HOME

Egypt
Origins

The Great Pyramid Pi Chamber

Copyright, Ernest P. Moyer, Author
February, 2001
Revised January, 2003

Introductory

The small antechamber located between the top of the Grand Gallery and the King's Chamber in Giza I is the most interesting of all chambers in spite of its small size. Sir Flinders Petrie, with his exquisite measurements of the Pyramid, expressed disappointment in the crudity of the construction.

See The Pyramids and Temples of Gizeh, Field and Tuer, London,1883, First Edition.

"In the details of the walls, the rough and coarse workmanship is astonishing, in comparison with the exquisite masonry of the casing and entrance of the Pyramid; and the great variation in the measures shows how badly pyramid masons could work."

Surprisingly, Petrie did not recognize its unique dimensions. In spite of its crude workmanship the antechamber exhibits highly distinctive properties.

The walls, floor, and ceiling are made from mixed blocks of limestone and granite.

Refer to the discussion on page 174 of Our Inheritance in the Great Pyramid, Piazzi Smyth, W. Isbister, London, 1880. This illustration is from his Plate XV  The different slabs are marked G for granite and L for limestone.

On the south wall above the entrance to the King's Chamber are four grooves running vertically from the entrance to the top of the chamber. The purpose of these grooves is unknown.

Along the east and west walls are two granite linings, or wainscots, of different heights; the one on the east side is 5.0 cubits; the one on the west side is 5.42 cubits. Three vertical grooves are cut into each wainscot beginning at the south end of the chamber. These grooves measure 1.0 cubit in width. At the top of each groove on the west side is a semicircular notch, as though intended for a circular beam or wooden log to be placed across the chamber. However, since no notches are cut into the grooves on the east side, any hypothetical beams would have to rest on the top. The grooves are separated from one another by a distance of about 2/7 cubits.

The thickness of the wainscots are such that the distance between them across the chamber is 2.0 cubits, the same as the width of the entrance from the Grand Gallery, and the exit to the King's Chamber. Note that Smyth shows the crude shape of the wainscots and that they are thicker at the north wall of the chamber, diminishing somewhat at the south wall.

A fourth groove near the north end of each wainscot does not extend completely to the floor; the rectangular bottom of the groove stops at the height of the passageway entering the antechamber from the Grand Gallery. Placed in this northern-most groove and held in position by the granite wainscot of both sides of the chamber are two slabs of granite, one resting upon the other. The purpose of the slabs has been accepted almost universally as a design for blocking entrance to the King's chamber. How this was to be done is not known; the slabs rest solidly within their rectangular grooves. Refer to discussion below.

The measurements of the granite wainscots, the grooves, and the two slabs were not reported in detail by Petrie because of their crude state.

But the dimensions of the chamber offer mathematical relationships heretofore unrecognized.


The Dimensions of the Chamber

Following is a tabulation from Petrie's measurements. All dimensions are in the English inches he used.

48. Taking the Antechamber alone, we may say that its dimensions above the granite wainscot of the sides, are as follows:

  Length N. to S.
Height
above floor
2 from W. Middle 12 from E. E. side
147 116.85 116.22 116.05 115.65
129 117.00 116.18 116.03 115.37
114 117.00 116.11 115.73 114.07
95 116.55 . . . 115.91 . . .
70 116.58 . . . 115.93 . . .
45 115.91 . . . 116.12 . . .

 
 
  Breadth, E. to W.
Height
above floor
2 from N.  40 from N.  76 from N.  2 from S.
147  64.80  64.48  64.96  64.76
129 64.72  64.98  65.26  65.25
114  65.06  65.00  65.48  65.21

The height was measured as follows:


 
  Near N. wall  14 from N.  59 from N.  61 from N.  S. wall
At E. side  149.47  149.09  149.17  149.62  149.63
Middle  149.53  . . .  . . .  149.64  149.64
At W. side  149.32  149.01  149.10  149.65  149.57
Above
gallery end 
153.04  152.95  152.83  152.84  152.61

From Petrie's measurements on the several pyramids he calculated that the value of the Royal Cubit was 20.62 to 20.63 inches. I use 20.625. Converting Petrie's inch measurements to cubits, we obtain the following for the range of each dimension, the mean, and the median.

  Length - North to South Width - East to West Height
Range, inches 114.07 - 117.00 64.48 - 65.48 149.01 - 149.65
Mean, inches 116.07 65.00 150.37
Median, inches 116.08 64.99 149.62
Range, cubits 5.531 - 5.673 3.126 - 3.175 7.225 - 7.256
Mean, cubits 5.628 3.151 7.291
Median, cubits 5.628 3.151 7.254

 
   

The Pi Dimension

The most obvious measurement is the width of the chamber between the east and west walls above the granite wainscots. (Note that the granite wainscots prevented measurement of the chamber width except above their height. If such measurements were possible the results might somewhat modify the mean measure.)

The width of the chamber in cubits is the numerical value of Pi.

The mean and the median of the measurements is 3.151 cubits. This is +0.09 or +2.8% from the ideal of 3.142. The individual values range around the ideal from 3.126 to 3.175, -0.016 to +0.033 cubits, or errors of -0.51% to +1.05% from the proposed model.

In this single dimension the pyramid builders demonstrated that they knew the numerical value of Pi, within the their ability to control a work of stone. The dimension has withstood more than 4500 years of time, with earthquakes and the ravages of later generations. The errors in construction and methods do not permit us to deduce that they knew the theoretical value.

The Height and Length Dimensions

The builders did not depend entirely upon this single dimension to show their mathematical knowledge. As I shall now show:

They designed the chamber to show knowledge of circular and spherical geometry.

Consider Petrie's mean measurements at L = 5.628, W = 3.152, and H = 7.291 respectively.

If we multiply these numbers we obtain:

W times L = 17.739 cub2

W times H = 22.981 cub2

L times H = 41.034 cub2

The calculations show that:

        Equation 1:         WL + WH = LH.

17.739 + 22.981 = 40.720 calculated from the first two terms vs. 41.034 measured. The difference is 0.314, or 0.76%.

Now consider the relationship of the total chamber surface area to the surface area of a side wall.

The total surface area is

        Equation 2:         Surface Area = 2(WL) + 2(WH) + 2 (LH).

Taking Petrie's mean values we obtain:

2(17.739) + 2(22.981) + 2(41.034) = 163.508 cub2 calculated for the total surface area.

If Equation 1 is true then 2(WL) + 2(WH) = 2(LH).

Substituting into Equation 2 we obtain 2(LH) + 2(LH) = 4(LH) = Surface Area.

        Equation 3:         2(WL) + 2(WH) + 2(LH) = 4(LH)

Therefore an alternate method for calculating the total surface area is 4(LH) = 4(41.034) = 164.136 from the measured values.

Compare the two:

The area of 163.508 cub2 calculated from Equation 2, and

The area of 164.136 cub2 calculated from Equation 3.

This is a difference of 0.63 cub2, or 0.38%, using Petrie's mean values.

Note how well the builders were able to control the dimensions to obtain this close comparison.

If the pyramid builders intended to demonstrate this relationship among the dimensions what was their purpose?

The meaning becomes evident when we recognize that

the total surface area of the chamber is related to the surface area of one side wall by a ratio of 4:1.

This is expressed directly by Equation 3.

163.508 divided by 41.034 = 3.985 (Using the alternate numbers we obtain 164.136/40.72 = 4.031.)

The error from 4.0 is 0.37% in the first case, and 0.78% in the second.

This is highly intriguing for the ratio of the surface area of a sphere to the area of a circle of the same radius is also 4.0.

                Equation 4:         Ssphere = 4 X Pi r2

From these facts we can propose that the builders intended to demonstrate knowledge of this relationship by a unique method.

The builders used rectangular geometry to represent spherical geometry.

If so:

The total chamber surface area represents the surface of the sphere; one side wall represents a circle of the same radius.

From this knowledge we can calculate the value of that radius.

                LH = Pi r2

                r2 = LH/Pi

                r2 = 41.034/3.1416 = 13.062, based on Petrie=s mean values.

                r = 3.614

This would be the radius of a sphere with the same surface area as the total area of the chambers walls.

But another property is more interesting.

If we take the volume of the chamber and equate it to a sphere we obtain different results:

                V = W X H X L =  3.152 X 7.291 X 5.628 = 129.338 cubic cubits.

The volume of a sphere is 4/3 Pi r3.

Equating the two expressions we obtain r3  = 129.338 X 3/4 X 1/ Pi.

This gives a value of r3 = 30.877.

Hence r = 3.137.

This is different from the value of Pi by 3.1416 - 3.137 = 0.0046 or 0.14%.

Hence the chamber has an equivalent radius, based on volume, that is equal to Pi.

This is an example problem in spherical calculations. It could be given to any pupil in a class on solid geometry. What is the volume of a sphere whose radius is equal to Pi?

If one were familiar with this problem he should be able to immediately recognize the volume obtained by calculation of the chamber dimensions as equivalent to the volume of a sphere of radius Pi.

The width of the chamber equal to Pi, and the volume equal to this simple illustration in solid geometry are methods to attract the attention of anyone who might study the chamber. This tells us something about the psychology of the designer and how he thought someone might later recognize his design intent.

Note that the radius calculated from the surface of the chamber differs from the radius calculated from the volume. This is due to the fact that the chamber has rectangular dimensions, and is not a sphere.

We can look at the relationship among the dimensions another way.

Dividing both sides of Equation 1 by WLH we may obtain:

                Equation 5: 1/H + 1/L = 1/W.

This calculates as follows, using Petrie's mean values:

1/7.239 + 1/5.637 = 1/ Pi (approximately).

0.1381 + 0.1774 = 0.3155 summing the first two terms.

Compare with 1/Pi = 0.3183.

Inverting the sum we find that the calculated value for Pi from L and H is 3.167.

This value differs from mathematical Pi by 0.81%.

Not bad for such a crude chamber.

Thus it is possible to calculate an approximate value for Pi, within one percent, from the length and height dimensions.

This value is not obvious; it shows only in analysis of the dimensions.

The design led to our evaluation of the length and height dimensions, and recognition of the circular geometry. By making one dimension equal to Pi, and by making the total chamber surface area four times as great as one side wall area they tied together their knowledge of Pi as well as circular and spherical geometry.

From these facts we can deduce that:

The pyramid designers had to know the theoretical relationship between a circle, and a sphere of the same radius.

They certainly did not achieve such happy results by sheer accident.

This means they also had to know the theoretical value of Pi.

Note also that the area of the two side walls sum to the hemispherical surface area. Using the value from LH, this is equal to 41.034 cub2. Twice this value is 82.068 cub2. Compare to 1/2(163.508 cub2) = 81.754 cub2 calculated from Equation 2.

Therefore, the area of the north and south walls, floor and ceiling also sum to the hemispherical area.

The Logarithms of the Dimensions.

While all of this is quite astonishing, the numerical values of the chamber length and height are even more astonishing.

From Equation 5 we can see that if W is held constant, the apparent first step for the designers, then L and H can vary in constant ratio to one another to preserve the value of W. What determined the choice of the designers?

The mean of Petrie's height measurements is 7.291 cubits. The median is 7.254.

This number is also unique and common to the physical sciences we practice today:

Pi (ln 10) = 7.234,

where (ln 10) means the logarithm of the number ten to the natural base Ae,@ using commonly accepted symbols and designations.

The theoretical value differs from Petrie's mean by 0.79% and Petrie's median by 0.28%.

Hence, we can postulate that the builders intended for

            Equation 6:        H = Pi (ln 10).

Rearranging Equation 1:

            Equation 7:         L = (WH)/(H-W)

Substituting Equation 6,

            Equation 8A:        L = {Pi[Pi (ln 10)]}/[Pi (ln 10) - Pi] or, simplifying

            Equation 8B:        L = [Pi (ln 10)]/(ln 10 - 1) or, expressed differently

            Equation 8C:        L = H/(ln 10 -1).

If the height truly represents Pi (ln 10) then L is forced.

An alternate way of expressing the relationship is

            Equation 8D:        L = H X (H/Pi - 1).

Calculating from the mean of Petrie's measured values, with the numerical value of (ln 10) = 2.303

L = 7.291/1.303 = 5.596 cubits.

Calculating the theoretical value from Equation 8B:

(3.142 X 2.303)/(2.303 - 1) = 7.234/1.303 = 5.553 cubits.

This theoretical ideals differs from Petrie's mean of 5.596 by 0.77%.

These are all rather astonishing results for such a rough and crude chamber.

The extraordinary relationships calculated from Petrie's measured values and compared to theoretical values may be readily observed in the following diagram. The vertical lines show the range of Petrie's measurements. This is a scaled drawing to emphasize the astonishing results.

 
 

 

We can see that the builders did not use more precision in construction than was necessary to convey their intent. Petrie may have been disappointed, but analysis, within the errors he reported, is adequate to show that intent. In spite of Petrie's assessment, the control of the chamber dimensions was tight, in the face of an apparent crude construction with huge blocks of stone, but certainly adequate to determination of the design.

Degrees of Freedom

A natural question arises about the degrees of mathematical freedom used in the various dimensions and relationships. Is there not a conflict between the volume of the chamber relating to Pi, and the use of the exponentials functions?

Yes there is. We cannot have conflicting definitions without some means to bring them into reconciliation with one another.

From calculations above we know the volume of the chamber is W X H X L = 129.338 cubic cubits from the mean of Petrie's measured values. But if W, H, and L are defined by the logarithmic relationships are the two results compatible?

For W = Pi, H = Pi (ln 10), and L = H/(ln 10 -1) these numbers multiply to a theoretical value of

        3.1416 X 7.234 X 5.552 = 126.172.

This is where the true genius shows. The numbers differ by 2.5%. The designer did not deal with unknown conflicting definitions; rather he knew his numbers and recognized that he could bring the definitions into numerical reconciliation with one another.

Why This Design Choice?

The genius of the designer lies in using cubit measures to express sophisticated mathematical knowledge.

When we discover the Pi dimension of the width, we are naturally drawn to examine the other dimensions. We might know the numerical value of Pi from more elementary mathematics, as has been known at least since Babylonian and Greek days, but the knowledge of logarithms is a recent development.

We would not recognize Pi (ln 10) unless we had a similar level of knowledge.

Stating it otherwise,

The designer of the Great Pyramid had a level of mathematical knowledge unequaled until the past three hundred years.

Furthermore, the designer found a method for talking to us without the symbols of language. He used mathematics. Since mathematical reasoning is universal, beyond linguistic symbolism,

the designer found a way to communicate with the future where linguistic symbolism would be unknown.

The question that naturally arises is this:

Did the designer expect his level of knowledge to become lost, and did he used the Great Pyramids of Egypt to make a record of his existence?

The answer to this question is a resounding, Yes!

Remember, the builder sealed the interior of the pyramid and this chamber from prying eyes until Al Mamun of the Arabs burrowed a crude entrance in 820 AD. With a team of engineers and masons Al Mamun was unable to find the entrance passage. Not until he broke in did anyone know where the entrance was located.

It is easy for us to deduce that the builder expected the intervening millennia would bring no disturbance to his great creation, but that at some time the structure would be opened up. The building is so massive that even modern stone robbery, with the limestone casing used to build parts of Cairo, was unable to seriously damage its form. Then the secrets of the chambers would be revealed with accurate measurements, and logical examination. The sequence was first Al Mamun, 820 AD, then a series of explorations and attempts at measurement by various men from the seventeenth until the nineteenth century, then Petrie's exquisite measurements in 1883, and then finally this analysis.

The question then becomes one of how good was the builders perception of the future?

The genius lay in his conception of a stone structure that would reveal advanced mathematical knowledge, from Pi, to circular geometry, to exponential forms, and logarithms. For his expectation to be realized the future had to advance in technical knowledge until such time that all of these steps could reach fruition, that the future had to reach his level of knowledge. He had to recognize these steps. But this also means that he recognized a great decline in knowledge from his stage in history, that the world was running down genetically and socially. He anticipated such decline, and the consequent redevelopment of man's knowledge. He may not have known what form it would take. Perhaps he expected an uplift of genetic stock, and mental capacity equal to the task. Or, he may have perceived it as a general social development without that uplift.

As we can now see, all dimensions represent fundamental geometric and physical constants, recognizable by any competent modern mathematician, physicist or engineer. They are constants that derive from applied mathematics and the laws of nature. No other numbers satisfy the values so simply, so uniquely, and so closely.

Our major difficulty in accepting this solution lies in a measurement related to a mathematical base of natural numbers. There is no evidence for such knowledge until modern times. None of the preserved records of ancient Greece, Mesopotamia, or Egypt show such knowledge. How could a more ancient people possess it?

In spite of this overwhelming tradition, the values are unique. If the designers were familiar with natural number systems and they wished to express their knowledge in a work of stone they made an excellent choice for the dimensions. Pi (in 10) shows knowledge of three distinct mathematical forms: circular geometry, base ten number systems, (10),and natural base number systems, (ln).

But these results do not sit in isolation. Other dimensions provide confirmation in all of the four great pyramids.

Other factors are important.

Petrie noted that earthquakes had disturbed the north-south width measure of the King's chamber. The large granite ceiling beams were cracked by seismic disturbances sometime in the past. The same problem may affect this small chamber, since the greatest dimensional error is in the length along the same line as the greatest error in the King's chamber. If the side walls rest upon the floor stones, and the ceiling rests upon the side walls, we should expect the least error in the height, since seismic disturbances would not change the height if the chamber were not tightly bonded to the core masonry and was left floating in the structure, unless the earthquakes were violently severe. Evaluation of the floor levels shows the effect of seismic disturbances.

I shall not discuss those effects here, since they do not impact upon this study.

This line of thinking leads to the possibility that the granite wainscots and slabs in the antechamber have a practical purpose other than inhibiting entry to the King's chamber.

The two side linings with the stone slabs inserted between them would prevent the north and south walls from collapsing inward under seismic disturbances. Only outward shifts could enlarge the chamber. The measurements show a median larger than the ideal, not smaller. This postulate may be verified by searching for space between the two wainscots and the respective walls.

However, this does not explain the other three grooves in the wainscots. Perhaps their purpose relates to other mathematical properties, or perhaps the builders changed their minds. Further observation and measurement seem necessary. If this part of the design was left in an unfinished state, shown by the different heights of the wainscots, we may never be able to determine the purpose of those parts.

In fact, the builders may have intentionally left the wainscots in an unfinished and contrasting condition to force our attention away from them, and on to the walls of the chamber.

HOME BACK TO TOP