22nd February, 2012
The VIRTUAL APEX METHOD for Accurate Pyramid Construction
The provision of a reference for accurately forming the sides of a pyramid without the apex being in place can be achieved by exploiting the geometry of a symmetrical pyramid.
A pyramid has the same proportions throughout its height, in either its profile or diagonal cross section.
The centrepoint-corner cross section of a pyramid forms a vertical right triangle, with the hypotenuse forming the corner edge. This triangle is either a right isosceles triangle, when the base centrepoint-corner dimension equals the height, or a right triangle with a base plus or minus a fraction of the height. This relationship can be described as the height:centrepoint-corner ratio and in a symmetrical pyramid, the height:centrepoint-corner ratio found at the base, is repeated at any height above the base.
This consistent geometry if systematically applied can provide the corner positions of a pyramid at selective heights and from these the corner edges. Straight lines running between the corner edges will automatically define the pyramid faces at any height.
Pyramids of any shape or dimensions can be formed using the virtual apex method, by employing different height:centrepoint-corner ratios in the same way.
In fact the method could be applied to form 'pyramids' with any number of sides of any length, to any chosen height, with an apex above any fixed point on the base.
However if the chosen dimensions for a symmetrical four-sided pyramid create a height:centrepoint-corner ratio based on whole numbers, both the reference heights and corner edge positions can be calculated using only basic arithmetic.
This is the case with all Egyptian pyramids.
When a method is known for providing an exact corner edge without the need for an apex to be in place, an accurate pyramid can be formed using this as the primary guide. In a solid pyramid, this can be achieved by first constructing a core structure to a reference height based on the chosen height:centrepoint-corner ratio, which will then act as a platform from which the pyramid corner positions and corner edges to this height can be found and placed.
A façade can then be attached using the corner edges as the primary means for maintaining the external shape to the reference height. The corner edges will automatically form the four sloping sides when a straight line is stretched between them at various heights. The same procedure is repeated for each subsequent reference height.
After preparing the base square, the construction order is therefore:
- Build a solid core to a reference height.
- Mark the pyramid corners on extended diagonals at this height.
- Connect the upper corner positions to the base corners with a straight line to provide an exact corner edge reference.
- Lay the first course of façade blocks with the casing aligned to the base square and cut their top surface level and flat.
- Cut the corner edge on the casing corner blocks to fit the reference line.
- Mark the top of the casing course with the face edge, based on a straight line running between the pyramid corner edges at this height.
- Lay the following course with the base of the casing blocks aligned to the face edge line marked on the top of the course below.
- Cut the corner edge on the corner blocks and from these mark the face edge position on the flat top of the casing course.
- Add further courses of the façade in the same way up to the height of the reference core.
- Construct the core to a new reference height and repeat the process of forming the corner edges and adding the façade.
- Core and façade construction continues to a height at which the pyramid corner edges can be readily extended to the apex with straightedges.
- From the capstone down, cut the pyramid faces to fit straightedges running between the face edge reference-lines marked on the top of each course of casing blocks as they were laid and described in Steps 6 and 8 above.
More detailed information about the method and its application as a fundamental component in Ancient Egyptian Pyramid building can be found in my book
