CHAPTER 7

Geometric Construction of a Pyramid with Proportional Dimensions to the Great Pyramid

It is simple to construct a pyramid with proportional dimensions to the Great Pyramid, without the use of mathematical calculations. The resulting pyramid will be its scaled representation.

The pyramid can be constructed by two methods. In one of them, the height of the pyramid is fixed, in the other; the sides of the base of the pyramid will be fixed. The drawing could be traced, over a paper, wood, or any other material to be used. If it will be constructed in the backyard, over a level terrain, it can be marked with suitable markers, as sticks of wood, nails, etc.

1. Construction of a Pyramid of a Determined Height:

To locate the reference points necessary to construct a pyramid with a determined height, and oriented to the North, proceed as follows:

1. Over the horizontal plane where the pyramid will be built, establish a line of length A’B’ oriented North. Use a map, compass, or a reference direction, for its orientation.

A’ ____________________________________________________ B’

North >

 

If the pyramid will be constructed in the backyard, you can use a string, preferably a long ruler, to set the marks and dimensions. If it is over wood, paper, you only need a straightedge to trace the lines and a compass to trace the circles.

2. Select a point in line A’B’. For example, point O in figure 103. From point O, trace a circle with a radius equivalent to the height that you have determined for your pyramid. Set the points A and B, at the intersection of the circumference of the circle, and the line A’B’. The distance between point A and B corresponds to the circle’s horizontal diameter.

Figure 103

3. Trace the vertical diameter of the circle. As shown in figure 104, from point A, trace an arc with a radius larger than the circle’s radius. With an equal radius, trace another arc from point B. The line that joins points (a) and (b), at the intersection of the two arcs, will be perpendicular to the diameter AB. Mark points Q and K at the intersection points of the line with the circumference of the circle. The line QK defines the vertical diameter of the circle.

 

Figure 104

4. With Q as center and QO as radius, trace an arc as shown in figure 105. From point B, trace another arc with equal radius, and mark point C at the intersection point of the circumferences of the two arcs. Join point K and point C. Now, with C as center, and radius equal to CQ, or CB, trace and arc from Q to B. Mark point M’ at the intersection of the circumference of this arc and line KC.

Figure 105

5. With K as center and KM’ as radius, trace an arc to cut both sides of the circle’s circumference, as shown in figure 106. Mark points P and N, at their intersection points, respectively. Join points P, Q and N to create the triangle PQN.

Figure 106

6, Mark point H, and point T, at the intersection of the lines PQ and QN with the horizontal diameter AB. The triangle HQT defines the vertical cross sectional plane of the pyramid, as seen through the center of its faces.

Figure 107

7. The four corners of the base of the pyramid can be established as follows: From center O, and radius OH, or OT, trace the circumference of a circle to cut the vertical diameter QK. Set the points S and U, at the intersection of its circumference with the vertical diameter. Since HT represents the length of the base, the lines that form the square pass through points H, S, T and U.

 

Figure 108

8. Using points H, S, T and U, already set, the four points that define the corners of the square can be established, as shown in figure 109. From point H, and radius HO, trace an arc as shown in the figure. Repeat the same procedure from points S, T and U. The intersection points between the arcs, identified as I, II. III and IV, define the four corners of the square base of the pyramid.

 

Figure 109

9. Join points I, II, III and IV to establish the square that represents the base of the pyramid.

Figure 110

If the paper where the drawing is made is cut between the points H and Q, and between Q and T, and raised vertically, using the line HT as its base, it will show a scale model of the cross sectional view of the pyramid. Its dimensions and angles will be proportional to those of the Great Pyramid.

Figure 111

10. To project the pyramid’s faces over the horizontal plane as shown in figure 112, from point H, project the distance HQ to the horizontal diameter AB, and set point Y’, at the intersection.

Figure 112

11. Join point IV and I with point Y’, as shown in figure 113. The triangle between points I, Y’ and IV, represents a projection of the pyramid’s face lying over the horizontal plane. The distance from point H to point Y’ represents the inclined distance from the center of the base to the top of the pyramid, that is, the pyramid’s apothem.

Figure 113

12. If you cut four pieces of paper, wood, or of the material to be used to build the pyramid, equal to the size of the triangle formed between the points I, Y’ and IV, and lay them over the already established square base (as shown in figure 114), placing point Y’ as the top, the pyramid will be formed. Its dimensions will be in proportions to those corresponding to the Great Pyramid.

Figure 114

Figure 115 shows how the four faces of the pyramid will look over the drawing board projected to the contrary side in the horizontal plane. When the four points 1, 2, 3, and 4 are brought together to form the apex, the pyramid will be formed.

Figure 115

2. Construction of a Pyramid having a Determined Side Length:

Follow this procedure to construct a pyramid, based on a determined dimension for the length of its base, without the use of mathematical calculations. The pyramid’s dimensions will be proportional to the corresponding dimensions of the Great Pyramid.

1. Establish the line A’B’, with direction due North, over the level plane where the pyramid will be constructed,

 

Figure 116

2. Select a point in line A’B’, as shown in figure 117 as point O. From point O, trace a circle with a radius equivalent to half the length of the require sides for the base of the pyramid. Mark points H and T, at the intersection of the circumference with the line A’B’. The distance between point H and T corresponds to the circle’s horizontal diameter.

Figure 117

3. Trace the perpendicular diameter of the circle. To construct it, from point A, trace an arc with a radius larger than the circle’s radius, as shown in figure 117. With an equal radius, trace another arc from point B. The line that joins points (a) and (b), at the intersection of the two arcs, will be perpendicular to the diameter AB. If points (a) and (b) fall inside the circle area, as shown in the figure, extends the line at both sides to intersect its circumference. Mark points S and U, respectively, at their intersection points. The line SU defines the vertical diameter of the circle.

Figure 118

4. From point H, with HO as its radius, trace an arc as shown in figure 119. With an equal radius, repeat the procedure at points S, T, and U. The intersection point between the arcs, set as II III, IV, and I, represents the four points that define the corners of the square base of the pyramid. The intersection points between the arcs, set as II III, IV and I, represent the four points that define the corners of the square base of the pyramid.

Figure 119

5. Join points I, II, III, and IV, to establish the square of the base. Trace a line from point IV to point S, and set point (g) in the intersection of this line with line HT. Point (g) represents the midpoint between H and O. From point (g) project the distance (gS) to the line HT and set point Y’.

Figure 120

6. Join points I, Y’ and IV to form a triangle. The dimensions of this triangle will correspond to the incline measurements of the faces of the pyramid. If you cut four pieces of paper, wood, or of the material to be used to build the pyramid, equal to the size of this triangle and place them over the square base, placing point Y’ as the top, the pyramid will be formed and its dimensions will be in proportions to those corresponding to the Great Pyramid.

Figure 121

7. To obtain the height of this pyramid, from point H, project the distance HY’ to the extension of the line US. Set point Q at the intersection. The distance OQ represents the vertical height of the pyramid.

Figure 122

8. From point Q, trace a line to point H and another to the point T.

Figure 123

9. The triangle formed between points HQT represents the vertical cross sectional view of the pyramid, as seen through the center of its faces. The square between the points I, II, III, and IV symbolize the projection of the square base as seen in the vertical plane.

To obtain the complete geometric configuration, from point O, and using OQ as the radius, trace the circumference of a circle. Extend lines QH and QT to intercept the circle’s circumference. Set point P and N, respectively, at their intersections. Trace a line from point P to point N. Point X will be located at the intersection of this line with the vertical diameter QK.

Figure 124

This is another method to trace the basic geometric arrangement of the circle, the triangle and the square, used to trace the outline of the Great Pyramid’s design. However, this particular method, using the side length as a basis for the pyramid’s design, is not applicable to other pyramid’s geometric configurations.

The use of the radius of the circle as the pyramid’s height to define the pyramid, instead of the length of the sides is more convenient. The use of the radius allows creating an indefinite amount of geometrical configurations to build pyramids,

 

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