = 154). Extended
downward, the northern shaft intersects ground level 77 cubits north of the
southern edge of the pyramid and 121 cubits south of the intersection of the
shafts. The northern shaft intersects the southern face of the pyramid 77
cubits south of the southern edge of the pyramid. Extended upward, the southern
shaft intersects the height of the pyramid 72 cubits north of the intersection
point of the northern shaft and the southern face of the pyramid. Extended
downward, the southern shaft intersects the level of the intersection of the
northern shaft and the southern face of the pyramid 144 cubits south of the
intersection of the northern shaft and the height of the pyramid. The shafts
exit the king's chamber and begin their angle of ascent 84 cubits above ground
level, or 196 cubits below the apex. 84 cubits above ground level, the half
base of the pyramid is 154 cubits. The intersection of the shafts, extended
downward vertically, divides the 440 cubit baselength of the pyramid at ground
level in the ratio of 198/242, or 9/11.**

**QUEEN'S CHAMBER SHAFTS GEOMETRY**

**The south wall of the queen's chamber is five cubits
south of the midline of the pyramid and the north wall is five cubits north
of the midline. The ceilings of the queen's chamber shafts are 44 cubits above
ground level at the walls of the chamber. According to Gantenbrink, the southern
shaft begins it's angle of ascent 1.96 meters south of the southern wall and
the northern shaft begins it's antle of ascent 1.93 meters north of the north
wall. ****Based on 24 measurements over a distance of 28 meters,
Gantenbrink found the southern shaft to be inclined at a mean angle of 39.6°.
After approximately 18 meters, the northern shaft bends to go around the grand
gallery. Gantenbrink's robot was only able to measure this part of the northern
shaft because the robot could not traverse the bend. Gantenbrink found a range
of angles from 33.3° to 40.1° for the inclined part of the shaft below the
bend. Gantenbrink gave an estimated mean angle for the northern shaft of 39.1°,
but he gave a margin of error of two degrees for his estimate of the southern
shaft and concluded that the angle of the shaft would need to be surveyed
above the bend to determine if the angle of both shafts was intended to be
the same. According to Gantenbrink, the first stone blocking the upper end
of the southern shaft is 59.4 meters from the queen's chamber.**

**In 2003, Zahi Hawass conducted further robotic explorations
of both of the queen's chamber shafts. Hawass found another blocking stone
in the northern shaft. Hawass reported that the distance from the queens'
chamber to the blocking stones was 208 feet for both shafts. Hawass also reported
that the blocking stones were both in the same relative locations inside the
pyramid.**

** A slope of 9/11 equally divides the difference between
the slopes of the king's chamber shafts:**

**The overall east-west and north-south distance of
the Giza site is in the ratio of 9/11:**

**The exit points of the king's chamber shafts divide
the height of the pyramid in the ratio of 9/11:**

**The queen's chamber shafts are in the north and south
walls of the chamber. The north and south walls of the chamber are 9 cubits
high and 11 cubits long. The doorway, the shaft and the course heights of
the north wall of the queen's chamber are diagramed below according to WMF
Petrie's measurements. The diagonal of the north wall shows the 9/11 ratio
between the height of the first two courses and the width of the doorway.**

**In the diagram below, the slope of the queen's chamber
shafts is 9/11 or an angle of 39.29°. John Legon and Robin Cook both suggested
that the queen's chamber northern shaft, extended upward, intersects the upward
extension of the king's chamber northern shaft at the height of the apex of
the pyramid. According to Cook, this requires an angle of 39.3° for the queen's
chamber northern shaft. The ceilings of the queen's chamber shafts begin their
angle of ascent 8.6 cubits horizontally from the midline of the pyramid. The
king's chamber northern shaft intersects the height of the pyramid 297 cubits
horizontally from the apex of the pyramid. The horizontal distance from the
point the queen's chamber shaft begins its angle of ascent to the point the
king's chamber shaft intersects the height of the apex of the pyramid is 288.4
cubits (297 minus 8.6). The vertical distance from the height of the ceiling
of the queen's chamber shaft to the apex of the pyramid is 236 cubits (280
minus 44). The height of 236 cubits, times 11, divided by 9, equals 288.4
cubits.**

**The sidelengths of the red square in the diagram are
equal to the 198 cubit horizontal distance from the exit points of the king's
chamber shafts. Extended downward, the king's chamber southern shaft marks
the 280 cubit diagonal of the square, equal to the height of the pyramid.
Cook suggested that the southern queen's chamber shaft, extended downward,
also intersects the corner of the square. Given that the queen's chamber shafts
are symmetrical, the queen's chamber northern shaft, extended downward, intersects
the opposite lower corner of the square.**

**In the diagram above, the queen's chamber shafts terminate
directly below the exit points of the king's chamber shafts, 118 cubits above
ground level and 99 cubits horizontally from the midline of the pyramid. Given
the height of 44 cubits for the shafts in the queen's chamber, the rise of
the shafts is 74 cubits. The run of the ascending portion of the shafts is
equal to the 99 cubit distance from the midline, minus the five cubit width
of the chamber from the midline, and minus the 3.6 cubit horizontal run of
the shafts from the chamber, or 90.4 cubits. A rise of 74 cubits over a run
of 90.4 cubits produces a 116.8 cubit length for the ascending portion of
the shafts. Adding the 3.6 cubit horizontal distance gives a total length
of 120.4 cubits for the queen's chamber shafts. The distance of the shafts
reported by Hawass is 208 feet, or approximately 121 cubits. ****Extended
downward, the queen's chamber shafts intersect at the vertical midline of
the pyramid. The shafts begin their angle of ascent 8.6 cubits horizontally
away from the midline. 8.6 times 9/11 equals 7.0 cubits. Given the height
of 44 cubits for the shafts at the walls of the queen's chamber, the intersection
point is 37 cubits above ground level. From the upper end of the shafts to
the intersection point at the midline of the pyramid the vertical distance
is 81 cubits and the horizontal distance is 99 cubits.**

**The vertical distance from the floor of the king's chamber
to the exit points of the king's chamber shafts is 72 cubits (154 minus 82).
The 72 cubit vertical distance is divided in half by the upper ends of the
queen's chamber shafts. Extended upwards, the queen's chamber southern shaft
intersects the height of the pyramid 72 cubits south of the point the king's
chamber southern shaft intersects the height of the pyramid.**

**The queen's chamber shafts begin 44 cubits above ground
level, 110 cubits vertically below the exit point of the king's chamber shafts,
equal to the 110 cubit horizontal distance of the king's chamber northern
shaft. The vertical distance from the intersection point of the queen's chamber
shafts to the termination of the queen's chamber shafts is 81 cubits, equal
to one-half of the 162 cubit vertical distance from the termination of the
queen's chamber shafts to the apex of the pyramid. The intersection point
of the queen's chamber shafts is also 81 cubits above the lower edge of the
198 cubit square in the diagram. Extended downward, the queen's chamber northern
shaft is 132 cubits horzonally from the intersection point of the south face
of the pyramid and the king's chamber northern shaft, and one-half of this
distance, or 66 cubits horizontally from the exit points of the king's chamber
shafts.**

**ASTRONOMICAL ALIGNMENTS**

**Kate Spence proposed that the pyramid builders used
the simultaneous transit of Kochab and Mizar, in vertical alignment above
and below the north polar region, to orient the sides of the pyramid to the
cardinal directions. In 2467 B.C. the vertical alignment of Kochab and Mizar
was due north. As a result of precession, the vertical alignment of Kochab
over Mizar slowly moved east of due north after 2467 B.C. The vertical alignment
of Kochab over Mizar was west of due east, slowly moving east towards due
north, prior to 2467 B.C. The great pyramid is oriented approximately three
arc minutes west of due north. Kochab was vertically aligned over Mizar three
arc minutes west of due north around 2480 B.C. One problem with Spence's theory
is that the second pyramid at Giza is aligned approximately six arc minutes
west of due north. Kochab was vertically over Mizar six minutes west of due
north around 2486 B.C. This contradicts the conventional chronology that the
second pyramid was built after the great pyramid. Spence explains this discrepancy
by suggesting that the second pyramid could have been oriented by the simultaneous
transit of Mizar over Kochab. In 2467 B.C. the vertical alignment of Kochab
over Mizar and the vertical alignment of Mizar over Kochab were both due north.
But as the vertical alignment of Kochab over Mizar moved east of due north
after 2467, the vertical alignment of Mizar over Kochab moved west. The simultaneous
transit of Mizar over Kochab, six minutes west of due north, occurred around
2448 B.C. This is Spence's proposed date for the orientation of the second
pyramid.**

**Giuglio Magli argues that while Spence may be right
about the date of the orientation of the great pyramid based on the simultaneous
transit of Kochab over Mizar, the idea that the builders would have used opposite
simultaneous transits for the first and second pyramid at Giza is unlikely.
In the alternative, Magli suggests that the simultaneous transit of Kochab
over Mizar was also used to orient the second pyramid, meaning that it would
have been oriented first, around 2486 B.C.**

**A third alternative is that both pyramids were oriented
by the simultaneous transit of Mizar over Kochab. In this case the date of
the orientation of the great pyramid would be around 2454 B.C. and the date
of the orientation of the second pyramid would have been around 2448 B.C.,
preserving the conventional chronology of the two pyramids and applying the
same method of orientation for both pyramids.**

**Virginia Trimble and Alexander Badawy proposed that
the southern king's chamber shaft targeted Orion's belt and that the northern
king's chamber shaft targeted the area of the north celestial pole. Robert
Bauval used Gantenbrink's data on the angles of the shafts to calculate the
approximate dates that the southern king's chamber shaft targeted Orion's
beltstar Alnitak and the northern shaft targeted Thuban, the closest star
to the north pole during the pyramid age. ****Bauval also used
Gantenbrink's estimates of the mean angles of the queen's chamber shafts to
calculate the approximate dates that the southern shaft targeted Sirius, and
the northern shaft targeted Kochab. However, Gantenbrink only measured the
angle of less than half of the southern queen's chamber shaft and he was only
able to penetrate the northern queen's chamber shaft to the bend around the
grand gallery. The report of the more recent survey by Hawass, indicating
that the total length of both of the queen's chamber shafts is the same, and
that both shafts terminate at the same relative locations inside the pyramid,
supports Gantenbrink's earlier speculation that the shafts were intended to
be symmetrical.**

**Sources of error in dating the pyramid from calculations
based on the surveyed angle of the shafts include: The ability of the ancient
Egyptians to accurately measure the angles of the stars above the horizon
at culmination; the ability of the ancient Egyptians to construct the shafts
to accurately match their measure of the angles of the stars; and the ability
of modern surveys to accurately determine the angles of the shafts. In addition
to the 14/11 slope of the pyramid and the 1/2 slope of the entrance passage,
the pyramid builders may have used the simple slopes of 1/1, 7/11 and 9/11,
as close approximations to the angles of the culminations of the stars, in
planning the locations of the chambers, passages and shafts, and in constructing
the pyramid. If so, we can only guess at the precise angles that builders
were approximating with the whole number slopes used in construction.**

**The queen's chamber shafts offer a simpler and more
accurate method for astronomically dating the pyramid, if the northern queen's
chamber shaft was intended to target Kochab and the southern queen's chamber
shaft was intended to target Sirius, and if the queen's chamber shafts were
intended to be symmetrical. Kochab is believed to have been an important star
to the ancient Egyptians of the pyramid age. As Thuban was slowly moving away
from the north celestial pole, Kochab was moving closer to the pole. If the
proposals of Spence and others are correct that Kochab was used to orient
the sides of the pyramids to the cardinal directions, this also indicates
the importance of Kochab to the ancient Egyptians. Sirius is known to have
been the most important star in the sky to the ancient Egyptians. Sirius is
the brightest star in the sky and the ancient Egyptian calendar was based
on the heliacal rising of Sirius. The heliacal rising of Sirius also signaled
the beginning of the annual flooding of the Nile that the ancient Egyptian
civilization depended upon.**

**During the pyramid age, Kochab was moving closer to
the north celestial pole. As a result, the circle described by Kochab as it
rotated around the pole was getting smaller and Kochab's upper culmination
was getting lower in the northern sky. While the upper culmination of Kochab
was getting lower in the northern sky, the culmination of Sirius was getting
higher in the southern sky. In the sky above Giza, the upper culmination of
Kochab was higher than the culmination of Sirius, prior to 2450 B.C. The culmination
of Sirius was higher than the upper culmination of Kochab after 2450 B.C.
The upper culmination of Kochab and the culmination of Sirius were the same
angle above the horizon in the sky above Giza in 2450 B.C. Given the significance
of Kochab and Sirius to the ancient Egyptians, it is likely that they were
aware that the angles above the horizon of the culminations of Sirius and
Kochab had been getting closer to each other for hundreds of years, and one
of the purposes of the shafts may have been to memorialize the time when the
culminations of both stars were at the same angle above the horizon. The angle
of both stars above the horizon was approximately 39.35° in 2450 B.C. This
is close to the angles proposed by Gantenbrink from his survey and close to
the angle produced by the 9/11 slope proposed above. 2450 B.C. is also close
to the astronomically derived orientation date given by Spence based on the
simultaneous transit of Kochab over Mizar, and even closer to the astronomically
derived orientation date if the great pyramid and the second pyramid at Giza
were both oriented based on the simultaneous transit of Mizar over Kochab.**

**See Also:**