In Control at the Pyramids
- an architect’s and builder’s answer
to how the pyramids were built
Domenic A. Narducci III and Michael T. Lally
All Content Copyright © 1997-2012. Domenic A. Narducci III and Freestone Inc. All Rights Reserved. Reprinted with Permission
In its purest form, a true pyramid consists of four identical triangles, which “spring” from a square base and culminate at a common point. When reduced to its plane geometric components, it is an easy form to comprehend. It is a far more daunting task to create the form in the built environment.
Imagine that you are charged with the construction of a true pyramid, whose square base will cover 13 acres. Perhaps your most daunting task will be ensuring that after decades of construction, the pyramid’s four sides will meet precisely at a point almost fifty stories above the ground. Imagine further that as the builder you have no laser levels, transits or other sophisticated measuring devices at their disposal to aid in the construction process. Well, don’t worry; it’s already been done! Almost 5000 years ago, the ancient Egyptians accomplished this very feat at the Great Pyramid in Giza.
This paper discusses the construction control procedures used in modern building and how important the same control would have for the building of the Egyptian pyramids. Also included is a step-by-step formula that puts forth a set of specific, simple control procedures that illustrate how those ancient Egyptians could have done. This paper is keenly focused on how and where to position the stones in order to create the pyramidal geometry. It is not interested in the quarrying, transporting or handling of stones or in the myriad of other issues regarding pyramid development.
To date, many pyramid researchers have acknowledged in their work that “control” of the pyramidal form would have been a most important aspect of the construction process. I.E.S. Edwards wrote “… imperfections in the setting of the stones would not only mar the outside appearance, but, unless counteracted, would lead to irregularity in the pyramidal form”1. Yet, few have put forth any theories which “standup” when subjected to practical construction analysis. Besides presenting the fundamental concepts, our step by step “how to” formula specifically shows the control procedures which “counteracted” the potential affects of the “imperfections” as stated by Edwards. These solutions are grounded in an understanding of the construction process (which basically has not changed from ancient times to today) and an enthusiasm for problem solving. Surely, those are human qualities, which were as prevalent among the Egyptian pyramid builders as they are among the builders of today.
What is Geometric Control?
Unlike the concrete realities of the huge machinery and impressive quantities of materials used in construction, geometric control (GC) is an abstract. It is part of the building process. As such, it is not visible and its significance is rarely understood. GC can be defined as a system of procedures that establishes and then maintains the geometry of a structure during the course of the construction process.
This paper’s mission is to impress upon the reader the significance of GC and how mastery of specific control procedures allowed the Egyptian builders to construct true pyramids.
Buildings are composed of many individual elements that are typically organized into systems. The combination of all the systems results in the completed building. For example, a brick veneer system may be composed of hundreds of thousands of individual bricks, set in mortar, laid in horizontal courses and mechanically fastened to the building’s structure. The spatial orientation of each brick, that is, each piece of the brick veneer system, needs to be controlled. Each brick must be set level with adjacent bricks, and its exposed face must align with the vertical plane established by previously placed bricks. Those steps represent GC procedures, which are applied to the setting of each and every brick. In a similar way, the entire brick wall, the system itself, must be properly oriented in the building and have the correct shape as prescribed in the construction drawings. It is the sum of the GC procedures for the individual elements, as well as the GC procedures for each of the systems, which results in the building being constructed to its planned geometric form. That form may be a square, a rectangle, a circle or even a pyramid.
GC is organized on two levels: “primary control” and “secondary control”. Primary control consists of those procedures that establish and then maintain the geometry of an entire building system (such as the brick veneer system) or several building systems. Secondary control maintains the orientation of the individual elements or pieces within each building system (such as a single brick in the brick veneer system). GC procedures can be further classified as external or internal. External procedures are those that use references, or control data, which are outside of the building. Internal procedures are applied using control data that are inside or on the building. Both internal and external procedures can be either primary or secondary.
It is important to note that the basic procedures for GC have not changed in 5000 years. However, technology has advanced to the point where GC, for today’s typical buildings, is a much more easily accomplished part of the construction process.
The more complex a building’s form, the more important it is to maintain GC at every step of the construction process. The Egyptians, with much simpler technology, managed to control a far more complex form than the common geometries that we construct today. You might ask, “What’s so difficult about building a simple, solid geometric form?” That is the tough part! Today’s buildings are “hollow” structures that create interior spaces for human occupancy. The open nature of today’s buildings allows for internal control procedures to be applied during construction from within the spaces of each floor, or vertically from basement to rooftop. The Egyptian pyramids, however, were solid masonry monuments (except for the minor volume occupied by chambers and access passageways) that were not intended for daily occupancy. The lack of internal open space resulted in the need to apply GC procedures from outside or on the building. The inclined, finished sides of the pyramidal form made the use of typical GC procedures (such as use of plumb lines) impractical. The sloped sides also made the installation of scaffolding and construction from outside the form unrealistic.
Without GC, many problems would arise in today’s buildings projects. For example, without adequate site controls, buildings would be incorrectly placed causing zoning, drainage, grading, access and solar orientation problems. Without internal building controls, walls would not be plumb (vertical), ceilings and floors would not be level, windows and doors would not fit into openings, chimneys would not stand straight, roofs would not meet at ridges, etc. Similar problems would have plagued the pyramid builders, perhaps with the exception of zoning and solar orientation issues. If GC procedures were not successfully implemented, it would have been impossible to accurately maintain the overall geometry, i.e. the level and orientation of the series of diminishing squares that occur at each successive course of a pyramid’s outer casing stones. Loss of control would have resulted in unsmooth faces, “hips” which were not straight and an apex that was not centered over the base below. Simply put, in ancient or current times, without carefully implemented GC, planned squares would not be actual squares, rectangles not rectangles, circles not circles and pyramids not pyramids.
GC Procedures in Today’s Buildings
GC procedures involve the “repetitive reference” to an established set of points, lines or planes, thereby allowing for the true vertical and horizontal orientation of the building’s component systems. In surveying, the parallel concept is the “point of beginning”. That is the acknowledged point from which all subsequent map points and directions begin. For a building it is much the same. For modern buildings, it is typically the foundation system that is oriented on the site through reference to established external “datum” information. Floors, walls, roofs and eventually finish systems all “build-on” and reference to the newly constructed foundation, not to any external information. This is an important concept. Once foundation (or base superstructure) is completed it effectively becomes the primary control reference. Builders then utilize internal control procedures (both primary and secondary) for construction of the subsequent component systems. These internal procedures not only provide for the locations of all the building’s systems but also maintain each system’s verticality (plumb), horizontality (level), squareness and orientation.
For illustration, examine these control principles and their application in the construction of a simple “stick-framed” single-family house. After initial site mobilization, the excavation for the foundation is the first major activity. But before the excavator is even delivered to the site, a surveyor must stake out the exact location of the foundation walls, as shown on the site plan. Using modern surveying equipment, the layout crew will establish one or more “benchmarks” at secure locations in close proximity to the planned house location. These benchmarks are new reference points whose exact locations are known relative to established more “global” control data. A benchmark may be established by driving a nail into the base of a large tree (scheduled to remain) and then documenting the nail’s exact location (both vertical and horizontal distances) relative to an established concrete monument at a property corner (or some other established point). From this benchmark, using a transit, the new foundation’s corners can be precisely marked. Wooden stakes are typically driven into the ground at offset distances from the proposed new foundation’s corners. See FIG.1.
The stakes are offset from the actual corners so that the excavator has room to work around the foundation perimeter without disturbing the stakes. If by chance a stake is disturbed, or even lost during the excavation work, the point it marks can be easily re-established from the benchmark. Once the excavation is complete, footings and foundation walls can be precisely placed by “measuring back” from the offset stakes. Vertical placement (elevation) of the bottom of the footings can be precisely set by direct reference to the benchmark. For a poured concrete basement foundation wall, the top of the pour is usually marked by a chalk line “snapped” on the inside the formwork. This line is located by repeated reference to a benchmark. Before the concrete walls are poured, a secondary level of GC occurs internally within the basement space to ensure that the concrete forms are plumb and square. Using simple tools and old techniques, this might involve checking diagonal measurements.
Some of the hand tools employed by today’s builders, for secondary internal control procedures, are shown in FIG.2. For comparison, they are illustrated adjacent to speculative tools used by the Egyptians for similar purposes. They are all simple devices. One can hardly fail to notice that in 5000 years, these basic tools have not changed very much.
Once the building’s foundation is complete, a significant change occurs in the GC process. The two-tiered, primary and secondary reference system still controls the geometry. However, the primary reference data will no longer be external to the structure. Framed floors, walls and roofs will not use the foundation offset stakes, or the benchmark, for their primary reference. These systems will rely on the foundation for their primary GC. The foundation structure itself will provide the primary control data that ultimately maintains the final geometry of the entire building. Secondary control will occur internally from within the space being created. This secondary control is achieved with the use of levels, measuring tapes, chalk lines and squares. For example, the first floor framing is placed directly on the top of the foundation walls (new primary control data), with perhaps minor perimeter adjustments for square (achieved applying secondary GC procedures). The height of the first floor walls is measured directly off the top of the first floor deck (which is now primary control data) and openings are placed in exterior walls and interior walls (employing secondary GC procedures). With the aid of levels, exterior walls are erected plumb (secondary GC procedures) and aligned with the outside face of the floor deck (primary control data). The process repeats itself for the rest of the construction. For this simple building, each system is built sequentially directly on the previous system. But all are predicated on the foundation system.
GC Procedures for Pyramid Construction
As stated above, the basic concepts for control of a building’s geometry have not changed in 5000 years. This is not an unreasonable conclusion since the construction process itself has not changed, despite significant changes in equipment and materials. With an understanding of the procedures and techniques for GC in today’s buildings, it is possible to appreciate the purity and practicality of the GC solutions presented in this paper for pyramid construction.
There are many unanswered questions concerning pyramid construction. Most start with “How did they….. Lift all that stone?… Get all that stone to the site?…. Cut the stone so precisely?. The list goes on. This paper answers a less frequently asked, but equally important question: How did they build that form so precisely? The simple answer: Establishment and successful application of GC procedures. Now let’s look closely at a set of GC procedures that those Egyptian builders could have used.
One of the first steps during the pre-construction phase of a pyramid project would have been site selection. That process is not relevant to the goals of this paper and is therefore not discussed here. Once selected, the site would need to be prepared for the construction of the new pyramid. After the leveling operation was complete, one of the first activities would be the establishment of the exact location of the pyramid’s foundation.
As we know from field documentation of existing pyramids, the four sides were precisely oriented perpendicular to the four compass axes. Much has been written on the technique by which a false horizon can be constructed and, through star sighting, a highly accurate North/South axis can be established. There is no reason to further discuss those theories here. But it is important to note that, just as today’s surveyors “stake-out” proposed buildings using external control data (benchmarks), so too did the pyramid builders. The stars were the external control data that they used to establish of a true North/South axis to which the entire pyramid foundation was oriented.
Once the true North/South axis was marked on the level pyramid site, the four corners of the pyramid’s foundation could be determined using simple geometry. Bear in mind that these four corners were not the corners of the final geometric form one would see after the construction was complete. Instead, these corners marked the corners of the structural foundation system for the pyramid, a stepped core.
From examination of extant pyramids, we know that pyramids are composed of four primary masonry parts;
- the stepped core consisting primarily of rough stone,
- the finished faces of the core, consisting of dressed stone,
- the backing stone behind the casing stone and
- the dressed casing stone.
The core blocks and accretion walls, whose final form represented a stepped pyramid, comprise the foundation system or the core. This initial-phase structure was constructed in platforms of diminishing size using horizontally laid blocks, or as accretion walls with inclined stone coursing. In either case, we propose that the completed foundation, a stepped core, became the primary control data for the final true pyramid form that would eventually encase it.
Evolution of the Pyramidal Form
The first “stepped” pyramid (Pyramid of King Zoser) appeared at Saqqara in the Third Dynasty. Religious or other ceremonial requirements may have initially dictated the stepped form. Or perhaps the stepped form was simply the “next generation” of the mastaba form, in use since the First Dynasty.
It was at King Sneferu’s pyramid at Meidum (Fourth Dynasty c. 2610 BC) that the Egyptians first applied the construction procedures that allowed construction of the true pyramid form over a stepped pyramid. FIG.3. shows the many phases of construction that shaped the pyramid of Meidum. The E1 phase (as noted by L. Borchardt) produced a 7-step pyramid. Builders of the E2 phase enlarged the structure to an 8-step pyramid.
We theorize that during the process of the E2 work, the builders realized that the same basic principles of geometric control that they were applying to construct the additional step, would also allow for the application of sloped sides…the creation of a true pyramid!
The builders of E2 devised and applied “measure-out” control procedures, using the existing E1 stepped form as their foundation (primary control). Perhaps, with this viable construction method in hand, the pyramid builders moved to the Dahshur site to start in earnest on what would be the first planned and executed true pyramid. Of course, problems at what became known as the Bent Pyramid ruined their intentions. The E3 phase, the true pyramid form at Meidum, may have been built after the disappointments of the Bent Pyramid, and during or after the Red (North) Pyramid was constructed at Dahshur.
The control procedures devised and perfected at Meidum precipitated the “boom” of true pyramid construction that marked the Fourth Dynasty.
How Did They Do It?
Let’s do it! We’re actually going to build two pyramids.
Below you’ll find a step-by-step formula showing the GC procedures necessary to build a stepped-pyramid, which we’ll call our model core (Phase I). Then, by using it as a foundation, we’ll build our model true pyramid over it (Phase II).
Phase I – The Model Core
This simple model is intended as a tool for understanding the application of the GC concepts to a real project. Assume that the model core is built on a perfectly level base. It will be composed of three platforms. At the 1st platform it will measure 22 FT x 22 FT. In three 4 FT steps, will rise to a total of 12 FT. The 2nd and 3rd platforms will each setback 4 FT.
Our model core’s incline, or the slope of the sides will be 45 Degs. This slope is the angle created by a line connecting the center points at the edges of the platforms, and the plane of the base. Refer to FIG. 4.
The incline of our model true pyramid (Phase Two) will be the same as the incline of the core. At the pyramid of Meidum, the angle of the core sides (E2) is approximately 52 Degs, a 14 to 11 ratio of rise to run. It is not surprising to us that Meidum’s true pyramid faces (E3) share the same 52 Deg. angle. Interestingly, most subsequent pyramids also show the same angle. We believe that the “measure-out” control procedures used by the E3 builders at Meidum (the same procedures we will use in building our model true pyramid), resulted in the 52 Deg. angle. There was no magic in the selection of 52 Degs.; it was simply the angle that was dictated by the already existing core geometry. A more complete analysis of pyramid angles is planned for a future discussion. We can state within the context of this discussion that the principles of control, as illustrated below in our model true pyramid formula, are applicable to a true pyramid of any angle.
1.1: Orientation and Base Layout
Using the North/South axis established by external astral reference, the length of the West side of the 1st platform is marked along the North/South axis line. Then, using diagonals, the other three sides of the square are established. Refer to FIG.5.
Lines AB, BC, CD and DA are permanently marked by chiseling a narrow slot in the bedrock base. These four lines, which represent the square of the base of the core, become the primary control data for the entire pyramid. All subsequent control procedures will make reference, directly or indirectly, to this square.
1.2: Construction of the 1st Course of the 1st Platform
For our model core, the stones that compose each of the platforms are laid in level courses, and the perimeter stones of each course are stacked such that their outside faces create vertical planes. The outside bottom edges of the perimeter stones for the 1st course of the 1st platform are now set along Lines AB, BC, CD and DA. Secondary control procedures, as illustrated in FIG.6.B., are then applied at the outside faces of all perimeter stones to ensure plumbness. After placing all perimeter stones, the interior area is infilled with “backing-stone”.
1.3: Completion of the Last Course of the 1st Platform
In our model, each platform is composed of two courses, each two feet high. The 2nd course begins with the placement of the four corner stones. Refer to FIG.7. Once the two outside edges of the corner stones are aligned with the outside edges of the stones below, the plumbness of the outside faces is checked. This secondary control procedure is accomplished using one of the “plumb rules” shown in FIG.6.B.
Then, between the corner stones and in line with their outside top edges, four string lines are “pulled”. If necessary, the string lines are supported at intermediate points to prevent sagging and other movement. These vertical supports are called “triggings” and can be secured in small sockets cut into the stones below. The remaining perimeter stones are now placed between the corner stones by aligning their two outside edges: the outside bottom edges are aligned with the outside top edges of the course below; the outside top edges are aligned with the string lines. These last two procedures represented internal secondary control. When the entire perimeter is enclosed, except for access points, the interior area is filled with backing-stone as per Step Two. (NOTE: This “string technique” could also have been used in Step1.2, after the corner stones were placed) This course results in a new, “parallel” square being constructed over the lower square. Now completed, this 1st platform is a perfect square in plan, four feet high. It is now necessary to set controls and begin construction of the 2nd platform.
1.4: Construction of the 2nd and 3rd Platforms
A new control procedure is employed before construction of the 2nd platform begins. A new square must be marked on the top of the completed 1st platform. This is done by dropping a plumb-line down to the initial square marked on the base (at the face of the 1st platform), or to points off-set from the initial square, then “measuring-back” horizontally along the top of the completed platform (See FIG.8.B). The rule used for measuring back must be held normal (perpendicular) to the face of the completed platform. The setback dimension from the face of the 1st platform is 4 FT. The “measure-back” procedure is done at two locations on each of the sides, near the corners of the completed platform (See FIG. 8.A). Once the setback dimensions are marked on the top of the lower platform, all four sides (See Fig. 8.B.) of the new platform can be established by simply connecting points.
These new lines, defining a new square, are now chiseled into the top of the 1st platform. This newly marked square now serves as the primary control data for the construction of the 2nd platform.
Careful placement of the 1st platform’s backing-stone ensures a level surface in the area where the new control square for the 2nd platform is chiseled. The top surface of the 1st platform’s final perimeter course is used as a reference for maintaining level.
Now that the 2nd platform’s square is marked, construction of its 1st course can begin with the placement of the corner stones (See FIG.8.C). Repeat Steps 1.2 and 1.3 to complete this platform. Repeat Steps 1.4, 1.2 and 1.3 to complete the 3rd platform.
NOTE: Future placement of the true pyramid’s casing stones, or the construction of a subsequent platform, does not require that the faces of completed platforms be planar (vertical surfaces). In other words, within a given platform, it is not crucial for the outside faces of each course of the perimeter stone to align with the outside faces of the course below (See FIG. 9). However, it is critical that each course be parallel with the one below, and also be a true square. Therefore, if there is a setback at each course, it needs to be the same on all four sides. Furthermore, control of the placement of each setback course can be achieved by reference to the primary control square marked on the platform below. In this way, cumulative error can be avoided.
For simplicity, this model shows the perimeter stones of each platform in a vertical plane, as indicated in Fig. 8.B.
An additional secondary control technique is employed when each new smaller primary control square is marked at the start of each new platform (Step1.4). That procedure is called “checking of the square”. The check is accomplished by measuring the diagonals of the square. A perfect square has diagonals of the same length.
Our model core is now complete- One down, one to go! See FIG.10. Quick summary: By transferring the base primary control square onto each new platform and then controlling the placement of the stones in each course, the geometry of the form was controlled during the entire process. It is now possible to begin Phase II, the construction of the sloped sides that will result in the creation of our true model pyramid.
Phase II – The Model True Pyramid
The model core, produced in Phase I, will now serve as the model true pyramid’s foundation system. It will become the primary control data for the GC procedures that will be employed to ensure the building’s final geometry. The pyramid’s outer casing stone (the finish stones you see on a pyramid) and backing-stone represent the finish system which “builds-on” and references to the foundation, much the same way the previously discussed house framing and subsequent finish systems “build-on” the house’s concrete foundation. Secondary internal control procedures control the placement of each of the casing stones, similar to the way individual bricks are set into a brick veneer system. Some of the simple tools that are used for secondary control are shown in FIG.2.
2.1: Layout of the Base Square Outside of the Core
At a distance “X” from a point near corner A on the primary control base square of the 1st platform, a new point A1 is marked. It would also be possible to measure out from an offset point near corner A, if we had previously established offsets. The measure-out must be a along a line that is normal to the face of the platform. From a point near corner B, repeat the same procedure. See FIG. 11. A string line is now “pulled” between points A1 and B1 establishing new Line A1B1, which is parallel to primary control Line AB. The distance “x” is determined based on the true pyramid’s final angle and the space required for working and moving stone in the area between the proposed finish face and the existing platform’s faces (See “Work Area” FIG. 4.A & 4.B). For our model true pyramid, “X” equals 7 FT.
Line A1B1 marks the outside edge of all the final casing stones for the model true pyramid’s first course along its West side. Repeat this procedure at the other three sides of the pyramid. Be sure to extend the new lines far enough to intersect and identify the new corner points. Extended lines B2C2, C1D1 and D2A2 establish the outside edges of the casing stones for the North, East and South sides. Again, this new square represents primary control for the construction of the true pyramid’s finish faces. It’s important to note that with the core in place, it was impossible to use the new square’s diagonals as a check on their accuracy. You can’t “pull a line” through solid mass! Instead, the measure-out procedure was used from a previously establish primary control system (the base square of the core). This procedure and its significance in building a solid geometry such as a pyramid is the very point of this paper.
2.2: Horizontal Control Lines on Platform Faces
Now we are ready to start placing casing stone. We need a procedure to control the placement of each course so that the outside face of the pyramid stays planar. The solution is to “measure-out” from the face of the core’s platforms. To start, we need to mark horizontal lines on the platform faces. These lines will be spaced at an increment that is equal to the height of the casing stones. Similar to the procedures followed in Step2.1 (but now on a vertical surface), “measure-up” from the base and mark points on the face of the 1st platform. Pull lines between the points thereby creating horizontal lines that are parallel to the base and each other. See to FIG.12. This procedure is repeated on all faces of all platforms. Since all platform surfaces at their primary control squares are level, they serve as the control data from which to measure-up and create the horizontal lines. It is important to note that each horizontal line is measured up from the level base surface below. This avoids the incremental errors that would have result if measurements were made from one horizontal line to the next. If the platforms faces are non-planar (per FIG.9.B), a similar, but lengthier procedure is used to create the horizontal lines.
2.3: Placement of the 1st Course of Casing Stone
The outside bottom edges of these first casing stones are now placed along Lines A1B1, B2C1, C2D1 and D2A2. The outside finish face of each casing stone was precisely dressed to its finish angle before placement (See FIG.14 “Typical Model Casing Stone”). This is important because, once placed, the ashlar line of each stone plays an important role in the control of the pyramid’s geometry. Even though the base surface is level, secondary control procedures (similar to those used at the core, as shown in FIG.6.B.), are then applied at the outside faces to ensure that the final slope angle is maintained. The set triangle shown in FIG.12.A. is used.
(NOTE: The “string technique”, previously presented in Step1.3, and again discussed in Step2.4, could also be used after the corner casing stones were placed.). With a level base, the base square, the horizontal lines at the platform faces and the secondary control tools we’ve previously discussed, there are numerous ways to maintain control at this 1st course. Once all the casing stones are in place, the backing-stone is placed. See FIG.12.
2.4: Placement of 2nd Course and Hip Construction
The first two casing stones of the 2nd course are placed at the locations shown in FIG.13.A. Their outside bottom edges are set precisely on the ashlar line of the course below (See FIG.13.B.). The ashlar lines of the lower course represent the square that serves as primary control data for the placement of this new course. The final orientation of the finish faces of these first two stones is checked by using the set triangle. This internal secondary control procedure ensures that the true design slope of the pyramid is being maintained. If necessary, minor adjustments to the surfaces onto which the stones are set, or to the bottom of the stones themselves, are now made to maintain the correct outside angle. The only other method by which to control the angle is to modify the dressed finish face. But that approach seems impractical. This is not to say that finish polishing of the outside face was not done after the stones were set.
With these two stones in place, parallel to the AB side of the core, a string line is stretched between the top outside edges of the newly placed stones to establish the new ashlar line. Since it is difficult to “pull” a line very far without the problem of sagging, intermediate “triggings” are set at regular intervals to support the line. The triggings are set at precisely the same “measure-out” distance from the face of the platform as the top outside edge of the first two stones of this course. The primary reference is one of the horizontal lines on the face of the platform that we marked in Step2.2. Now the rest of the casing stones along the AB side of the core are set by placing the outside bottom edges precisely on the ashlar line of the lower course and the top outside edges along the new ashlar line defined by the string line. Then as was done with the first two stones, modifications are made to the surface below the stones, or the bottom of the stones themselves, to ensure alignment with these two references. Repeat these same procedures along the other three sides of the pyramid.
At this point, we need to place the corner or hip stones. Like the other casing stones, the hip stones are already dressed to the pyramid’s final angle before being set. The control of the placement of the four hip stones was done in conjunction with the two initial stones that were placed on each side of the pyramid as described above. The hip stones are set at each corner by first aligning their bottom outside edges precisely along the perpendicular ashlar lines of the lower course. Refer to FIG.14.
Then by extending the ashlar line, as established by the “measure-out” dimension “M”, the corner point of the hip stone is controlled. As is the case with all other casing stones, the surfaces onto which the hip stones are placed, or the bottom of the hip stones themselves, are modified to maintain alignment with the control data.
Then, with all the sides and hip stones in place, the area between the newly placed casing stones and the platform is infilled with backing-stone on all sides.
2.5: Placement of Casing Stone for all Remaining Courses (Up to the Top Platform)
The procedures outlined in Step2.4 are now repeated for all subsequent courses of casing stone until the top platform is reached. The “measure-out” dimension, as shown in FIG.13.B and 14.B, varies at each course but is the same on all sides for any given course.
2.6: Placement of Casing Stone Above the Top Platform
At this point, the only stones left to be placed are those above the top platform. In this model, those stones represent about 3.7% (288 CF) of the pyramid’s total volume (7776 CF). Since there are no vertical platform faces to measure-out from, new control procedures are now employed to maintain the geometry. But because the entire work area is open, and completely accessible from the inside, control procedures are much simpler.
Diagonals are easily measured and compared to maintain parallel, diminishing squares as each new course is placed. In addition, the center point of the entire structure can now be projected up to the apex. For our model true pyramid, the dimension from the top platform up to the apex is 6 FT. With simple bracing, we’ll now set a wooden pole whose top point will mark the apex. From that point, four string lines are stretched to the uppermost four hip stones. By employing these simple internal control procedures and the others we’ve used up to this point, the square geometry of the remaining courses is maintained. And the four hips of the pyramid are sure to meet at the pyramid’s designed apex.
Before the last stone, the pyramidion, is placed, we’ll remove the wooden pole and “hip strings”. Now we can set the last stone, and the construction of our model true pyramid is complete. See FIG.15.
The Phasing of the Construction
The process put forth in this paper defines two separate and distinct construction phases: the first, for the building of the pyramid’s foundation, and the second, for the installation of the casing stone, true pyramid’s finish system. But, is it possible that both operations occurred at the same time, with the core construction proceeding just in front of the casing work? Although not discussed here, the answer may lie in the constraints and dictates of the ramping system(s) utilized to move the stone. We theorize that with an internal ramping system (perhaps in the area between the platform faces and the casing stones), it would be possible for both operations to occur simultaneously. This subject of ramps is indeed a significant part of the construction process and one that will be dealt with in greater detail in another paper
Evidence and Summary
Field research reveals that the faces of the cores of several pyramids are composed of dressed stone. The question has been asked: Why dress these outer faces if the construction of the final true pyramid would completely bury all these surfaces? The one valid explanation is that these faces were part of the control system that was employed by the builders in constructing the sloped sides of the pyramid. The dressing of the faces provided a more accurate surface from which to “measure-out” and maintain geometric control. Such field documentation strongly supports the concepts presented in this paper.
With regard to the distinct and separate phases of construction, I.E.S. Edwards has recorded two dates that were found on casing stones at Dahshur 2. The first indicates the twenty-first year of the reign of King Senefru and is located at the Northeast corner of the pyramid. The second stone bears a date of the following year and is located halfway up the structure. If both core and casing stone were done at the same time, it would have been impossible for the completed pyramid structure to have reached that height in such a short period of time, especially considering the fact that at the half-way point more than 85% of the stone would have been placed. The one-year period is realistic if the only work completed during that period was partial installation of the finish casing stone system. This field data seems to support the “two phase” approach, but not very strongly. Perhaps the noted dates were not marked on those stones when they were installed, but rather when they were produced at the quarry well in advance of their arrival at the job site.
In summary, the principles put for in this paper focus on the significance of the geometric control procedures that were used by the builders of the pyramids. The specific procedures described above would have been easily accomplished in the field with the use of simple tools. Many authors have described procedures by which the construction followed a “one course at a time” approach to the entire pyramid. These theories are weak. Through analysis one can easily see that the ability to maintain geometric control with that method of construction would have been impossible. Without utilizing internal primary control data for reference, the incremental error that would have resulted by ratio measurements on the outside of each course (or several courses) would have been disastrous. And the idea that one could control the entire geometry of this complex form by sighting along the “hips” to a distance of more than 700 FT (as would be the case at the Great Pyramid), or along the sloped sides, is naive as best.
Accurately constructing diminishing, parallel and centered squares is the key to successful pyramid construction. And based on what the Egyptians had for tools and technology, that success could only have been achieved by employing the concepts presented here: construction of a core that served as the primary geometric control reference for the true pyramid.
We are confident that subsequent excavations at pyramid sites will add information supporting the ideas presented here. Future papers are intended to complement this presentation and provide additional construction theories regarding the movement and placement of stone, ramping systems and the angles of construction.
Lastly, this is a drastically different approach to solving pyramid construction questions than is typically applied by academics. That approach is somewhat like “reverse engineering”, i.e. gathering field data and then guessing at its intended purpose. But ultimately, no amount of field data will solve the challenge of geometric control, because the answer lies in an abstract, a methodology. And that simply can’t be gathered, measured, excavated or photographed. As architect and mason, we have proceeded to solve the unique construction challenge of this type project, as we have done for other projects throughout our careers, by applying age-old, proven construction concepts. And we have done this absent advanced degrees in Egyptology and reams of archeological data.
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Architect and Author Domenic A. Narducci III practices in Connecticut, USA.
- I.E.S. Edwards, The Pyramids of Egypt (Harmondsworth, Middlesex 1961) 261
- Edwards, op. cit. 230