Copyright EDF free. Added drawings to the image not authorieed

Do Satellite photos confirm theory of degrees?

Figure 1: Satellite photo of the Cheops pyramid (License GeoEye to H.D. Bui)
Figure 2: Inverse 3D Building
Figure 3: Observed bands and the Petrie sequence of stones (Drawings after J. Rousseau, Construire la Grande Pyramide, L'Harmattan, 2001)

Satellite photos are displayed in a 3D view when the Satellite is nearly, but not exactly, at the apex of the Pyramid. The faces of the pyramid are non-isosceles triangles, Figure 1. Even if the photo has a very high resolution, about 50 cm, it does not reveal anything particular, except the strata. For a comparison between the four faces, in order to have the same contrast and brightness, it is more convenient to have a 2D view of the Pyramid as a perfect square with its diagonals, or a 90° 3D view. We need the inverse of the so-called “3D building” of Google Earth. The ideal case occurs when both the Satellite and the Sun are at the apex. Google Earth technique is similar to the operation used in Chapter III of my book when I raised the 2D densitogram to obtain 3D views with hidden faces.

3D building

3D Building with a camera looking at all faces of the pyramid, as well as its inverse, consists of successive transformations of each triangular face, by dilatation or compression along an edge, then along the transformed base and along its orthogonal, and finally a rotation, Let us transform first each perspective face of Figure 1 into a 2D view by the use of an inverse 3D Building operation. The three-dimensional perspective of Figure 1 becomes a flat view in Figure 2. However, because of limited software, these successive transformations significantly alter the image quality even if we still distinguish the presence of bands.
We can also directly change the contrast and brightness of each pyramid face of Figure 1 and then assembly the results. The result is better than that given in Figure 2. One clearly sees bands in Western and Southern faces of Figure 3.

The Petrie sequence observed by Satellite photos
Figure 3 shows evidence of bands of unequal width, unequal contrast or brightness. Perhaps this is due to stones from different quarries. But a careful examination of the Petrie sequence of thick stones, labelled as a, b, c, d .. , shows that they correspond to sharp lines of the band edges in the West and South. Various combinations of intermediate stones layers between these thick layers can explain the difference of contrast and brightness.
Petrie (1892) reported the thicknesses of visible layers of stones, which vary from 0.50 m to 1.5 m (at the foot). The 35th layer (label b) has 1.25 m thick. Since he reported the thickness of each layer in abscissa as function of its number in ordinate (N°202 for the top at 138 m), there may be a little mismatch between the position of thick layers in the medallion with their true position in the pyramid. Globally, the correspondence between bands and the Petrie sequence seems to be good. These results highly suggest the existence of degrees and horizontal degree cornices. Thick layers a, b, c, d.. would be used for updating the horizontal level of cornices and also for the foundation of degree walls, as suggested in Chaper III, to reinforce the building. Degree wall heights would not be constant. It raises the question on how the “constraints” revealed by Satellite observations, Petrie’s work (1892) and our microgravity measurement results can be respected by theories on the Cheops pyramid construction: Goyon (1977), Holscher (1912) or Borchardt (1922), Guerrier (1981, 2006), Houdin (2002) or Skillern (2012)? Skillern’s paper presents a theory which might overcome some difficulties related to the outgrowth of stones in horizontal cornice theories (K. R. Skillern, New Satellite Perspectives on Egypt’s Great Pyramids - With A Twist. The Significance of Exposed Rock Bands to Deduction of Exterior Construction Techniques, Private communication, to appear. The author’s email: geezeronpyramids@centurylink.net). According to him, thick stones would be put in a horizontal layer after the achievement of the main work, by the completion of the cornice, backwards. He considered two intertwined ramps of 3.5 m width, whirling around the pyramid along inclined cornices of very low slope 2.2%. The “screw dislocation” height corresponds to the degree wall would be hidden by the infillings and coatings. His theory is another version of D. Duham’s model with four intertwined ramps (“Building an Egyptian Pyramid”, Archaeology, 9, 1956). A maquette of his “four ramps”, each ramp starting at a corner, is displayed in the Science Museum of Boston.
Finally, what theory to be considered? I don’t know.

Definitely the Cheops pyramid will keep its mystery for long time.

Books

1. My book Fracture Mechanics, Inverse Problems and Solutions, Springer 2006, contains method for solving exactly crack inverse problems in elasticity, acoustics, elastodynamics (with application to the earthquake inverse problem), heat diffusion equation. It has been translated into Russian under the title Fracture Mechanics, Fyzmalit Publisher 2010. The Russian book contains an additional topic on "Renormalizations in Solid Mechanics"

2. The book Duality, Symmetry and Symmetry lost: Selected works of H.D. Bui , Alain Ehrlacher and Xanthippi Markenscoff (Eds.), Presse des Ponts 2011, contains some of my papers in these topics, particularly the ones which give exact solutions to some inverse problems in Fracture Mechanics.
See the contents in:
http://www.presses-des-ponts.fr/notre-librairie/299-duality-symmetry-and-symmetry-lost-in-solid-mechanics.html

3. Series: Solid Mechanics and Its Applications, Vol. 182
Imaging the Cheops Pyramid
Bui, H.D.
2012, 2012, XVII, 83 p. 51 illus., 39 in color.
Hardcover, ISBN 978-94-007-2656-7
Due: December 31, 2011-11-13
• About this book
• The author's densitogram of the Cheops Pyramid will for the first time be published in book form
• The gravity inverse problem is for the first time considered for a historical monument
• Illustrated with color figures
In this book Egyptian Archeology and Mathematics meet.
http://www.springer.com/engineering/computational+intelligence+and+complexity/book/978-94-007-2656-7

Densitogram

Density imaging of the Cheops pyramid. A 3D view

Figure 1 : A 3D view of the surface density.

Figure 2. The SE view. Houdin's theory of internal tunnel ramp and theories of degrees.

The surface density is a mean value of the surface finite element of the pyramid, through 10m depth at the bottom and 3m at the top. The density imaging is obtained by discretizing the pyramid into 2000 elements and using 750 measurements inside and outside the pyramid. Methods of inverse problems for the gravitation equation were used with constraints on the density (see our 1988 paper). The vertical projection can be raised up to the four faces. Then we can display a 3D view of the pyramid in different angles, Figure 1.

We select the SE view and make a comparison with Houdin's theory of internal tunnel ramp Fig. 2d and theories which involve degrees. The cornices of these degrees may be filled of stones with many void so that the density can be low (green colour), Figure 2. With degrees theories (Holscher, Borchart, Guerrier etc) the image patterns of degrees are horizontal Fig. 2c.

Errata of Fracture Mechanics, Inverse problems and Solution, H.D. Bui, Springer 2006

Kheops Pyramid

My interest in the Kheops Pyramid dated about twenty years ago. At this time (1987), working at EDF on inverse problems for the assessment of nuclear structures, my Director Marc Albouy asked me to supervise a study on the Kheops Pyramid for searching an unknown tomb of the Pharaon, from microgravity measurements. At this time, two French architects, G. Dormion and J.P. Goidin remarked some architectural details of the tunnel walls which should hide some cavity behind, may be an unknown tomb of the Pharaon? It was a fantastic challenge for all of us to work on the Great Pyramid, which was more exciting than to detect a dangerous flaw in a nuclear vessel !
As a sponsor for scientifc works devoted to archeology, EDF granted financial supports to the French Company of Geophysical Prospection (CPGF) for microgravity measurements, done by Jacques Lashkmanan and his team. I was in charge of the mathematical and numerical team at EDF for the inversion of measurements in order to obtain the density distribution near the tunnels. The result obained for the unknown tomb was rather negative. What a disappointment to all of us.
We then decided to modellize the entire Pyramid with regular meshes and to look at the density distribution. The results for the second study are very interesting. First of all, we found that the pyramid has a high density about 2.4 near its base and a very low one at the top, about 1.87. Second, the misfit of density presents some special structure. In the Proceedings of the International Symposium : "The Engineering Geology of Ancient Works, Monuments and Historical Sites", The Application of microgravity survey in the endoscopy of ancient monuments by H.D. Bui, J. Montlucon, J. Lahksmanan, J.C. Erling, C. Nakla, pp. 1063-1069, Athens 19-23 September, 1988, Marinos & Koulis (Balkema, Rotterdam), we wrote in the conclusion : ".. the Pyramid density does not present a simple symmetry other than a certain spiral structure". An altogether incredible story is that in 2002, Henri Houdin (Engineer) and his son Jean-Pierre Houdin (architect) proposed a new theory on the construction of the Kheops Pyramid which should involve an internal tunnel spiraling up counterclockwise, exactly as given in our numerical result (1988) !

For more information about Houdin's theory, see: www.construire-la-grande-pyramide.fr

The densitogram represents the mean density distribution on the surface of the Pyramid over 10m thick at the bottom, 3-4 m at the top. It has been worked out in 1988 by H.D. Bui (EDF), J. Lakshmanan (CPGF) and coworkers. It is the property of Fondation EDF. (Copyright EDF free, additional drawings to the image not authorized)