Thursday, 19 March 2009
The Great and second Great Pyramid Geometry
This figure as found in Tompkins, pages 260-261, basically a four square with (1 by 1) and (2 by 1) diagonals, contains the necessary information to construct both of the great pyramids at Giza.
I've been a bit pre-occupied for the past few weeks, re-reading 'Secrets Of The Great Pyramid, by Peter Tompkins, and the appendix on Ancient Measure by Livio Stecchini. Slowly it dawned on me that the basic geometry the Great Pyramid is based upon is so simple, yet I have not seen it explained previously. I also came to realize that the Second 'Great' Pyramid is also derived as simply from a (2 by 1) rectangle, or double square.
Tompkins in a discussion about the above figure describes the idea of Tons Brunes who in The Secrets of Ancient Geometry
who 'shows that the Great Pyramid, like most of the great temples of antiquity, was designed on the basis of an advanced but hermetic geometry known only to initiates. only fragments o which percolated to the classic and Alexandrine Greeks.'
Tompkins states: Brunes shows how the ancient Egyptians used the basic design of a circle inscribed in a square to divide both circle and square geometrically into equal parts from 2 to 10, and all their possible multiples, without recourse to measuring or arithmetical calculations, with the aid of nothing but a straightedge and a compass - common emblems, along with the Pyramid, of the Masonic orders of yesterday and today.
He continues: In Brunes' reconstruction o the secret geometry, the cross emerges as the first geometric addition to the circle and square, and is the key not only to the solution of geometric problems but to the development of numerals and the alphabet.
By including the diagonals, every number both Latin and Arabic and all the letters of several alphabets may be obtained.
According to Brunes, both mathematics and the alphabet sprang from geometry, not the reverse. He says that nowadays we use numbers as the primary factor in our calculations, and geometry only as a subsidiary, whereas he believes the Egyptians reversed the order. He uses a detailed analysis of the Rhind Mathematical Papyrus to demonstrate that the ancient Egyptian system of counting was directly governed by geometric factors and that their ideas and theories were bound in geometric rules.
Brunes found that the circle was indeed considered sacred by the Egyptians, as were the square and the cross and the triangle, all of which are intimately incorporated into the Great Pyramid with its square base and triangular faces designed to represent the ''sacred'' circle.
Brunes demonstrates how the circle inscribed in a square and quartered by a cross enabled the ancient Egyptian geometer to inscribe in a circle the basic figures of pentagon, hexagon, octagon and decagon.
Tompkins points out that the golden section is formed automatically between the sides and chords of the pentagon. This is so!
What is also true though is that very simply the golden section can be deirved directly from the (2 by 1) rectangle, as the diagonal is square root five, and if the unit side is added and the whole length halved, this is Phi in relation to the side. (1.618034 : 1)
It was musing on this and that the apothem of the Great Pyramid is Phi in relation to half of the base that the geometric construct of the Great Pyramid became apparent, as described below.
Right click on images, and 'Open in New Window' to see at full size:
figure 1: The basic (2 by 1) rectangle a circle is drawn centred on the common side with radius half the base. The points on the diagonals cut by this circle are 1.618034 units, with square side being 1 unit. Aa = Bb = Cc = Dd = 1.618034, Phi.
Diagonal = Square root five, 2.236068.
2.236068 / 2 = 1.118034
1.118034 + 0.5 = 1.618034.
Figure 2 is the construct of the Great Pyramid geometry. Figure 3 is labelled.
From points A and B with length Aa and Bb, or Phi, arcs are scribed to cut the axis, at P. The figure so formed is a cross section of the Great Pyramid, with base 2, apothem or slant height, Phi, and height square root Phi.
The angles PAQ and PBQ equals 51.8273 degrees.
Figures 4 and 5 show how the Second Pyramid geometry is found.
Stecchini, pages 378/379, gives Petrie's figures for the Second Pyramid at Giza, which he reckons best. He shows that the cross sectional triangle, half base, height and apothem are in the relationship, 3,4,5, which gives a base angle of 53.13 degrees.
In figure 4 the angle formed by the intersection of the two diagonals at O, is 53.13 degrees, so a construct of the Second Pyramid is possible from this fact alone. Angle BOM is 53.13. An arc is scribed from point B with length BO, to the intersection with the diagonal extended to M.
Figuire 5 shows a variation using the length Phi, Bb, to construct a triangle with the same apothem as the Great Pyramid, namely Phi.
Triangle BKL is a 3,4,5 triangle. KL is 3 units, BK is 4 units, BL is 5 units.
As the two pyramids are constructed I can't find any convincing correlation in dimensions, the second Pyramid being some 97% roughly smaller in base and height. Apparently, the Second Pyramid is on a higher level than the Great Pyramid so appears to be somewhat higher.
I find it fascinating that both can be found so simply from the same basic figure, namely figure 1 above.
Finally, the (2 by 1) rectangle diagonals form angles of 26.5650512 degrees, and 63.43495 degrees. The angle 26.5650512 degrees is very close to the angles given for the Ascending and descending passages of the Great Pyramid, adding further, perhaps, to the links shown here.