The Washington Monument: Icon by Design © 2006 Bernard I. Pietsch and S. Thompson
To the public eye, the towering blocks of masonry of the Washington Monument commemorate our first President and the aspirations of the republic. But to an inner eye, the colossal obelisk conceals an invisible–and some might say sacred, realm of value and meaning. Though not hidden from view, this higher aspect of the monument’s design expresses itself indirectly and by implication. It is not discerned by logic alone--one must be educated to its perception. To fully enter the mystery of the Washington Monument, it is necessary to go beyond that which we see and attempt to apprehend its subtle and sublime intended form. By deconstructing the form, we will better understand the monument itself, and also the ancient canonical ideal which inspired the work.
Quomodo Legis? How do you read? In order to “read” the invisible aspects of the monument, we will apply the same analytical methods I have used to decipher ancient works of art and architecture around the world. The first step in the process is to define the dimensions of the visible form. This includes all the lengths, angles, areas, surfaces, and volumes of the existing structure. These attributes are the vocabulary of the language used to encode/decode its meaning. For the Washington Monument there are three geometric components: a slightly tapering tower or shaft, a truncated four sided pyramid atop the shaft called the pyramidion, and a smaller pyramid sitting atop the pyramidion called the capstone. The physical details of these features are listed in Appendix A. Both dimensions from the U.S. Park Service and those I adopt as intended are given.
Actual and Intended: A note on accuracy It is apparent from Appendix A, that some of the measures I use depart slightly from the official dimensions suggested by the Park Service. Because no measuring of the physical world can ever be completely accurate and because perfection in the physical world can only be approximated, the reader is asked to tolerate a margin of inexactitude. For the purposes of reading a monument, the virtue of any single measure is determined by the extent to which it accords with the overriding harmonics of the work. By making minor adjustments to the figures, the artful statement of a work becomes visible, and its metaphors surface. Consequently, I have adopted those values I conclude as
Washington Monument
10/03/08© B.Pietsch / S.Thompson
“The Washington Monument: Icon by Design”
<bernard@sonic.net> Page 2 of 18
intended by the architect. For example, my adopted measure of 555.555 feet for the monument’s intended height is more congruous with the overall scheme of the structure than the published figure of 555.427 feet--a difference of only 1.54266 inches. I submit that the profit gained by this approach outweighs any supposed error in metric accuracy. The second part of the deciphering process involves identifying the underlying harmonies of the structure. Some correspondences are direct, others subtle. But it is in the inferential realm where the characteristics that distinguish ordinary from iconic, and sacred from mundane, reveal themselves. How the Washington Monument fulfills the following ten elements of iconic architecture proves this form to be among the world’s most extraordinary structures. The reader is invited to participate in our process of investigation. Just as literature utilizes metaphor to convey inferential meaning, we will similarly use the languages of number, geometry and measure to articulate the deeper levels of the Washington Monument’s design and give voice to its otherwise mute stones. Through discovery, that which was invisible will become, by demonstration, self-evident.
Elements of Iconic Architecture I.
The design is generated from a singular idea. Those architectural achievements most worthy of note are generated from a singular principle that pervades and unifies the structure. As I have found in my work on ancient monuments, this is especially true in the case of the Egyptian obelisk, after which the Washington Monument is modeled. For the Washington Monument, that singular notion is the number 10 – a number used in many traditions to represent “unnamable perfection.” The following is a partial listing of the ways in which the number 10 manifests itself in the structure of the Washington Monument. 1. The entire monument is 10 times the height of its pyramidion. 2. The height is 10 times the length of one base side of the monument. 3. The number 10 generates the volume of the perfect pyramidion:
The volume of the pyramidion in cubic feet is the same number as the anti-natural logarithm of 10. The volume of the pyramidion: 22026.46579 cubic feet.
10/03/08© B.Pietsch / S.Thompson
“The Washington Monument: Icon by Design”
The anti-natural log of 10:
<bernard@sonic.net> Page 3 of 18
22026.46579
4. The number designating the cubic volume of the capstone, 94.25959089,1 is very nearly 10 times the number designating the length of the apothem (plane face) of the capstone, 9.423579201 inches. The shape of the Washington Monument entreats the observer to look beyond its exterior, where the truly sublime elements of the obelisk’s geometry lie. The identification of this geometry, which is external to the structure, requires the active participation of the observer--the monument’s visible form must be projected to its geometric conclusion. In this example, we consider how the apothem or face angle of the pyramidion (Fig.1) relates to the whole obelisk, and how again the number 10 by extension is an integral part of its design. 5. When the axis of the face plane of the pyramidion is projected to the ground (Fig. 2), it strikes the earth 172.4 feet from the center of the obelisk at its base level. 2
Fig. 1 Apothem Angle
172.4 is 10 times the number of the decimal angle of the pyramidion’s face, 17.2 4 degrees. 2 Fig. 2 Apothem angle projected to base level.
Observation: At a height of 555.555 feet, only a face angle of 17.24 degrees could generate this convergence. 6. When the corner edge or arris angle of the pyramidion (Fig. 3) is likewise projected to the base plane of the monument, it converges with the ground at a point 243.8 feet from the center of the obelisk (Fig. 4). See that the length of the hypotenuse projected along the arris angle from the top of the pyramidion to the ground is 606.72 feet3. Observation: 606.72 feet is 10 times the length of the arris corner of the pyramidion, 60.67 feet. II.
The smallest component informs the whole: the Capstone as initiator. 1
94.25959091 (the reciprocal of .010609) X 3 = 282.7787727 282.7787727÷9 = 31.41986363, which ÷ by 10 = canon π . The √ of 31.419 is 5.605342419, read as inches is the length of one base side of the capstone. To find the angle, divide the length of the base by 2 = 2.80267121.Then divide half the base by the height, 9” to get .311407912, the tangent of 17.29700181degrees. The cosine of 17.297 degrees is .994776359. Divide the height, 9” by the cosine of 17.297° = 9.426291206, the length of apothem in inches.(Slight departures in the decimal places occur due to the indeterminate height of the capstone and the use of canon π.) 2 3
17.2439725 degrees 606.7235226 feet
Fig. 3 Arris edge of Pyramidion 60.67'
Fig. 4 Arris angle extended to base.
10/03/08© B.Pietsch / S.Thompson
“The Washington Monument: Icon by Design”
<bernard@sonic.net> Page 4 of 18
Herodotus, the early Greek historian, was told that the Great Pyramid was “built from the top down.” This is veiled language for the notion that the design of the Great Pyramid is explained and driven by the geometry of its top portion. (Alternately stated, the last detail determines the first.) Likewise, in the Washington Monument, it is the 9” capstone – the topmost and smallest component of the monument -- which similarly illuminates the monument. The capstone is both the geometric conclusion of the monument’s form and also the “seed” from which the rest of the monument may be generated. The capstone has a square base, a height, an apex angle and a weight. With even the slightest adjustment of any one of these attributes, the obelisk projected from the capstone would be altered. By selecting a perfect height of 9” and a base area of 31.41 square inches, an angle of 17.24° is assured.
Laying of Capstone Ceremony December 6, 1884
7. The ratio between volume of capstone and the volume of the pyramidian can be closely derived from the capstone:
The capstone, a 9 inch pyramidal form with a base side length of 5.605342419 inches has a volume of 94.25959091 cubic inches.
The pyramidion (including capstone) is 666.6666 inches high with a base side length of 413.8572194 inches and volume of 38,061,732.89 cubic inches.
The ratio between volume of pyramidion and capstone is: 38,061,732.89 cubic inches in pyramidion 94.25959091 cubic inches in capstone
Capstone Replica
Formula for finding the volume of a pyramid: (Height x Base2 )
= 403,796.9242
Observation: If the number of inches in the base side length of the capstone (5.605342419), is used as a common logarithm--we derive, with a small departure, the same number as the ratio between the two volumes: 5.60534241910 = 403,034.6813
The common log of the actual ratio number of the two volumes, 403,796.9242, would render a side length of the capstone at 5.606163007”, a difference of only .00999853627 from the mathematically determined figure above.
10/03/08© B.Pietsch / S.Thompson
“The Washington Monument: Icon by Design”
<bernard@sonic.net> Page 5 of 18
The alchemy of the capstone We know that the capstone was fashioned of aluminum – an unusual choice of material, as in its time it was a rare metal and difficult to manufacture. Beyond its value and utility in protecting the tip of the pyramidion from lightning, there are non-linear correspondences which intimate an integral relationship between the capstone's geometry and its aluminum complement. For example: 8. The number of pounds in one cubic foot of aluminum can be extracted from the dimensions of capstone:
Setting Capstone
Divide the length of the side of the capstone’s base by 2: 5.605342 = 2.802671209 2
Find the anti-natural log, e of 2.802671209 and multiply by 10 to find: 2.802671209e = 16.48863259 16.48863259 x 10 = 164.8863259 is also the number of pounds in one cubic foot of aluminum. Divide the number of pounds per cubic foot of aluminum by the number of cubic inches in one cubic foot (1728): 164.8863259 ÷ 1728 = .095420362 lbs/cubic inch of aluminum.
The aluminum capstone is reported to weigh 100 ounces. But given that the weight of aluminum is 164.8863259 lbs. per cubic foot, or 1.5267253 ounces per cubic inch -- the weight of a 94.25959091 cubic inch aluminum pyramidal form ought rightly to weigh 144 ounces. Based upon its reported weight of 100 ounces, we must assume that the capstone is hollow or perhaps designed with an aperture for affixing it to the top of the pyramidion. This missing 44 ounces however, is not inconsequential; and as shown below, the missing aluminum portends alchemy invoking the highest levels of intuition. 9. One method for deriving the cubic volume of the missing mass of the capstone uses the reciprocal of the number of times one cubic foot of aluminum can be divided by 44 ounces. To find the number of missing cubic inches see that:
Aluminum 1 Cubic Inch = .095420362 lbs. 1 Cubic Inch = 1.5267253 oz. 1 Cubic Foot = 164.8863259 lbs. 1 Cubic Foot = 2,638.181241oz.
10/03/08© B.Pietsch / S.Thompson
“The Washington Monument: Icon by Design”
<bernard@sonic.net> Page 6 of 18
In one cubic foot there are 2,638.181241 ounces of aluminum. 44 ounces of aluminum, divides into one cubic foot 59.5866396 times. 1 x 1728 = 28.81985498 cubic inches 59.5866396 28.81985498 cubic inches of aluminum weighs 44 ounces. By this method the difference in the volumes of a capstone weighing 144 ounces and one weighing 100 ounces is 28.81985498 cubic inches of aluminum.
Aluminum to Gold
Gold as a standard of value is referenced in many ancient works. In several monuments and artifacts we have observed synchronistic crossovers between the volume of an object itself and the weight of that volume if it were transformed to gold. Although aluminum and not gold was used in the Washington Monument’s capstone, the capstone nonetheless obliquely references gold in a remarkably synchronistic way.
See that: 100 ounces of aluminum, as reported, has a volume of 65.45804925 cubic inches and that,
144 ounces of aluminum, as derived, would have a volume of 94.25959091 cubic inches.
Notice that the weight of 94.25959091 cubic inches of gold would be 65.45804925 pounds troy -- the same number as the number of cubic inches of aluminum in a 100 ounce capstone.
10. Another referent to gold is indicated by the length of the mean width of the shaft of the monument. Here the number of grams in one cubic foot of gold is designated. Although the correspondence is not exact, we are reminded that the object itself is an artful statement—its ability to indicate many different levels at once is to be appreciated.
The mean width, 44.77648678‘, multiplied by 10,000 renders 447,771.3305, very nearly (.998613) the number of grams in one cubic foot of gold. Divide the mean width of the shaft by 1200, the number of pounds in one cubic foot of gold:
The base side length of the shaft + the base side length of the pyramidion, divided by 2 = the mean width of the shaft.
55.06616449’ base side length +34.48680907’ pyramidion side 89.55297356’ 89.55297356’ = 2 44.77648678‘ mean shaft width.
10/03/08© B.Pietsch / S.Thompson
“The Washington Monument: Icon by Design”
<bernard@sonic.net> Page 7 of 18
447,771.3305 gr 1200 lbs
= 373.1427754 grams per pound
373.1427754
= 31.09523129 grams to one ounce troy gold
12 oz./troy lb.
(The actual exact number of grams in one cubic foot of gold is 447,833.4407--approximately 62.11 grams per cubic foot departure from that derived from the mean width of the shaft.) III.
Design elements are self-reinforcing and redundant. In the best-crafted monuments, every detail holographically or holistically supports and enlightens the whole: design, measure, and material are congruous with the concept. Such mutual relatedness results in redundancies in the underlying relationships, which manifest as repeating geometric patterning or reiterated numeric values. These redundancies reinforce and recapitulate the intention of the overarching design. Some of these redundancies have been noted above, particularly in the section on the singular idea of the monument. But I suspected the existence of other redundant indicators in the Washington Monument, and in fact discovered several meaningful instances in the course of my work.
Fig. 6 Apothem angle projected to ground as radius.
11. Observation: The perimeter of the shaft’s base restates the volume of the pyramidion. There are 220.26 feet in the perimeter of the base of the obelisk’s shaft. This number, times one hundred approximates the number of cubic feet in the pyramidion: 22,026.46579 cubic feet 12. Return to demonstration Number 5, where the line along the apothem or face angle of the pyramidion is projected to the ground. (Fig. 6) This line would strike the base plane 172.4405081 feet from the center of the monument. Let this point of intersection create the radius of a circle. The area of this circle A, 93,429.25446 square feet, is nearly equal to the surface area of the entire obelisk. (Fig. 7) 13. The Area of B (Fig. 8) created by extending the line of the arris angle of the pyramidion to the base plane of the obelisk is 186,858 square feet – approximately double Area A and double the number of square feet in the surface area of the entire obelisk. (Fig.7)
Fig. 7 Surface Area
Fig. 8 Arris angle projected to ground as a radius of outer circle B.
10/03/08© B.Pietsch / S.Thompson
“The Washington Monument: Icon by Design”
<bernard@sonic.net> Page 8 of 18
14. Observation: The ratio of the circumference of circle A (1083.61145’) to the circumference of Circle B (1532.458008’) is also the same number as the cosine of 45 degrees: .7071067816. 1083.611450 1532.458008 IV.
A B
= .7071067816
Iconic architecture may reference its location. Ancient monumental work I have investigated, especially obelisks, often show themselves to be geodetically site specific, operating as markers on the earth. The Washington Monument announces its own latitude in at least two different ways; and as we will see below the evidence suggests that its longitudinal positioning may have been consciously selected as well.
Cosine 45° .7071067816
From my own measurements I determined that the sides of the monument are actually slightly skewed--the base is not exactly square. Though slight, the difference in the two major diagonal angles is an important geometric signifier. 15. Observation: The length of the Southeast/Northwest diagonal of the monument’s base indicates the geographic latitude of the site. The diagonal of the base of the Washington Monument in feet, divided by 100 and taken as the cosine of an angle, derives almost exactly the latitude of the site.
The South East to North West diagonal measure of the base of the obelisk is approximately 77.836 feet. .77836 is the cosine of the angle 38º 53' 21" 77.836 feet ÷ 100 = .77836 which is the latitude of the site of the Monument.
This observation is confirmed and reinforced by another method of identifying latitude from the form of the obelisk as in the following: 16. The latitude of the site of the Washington Monument can also be derived from the capstone’s apothem length.
The length of the capstone’s apothem face length, 9.426 inches, approximates the number of degrees of
Fig 9. Latitude of Washington Monument
10/03/08© B.Pietsch / S.Thompson
“The Washington Monument: Icon by Design”
<bernard@sonic.net> Page 9 of 18
precession of a Foucault pendulum on the latitude of the monument4. The precise height of the capstone is properly indeterminate because of the absence of an apex. If the capstone were taken to be exactly 8.99465 inches in height, (a difference of .00535 inches,) the degrees precession derived from the apothem’s length, 9.417 inches activated as a pendulum, would indicate the monument’s actual latitude of 38° 53’ 21”. V.
Iconic architecture references universal constants In many instances, great architecture commemorates and encodes the great constants of the universe, both mathematical and physical. The Washington Monument participates in this respect through its referent to the constant Pi.
Fig. 10 Photo of replica from Smithsonian Institution.
17. Observation: The area of the base of the capstone is 10 x
π
or 31.41 square inches5.
18. The shaft of the Washington Monument is actually a truncated pyramidal form. (The sides of the shaft, were they not interrupted by the pyramidion, would come to their geometric conclusion 1,333.333 feet above the base of the monument, or 777.777 feet beyond its tip.) (Fig. 11) The Great Pyramid is also truncated--its top does not form an apex or point, 6 but rather a rectangular platform. (Fig. 12) The top of the shaft of the Washington Monument, (coincident with the base of its pyramidion) is 1,189.429 square feet in area. The top of the Great Pyramid is 1,892.798083 square feet in area. The relationship between the two areas can be represented by a simple ratio with an interesting correspondence with pi . Again the elegance of communication exercised by consummate architectural design is affirmed. 1892.798083 sq’ 1189.429153 sq’ 4
= 1.59135
__1__ X 10 = 1.59135 πx2
The formula for deriving the precession of a Foucault pendulum is: 15º x sine of the latitude = degrees precession per hour. The actual latitude of the WM is 38.88948º (TerraServe Web Map Service), the sine of the latitude would be .6278190681 x 15º = 9.417286022º precession per hour. See that 9.417 is not 9.426, the inches of the capstone’s apothem length derived trigonometrically from a 9” capstone height. Working backward from the apothem’s length number derived from the 9” capstone, would render a latitude of 38.93360578 or 38° 56’ 00”.98, a location slightly to the north. 5 6
Fig. 11 Sides extended to apex
3.141986363 is the figure we use for π in our computations in ancient metrology.
We hold the 206th to have been the topmost and last stone course intended on the physical pyramid. More information about the significance of the 206 stone courses or levels can be found in the article, "New Perspectives on the Great Pyramid," on our website: www.sonic.net/bernard.
Fig. 12 Truncated tip of Great Pyramid model from above.
10/03/08© B.Pietsch / S.Thompson
“The Washington Monument: Icon by Design”
<bernard@sonic.net> Page 10 of 18
19. Another way to find the area of the top of the shaft (or the area of the base of the pyramidion) is derived by the following formula which again summons e, the base of natural logarithms and the height of the pyramidion: 10e x 3 ÷ 55.5555 = 1,189.429153 square feet. 20. Observation: A second novel derivation of a number close to the base of natural logarithms can be conjured from the number for the feet in the base side length of the obelisk. The mean base side of the obelisk is 55.06616 feet. Square the side, divide by 2 and then divide by 100. Now find the natural log to get a number which approximates e, the base of natural logarithms7: 55.066164492 = 3032.282471 3032.282471 ÷ 2 = 1516.141236 1516.141236 ÷ 100 = 15.16141236 15.16141236-e = 2.718753539 (2.718281828 is the precise number of the base of natural logarithms) 21. The line projected from the apex along the apothem face of the pyramidion to the ground would be 581.7 feet in length. Multiply this number by 10 and read it as inches: 5,817 inches. 5,817 inches is also 484.75 feet – an approximate indicator for the tangent of one second of arc: .000004848136.8 22. Observation: In a perfect world, one could expect that the longitude of the Washington Monument would exactly indicate the reciprocal of the number of seconds of arc in a 360° circle, divided by one billion: 1____ 1,296,000 seconds ÷ 1,000,000,000 = 77.16049383
Read sexigesimally, 77.16049383° minutes 37 seconds.
is
77 degrees 09
The actual longitude of the Monument is: 77.03503° or 7
e (2.718281828) A mathematical constant used in describing rates of change in natural growth and decay behaviors. The function of e might be used for example to calculate the rate of decay of a radio active isotope, or the rate of growth of bacteria, or the spread of a disease in an epidemic, or the pattern of a bouncing ball as it diminishes to a halt. 8
.000004848136 is the trigonometric tangent of one second of arc: 00º 00' 01"
Apothem angle extended to ground 581.7 feet
10/03/08© B.Pietsch / S.Thompson
“The Washington Monument: Icon by Design”
<bernard@sonic.net> Page 11 of 18
77 degrees 02 minutes 06.108 seconds. Unfortunately, the constraints of the swampy terrain surrounding the monument precluded it from perfect placement. VI.
Iconic architecture actualizes the co-in-cidence of a length of a line with a period of time. Architecture can only indicate a length—perhaps in the height of a doorway or in the perimeter of a building. It is contingent upon the observer to take what is given one step further and ask, “What does this length signify? Is there another level to the information presented here? The answer to the question and the model for the architectural metaphor lies with a simple instrument: the precessing pendulum. This most elementary apparatus enables communication from one dimension to another—from length of line to length of time. Through the pendulum, architecture can invoke time and all the dynamics associated with notions of frequency and duration from a simple length. 9 23. Observation: Let the perfect indicated height of the obelisk, 555.555’ represent the length of a Foucault pendulum. A pendulum this length would swing back and forth one time in a period of 26.12011309 seconds. Find that the square root of the height 555.555 is 23.57022604. The ratio of the square root of the height to the period of its time as a pendulum generates another “perfect” number: 1.228068554. 26.12011309 23.57022604
= 1.108182545
1.1081825452 = 1.228068554 1.22806554 is a primary number in the most synchronistic sense. It represents the node of unification between a measure of time and a measure of length. Using this perfect unit, a builder can indicate a period of time using a linear measure of length: a pendulum length of 1.22806554 feet beats a period of time lasting 1.22806554 seconds. At this node, time and length are commensurate –length in time and length in feet are the same number. 9
Formula for determining the time a certain length would beat as a pendulum: L= Length in feet, G = gravity, 32.15481179' , T = Time in seconds.
2π √_L_ = T G
10/03/08© B.Pietsch / S.Thompson
VII.
“The Washington Monument: Icon by Design”
<bernard@sonic.net> Page 12 of 18
Iconic architecture incorporates astronomical orientation. In many instances I have found that links to Saturn ђ (Kronos) as source of measure are never far from the roots of iconic architecture.10 24. Observation: A pendulum the height of the pyramidion, 55.5 feet high, would beat 10609 times in one canon day. 11 10609 references the number of canon days in the orbit of the planet Saturn. 25. A strange and wonderful relationship exists between the number representing the cubic volume of the capstone, 94.25959091 cu” and the reciprocal of the number representing Canon days 12 in the orbit of the planet Saturn. Note that this figure diverges only in last two decimal places from that of volume derived mathematically. ___1____ .010609 ђ
= 94.25959091 as cubic inches in volume of capstone.
26. Observation: If the angles at the base of the shaft were to be extended upward beyond the pyramidion to their geometric apex, (Fig.13) they would converge at a point 1,333.333 feet above the base. In another synchronistic correspondence, that height in a process coupled with 10609 (mentioned earlier as a number referencing Kronos or Saturn) indicates the diameter of the Earth, a necessary referent from which to reckon all other measures. 13 _10609_ = 7.9567502 x 1000 = 7,956.75 miles 1,333.333 diameter of earth
27. Observation: An indicator of the rate of the precession of the equinoxes can be extracted from the pyramidion’s arris length, 60.672 feet: 10
For more background on this most significant correlation, see our article “Saturn, Source of Measure” online @ The Philosopher’s Stone: www.sonic.net/bernard/ 11
10609.80724 times in 87636 seconds or one sacred day of 1460.6 minutes (360 sacred days to one year). 12
A canon (sacred) year is adjusted to 360 Earth days, as opposed to 365.25 astronomical days. 13
In the canon system, the circumference of the Earth is 25,000 canon miles. There are 5,258.16 feet in one canon mile, commensurate with the number of minutes in one Sacred Year of 360 days: 525,816 minutes.
Fig. 13 Sides converging to apex.
10/03/08© B.Pietsch / S.Thompson
“The Washington Monument: Icon by Design”
<bernard@sonic.net> Page 13 of 18
60.672 feet = .60672 100 .60672e = 1.8344047 1.8344047 = .0050956 360 3 .0050956 x 10 = 50.9 seconds of arc precession/year. VIII.
The form is contained by a geometric receptacle: the box it came in. The physically existing Washington Monument has an "essential" counterpart ideationally disclosed through geometry. This invisible or complementary form reveals information about the existential form not otherwise apparent or available. Just as a block of stone contains a sculpture before it is released by the sculptor, so too can the geometry of the shaft and pyramidion of the Washington Monument (exclusive of the capstone)14 be conceived as contained in an imaginary rectangular box. (Fig.13a) We call this imaginary solid "the box it came in." The square base of the monument (55.06' per side)15 is the bottom of the box, and the distance to the top of the pyramidion, minus the capstone16, its height. The volume of this rectangular container would be:
Fig. 13 b
Fig. 13 a
base2 x height = volume or 3,032.28 sq feet cubic feet
17
x 554.81 feet
18
= 1,682,363.274
The process of containment in a rectangular form can be carried further by reducing the rectangular volume to a perfect cube that would exactly encase the form’s volume. (Fig. 14) This process of “homogenizing” or “coercing” a form into a perfect cubic counterpart allows it to be compared to other cubes of different media – so that, for example, the densities of different materials can be correlated, as between gold, aluminum, water, jade, granite etc. 14
The aluminum capstone is categorically different than the rest of the monument. Conceptually, it is in the world as part of the monument, but it is not “of” the world nor of the same substance. The capstone informs from without. Note that the height of the shaft, 500 feet divided by 9 = the height of the pyramidian: 55.55 feet. 15
55.06616449'
16
See that by "tempering" the height of the capstone to 8.85708726", the perfect box "coerced" to a cube with 206' diagonal can be constructed as shown in the next exercise. 17 18
3,032.282471 sq. feet 554.8174649 feet
Fig. 14 Cube of Volume
the
10/03/08© B.Pietsch / S.Thompson
“The Washington Monument: Icon by Design”
<bernard@sonic.net> Page 14 of 18
28. The rectangular box (Fig. 13b) which would contain the shaft and pyramidion of the Washington Monument (minus capstone), would have a volume of 1,682,363.274 cubic feet. If the rectangular volume of 1,682,363.274 cubic feet (Fig. 13b) were reshaped into a perfect cube, that cube would have a major diagonal of 206 feet 19 -the same number as the 20.6 inches of the common cubit multiplied by 10. (Fig.15)
Fig.18 Angle of Great Pyramid: 51º 51’ Fig. 15 Cube with 206’ major diagonal
29. The 118.934-foot vertice edge of such a cube (Fig. 15) multiplied by 10 would be nearly the same number as the number of square feet in the area of the base of the Washington Monument’s pyramidion: 1189.34. The square root of that number being an indicator for the side of the pyramidion’s base: √1189.34 = 34.48680907 IX.
Iconic architecture often makes reference to other iconic works. Because the Washington Monument is modeled after the Egyptian obelisk, it is circumstantially linked to the tradition of ancient architects, and thus certain correspondences with the Egyptian system are insured. In the following analysis we find yet more homage to the Great Pyramid of Cheops. 30. Observation: A referent to the Great Pyramid is derived from the length of the apothem (face) of the pyramidion (Fig. 16) of the Washington Monument which is 58.17 feet.
Fig. 16 Length of Apothem 58.17’
58.17 x 100 = 5,817. Read as inches, 5,817 is coincidentally the geometric height of the completed Great Pyramid. 31. Observation: Another similar reference arises when the apothem angle of 17.24º, is extended from apex (peak) to the level of the base of the obelisk. ( Fig. 17) As an hypotenuse striking the ground, it will be 581.7 feet in length, again one tenth the number of the number of inches in the height of the Great Pyramid. 581.7 x 10 = 5817 32. Observation: The height of the entire Washington Monument is 555.555 feet, or 6,666.666 inches. If the height in inches is divided by the number of inches in the Canon Remen20 of 17.4 inches, we arrive at a height of 381.4814813 Remen. See that this figure divided by 10 19
206 is a germinal number generating other canon numbers: as 20.6 inches in the common cubit, as 206 courses in the Great Pyramid, 20.6 minutes difference per day between the solar year and sacred year of 360 days.
Fig. 17 Apothem angle extended 581.7 feet to ground.
10/03/08© B.Pietsch / S.Thompson
“The Washington Monument: Icon by Design”
<bernard@sonic.net> Page 15 of 18
and read as an angle becomes 38.1481 degrees. 38.1481 degrees is the complement of 51.8518587 º, the angle of the side of the Great Pyramid: 51º 51’ 06" .69 (Fig. 18) X.
Iconic art transcends time and culture. Monumental and iconic works of art awaken the recognition of the sacred within and are accessible to the perceiving individual regardless of his or her culture or epoch. The deepest expression of iconic form is perennial. Its intention can be read and understood through its essential geometry. More than an image, an icon suggests potentiality beyond the image it represents. As an icon of the nation, the Washington Monument points to a higher state of evolution, and ultimately to a “more perfect” union. We know from history that the founding fathers were committed to the establishment of a national system of weights and measures. Their dream was to bring into the world a system wherein all units assigned as standards would have relevant relationships to all other standards in the system--weight, temperature, distance, volume—all unified by congruent principles. It was a vision not realized but the intent was clear. The ideals of truth, justice, liberty and individuality were to be epitomized by the notion of commensurability which, in its highest form, is the understanding of harmony. As a public structure, the Washington Monument may ultimately prove of greatest benefit and utility as an instrument of learning. In the future, school children might come not only to experience the Washington Monument’s grandeur, but to be energized by the adventure of discovering its integrity – the service of each attribute to the whole. Working collaboratively with protractor, ruler, and calculator, the intellect and intuition of each student would investigate our national icon as form, metaphor and consummate demonstration of e pluribus unum. The preceding observations are a beginning. A great work of art, be it music or literature or architecture, continues to enlighten the inner landscape of those whom it engages, even as she or he evolves to deeper levels of awareness. The Washington Monument is just such a work–singular and articulate among the world’s most noble structures. The authors welcome comments, corrections and suggestions. You may contact Bernard by email at bernard@sonic.net or visit his web site “The Philosopher’s Stone” at 20
Our survey defines the Canon Remen perfectly at 17.47572816 inches. We have found numerous application and reference to this measure in Egyptian/Hebrew works.
10/03/08© B.Pietsch / S.Thompson
“The Washington Monument: Icon by Design”
<bernard@sonic.net> Page 16 of 18
www.sonic.net/bernard/. You can also watch a short video presentation of these ideas at www.youtube.com/bernardpietsch. Appendices follow.
The following chart lists the published dimensions of the site and the reconciled measures utilized for this treatment. Note: the decimal numbers are derived trigonometrically. 2/17/2006
Appendix A: Dimensions of Washington Monument Adopted figures
Published figures
Height of shaft 500’ 500' Height of pyramidion 55.555’ 55’ 5 1/8” Height of capstone 9” 8.75”-8.875”-8.9” Total Height of the Monument 555.555’ 555' 5 1/8" Av. width of base side of shaft 55.06616449’ 55' 1½" Perimeter of base of shaft (4 sides) 222’ Diagonal length of base of shaft 77.87531665’ Surface area of one side of shaft 22,357.10195 sq ‘ 22,354.22301 sq’ Total surface area of shaft (4 sides) 89,428.4078 sq ‘ 89,416.89203 sq’ Face angle of pyramidion (apothem) 17.2439725° 17° Length of pyramidion apothem 58.17024188’ Arris angle of pyramidion 23.69964257° Arris length of pyramidion 60.7065337’ 60.67235226’ Width of one base side of pyramidion 34.4881’ 34' 5½" – 34’6” Perimeter of base of pyramidion (4 sides) 137.9524065 ‘ 138’ Surface area of one side of pyramidion 1,003.090607 sq’ Surface area of pyramidion (4 sides+capstone) 4,012.362428 sq ‘ Width of one base side of capstone 5.605342418” 5.6" Perimeter of base of capstone (4 sides) 22.42136967” Surface area of one side of capstone 26.32524814 sq “ Surface area of capstone (4 sides) 105.3009926 sq “ Arris length of capstone 9.828913856 “ from 10 Apothem length of capstone 9.423583184” from 10
Total Surface area of the Monument
93,566.26612sq’
Volume of shaft 1,055,427.906 cu ‘ 1,002,495.822 cu’ Volume of pyramidion (includes capstone) 22,026.46581 cu’ 22,026.465 cu’ Volume of capstone 94.25959091 cu” Weight of capstone 100 oz. or 9 lbs.
Total Volume of the Monument
1,024,522.288 cu’
Surface area of pyramidion w/o capstone 4,011.631171sq’
10/03/08© B.Pietsch / S.Thompson
“The Washington Monument: Icon by Design”
<bernard@sonic.net> Page 18 of 18
Appendix B Some elements that distinguish iconic architecture from the mundane. The truly sacred in art and architecture is intangible. It is neither physical nor apprehended by the senses--it can only be indicated. It emerges into the world from a singular idea, encapsulating a grand view, incorporating a whole concept. Its meaning is conveyed through geometry and measure, material, location and time. In order to access the deeper levels of a work, participation from the observer is required. When understood correctly iconic art can be entered at any single point. It often references astronomical, metrological, and physical constants. It is accessible to future generations, regardless of culture or epoch. The intention of an iconic work can be reconstructed from a fragment of the original. The integrity of the work is characterized by redundant facets which fortify its intended meaning. It shares aspects with other monumental work in the world and has relationship to other monumental markers. Iconic forms may be contained or embedded in a larger geometric receptacle. Can be site specific and self-referencing via geographic/magnetic/astronomical frameworks. Iconic work often identifies itself in the context of historical time in some manner: aligning with or matching astronomical asterisks and events or biological growth patterns established over time e.g. tree rings, coral reefs, glacial deposits. The conclusion/origin of iconic design may be exterior to its physical architecture. Nothing is hidden or secret in iconic art—that which is revealed is always apparent when seen properly. The unified iconic field is guaranteed by the perennial emergence of a canon of measure wherein all units of time, distance, weight etc. are commensurate with one another. The fundamental unit is the canon inch. It is derived from and defined by its relationship to the second of time. Through the mechanism of the swinging pendulum, that unit of length which beats a matching unit of time demonstrates the convergence of categories in the domain of intuition "wherein all things may be considered." The iconic expresses itself elegantly—the intended referent is never far from the physical indicator. Iconic principles are expressed as geometric and numerical metaphors. Form is antecedent to physical expression. END.