A major component of my argument against the star-shaft idea has been
my
recovery of the geometrical conception of these shafts, showing that
the design
constituted a very simple and elegant extension of the geometry of the
Great
Pyramid as described in my previously-published articles. |

Long before Rudolf Gantenbrink obtained new data for the shafts as the
result of his exploration using the robot Again, because the Queen's Chamber is situated in the mid-plane of the
pyramid, the equality of the angles means that the shafts would have
emerged at
the same level in the casing on the north and south sides of the
pyramid - a
design of no consequence for star-shafts. The angles as determined by
Petrie
are: ^{
} |

Shafts from Queen's Chamber | North | South | Mean | |

Angle of Slope, Petrie | 37° 28' | 38° 28' | 37° 58' |

The small difference in these angles is easily explained by the difficulty experienced by Petrie when taking his measurements at the lower ends of the shafts, since the sloping sections were only accessible through the initial horizontal sections, each about 20 cm in square cross-section and some two metres in length.
^{
} |

Shafts from King's Chamber | North | South | Mean | |

Angle of Slope, Petrie | 31° 33' | 45° 14' | 38° 23' |

Once again, therefore, the mean angle is close to the value required to give the shortest distance from the King's Chamber to the outside of the pyramid; while the difference in angle brought the outer openings of the two shafts to the same level on the north and south sides, as we will see.
Notwithstanding Petrie's careful work, his observations were limited to
relatively short sections of the shafts, and precedence must now be
given to
Rudolf Gantenbrink's findings for the angles of slope. |

Shafts from King's Chamber | North | South | Mean | |

Angle of Slope, Gantenbrink | 32° 36' | 45° 00' | 38° 48' |

Since both angles can be ascribed to an elementary geometrical origin,
Robert Bauval has sought to explain the design as an example of 'sacred
mathematics' being used to achieve a religious function; and he has
drawn an
analogy with the orientation of churches and cathedrals in relation to
the
eastern horizon.
According to the air-shaft theory, however, the architect was free to
determine the angles of slope which best suited his development of the
geometrical design, within the broad framework dictated by the need for
a short
route to be taken to the outside of the pyramid. The measurements now
published
by Gantenbrink have made it possible to establish this design,
revealing a very
interesting geometrical conception which explains why the shafts were
so
carefully constructed. It turns out that the geometry of the air-shafts
is a
function of the meridian cross-section of the pyramid itself, and can
be
developed simultaneously with the geometrical placing of the King's
Chamber
which I have previously described. |

The north and south shafts from the King's Chamber are now reported by
Gantenbrink to have both opened in the casing at the height of 80.63 ms
± 4
cm above the base. |

We can therefore now define the positions of the outlets as indicated in my geometrical diagram, by first marking off the height of the pyramid of 280 cubits along the diagonal of a square constructed on the height. The King's Chamber is thereby placed at the level of 280 ÷ √2 equals 198 cubits below the apex of the pyramid, as shown by Petrie's data. By forming a square on this height, with the base of 198 cubits centred on the central axis of the pyramid, the positions of the outlets on the north and south sides of the pyramid are immediately obtained, and the slope of the southern shaft can be drawn along the 45° diagonal of a square.
This angle agrees closely with Gantenbrink's finding of 39°
36' 28"
for the southern shaft, given a reported uncertainty of 1/5°;
while the
stated angle of 39° 7' 28" for the northern shaft is said to
be
uncertain within 2° at present, since only a short fractured
section at
the
lower end could be measured. |

Shafts from Queen's Chamber | North | South | Mean | |

Angle of Slope, Gantenbrink | 39° 7' 28" | 39° 36' 28" | 39° 22' |

While the King's Chamber was placed at the exact level in the Great Pyramid at which the diagonals of the horizontal cross-section measure 440 cubits, and equal the sides of base, the level of 131.5 cubits for the outlets of the shafts from the Queen's Chamber is that at which the diagonals of the cross-section measure exactly 330 cubits or 3/4 of the sides of base. Again, at the level of 154 cubits now obtained for the outlets of the shafts from the King's Chamber, the diagonals measure just 280 cubits, and equal the height of the Great Pyramid. It should be more obvious than ever, therefore, that the design of these shafts was determined by considerations of geometry, symmetry, and the desire for a coherent dimensional design, and had nothing to do with the conjectured astronomical alignments. |

Notes[1]. J.A.R. Legon, 'The Air-Shafts in the Great Pyramid',
27 (1993), 35-44;'Air-Shaft Alignments in the Great Pyramid', DE
28 (1994), 29-34. 'The Orion
Correlation and Air-Shaft
Theories', DE 33
(1995), 45-56 DE[2]. J.A.R. Legon, 'The Design of the Pyramid of Khufu', 12 (1988),
41-48; 'The Geometry of the Great
Pyramid', DE 108 (1989), 57-64
GM[3]. W.M.F. Petrie, The Pyramids and Temples of Gizeh
(London,
1883), 71. [4]. Petrie, Ibid. 83. [5]. R. Gantenbrink in R. Stadelmann, MDAIK 50
(1994), 285-294.
[6]. In conversation with the present writer. [7]. R.G. Bauval, DE 31 (1995), 5-13, 6. [8]. Ibid., Fig. 3. [9]. J.A.R. Legon, DE 12 (1988), 41-48, 43. [10]. I am grateful to Robert Bauval for supplying me with this data. Now see also R. Gantenbrink's web site, www.cheops.org [11]. Petrie, op.cit., Pl. VIII. [12]. Gantenbrink, op.cit., 293. |