Egypt

Origins

Origins

Part Three

Figure Three shows a plot of the logarithm of the thickness of Set #2 for courses nineteen through thirty-four, except for courses twenty and twenty-one, which are part of the first decay curve.

You can see that the calculated regression line again is logarithmic, with a calculated slope of 1/54.9 ln (cubits)/cubit height. I show a regression line slope of 1/55. The vertical scale is expanded from Set #1. Because this Set is farther removed from ground level, the intercept on the Y axis at zero course height becomes much larger at 1.215 (ln), 3.37 actual value, and hence not meaningful

When I first examined this Set I estimated the slope at 1/50. This would be logically connected to the slope of 1/40 for Set #1. Also I was sketching the third decay curve at the same time and estimated that slope at 1/30, also seemingly connected logically to the first two Sets. However, I noticed that my estimated regression line did not center on the two sets of three data points at the bottom of the curve. Further, an estimated three-sigma band limits was much wider to include these data points. After some reflection I decided my estimates were not adequate and hence performed a rigorous calculation of the slope, and the three-sigma standard deviation band limits. (I show the 1/50 slope for comparison.)

After performing the calculation I discovered that the three-sigma band limits were more narrow than my estimated band while still including all data points, especially the two sets of three data points at the bottom of the curve. Further, the pair at Course #19 now just straddled the upper three-sigma band limit.

These three-sigma band limits are +/- 0.056 on the logarithmic scale. This is a difference of +/- 0.7 cubits, or +/- 1.5 inches. Thus the dispersion of differences between the SW and NE corners for this group was held to better than one half of that for Set #1.

If we analyze the two sets of three data points at the bottom of the curve we can grasp how tightly the builder could hold control. The maximum spread in course thickness from the SW to NE corners of six data points for courses #29 to #31 from Petrie's actual measurements is 0.025 cubits, +/- 0.25 inches, or +/- 6 millimeters. The maximum difference for the six data points of courses #32 to #34 is 0.15 cubits, +/- 0.15 inches, or +/- 4 millimeters.

This is exceedingly tight, phenomenal, control.

You can see from the plot that we are led to deduce that the builder intended those six data points to define the calculated three-sigma band limits, with the outer most of each set falling directly on the three-sigma band, and the middle-most falling on the calculated regression line. Some variance is seen in the six points but the control is still exceptionally tight.

Obviously, if the builder wanted to hold all courses to such tight control he could have done so. The fact that he could do so over six courses strongly demonstrates his ability at production control. But he was practical minded. With his control ability he could intentionally permit wider spreads to display his finesse, as he demonstrated in Set #1.

We are now justified in our conclusion that the thickness variations were deliberately cut in the building stones to provide the dispersion of the data points.

We also become acutely aware of how problems in stone logistics are multiplied by such tight control.

The designer/builder is once again leading us into insights of his abilities, discernible only to later students who are willing to give proper respect to his data display.

I did not include the course height delimiter at Course #35 in my calculations of the slope and intercept.

Calculations provided a slope of 1/25.2 with an Y axis intercept of 2.885 (ln), 17.9 actual. The three-sigma standard deviation band limits were again greatly reduced at +/- 0.0247. The reader can observe how tightly the data points were held to provide such tight band limits. The lowest data pair appear to be cut and trimmed to define the three-sigma band limits.

I now understood the reason behind the puzzle of the slope of Set #2. It was one-half of Set #3. Since the calculated value for this Set was exactly 1/27.2 I naturally concluded the designer/builder was showing us a slope of 1/10e, the natural base log. Then the slope of Set #2 was intended to be 1/20e.

Really quite striking.

I had now rigorously defined the first three decay curves, and in the process obtained a much better handle on the stone cutting abilities of the builder.

Another curiosity is that the spread of courses for each of the curves is in multiples of seven: 21. 14, and 7 to define the regression lines. He could go to fewer courses to define the regression lines only if he held the courses to sufficient tight control.

I tabulate the parameters for these first three decay curves below. All varibles are in logarithmic numbers.

Note that the sigma band limits are in ratios to one another of approximately 2.54 for 1:2, and 2.24 for 2:3.

Set # |
Spread of Courses |
1/m Slope |
C Y Axis Intercept |
r Correlation Coefficient |
Three-Sigma Band Limits |
---|---|---|---|---|---|

1 |
21 (19 + 2) |
40.32 (40) |
1.02 (1.0) |
0.981 |
0.142 |

2 |
14 |
54.95 (54.4) |
1.215 |
0.986 |
0.056 |

3 |
7 |
27.2 |
2.885 |
0.998 |
0.025 |

Mathematical equations for the regression lines may be expressed by the general form:

P = Ce^{-(h/m)}

where

P is the point on the vertical scale

C is the vertical intercept

h is the pyramid height of the respective courses.

m is the slope of the line.

Taking the natural logarithm of both sides this equation may be written in the form:

ln P = ln C - (h/m)

For example, the first Data Set, Figure 2, has the following values, with C = 1.0, m = 40:

h | h/m | ln P |
---|---|---|

0 | 0 | 1.00 |

10 | - 0.25 | 0.75 |

20 | - 0.50 | 0.50 |

30 | - 0.75 | 0.25 |

40 | - 1.00 | 0.00 |

From this simple computation we can see why the designer chose such neat numbers for this first curve. He made it more obvious to our investigation.

The second Data Set, Figure 3, has the following values, with C= 1.215, m = 54.4 (20e):

h | h/m | ln P |
---|---|---|

30 | - 0.546 | 0.669 |

40 | - 0.728 | 0.487 |

50 | - 0.910 | 0.305 |

60 | - 1.103 | 0.112 |

66.1 | - 1.215 | 0 |

Note that the intercept on the horizontal axis at one cubit is at 66.1 pyramid height.

The third Data Set, Figure 4, has the following values, with C = 2.885, m = 27.2 (10e):

h | h/m | ln P |
---|---|---|

60 | - 2.206 | 0..679 |

65 | - 2.390 | 0.495 |

70 | - 2.574 | 0.311 |

78.5 | - 2.885 | 0 |

Here the intercept on the horizontal axis at one cubit is 78.5 pyramid height.

These calculations can be followed on the data plots.

A word is necessary at this point. Our analysis depends on the comparison of opposite corners of the pyramid, SE and NW. But the builder had no ability to know which two corners, or which points on the courses, we would take for analysis. It must be more than fortuitous that Petrie chose those two corners to measure and publish, to permit this data display. We can only reasonably conclude that each entire course was laid at the proper level and with proper thickness control to provide such information. Otherwise the builder put his astounding display in jeopardy.