A New Perspective
The Geometry of the
Washington National Monument
by Bernard I. Pietsch
The two questions I most often receive from audiences after having given a lecture about the communication systems found in ancient architecture is, "What happened to the tradition of the great builders?" and "Are there modern structures that have meaning like the ancient monuments you describe?"
These are questions I too have asked in the last twenty-five years of my research in archeo-metrology,the recovery of measuring systems used in ancient times. My endeavor has been to understand something of the mind of the ancient builder as expressed in his works. How did the artist or architect exterioralize his inner vision of harmony in the form of his creation? What was his or her view of the world?
After studying the remnants of many of the great monuments of the world--their dimensions, geometry, alignments, coordinate positions etc., I have concluded that there is more to the builder's art than the mere fashioning of stone. The best builders used a remarkable communication system to encode their knowledge. The system is specific. It is readable. The medium is stone and the key to its decipherment is measure. Measure articulates form. Even though form is silent, if the mind can move, stone can be the instrument of communication.
Ancient metrologies emerged from the observation of natural forms and events. Measures had meaningful correspondences to natural phenomena. For example, a pendulum is a simple swinging weight on a string until the string is defined by its length and the beats of its swing are counted as frequencies. Immediately the elements of a system of time and timing are initiated. The pendulum expresses the laws: the length of the string correlates with the law. By observing the pendulum one can discover things about location in latitude, about gravity, about the value of pi. Geometry, mathematics and time converge in a simple demonstration, and measure is the bridge. By using a length found in an art form as a pendulum's length, a statement about dynamics, or time or location can be encoded. Through the process of co-relating measure with meaning, sculpture or works of architecture can go beyond the realm of art and communicate ideas from the domain of physics. Again, the observer must participate in order to see beyond the obvious.
Many of the stone temples, tombs, and megaliths of great civilizations incorporate this methodology integrating geometry, mathematics and time. Until now however, no one has suspected that the Washington National Monument, built in a modern epoch, integrates in its form, these diversities. It is truly an architectural commemorative to the American ideals of individuality, diversity and integrity.i [ footnote ]
Mr. Pietsch Goes To WashingtonOn June 21st, 1988, the day of the Summer Solstice, I visited the Washington Monument. My intention was to verify the measures given in various publications and to confirm an intuition I'd had about the sun angles and shadows created by the obelisk. Since measure is my avenue of discovery, I measured the base on the four sides. At noon I also observed the path of the shadow of the obelisk, and recorded the point where its apex shadow touched the ground. As it turned out, because of the conformity of the surrounding terrain I was not able to complete an accurate survey of the shadows.
Nevertheless, as a result of my investigation I discovered several fascinating metrological relationships about the monument.ii [ footnote ]
The Art of Indication: Real Vs IdealThe Washington Monument is as articulate as any I have investigated in the entire ancient world. The synchronistic aspects demonstrated by its form can actually take us beyond its architecture to the domain of the perfect and sublime.
In the sensible world objects can never be perfect--the notion of perfection can only be entertained intellectually. However to penetrate the mystery of this monument, it is necessary to go beyond the imperfect form which we see and apprehend its perfect geometric form. This perfect form can only be indicated. Even though invisible, it is this indicated measure we contend which is the intended communication.
There is a small price to pay for entrance into the unseen realm. Some exactness must occasionally be sacrificed. But the reward, as will be shown, is great.
What I physically measured at the site for example, varies slightly from the dimensions published by the National Park Service. The measurements I adopt as indicated depart as much as one eighth of an inch per side at the base of the Monument.iii [ table ]
I also adopt an indicated height of 555.555 feet. The published figure is 555.427 feet, a discrepancy of 1.54 inches. As in the real world, perfection is not always possible in the final form, but if it's close, we can get the message.
Art of the Ancient BuildersAncient builders, I have observed, were masters of the sychronistic metaphor: the building of artful correspondence between divergent categories. In their universe of harmony, the relationships between mathematics, geometry, astronomy, biology, physics,and Time, were not separate and discrete, but aspects of the whole. Units of measure, angles, volumes, weights and time could be meaningfully interchanged, interwoven and interconnected.
It is not that the ancients knew more than we do, but rather that they knew more of themselves than we do. Hence their products were the integrated expression of an inner state of harmony. An examination of the elements of this integration will bring us closer to understanding a creation by revealing the mind and values of its creator.
Because the Washington Monument is patterned after the form of Egyptian Obelisks, it is circumstantially linked to the tradition of ancient architects. By being true to the Egyptian model, the correspondences which we will unfold are insured. To speculate on a connection to any secret school of builders is beyond the scope and purpose of this paper. Regardless, the correlation's which follow, are self-evident.iv [ footnote ]
"Math-Magic" of the Washington Monument
1. The Southeast / Northwest diagonal length of the base in feet is an "indicator" of the cosine of the geographic latitude of the site.
2. The number 10 generates the perfect pyramidian.
The rectangular shaft of the obelisk is capped by a four sided pyramid. The number of cubic feet in the volume of the pyramid is an indicator of the number that is also the anti-natural logarithm of 10.
Comment: In many traditions the number 10 is used as the representation for perfection-the unnamable-or for God. Art can indicate perfection, but perfection cannot be executed in this world. Only in the mind can the concept of perfection reside.
3. The number of cubic feet in the pyramidian, divided by 100 is the same number as the number of feet in the perimeter of the obelisk at the base.
4. When the axis of the face plane of the pyramidian is extended to the ground, it strikes the earth 172.5 ft from the center of the obelisk at its base level.
The sychronicity emerges when it is seen that the number 172.5 is 10 times the number of the decimal angle of the pyramidian's face: 17.25 degrees.
5. When the arris (the corner edge) of the pyramidian is likewise extended to the base plane, it converges with the ground 244.07 ft from the center of the obelisk. One tenth of this length is exactly one-half the length of the pyramidian's base diagonal.
6. The area of a circle created from the extended arris edge is two times the area of the circle created by the pyramidian's plane angle.
7. The area of circle A has the same number of square feet as the surface area of the entire obelisk.
Comment: Only a face angle of 17.25 degrees at a height of 555.55 feet could generate this convergence.
8. Let the perfect indicated height of the obelisk, 555.55' represent the length of a Foucault pendulum. A pendulum that length would swing back and forth one time in a period of 26.12011309 seconds. Comment: This is a primary number in the most synchronistic sense. It represents the unification node for a measure of time and a measure of length. Using this perfect unit, the builder could indicate a period of time using a linear measure of length.
9. Another referent to the number 10 is found in the ratio between the period of a pendulum the length of the entire monument, and the period of a pendulum the length of the height of the pyramidian. The ratio is the same number as the square root of 10: 3.1622726.1201 seconds 8.2599 seconds 10
10.The base of natural logarithms can be derived from the base side length of the obelisk.These are a few of the many interesting geometric and mathematical correspondences I found demonstrated in the Washington Monument. The author welcomes inquiry from the interested student seeking clarification or additional information regarding the framework out of which these observations arise.
55.053178372 = 3030.852448
3030.852448 ÷ 2 = 1515.426224
1515.426224 ÷ 100 = 15.15426224
15.15426224-e = 2.718281828
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Comments and suggestions welcome: Bernard Pietsch.