A TECHNOLOGICAL CONTINUUM

OF THE LOGISTICS ALGORITHMIC ?

Proposed publication to the Scientific Review SCIENCE - USA

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AUTORS

Pierre Crozat* PhD. – Independent Researcher^[1]Note [1] – Pierre Crozat* - French - Architect-Urbanist graduated from the Ecole Polytechnique Fédérale de Lausanne - EPFL (1971) (Prize of the Society of Engineers & Architects - SIA (CH) (1971), Dr. Engineer in Civil Engineering of the National Polytechnic Institute of Lorraine -INPL (F)/ National Higher School of Mines of Nancy - ENSMN (2002) under the title “Le génie des pyramides” (Congratulations of the Jury Chaired Dr. Hany Helal); Thesis Director: Prof. Jack-Pierre PIGUET Director ENSMN and Prof. Thierry Verdel, Director of the Environment, Geomechanics and Work Laboratory - LAEGO.
Architect-Urbanist graduated from the Ecole Polytechnique Fédérale de Lausanne – EPFL (CH). Dr in Civil Engineering at Université de Lorraine (former National Polytechnic Institute of Lorraine) (F). Address: 54 Rue du Val d’Amour, 39380 LA LOYE (France)
pierre.Crozat@free.fr

Prof. Thierry Verdel PhD.^[2]Note [2] – Thierry Verdel - French - Prof. Department of Geoengineering /ENSMN (F), currently Rector of the French-speaking University Leopold SENGHOR of Alexandria (Egypt).
Georessources, Université de Lorraine, CNRS, GRECU, Mines Nancy (F).
Engineer in Civil and Mining Engineering, PhD in Civil Engineering, Professor at Mines Nancy, School of Engineering.
Address: 1, Place Ahmed Orabi, Al Mancheya BP 415, 21111 Alexandria (Egypt)
thierry.Verdel@mines-nancy.univ-lorraine.fr

Collaboration: Dr Pierre Crozat obtained his PhD (2002) about the Engineering of the Pyramids under the co-supervision of Pr. Thierry Verdel

TITLE

GEOLOGICAL EVIDENCE FOR AN ALGORITHMIC CONSTRUCTION OF THE GREAT PYRAMIDS

ABSTRACT

We show here that the great pyramids sit in the middle of their respective quarry, their constituent blocks being extracted at their immediate perimeter, as demonstrated by the natural fracturing networks of the rocks and the 3D simulation of the paleo-topo-stratigraphy of the Giza Plateau. Geology also explains their location and orientation.

The proposed pyramidal increase method is to overlay successive envelope cones (in onion peels). The “wooden machines multiplied and/or moved” (dixit Herodotus, 5th century BC) allow -simply- to raise and lay a block on two others. This construction method is not in the realm of external geometry (with ramps), but in the realm of internal algorithmic logistics (movement-machine) of labor practice. The final pyramidal volume corresponds to the regularities of pre-Pythagorean geometry.

TEXT

I. STATE OF THE ART REGARDING ANTERIOR WORKS ON THE CONSTRUCTION MODE OF THE EGYPTIAN PYRAMIDS.

1. Numerous authors have theories on the construction of the great pyramids of Giza, from antiquity to the present days: (according to J-Ph. Lauer)^[3]Note [3] – Specially the pyramid of Cheops, “the most enigmatic” because of its interior features: corridors, rooms and large gallery, which, as a result of our study, will be "the most revealing” of the Constructive System of the great pyramids.
   1. Mystical theories:
      1. Biblical theories^[4]Note [4] – Biblical theories: John Taylor, ”The great pyramid : Why was it built and who built it?“, 1859; Piazzi Smyth, ”Our inheritance in the great pyramid“, 1864, and ”Life and Work in the great pyramid“, 1865; Morton Edgar, ”The great pyramid : its scientific features“; D. Davidson; Adam Rutherford, ”Pyramidology“, 1972; Georges Barbarin, “Le secret de la grande pyramide ou de la fin du monde adamique“, 1936
      2. Theosophical theories^[5]Note [5] – Theosophical theories: Marsham Adams, “The house of the Hidden Places” et “The book of the master”, London, 1845. Ralston Skinner, “The source of measures”, qui fait un rapprochement avec la cabale juive, clef ésotérique de la Bible; H.P. Blavatsky, “Isis Unveiled”, 1931; Edouard Schuré, “Les grands initiés”, Paris, Perrin, 1961; Georges Barbarin, “Le secret de la grande pyramide ou de la fin du monde adamique”, Hadyar, 1936; André Pochan, “L'énigme de la grande pyramide”, 1971; Michel Claude Touchard, “Les pyramides et leurs mystères”, 1966.
   2. Pseudo scientific theories:
      1. Astronomical theories^[6]Note [6] – Astronomical theories: Jomard, “Description générale de Memphis et des pyramides, accompagnée des remarques géographiques et historiques, ainsi que Remarques et recherches sur les pyramides d'Égypte, et Exposition du système métrique des anciens Egyptiens, contenant des recherches sur leurs connaissances géométriques, géographiques et astronomiques et sur les mesures des autres peuples de l'Antiquité”, 8 tomes en 9 volumes, Paris, Imp. impériale, 1808-1822; Le Père et Coutelle, “Observations sur les pyramides de Guiseh et sur les monuments qui les environnent”, Antiquités, Mémoires , t. 2, 1818; Abbé Moreux, “Les énigmes de la science”, 1941, et “La science mystérieuse des pharaons”, 1943; Richard A. Proctor, “The great pyramid”, 1888; Duncan Machaughton, “A scheme of Egyptian chronology”, 1932; Cotsworth, “The national Almanac”, 1902.
      2. Mathematical theories^[7]Note [7] – Mathematical theories: Richard Lepsius, “Über den Bau der Pyramiden”, 1843; L. Borchardt, “Die Pyramiden, ihre Entstehung und Entwicklung”, 1922; Jarolimek, “Der mathematische Schlüssel zu der Pyramide des Chéops”, 1890; Flinder Pétrie, “The building of a pyramid”, 1930; Hermann Reikes, 1907; K. Kleppisch, “Die Chéops pyramide, ein Denkmal mathematischer Erkenntnis”, 1921; F. Noetling, 1921; I.E.S. Edwards, “Les pyramides d'Egypte”, Tallandier, 1981.
   3. Original theories, including:
      1. Locks
      2. Water pressure
      3. Artificial stone
   4. Constructivist theories:
      Generally speaking, these various theories attempt to propose solutions for the construction (piling of the strata) by men organized in building groups (without involvement of God, of mysteries, of astronomy or mathematics). Some of them, following Diodorus of Sicily, involve advanced work sometimes larger than the pyramid itself (ramps or locks).
      1. Frontal ramps^[8]Note [8] – Frontal ramps theories: Prof. J. Davidovits, “Ils ont bâti les pyramides”, Paris 2002, “La nouvelle histoire des pyramides”, Paris 2004, “Bâtir les Pyramides sans Pierres ni Esclaves? La science défie les égyptologues”, Paris 2017; Prof. G. Demortier, 2004, et J. Bertho, “La pyramide reconstituée”, 2001: They rely on the mode of transportation of great weights with sleds gliding on ramps made of layers of silt, as shown on Egyptian bas-reliefs. These ramps can be numerous, long and with marked slopes, depending on the authors, and could be raised as the layers of the pyramid are stacked.
      2. Lateral or enveloping ramps^[9]Note [9] – Lateral or enveloping ramps (external or internal): Louis Croon, op. cit. (multiples rampes le long des gradins). Uvo Hölscher, “Das Grabdenkmal des Königs Chephren”, 1912 (quatre rampes superposées sur une face). Wheeler, “Pyramids and their purpose”, 1935 (quatre rampes). Georges Goyon, “Le secret des bâtisseurs des grandes pyramides”, 1983. (external or internal): Based on the principle of ramps put side by side, that get multiplied as the pyramid grows, or of ramps that surround the pyramid, which in turn serves as a support.
      3. Step pyramids and extra steps^[10]Note [10] – The step pyramids of the 3rd Dynasty have been little studied, except that of Djoser in Saqqarah by the Egyptologist-French architect J-Ph. Lauer, who thought it was an overlay of Mastabas. This vision is “formalistic” and not “constructive”. We have demonstrated in the thesis (P. Crozat - 2002) in the chapter “Technical Continuum of Tumular Works” that the appropriate method existed from the first rural cleansing of the Neolithic.to make it smooth^[11]Note [11] – Step pyramid turned smooth pyramid : L. Borchardt , E. Guerrier, “Le principe de la pyramide égyptienne”, 1981, “Les PYRAMIDES l’enquête 2006”., L. Borchardt (1863-1938): In spite of a formalistic and non-constructive vision that requires an enormous ancillary work, this author’s concept of three chambers (subterranean, King’s and Queen’s) has been considered solid and therefore taken over by numerous Egyptologists without any further proof (Fig. S1: The 3 rooms by L. Borchardt).
   5. Accretion and lifting machines:
      1. All preceding works are inspired by the final shape of the pyramids.
      2. By contrast, we are interested in the construction method that, in turn, generates the shape.
      3. Some authors following Herodotus look for an accretion system and for block lifting machines, without describing, however, a logical constructive process. In addition, they do not determine the origin of the materials (quarries).For example, description of a central core of total height ^[12]Note [12] – Accretion of decreasing of walls: H. Straub-Roessler, A. Choisy surrounded by walls of decreasing height.
   
   
2. We attempted to describe the construction process of the pyramids from the point of view of a builder: seeking solidity, easiness and economy.
   
   By contrast, in this article, we propose a system of algorithmic pyramidal growth^[13]Note [13] – Algorithmic pyramidal growth: Pierre Crozat- Ph.D (1990-2019), “Système constructif des pyramides” 1996, “Le génie des (grandes) Pyramides” 2002.: we based our proposal on Herodotus’ historical revelations and particularly in a passage in book IV that describes a mode of lifting with wooden machines, and an accretion process based on the movement of the machines. Starting with a primary core, the builders created a succession of enveloping cones made of blocks of local stone extracted around the building, in horizontal quarries, in successive corresponding volume phases (Cf. 1: CV of P. Crozat Searcher).
   

   We based our work on:
   

   1. Herodotus (450 BC): his description of the working mode of the machine made of short pieces of wood contains a solution: a real constructive “system” representing a technological and conceptual leap, the final outcome of the technological continuum of funerary tumuli.
   2. Strabo (I^st cent. BC): “Visiting the pyramids, we observed an extraordinary fact that deserves, we think, to be mentioned. It is a large heap of stone shards that cover the ground in front of the pyramids. One only has to search in these heaps to find small petrifactions with the shape and the size of a lentil, resting sometimes on a bed of other debris (petrified as well), that look like half-shelled vegetable peels” says the Greek geographer, who is interested in the stones of the pyramids and recognizes “lentils” (Eocene limestone: Nummulites of the Lutetian period).
   3. K.R. Lepsius (1810-1884):he was the first to relate the volume of the pyramid and the length of the reign of the pharaoh.
   4. A. Choisy (1841-1909): the author of a “History of Architecture” (1899), he studied the construction techniques since Antiquity, including “The Egyptian art of Building” (1904).
      He had the intuition of a building method via successive envelopes. However, he did not develop and demonstrate the full constructive system, which is what we attempted to do.
   5. M. Serres (1939-2019): historian of sciences and techniques, in “The origins of geometry” (Paris 1993) describes a specific pre-pythagorician period of evolution in Egypt that he called “logistics or algorism” and which, for us, corresponds to the technological and conceptual leap constituted by the construction period of the five great smooth pyramids (Cf. 2: Engineering of pyramids 2017 DIGEST).
      
   
   
3. We present a different approach, scientific, technical and operational
   
   
   1. Our experience as builders cannot admit “rampist theories” of any kind that require far too heavy ancillary work, such as:
      1. creating a horizontal platform by clearing a large volume of material (what to do with it, where to place it?),
      2. building via “horizontal seating” (increasingly reduced) that requires an annex (ramp) of stone or earth which in the case of the frontal ramp is more important than the future pyramid itself, and which will then have to be dismantled (double work),
   2. Starting from the builder’s point of view: as experienced builders, we understood the technical and operational value of the constructive system described by Herodotus.
   3. The technological level of block-picking equipment and the precision of the joints indicate a very advanced (almost contemporary) mastery of the know-how of Stone Trades.
   4. It is the vernacular principle of borrowing the constitutive materials of the work from the immediate surroundings^[14]Note [14] – Photograph of the Neolithic Maes Howe tumulus of the ORKNEY (ORCADES, north of ECOSSE) (more than 20 feet high and 100 feet in diameter) dates from the Neolithic (-5000 BP) is the most expressive prototype of the “vernacular” method: exploiting around the edge to abound to abound in the center. (Ed. NATIONAL GEOGRAPHIC - Review August 2014)., piling them in a conic form, and then keeping building from the center, with enveloping pyramidal cones that create the volume accretion. The pyramid is actually in the middle of its very quarry, with many blocks pre dimensioned by natural fracturation. This interpretation is typical of the technology of the time and belongs to the continuum of tumular works from the Neolithic period.
   
   
4. We put in relation the geological characteristics of the site with the construction
   
   Egyptologists and archeologists have not taken under consideration the materials, the geology (structure and stratigraphy), the engineers’ geology (study of the natural rock fracturation), the modeling, or the evolution of the constructive method that creates the shape, nor the interaction between the pyramid and its quarry.
   1. M. Lehner and Z. Hawass: In the topographical and geological rendering of the site (Mapping Project on the Giza Plateau, 2005) they have not taken under account the advantages of photogrammetry, nor the importance of geology (structure and stratigraphy), nor the importance of engineering geology (natural process of rock splitting) that we observed on the site at the time. The National Geographical Institute in Paris used photogrammetry as early as the 1960s to create an orthophotoplan of the Giza Plateau.
   2. the step pyramids are constructed with manu-portable stones (small apparatus) supplied by the Superior Eocene deposit of their site and with the constructive “degree” method (Fig. S2: Step Pyramids).
   3. the photograph of the great excavation of Zaouiet el Arian shows the surface the Upper Eocene layer and, below, the Middle and Lower Eocene layers: although the sarcophagus was installed, this pyramid was never built. Dated to the 4th dynasty (by G. A. Reisner) it belongs rather to the 3rd dynasty as suggested by its pit and the stratigraphy of the site (Fig. S3: Zaouiet el Arian).
   4. the great pyramids are constructed with “cyclopean blocks” coming from the Middle Eocene deposit of their site and by the appropriate constructive method of “successive benches” (see further) (Fig. S4: Great smooth pyramids).
   5. The definitive proof is provided by the 3 pyramids of Abousir (degree pyramids with addition of degrees, to make them smooth) which, dated to the 5th Dynasty (thus after the great pyramids of large apparatus), used the old “degree” method imposed by the Upper Eocene deposit (small apparatus) (Fig. S5: Abousir pyramids).
   6. text pyramids (of the type of Ounas in Saqqarah - 6th dynasty) have become “symbolic”, consisting of cyclopean blocks on the periphery (backing stones of the Middle Eocene) and an internal opus incertum of the Upper Eocene). The important part is only the burial chamber with the sarcophagus, the painted starry sky and the Texts of the Pyramid (Fig. S6: Text Pyramids).
   7. brick pyramids (Fig. S7: Brick pyramids).
   8. Our analysis is corroborated by the scientific thesis of M. WISSA^[15]Note [15] – Myriam Wissa: “Old Kingdom's Building Stone in the Royal Funerary Complexes from Memphis and Letopolis : Typological and Lexicographical Studies” (Paris IV Sorbonne – 1995). This dissertation is a multidisciplinary research dealing with archaeology-Egyptology and petrography and stratigraphy (geology). Focusing on the nature and the origin of building stones used in the royal funerary complexes of the old kingdom and the reason of choice of the materials. (1995).
   9. In fact it is the underlying deposit of the pyramid site that imposes the appropriate constructive method. The method, in turn, generates the shape. In other words, the shape of the Pyramids is not a Game of Styles but the result of the Method.
   10. The architectural geometric approach (inherited from the Greeks) has remained inoperative from Diodorus to this day that is for 2500 years, because it was not adapted to buildings made 2200 years earlier.
   
   
5. Some archeological proofs that corroborate our vision
   
   
   1. The bent pyramid is in the center of its own quarry, as shown by the picture of M. Bridges (“Egypt: Antiquities from above”, Little, Brown and company, Boston) (Fig. S8: Bent Pyramid).
   2. The 5 Great Pyramids (the rhomboidal pyramid (South Dasher), the red pyramid (North Dasher), and the 3 pyramids of Giza) are made of parallelepiped blocks, arranged as headers, (i.e. their length is perpendicular to the observed face) (Fig. S9: Arrangement of the blocks “in the back”).
   3. The blocks on top of Cheops’ pyramid have visible edges that have been disrupted a lot by various demolitions, fractures and renewal. Nevertheless, for each cut block one can see the “horizontal corbelled apparel”, the most economical according to A. Choisy (Fig. S10: Top of the pyramid of Cheops).
   
   
6. By contrast the reasoning presented here corresponds to a deep paradigm shift: it is based on the method, called “system” by Herodotus:
   
   Very ancient, it had to be recreated stage by stage as it applied to the pyramid. This system of reasoning, called “Algorithmic Logistics” needs a concrete support. It necessitates the learning of the theoretical algorithmic path, and its permanent confrontation with concrete and operational reality, and the know-how of stone works (Cf. 3: Simulation pyramidal increase - 1997).

II. CONSTRUCTIVE SYTEM OF THE PYRAMIDS: AN ALGORITHMIC LOGISTICS (?)

A. INTRODUCTION: THE PRINCIPLE OF THE METHOD

The great pyramids of Egypt are human constructions that belong to the technical continuum of major and minor tumular buildings (Fig. 1) which, all over the world, are built according to the size and weight of the local materials, and according to the techniques (by hand, with scaffolds, with lifting machines, with elevating engines) available at the time of their erection, or due to circumstances. This analysis explains the evolution of the constructive methods that produce and generate the specific shape of these works. The passage from step pyramids to the great smooth bench pyramids is not just a variation of shape, but the very result of the method chosen.

B. Herodotus’S ESSENTIAL CONTRIBUTION

The first historian, Herodotus of Halicarnassus (5th century BC) described very precisely the operating mode of piling the blocks of Cheops' pyramid, with a system of successive benches (but not steps), called alternately bomides or crossaï (there are several versions).Bomides meaning “resting table” and crossaï meaning “corbel”, two complementary, somewhat technical concepts belonging to the domain of construction, that the translators, literary minded Hellenists, did not fully understand and interpret, construction not being part of their skills (Fig. 2). This description is quite precise and efficient, worthy of development.
Later visitors, heirs of Greek culture, and practicing a geometric vision (Thales already), later Egyptologists, archaeologists and pyramidologists have stumbled on that text and have tried to pierce the mystery of the construction of the pyramids according to the other description, given by Diodorus of Sicily (1st century AD) and mentioning earth levees (ramps) (Fig. 3), which however represent more work than the pyramid itself, and need to be demolished afterwards. Since that period of antiquity, all kinds of ramp solutions have been proposed, without ever showing plausible proofs, and the solutions invoking machines have remained quasi absent.

C. SCIENTIFIC, TECHNICAL AND OPERATIONAL RESEARCH

The analysis and comprehension of Herodotus’ text has opened a new field of inquiry on the construction mode of the pyramids of Egypt, based on three domains:

1. The research on the evolution of the construction methods of tumular works, including the great pyramids of Egypt, belongs to the technical domain of building art.
2. Discovered by an experienced professional, it offers new perspectives, by contrast to the numerous interpretative and personal theories inspired by religious or story telling pretexts. It follows a rigorous path comprising hypothesis, research and development, verification and expertise, experimentation and simulation that is verifiable at every step of the way.
3. It comes from the description of the operating mode of the machine capable of lifting and placing a block from one layer onto the next that the know-how of stone workmanship has in part maintained to this day.
   The research presented in this paper is structured in 24 phases of successive hypothesis, according to a logical path based on the logistics of construction, based on personal experience, intuition, observation, study, comprehension that started in 1990 in front of the Sphinx, with a challenge to the construction professional: "how would I build a pyramid?”.

1. Hypothesis of the principles of vernacular techniques and effort saving: one builds with materials taken from the immediate surroundings, hence the choice of a site capable of providing them and according to the available technique.
2. Hypothesis of the "sand castle": how to create a big heap? Start with a small one and find how to increase its size (Fig. 4).
3. Hypothesis of construction with stairs^[16]Note [16] – A. Choisy, “The art of building among the Egyptians”, History of Architecture, Chapter II EGYPT - Ed. Inter-Books, then research and analysis of Herodotus’ text followed by hand drawn, then infographic modeling of the piling (Fig. 5) (see the infographic animation in the annex). Development of the system of “pyramidal accretion”, with stacked envelope-cones and demonstration of the utilitarian role of the great gallery (Fig. 6) in Cheops' pyramid: an extraordinary oblique lifting ramp, capable of hoisting to their destination the 50 granite monoliths of the King’s chamber, on a cluster of inclined surfaces created by the system itself. Thus, the most enigmatic pyramid is the most revealing of the constructive system of the great pyramids^[17]Note [17] – Premier livre “SYSTEME CONSTRUCTIF DES PYRAMIDES” Pierre Crozat - Ed. Canevas (F) & (CH) – 1996 - ISBN 2-8838-064-3- of Egypt.
   

   D. THE PART OF GEOLOGY

4. Application of the old vernacular principle, i.e. borrowing the materials from the very surroundings of the construction erected in the center, via successive corresponding phases: the pyramid is therefore in the middle of its quarry site, at every stage, which implies the geological and petrographical study of the erecting site.
5. Study of the geological characteristics of the Giza plateau (origins, structure, stratigraphy, karstic erosion) showing that the Giza plateau is an anticline fold with an NE/SO (axis, North 45°) with two networks of discontinuities^[18]Note [18] – M. Ruhland: “Researches on natural fracturing of rocks (1969 - 1972) - Method of study of natural fracturing of rocks associated with various structural models” Institute of Geology, Louis Pasteur University, F 67084 Strasbourg Cedex - Associate Research Team at C.N.R.S. “Structural geology and tectonic analysis”. Bulletin of the Geological Society, 26, 2-3, p. 91-218, Strasbourg, 1973 (LOUIS-JEAN Imp., 1973).: the main one made of parallel and transversal joints and the secondary one, diagonal to the previous one(alpha delta right and left (Fig. 7).This framework of diagonal (closed) joints dictates the orientation of the exploitation of the horizontal and slope quarries, today as well as yesterday, and therefore the orientation of the pyramids themselves. The three Giza pyramids are thus oriented by geology^[19]Note [19] – Public lecture given at the CULTNAT / Smart Village / CAIRO on the 26/09/2017 by Pierre Crozat.. As a proof, striated tectoglyphs oriented E/W were discovered on the North face of the rock bed of Chephren’spyramid in 2005 (Fig. 8).
6. The Sphinx is a quarry leftover (the plateau was dug all around it (Fig. 9) where one can read the stratigraphy^[20]Note [20] – J. Cuvillier, “Revision of Egyptian Nummulitics” – Ed. Cairo, imp. Schindler, 1930. In-4, 372 p., Pl. ( Memoirs of the Institute of Egypt.) ( p.538). and the slope of the plateau (Fig. 10). Its head belongs to the same (a) layer (Auversian) than the second story of KhentKawes’ tomb and the high temple of Chephren. Chephren's pyramid borrowed its materials from a single stratigraphic layer called (g) “building stone”, that provided good quality stone, 12 meters in thickness, dense and easy to exploit.
7. The 3D simulation (GoCAD)^[21]Note [21] – 3D simulation of the paleo-topo-stratigraphy of the Giza plateau was carried out with the assistance of the Geoengineering Department of the Ecole des Mines de Nancy (Prof. J. Sausse and student Th. Burlett) in 2015. of the paleo-topo-stratigraphy of the Giza plateau with the orthophotoplan^[22]Note [22] – Institute of National Geography - IGN, Paris (F), established by photogrammetric restitution, well the digitized topographic survey of the plateau GIZEH: Giza Plateau Mapping Project directed by Mark Lehner (AERA). as a background constitutes a novel approach (Fig. 11). It made it possible to situate all three pyramids within the stratigraphy: Mykerinos' at level 75 m; Chephren's at level 70 m; Cheops' at level 60 m (Fig. 12). Each pyramid used deeper and deeper geological layers, with different levels of hardness and density, which explains their various states of conservation.
8. Chephren's pyramid being built with the single (g) layer of building stone, quite dense but a little fragile, presents a great regularity at its base, and in its blocks and their assembly, but also a strong erosion (Fig. 13) noticeable on several of the envelope-cones. This type of erosion does not exist on Cheops' pyramid, because its envelope-cones came from lower (f) layer at 4 m, and (e) layer at 5 m and a portion of (d), that are much less regular but much more resistant to erosion. Only the final cover is in limestone from Tura, on the left bank of the Nile, and it disappeared because of erosion and human carrying, as did Chephren's, in part only.
9. In Cheops's pyramid, the Queen’s chamber appears situated on the pre-existing plateau^[23]Note [23] – As intuitively envisaged in “CONSTRUCTIVE SYSTEM OF PYRAMIDS” ( p.143-144) Pierre Crozat - 1996., at the center and at the beginning of the construction of the pyramid, and at the center and the beginning of the sub-horizontal peripheral quarries, as indicated by the median cut (N/S), despite a slope NW/SE of about 3,2%.
   

   E. THE VOIDS DISCOVERED BY SCAN PYRAMIDS MISSION OF 2017

10. The existence of the great and small voids in Cheops' pyramid, discovered by the Scan Pyramids^[24]Note [24] – SCAN PYRAMIDS is a Mission under the aegis of the Arab Republic of Egypt / Ministry of Antiquities, designed by HIP INSTITUTE Heritage-Innovation-Preservation and carried out by the Faculty of Engineering of the University of CAIRO. mission has not so far been explained by Egyptologists^[25]Note [25] – Committee (RAE) of famous Egyptologists-Archaeologists composed of Zahi Hawass (EG), M. Lehner (USA), R. Stadelmann (D) all following the rampant solution. (Fig. 14).
    By contrast, research on the construction mode of the pyramids allows a cautious interpretation^[26]Note [26] – See: OPEN LETTER (official) of P. Crozat at 10/11/2017 to Dr. Hany Helal, Prof. at the Faculty of Engineering at Cairo University, Dr. in Rock Mechanics and as such President of the thesis jury of Pierre Crozat about “The Engineering of the Pyramid” (2002) and Scientific Director of SCAN PYRAMIDS., based on statics and stone resistance: the voids could represent a protection system for the great gallery and the ascending corridor that had to remain functional until the entombing of the king at least, while the pyramid was only half built.
11. This cautious interpretation suggests that it is a relief duct, placed over the fragile great gallery and the ascending corridor, to protect them. This duct consists of horizontal bushels made of rafters (Fig. 15), whose static role would be to distribute the charge (of the future pyramid not yet constructed) on each side of the body already built. An image of this cautious charge distribution system can be seen on the North face, with the two overlapping rafters above the entrance – the outcome of this Chimney of discharge (Fig. 16), which show the combinatory organization of the overlapping bushels on the slope.
12. The existence of the voids revived our constructivist approach. Indeed, the use of “a machine made of short pieces of wood” for lifting (a lever on a tripod called Chadouf in Egypt), whose machine motion constitutes an algorithm: a simple concrete physical lifting algorithm, consisting of lifting one block to place it on two others, with a necessary foothold to achieve the action (as described 24 centuries later by Archimedes). Our research developed this vision as early as 1996.^[27]Note [27] – P. Crozat in the “Constructive System of Pyramids” This approach takes under account the coordinated use of the quarry and the erection of the pyramid, as well as the simultaneous achievement of the inner arrangements of the pyramid. Particularly the poorly named "girdlestones" that are successive abutments whose role is to block the great gallery and prevent it from slipping, since its floor blocks are arranged on a slope rather than horizontally (Cf. 4: Open letter to Scanpyramids).
    

    F. MODULAR COORDINATION

13. It made necessary the research of a modular coordination and a progression rhythm P. /Q. (Pyramid/Quarry). We used a theoretical design with a square plan where the center, common focus of both the pyramid and the quarry for each phase of accretion, is at level (0), on the pre-existing ground, and the top at level (6); the extraction floor is at level (-1), generating a volumic pyramidal accretion, with a ratio of 7, whether in plan or in elevation, whatever the ratio between the height and the half base^[28]Note [28] – Jean-Philippe Lauer (1902-2001) a French architect, versed very early in Egyptology-archeology, has worked throughout his life on the site of Saqqarah (step pyramid of Djoser), he is the author of the “theory of the frontal ramp” of the great pyramid of Kheops in his book “The mystery of the pyramids” (Ed. Press of the City - France - 1988 - ISBN: 2-258-02368-8) and the report Height / 1⁄2 Base = 14/11, an irrational number (anachronistic according to the History of Mathematics or even esoteric doubtful) in Chapter II “Scientific Knowledge - The Geometry of Pyramids” (p.227-237) that this study “CONSTRUCTIVE SYSTEM OF PYRAMIDS” fundamentally contests. An exchange of letters between J-Ph. Lauer (with his hand) and P. Crozat - 1997) explain the reasons for this confrontation; it has been preserved and can be produced. This ambiguity corresponds to the esoteric current of a so-called Sacred Architecture (Fig. 17).
    

    G. ALGORITHMIC HYPOTHESIS

14. This algorithmic hypothesis compares quite favorably with the surveys of Maragioglio & Rinaldi (1965) and of Dormion (1996) and with the stratigraphic data (Fig. 18):
    1. The queen’s chamber is situated on layer (g) (12 m) at 21m (level (0).
    2. The ascending corridor cuts through the limits of layers (g) (12 m) and (f) (4-5 m) at the very spot where the “local stone block” appears;
    3. It descends through this layer (f) (4-5 m) to its limit with the sub layer (e) (also 4-5 m), where it joins the descending corridor that goes through that layer and through the following one (d) (also at 5 m), etc. The limit between (c) and (d) corresponds to the butt of the built masonry so called “thieves passage”, at the bottom of the grotto thatmarks the final level of taking from the quarry and the beginning of built blocks at the foot of the pyramid.
    4. One finds the layer of pebbles in which the grotto was dug on the slope, above, at the foot of the G 2000 Mastaba, cleared away by the slope quarry.
    5. Al-Ma’moun sapping was dug at the bottom of layer (f) (4-5 m); its horizontal floor belongs to the inferior layer (e) (5m).
15. The double concept of “Logistique ou Algorisme” (M. Serres^[29]Note [29] – Michel Serres (1930-2019) “The origins of geometry” (Paris, Ed. Flammarion, 1993 ) is a philosopher, historian of science and mathematics, epistemologist and French man of letters, elected at the French Academy in 1990. His expression “Logistics or Algorism” which links the Logistics to the Algorithm (named after the Persian mathematician Al Kwritzmi of the Baghdad School - and his Egyptian colleague Al Kamal, under the Caliphate of AL MAMOUN - the patron of the undermining of the Cheops’s pyramid - in the 9th century) is a brilliant epistemological shortcut that suggests its source because, given the effectiveness of this undermines, it is very likely that Caliph was well informed!, 1993) confirms the ensemble of the mechanistic solution. It was in use at the time of the construction of the great pyramids and played a major part in the appearance of geometry in Greece, 2000 years later. The metrics of ancient Egyptians was misunderstood and rejected by the Greeks.^[30]Note [30] – Plato's grave mistake in thinking that Zeno represented the knowledge of the Ancient Egyptians.
    

    H. CHEOPS' CUBIT

16. Egyptologists^[31]Note [31] – It is mainly the Egyptologists of the biblical, theosophical and astronomical theories. have determined a measure unit to apply to the pyramids, called cubit. But at the theoretical level, both the size of the unit and the ratio between the height and the base of the pyramid have no influence whatsoever on what generates the pyramidal accretion. The accretion depends only on the plan, and on the algorithmic interplay established by the ancient surveyors (Harpedonapts, or string stretchers). Such an algorithmic construction logistics is necessary for any building at any time. It is obvious in all three great pyramids, and particularly Chephren's (Fig. 19).
    One must examine the rules of stone professionals^[32]Note [32] – These skills of the Stone Trades were brought through the collaboration with the Companion J.-P. Foucher, ex-director of the Higher Institute of Research and Training of the Crafts of the Stone - ISRFMP in Rodez (F), and The Companion R. Morel (Ϯ) Stonemason (Master of Art), members of the Association Œuvrière of the Companions of the Duty of the Tour de France AOCDTF (headquarters in Paris - France). (quarry workers, cutters, positioners, etc) to understand the process. One must also examine the orientation of the stone extraction depending on the diagonal network, the depth and width of anthropometric trenches, the technical ways to detach the blocks, the cutting of the stone base into the blocks, etc., as demonstrated by the traces left in situ. These operational practices are still in use nowadays in the last manual stone quarries in Egypt (Fig. 20).
17. To summarize, the ramp solutions are all based on the posterior geometric vision inherited from the Greeks. But the pyramids were built by the anterior know how, that we call algorithmic logistics, and represent its concrete demonstration.^[33]Note [33] – Pythagora of Samos (6th century BC) mathematician initiated in Egypt is the author of the theorem of the Right Angle that bears his name (which the Egyptians practiced long before him), but also Series of Figured Numbers (so-called “sacred”) or series of arithmetic suites (linear, triangular, square) that interest the subject of the construction of the Great Pyramids because they open to Pyramidal Growth and Algorism.
    

    I. PHOTOGRAPHY & COMMENTARY by Z. Hawass

18. The aerial photographs of Chephren's pyramid (M. Bertinetti) published by NGS^[34]Note [34] – NATIONAL GEOGRAHIC “EGYPTE between heaven and earth” ©French edition - 2004 - ISBN 2-84582-133-6. and by WhiteStar^[35]Note [35] – WHITE STAR LLP has been the exclusive publisher of National Geographic books and guides for Italy since 2001. A prestigious American non-profit institution, founded in 1888, the National Geographic Society aims at scientific disclosure and education. are all very interesting, but the most revealing (Fig. 21) deserves particular attention^[36]Note [36] – Photography reserved for the former Secretary General of the Supreme Council of Antiquities of Egypt, Dr. Z. Hawass for his album “Treasures of THE PYRAMIDES” Ed. White Star LLP - Vercelli (I) - ISBN 978-88-6112-382-3. because of the accompanying commentary (p.59): "Summit of Chephren's pyramid in fine limestone from Tura. The supporting masonry appears regular here, under the facing, but the depth of the blocks and of the steps varies considerably. The fragments of the supporting masonry adhere to the core lower, after removal of the facing".
    This statement does not take under account the natural disintegration of the limestone blocks by erosion. Also, the term “supporting masonry” does not correspond to 99% of the volume of the pyramid: it constitutes and generates its shape. This formalist vision concentrates on the surface and does not understand that the fragments are caused by the sliding of the eroded enveloping cones, clearing the one that is being observed: one can see four of five of them on a photograph of the ensemble, counting their projecting shadows (Fig. 13). Under this enveloping cone called, here "supporting masonry" there exist many others starting at the core of the pyramid.^[37]Note [37] – This comment denotes the same syndrome as in "formalist" architects, to speak only of pyramid architects, that is, a phobia of the notion of “system”, a system that they say would deprive them of their creativity.
    The 3D simulation of the Gizeh Plateau paleo-topo-stratigraphy explains the regularity and the normalization of the blocks of Chephren's pyramid. They were only taken from the (g) layer, i.e. “building stone”, to the exclusion of any other layer (Fig. 10). This geologically based observation leads to others of the same type, that explain, in turn the irregularity of Cheops' pyramid blocks, taken from the underlying (f) and (e) layers and the presence of a very regular core at the heart of the Cheops pyramid, precisely where all the inner devices are (Queen and King’s chambers, ascending corridors, great gallery and relief chimney). The algorithmic regularity allows quite precisely their execution.
    In the same manner, stratigraphy explains the quality and the color of Mykerinos' pyramid and of the second level of KhentKawes' tomb, as well as the Sphinx head and the top of the massifs of the high temple of Chephren, all coming from the Auversian(a) layer (partial “on-lap” surface of the plateau).
    Chephren’s pyramid is more perfect than his father’s Cheops. The construction minded analyst finds it strange that Egyptologists consider it the most simplistic, and we think that when it is submitted to muography, a new discovery will be made.
19. One sees a fundamental difference between commentaries focusing on the surface of the pyramids and the study “from the inside” summarized here. The constructor’s approach is necessary and gives way to new solutions. It should be taken under account and debated at the world level of scientific culture and technique. An international expertise could be led, independently from the current protagonists, since the Egyptian pyramids belong to the World Heritage, to the history of civilization, to the history of construction (geology, engineering, civilian engineering, mines and quarries), to the history of science and of mathematics.
20. The pyramid sits at the center of its quarry: the vernacular principle (P. Crozat, 1997) consisting of borrowing from the immediate surroundings to pile and amass in the center was corroborated by the paleo-topo-stratigraphy simulation. Additional studies are needed on the dimensional and quantitative coordination of the blocks through each phase of the pyramidal accretion. Also, one needs a modular coordination of the correspondence Pyramid/Quarry.
    One should seek the surfaces and depths of the quarry, except for the extraction trenches, that correspond to the volume and number of the examined enveloping cones. Theoretically, they should represent a geometric, volume progression, to reach the final phase of the pyramid before its cleaning, the cleaning consisting of breaking the edges of the steps of the last enveloping cone made of fine limestone from Tura: "the pyramid was finished starting with the top and the lower benches (not steps) until the base of the building", says Herodotus. The pyramid is finally completed and "dead" since there is no more support for the crossaï blocks to be put.
    

    J. ALGORITHMIC LOGISTICS THEORICAL MODEL

21. The solution proposed by this research on the modular coordination between the pyramid and the quarry is based on a network of extraction trenches with a width of a “royal” cubit (the width of the quarry worker) isolating square blocks of 6 cubits in both directions of the secondary diagonal network created by the natural fracturation of the rocks (see above part 5) geological hypothesis). The network is diagonal with respect to the NE/SW axis of the anticlinal fold of the Giza plateau, i.e. N/S and E/W:
    1. This provides 6 blocks of 2x3 cubits for each face of the pyramid, also oriented N-S-E-W. The axial design of this square network of trenches is therefore 7 cubits. The theoretical growth rhythm proposed here is of one "negative" (1n) extraction bed in the quarry providing the blocks (2x3 cubits) necessary for piling six “positive” (6p) beds built “in charge pile”, that is, a pseudo symmetrical triangular piling, as defined by the construction system shown.
    2. The ensemble of the construction processes in this way for each enveloping cone fitting over the preceding one: 2 negative beds providing the necessary blocks for piling 12 positive ones, then 3 negative ones for 18 positive ones, etc. (Fig. 22).
    3. The “theoretical” research of "algorism" (step by step) drawing on the concrete extraction of Quar. blocks (Fig. 23) requests coordination with the "pile-up" stacking of the Pyr. blocks. These blocks are pseudo-normalized because they are imposed by the manual extraction mode (and by the trench grid) and the limited handling-transport mode (at the most economical). It requires the relearning of this mode of reasoning or constructive logic dictated by the algorithm provided by the “machine-motion” which imposes its rigorous logistics (Fig. 24).
    4. In fine, whatever the height of the layers, the blocks must be extracted, moved and installed manually, through the multiplication [1n (negative) + 6p (positive) = 7] of the proposed pyramidal accretion. Each of the enveloping cones, at the end, will always be 7 times the height of its quarry, even if the sitting benches are not very regular; hence the necessity-invention of a horizontal assembling with recesses^[38]Note [38] – Auguste Choisy, Engineer Polytechician and Architect (1841-1909), “History of Architecture” - Ed. Inter-Books - Chapter II EGYPTE - Paragraph: Device, Details of The Execution of Walls and Massifs: p.28 and 29), describes the three processes used (with the corresponding sketch), in the construction of the pyramids- “the device by set seats A (which corresponds to Chephren -NDR); B the B-drop-off machine (found in Cheops and Mykerinos - NDR - the most work-efficient equipment," he will say); C the device by successive veneer (which corresponds to the dime pyramids of Djoser - NDR)”., that one can observe everywhere.
22. The most striking moment in the checking part of this research consisted (Fig. 25) of over imposing drawings of the inner arrangements (according to Dormion’s renderings, 1996) with the colored drawings of the strata of the quarry (colored to follow easily the logistics imposed by the algorithm, on a N-S section of Cheops pyramid, at a scale of 1/200th - 5 mm/meter). One can then see that the ensemble of the inner arrangements (King’s and Queen’s chambers, ascending corridor, great gallery and relief chimney) and the ventilation ducts come under the same algorithmic logistics proposed by the builders. In addition, the stopping of the ventilation ducts of the Queen’s chamber, 63m from the center, could correspond to an accretion phase of the Cheops’ pyramid of a different origin (borrowing from the West at level (g) building stone).
    In this first regular inner pyramid, we were able to model in details the progression of pyramidal growth that creates the access way to the two symmetrical voussoirs of each rafter constituting the plugs of the “careful” discharge chimney over the great gallery. Therefore, we propose a second use, their simultaneous erection, for the “extraordinary slanted elevator”, whose primary function is to put in place the King’s chamber monolith.
    

    K. COMBINATORIAL OF TWO ALGORITHMS

23. M. Bertinetti’s photographs are always of great quality, but they are also of great interest for our thesis, for they allow a detailed, in depth analysis. The one showing the top of Chephren’s pyramid was colorized by us for better understanding. It shows a new algorithm for the covering, ignored up to now, that combines with the construction algorithm of the pyramidal accretion modelised since 1997.
    The difference between these two algorithms is the following (Fig. 26): for the construction, the blocks were placed as headers, while for the covering they were placed as stretchers, which allow the crossing of the angular blocks on the edges, from one bench to the other. This archeological observation, available to all nowadays, involves definitively the solution of algorithmic logistics for the great pyramids.
    

    L. CONSTRUCTION’S SIMULATION - 2019

24. Following the various partial simulations already carried out (1996, 2001 and 2005, into P. Crozat’s web site www.crozat.damia.ch), the movie simulation of the construction of Cheops' great pyramid is now almost completed (Fig. 28). However, it necessitates competency, collaboration, technical, logistical and financial means to be achieved. The system of pyramidal accretion that was developed, beginning with Herodotus’ text, engage the domain of algorithmic logistics and corresponds perfectly to the concept of self-regulating system developed by Edgar Morin.^[39]Note [39] – Edgar Nahoumsay Morin (1921 - Paris) (“Introduction à la pensée complexe”, ESF Ed., Paris. 1990)is a French sociologist and philosopher. A thinker of complexity, he defines his way of thinking as “co-constructivist” by stating: “I am talking about the collaboration of the outside world and our mind to build reality”. It should be taken under account and debated.
    

    M. CONCLUSION

25. To conclude this communication on the constructive system of the pyramids, there is nothing sacred, mysterious or irrational there. Also, it is not “geometry”, which appeared in Greece 2.000 years later, contrary to what is commonly claimed. It is “an appropriate system” (Fig. 27) based on Algorithmic Logistics, the existence of which at the time of the pyramids was authentified by Michel Serres -1993).
26. So far, no one had taken geology under account: genesis, structure, stratigraphic, and Engineer’s geology (networks of natural fracturation of the rocks) although it is what explains rationally the positioning and the orientation of the three great pyramids of Gizeh; nor had the quarry extraction method in sub horizontal layers been studied, nor its clear traces, observable by everybody and nor had the evolution of the method used within the Technical Continuum of tumular edifices been discussed.
    Purely external observations of the shape of the pyramids cannot possibly clarify the fact that the shape of the pyramid is the very expression of the method used, itself adapted to the size and weight of the local material provided by the immediate surroundings, hence the choice of an adequate site (Cf. 5: Lettre circulaire à mes Pairs 2018 – in French).
    In fine, beyond time, space, and civilization difference, let us observe this photograph of the digging of a channel, by hand, in PRC, at the time of the Mao Zedong’s Cultural Revolution, for social re-adaptation (Fig. 29). It is the archaic technique that interests us, the only usable one in the absence of mechanical engines. Without contest, it goes back to the first agrarian civilizations of the great rivers (Igharghar, Triton, Indus, Tigris and Euphrates, Nile, Huang HO, etc.) and to the hydrological works to control the swelling waters and irrigation, primordial structuring features of the constitution of the first States. For our thesis, it represents a fine illustration of the Algorithmic Logistics, although spatially reversed vis à vis the pyramids. Thus, in Shaanxi province around 2300 BP numerous earth pyramids were erected (tombs) where one can see where the material was taken from, around them. It represents an “ontological” direct link between quarry and pyramid (Fig. S11: Pyramids of Shaanxi-PRC).

ACKNOWLEDGMENTS

Without the precious assistance of my friends, this communication could not have happened:

Ms. Soroor Ghanimati, Ph.D., architect and archeologist at NES/UC-Berkeley, USA, for following my research and suggesting publishing,
Ms. Brigitte Berthier, Information Technology professor, Lycée Pasteur in Dole, France,
M. Bruno Hostalery, engineer ESTP- Paris, France, Infographist and programmer, Lausanne, Switzerland,
M. Patrick Faivre, DPLG architect, for his help with document research, in Lyon, France,
Ms. Françoise Zurif, Ph.D, for translating and providing scientific redaction advice, La Loye, France.

List of Supplementary Materials

1. Publications: Publications produced as this research progressed have been filed or/and published, the following non-exhaustive list:
   
   
   1. “SYSTEME CONSTRUCTIF DES PYRAMIDES” 1996-97 - Ed. Canevas (F) et (CH) - ISBN 2-88382-064-3
   2. “Le Génie des Pyramides” 2002 - Ed. Dervy - Les lieux de la tradition - Paris - ISBN 2-84454-161-5
   3. Publication of the Doctoral Thesis “Le génie des pyramides” supported in 2002 in Civil Engineering.
   
   
2. Reviews and Deposition Sites for Researchers :
   
   
   1. POUR LA SCIENCE, édition française de SCIENTIFIC AMERICAN, no 265 novembre 1999, rubrique Perspectives Scientifiques, page 14, intitule : “Rampes ou enveloppes ? : de nouvelles hypothèses remettent en cause l'utilisation des rampes pour la construction des pyramides d'Egypte”.
   2. Journal officiel de la National Geographic Society, vol. 2.4, no 7, avril 2000, rubrique GEOGRAPHICA, intitulé : “NOUVELLES HYPOTHESES SUR LA CONSTRUCTION DES PYRAMIDES EGYPTIENNES” par Anne Hebert.
   3. SYSTEME CONSTRUCTIF DES PYRAMIDES : de la géologie à l’édification (P. Crozat et Th. Verdel) 2002 /JNGG et ResearchGate.
   4. “Les grandes pyramides”, Ed. Publi-Topex / Ordre des Géomètres - Experts, revue “mesure & grands chantiers - 4000 ans d’histoire” - 2002.Revue du Palais de la découverte - Paris : DÉCOUVERTE N°343 DÉCEMBRE 2006 - Article de P. Crozat intitulée “le génie des pyramides” – 2006.
   5. Journal de l’Association Ouvrière des Compagnons du Devoir du Tour de France - AOCDTF / Paris.
   
   
3. Depositions on: Academia.edu-ResearchGate- Pyramidales
   
   
   1. Diaporama Espace des Sciences de Paris
   2. Animation Pyramide
   3. CV P.C. searcher (in French and in English)
   4. Abstract 2017 (in French and in English)
   5. Digital invitation CULTNAT
   6. Engineering of Pyramids (digest in French and in English)
   7. Open letter (in French and in English)
   8. Lettre Circulaire à mes Pairs (in french)
   9. Communiqué 2019 (in French)