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In Pursuit of Sacred Science, Part I
Architectural Survey of Borobudur's Summit
by Mark Long
Table of Contents:
In 1976, University of Michigan researchers announced the results of a scientific study which suggested that the architect of ancient Cambodia's Angkor Wat had encoded calendrical, historical and cosmological themes into his architectural plan for the temple. Published in the journal Science, the study also demonstrated how Angkor Wat's architect had established solar alignments between the temple and a nearby mountaintop shrine that took place during the summer solstice.
"Astronomically, it (Angkor Wat) has built-in positions for lunar and solar observation. The sun itself was so important to the builders of the temple that solar movement regulates the position of the bas-reliefs. It is not surprising that Angkor Wat integrates astronomy, the calendar, and religion since the priest-architects who constructed the temple conceived of all three as a unity. To the ancient Khmers, astronomy was known as the sacred science." (1)
In 1998, one of the authors of the aforementioned University of Michigan study published Angkor Wat: Space, Time and Kingship, which further defines the calendrical, historical and cosmological themes contained in the temple's architectural plan. According to former University of Michigan professor Eleanor Mannikka, the process that eventually led to her discovery of Angkor Wat's ruling unit of measure began with an intuitive deduction.
"On close inspection, I noticed that the temple's measurements were extraordinarily precise along certain sectors. As an example of this precision, both the northern and southern corridors of the third gallery are 202.14 m long. The eastern and western corridors are 114.22 and 114.24 m., respectively. Why--and how--would anyone construct the circumference in such a remarkably accurate manner?"
"To find out why Angkor Wat was constructed so precisely, I started to search for the unit of measure used to build the temple. That unit had to be a cubit length--the distance between the elbow and outstretched fingertips--since no viable alternative existed in Khmer inscriptions...."
Diagram 1 (above right): Architectural plan of Angkor Wat.
"A standard cubit in Cambodia would range roughly between .40 and .50 m. I used this range to divide axes and circumferences at Angkor Wat until finally, after four months of trial and error, a very precise unit of .43545 m yielded the most consistent results." (2)
By applying Angkor Wat’s cubit to various dimensions within the monument, Professor Mannikka was able to decode numerical themes that have a calendrical, astronomical and cosmological significance. In addition, she discovered that Angkor's architect often duplicated numerical themes that had already been expressed dimensionally through the grouping of nearby architectural elements, such as pillars, windows, and steps.
For example, the western entrance bridge that connects the outside world with the temple grounds consists of a 200-meter horizontal span that is divided into two halves by a set of staircases that lead down to the water of the moat that surrounds the entire temple complex. Each half of the bridge measured 216 cubits in length. In addition, the architect had installed a total of 216 sandstone columns and balustrade supports as part of the bridge's overall structure. (3)
The expression of calendrical, astronomical and cosmological themes within the dimensions of Angkor Wat raises an important question. Was this purely a Cambodian innovation or had Angkor Wat's architect learned of the practice from an even earlier source representing a tradition that was practiced elsewhere?
According to the account written in 916 CE by the Arab trader Abu Zaid, the Maharaja of Zabag (Java) once sailed his fleet up the Mekong River to the Khmer capital for the express purpose of capturing and decapitating the Cambodian ruler. This legend receives a modicum of historical support from several stone inscriptions found on the mainland of Southeast Asia. Discovered on the Malay peninsula in what is today the kingdom of Thailand, the Stone of Ligor commemorates the victories of the Rajadiraja (King of Kings) Vishnu, who is described as "resplendent like the Sun" and born of the "Sailendra"--a Sanskrit word that means "Lord of the Mountain." Historians ascribe the construction of several of central Java's Buddhist temples to the Sailendra dynasty of kings that ruled over this part of the island during the late eighth and early ninth centuries CE. (4)
Other stone inscriptions that have been discovered in what is today southern Vietnam--also refer to sea-faring invaders out of islands who had conducted raids along the coastline during the latter half of the eighth century, with an inscription dating from 787 CE specifically referring to an invading army of Javanese origin. (5)
In 802 CE, the founding king of Cambodia's Angkor civilization participated in a ceremony that involved the installation of a linga--the phallic emblem of the Hindu deity Shiva--on top of Mount Kulen north of Angkor. This inaugurating act of King Jayavarman II, which is commemorated in the Khmer inscription of Sdok Kak Thom, states that "...a Brahman...well versed in magic, came...at the invitation of the king to establish a ritual in order that Cambodia might no longer be dependent on Java, and that there might only be one king ruling the country. This Brahman recited the texts from beginning to end, to teach them to the (king's) chaplain, and he instructed him how to initiate the ritual of the deva-raja (literally "god-king"). Each succeeding Khmer ruler during the Angkor period of Cambodian history followed Jayavarman II's example by building a sanctuary for the cult of the deva-raja in the form of a pyramid-mountain, either natural or artificial, that was located at the very center of the Khmer ruler's realm. (6)
Photo 1 (above): Bakhong temple near the modern city of Roulous in north-central Cambodia.
Despite the Khmer King's
commission of this symbolic declaration of
Did the extent of Javanese influence on Khmer architecture also extend to the practice of encoding calendrical, historical and cosmological themes into the dimensions of their temple designs? If so, can these practices still be discerned today in the form of the Javanese Buddhist temple of Borobudur?
2 (right): Bakhong temple architectural plan.
Following the collapse of the East Javanese kingdom of Mahapajit in the fifteen century CE, members of the Hindu royal court fled to the nearby island of Bali, carrying with them a number of palm-leaf manuscripts governing architecture. Ever since, the temple architects of Bali have continued to consult the texts of the Asta Kosala Kosali, which set forth the principles for deriving measurement units from various dimensions to be found in the human body, which are then used as units of measure for laying out various architectural dimensions. (8)
The Balinese believe that the architectural plans for temples and family compounds play a direct role in determining the fate of each structure's occupants. The goal of the Balinese architect is to harmonize the microcosmic forces that govern life in the human world with the macrocosmic forces that rule over the realm of the gods. According to this point of view, the extent to which any building fails to conform with an architectural plan designed to mirror the perfection of the cosmos can make the difference between a "living" structure that will allow benevolent spiritual forces to enter and take up residence and a structure that is considered to be blocked, closed off or "dead."
"Other frames of reference also operate, including the metaphorical representation of the compound and its various structures in terms of the human body. Thus, the family shrine is identified with the head; the sleeping quarters and pavilion for receiving guests with the arms; the central courtyard with the navel; the hearth with the sexual organs; the kitchen and granary, with legs and feet; and the refuse pit in the backyard, with the anus." (9)
In deriving the dimensions of a Balinese household compound, the Balinese architect directs the owner of the household to pace off the distances between the various structures to be constructed within the compound in units consisting of eight foot lengths a number that has a symbolic significance with regards to the eight points of the compass as well as the specific Hindu deities associated with each of these directions. (10)
Photo 2 (below right): Javanese carving of a lotus on the back of a tortoise, with each of the petals bearing the emblem of a Hindu divinity associated with one of the eight directions of space. National Museum - Jakarta.
The Javanese practices inherited by the Balinese were based on even earlier architectural principles developed in India, under which the temple's patron was responsible for appointing a priest to preside over the entire temple construction process. Called the Sthapaka, this priest was responsible for guiding the activities of the temple's chief architect, called the Sthapati, who had to be well-versed in all the traditional sciences, including mathematics.
The Sthapati’s primary disciple, often his son, was called the Sutragrahin. Charged with the responsibility of carrying out the Sthapati's orders, the Sutragrahin had to know the proportionate measurement by both cord (sutra) and rod (danda) as it applied to the entire building as well as its various parts. (11)
Although Borobudur's construction period lasted for more than fifty years and consisted of several distinct phases, the rules of Indian architecture required that the builders maintain continuity with regards to certain basic principles.
“The temple or any other (construction) begun by these two should be continued by them only and by no other. In case they should be not available, the work should be done by either their sons or disciples who are competent in the work.” (12)
The Faculty of Engineering at Indonesia's Gadjah Mada University believes that the temple architects of central Java used a basic unit of measure called the tala, which is defined as the length of a human face from the top of the forehead’s hairline to the tip of the chin. The tala is also equal to the distance from the tip of the thumb to the tip of the middle finger when the palm and thumb are at their maximum distance from each other. (13)
Photo 2 (above right): Bust of a Javanese monk found in the vicinity of the Buddhist temple of Candi Sewu, which is located some 35 kilometers to Borobudur's southwest. National Museum - Jakarta.
The tala measurement system originated in India, where it was more often employed by sculptors when determining the proportions for the statues of deities that they had been commissioned to carve. According to the Agni Purana, the sculptor used the tala to define the crown of the statue’s head as well as the regions of the face, neck and heart, the space between the two breasts, and the circumference of the forepart of the arm. In addition, the distance between the navel and the genitals equaled one tala, while the length of the thighs was equivalent to two talas. The tala in turn consisted of twelve smaller divisions called angulas (the width of a finger), which the artist employed in the laying out of smaller dimensions such as the width of the ears and mouth or the length of the bridge of the nose. (14)
Since the dimensions of the human body vary widely from one person to the next, Indonesian archaeologists have limited their examination of the dimensions for temples to the ratios that exist between building components. Without denying the above, the possibility remains that several temples could all have been based on the same measurement unit if they had all been founded under the patronage of a single person. In addition, it is even possible that a single royal dynasty might have elected to arrive at a standardized measurement unit based on a dimension derived from the body of an illustrious ancestor.
If just such a unit of measure could be determined, then the information that could be derived from this knowledge would give archaeologists an additional tool for discerning the architect's original intentions. This is what this writer set out to do in 1998.
My initial survey of the literature that describes Borobudur was disappointing. The reported dimensions, which varied widely from one text source to the next, were only rough approximations that typically had been rounded off to the nearest quarter of a meter. In addition, the dimension of certain components of the structure have, to my knowledge, never been disclosed in print. For example, each of the three terrace platforms at Borobudur's summit has a slightly oval shape that consists of two minor axes aligned with the cardinal points of the compass and two major axes aligned with the intermediate directions. None of the available sources provides the dimension of these platforms in the intermediate directions. With regard to the platform axes in the cardinal directions, A. J. Bernet Kempers has reported them as follows:
"Three circular terraces with a radius of 25.60 (D=51.2m), 19.2 (D=38.4m), and 13.40 (D=26.80) m, respectively, support three rows of stupas (diam. at the base 3.40-3.80, height 3.50-3.75m) with perforated walls and a space inside which contains a Buddha image." (15)
In 1998, I conducted my own ad hoc survey of Borobudur's summit--no simple task in a monument that receives more than one million visitors each and every year! After rising with the dawn, I would scamper up the monument's eastern staircase to beat the first wave of tourists to the top. The resulting measurements are provided in Figure 1 below, where the outermost figures in the cardinal directions represent the distances between the walls that surround the summit.
Figure 1: Summit dimensions according to the author's own survey (in meters).
Beginning in the 1970s, the Indonesian government, in cooperation with UNESCO, undertook a major renewal of the entire monument in order to prevent its further deterioration. In advance of the reconstruction, the length of each wall segment for Borobudur's four lower rectangular galleries were accurately measured and reported in Reconstruction Committee documents. However, the Reconstruction Committee's survey did extend to reporting the actual length of the monument's two axes in the cardinal directions.
To gather this supplementary information, this writer moved down each of the four axes, pausing to measure and record the distances between the various levels of the monument, from the outer rim of the main stupa platform at the monument’s summit all the way down to the tip of each makara nose on each of the monument's terminating staircases at ground level. In addition, I measured the length of the monument based on both the west and north sides of the monument in order to estimate the base perimeter, which is no longer in its original condition on the east and south side of the monument.
This particular report therefore represents the very first time that all of Borobudur's major dimensions have been reported within the confines of a single document. However, this is not to say that additional work is not needed. For example, a more detailed survey of the summit in which the individual circumference and height of each stupa is calculated would be most welcome, presented together with the spacing between the stupas as well as the precise position of their central axes with respect to the platforms upon which they rest.
This writer obtained the first potential clue for arriving at the precise length of Borobudur's governing measurement unit upon reading a casual observation of Dr. Kempers, who had noticed that the majority of Borobudur's stone masonry blocks were between 22 and 23 centimeters in height. (16) It is possible that Kempers was inspired to make this observation by Jan Rombout van Blom, who had observed more than 50 years earlier that the three temples of Borobudur, Pawon and Mendut were all constructed using stone blocks of about the same size.
Sadjiwan (Van Blom 1935:41) "...one finds stone layers with a
cross-section of 27 - 40cm, whereas the average height at Borobudur,
Mendut, and Pawon only amounts to 23cm."
According to the Borobudur Reconstruction Committee's survey of the lower gallery levels, the combined length of the fourth gallery walls that surround the monument's summit is 247.33 m. Presuming that this perimeter must have had a major symbolic significance, I examined possible units of measure between 22 and 23 cm in length that might enhance the symbolism of the structure itself. When I employed a "tala" of 22.9 cm as the defining unit, the perimeter equaled 1,080 talas. (17)
The composers of the Hindu scriptures often used the number 108 and its multiples as the basis for defining various time cycles. When the author later applied a tala of precisely 22.9 cm to other dimensions within the monument, other numbers are produced that play distinct astronomical and cosmological roles in the sacred sciences of India. The tala of 22.9 cm also happens to be a close approximation of one-half of the 43.545-cm “cubit” that Mannikka had previously determined was the measurement unit employed by Angkor Wat's architect. According to the temple building traditions of India, there is a class of temple that does indeed call for just such a half-cubit measurement system. (18)
In addition, when a tala of 22.9 cm in length was applied to the dimensions of the Buddhist temples of Candi Pawon and Candi Mendut located in Borobudur's general vicinity, similar calendrical themes became evident. (The relevant measurement results will be reported in part IV of this article.)
The number 108 appears in material form throughout Borobudur. Its multiple 10,800 represents the number of muhurtas (the 48-minute Hindu "hour") of the calendar year. The Satapatha Brahmana provides detailed instructions governing the construction of a sacrificial altar consisting of 10,800 bricks. The altar is considered to be the body of the Lord of Creation, Prajapati, “whose body is the year.” His presence in the altar is indicated by a golden effigy that is installed in the bottom of the altar on top of a golden plate that symbolizes the Sun, which rests upon a lotus leaf. This golden effigy of this "Man in the Sun" faces the sky, with its head in the east and its feet in the west.
The Menoreh ("Tower") ridge to the south of Borobudur presents the shape of an anthropomorphic figure that--like the golden effigy of the Satapatha Brahmana--faces the sky with the "head" in the east and the "feet" in the west. According to local legend, this is the body of Borobudur’s legendary architect Gunadharma, who has elected to remain in the area in order to keep watch over his creation. Since Gunadharma is a pure Sanskrit name, the Dutch scholar N.J. Krom thought that this local legend might actually be based on some historical figure. Javanese folk tales typically present figures that bear the names of local, rather than Indian characters.
Gunadharma legend may represent the forging of a link between
Borobudur's architect and his macrocosmic form, for it is Prajapati who
"...bears the measuring rod, knows division and thinks himself composed
of parts." The measuring rod symbolizes his role as the architect
of the universe, Visvakarman, who is the celestial model for the human
temple architect who is Prajapati's microcosmic manifestation and
representative on Earth. (19)
Photo 4 (above right): The "head" and "torso" of Gunadharma face the sky in the form of the Menoreh ridge to Borobudur's south.
The Purusha is mentioned in King Sanjaya's Gunung Wukir inscription of 732 CE, which calls the island of Java the footprint of the Purusha. Perhaps the natives regarded the anthropomorphic shape of the Menoreh hills to be yet another signature of the Purusha's local presence, one which clearly marked the site of Borobudur as a spot from which an ascent to heaven could be make.
In the later literature of India, Prajapati is the inheritor of the persona that earlier Vedic legends had assigned to the primordial giant called the "Purusha," whose undifferentiated body was said to initially fill the entire universe. In the Rig Veda, the Hindu gods conduct a primordial sacrifice during which they divide the Purusha's body into parts that become the discrete components of material existence.
In the Satapatha Brahmana, the building of the altar is a ritual process that reverses the division of material existence into discrete parts to provide the Prajapati/Purusha, in the form of the sacrificer who symbolically assumes his identity, to make a re-ascent to heaven to assume his primordial, undifferentiated state. This symbolic re-ascent to heaven would later be incorporated into the rituals surrounding the rituals surrounding the construction of the Hindu temple, within which the patron assumes the mantle of the Prajapati/Purusha and the temple foundation becomes the architectural body through which one undertakes a re-ascent to heaven to assume the "immortal" life. (20)
This belief in one's making of a re-ascent to heaven is reflected in the Javanese inscriptions, which at times have compared the ancestor-spirits of the kings of old to demi-gods "rushing along the firmament." They also warn those who might be tempted to interfere with the well-being of a religious foundation that they risk being thrown off the firmament. So let it not be said that you weren't warned to be courteous and respectful when visiting Java's cosmic pyramid-mountain! :-)
Figure 2: Borobudur's "Head, Body and Foot" in the Vertical Domain according to Professor Parmono Atmadi.
Figure 3: According to Parmono Atmadi, each of the three basic divisions of the temple also has its complementary segments of head, body and foot. The figure above is based on Professor Atmadi's analysis of Borobudur's "head."
The temple architects once employed ratio formulas to subdivide the temple's architectural plan into three major divisions that were linked with the "foot, body and head" of a human body. Modern recognition of this ancient practice has actually provided Indonesian archaeologists with valuable assistance in restoring several of the island's temples to their original form. This correspondence between temple and body is reflected in a famous poem by the Indian poet Basavanna:
"The rich will make temples for Shiva.
My legs are pillars,
Listen O Lord of the meeting rivers,
In 1977, Parmono Atmadi conducted a survey of Borobudur's dimensions for the express purpose of determining the ratio formula that the architect had used to lay out the dimensions of Borobudur's foot, body and head. The ultimate goal of this exercise was to determine the original height of Borobudur's main stupa pinnacle above ground level. (21)
To accomplish this task, Atmadi measured a large number of the monument's architectural elements in the vertical domain. From the resulting data, he was able to determine that a ratio formula of 4:6:9 had been employed by Borobudur's architect (See Figure 2 above). In addition, Atmadi reported that the identical ratio formula was employed at the nearby Buddhist temples of Candi Pawon and Candi Mendut.
Atmadi's study also states that the ratio formula of 4:6:9 had been used in the layout of each of the monument's three major subdivisions; i.e., the monument's "head" also incorporates three subdivisions that we might call the minor head, body and foot of the monument's entire head. This discovery allowed Atmadi to calculate that Borobudur's original height must have reached 41.81 m above ground level (see Figure 3 above). Using a tala of 22.9 cm, this dimension equals 182.576 talas, a number that closely approximates one half of a solar year of 365.25 days. (22)
The horizontal dimensions of the summit's second round terrace platform in the intermediate directions equals 41.8m (NE/SW) and 41.86m (NW/SE), respectively, which again present equivalent dimensions in talas that evoke the duration of one-half of a solar year. In addition, the mid-point of the entire monument based on Atmadi's height estimate for Borobudur, also corresponds with the mid-point in the vertical domain of the summit's second terrace platform.
In Hindu cosmology, a "day of the gods" consists of the half-year period from the vernal to the autumnal equinoxes, when the duration of daylight exceeds the daily period of darkness. This time period is also called the "northern progress" (uttarayana) of the sun because it is during this particular period of the solar year that the position of the rising sun on the eastern horizon progressively moves northward from one day to the next.
conventional method for laying out the temple in conformance with the
annual movements of the celestial realm is in terms of a square diagram
called the vastupurusha-mandala. In my article on the Essence of Buddhahood, I have described how
Borobudur's summit conforms with the essential characteristics of the
vastupurusha mandala, which is not only the residence of the
Prajapati/Purusha on earth but also the schematic for all Hindu temple
foundations within which the equinoxes and solstices are represented by
the four faces and four corners of the diagram. A Mahayana Buddhist
text discovered on Sri Lanka called the Mañjusri
Vastuvidyasastra makes use of the same types of diagrams are are
described in Hindu architectural texts such as the Mahamatam, but calls
these plans "candita" -- a word that may have serves as the point of
origin for the Javanese term candi.
On the second terrace platform at Borobudur's summit, the sum of the two axes in the intermediate directions equals 365.3 talas (t), nearly equaling the duration of one sidereal year of 365.2562 days. Therefore the combined dimensions of what might best be called the "solar core" at the center of the entire monument also represents both halves of the solar year. This core area resides at the center of the monument, surrounded by the 32 stupas of the first terrace platform; likewise the core of the abode of the Devas on Mount Meru is surrounded by 32 divinities according to Hindu cosmology.
After conducting my own survey of the monument's dimensions, it became evident that Borobudur also conforms with the same 4:6:9 ratio formula in the horizontal domain (Figure 4 below). The diagram presented below shows how the architect had expanded a vastupurusha mandala of 9 x 9 squares by gnomonic projection to include the entire temple within a larger diagram that consists of 19 x 19 squares (Figure 5 below). In other words, the central square at the center of the monument is only surrounded by 360 squares to symbolize the 360-degrees of the celestial circle that surrounds the Earth but the outermost tier of squares also numbers 72 in total. This particular design has been echoed in a Buddhist architectural text from Tibet, which delineates a procedure for mapping out the temple site within a square diagram that has been divided into four equal quadrants, each of which contains 90 blocks. (23)
On the monument's lower rectangular terrace levels, the architect has observed the principles of Indian architecture by avoiding the placement of structures on top of the of the intersections of the horizontal and vertical grid lines that define the overall structure of the plan. According to Indian architecture, the lines represent the immanent breaths (prana) of the temple and their intersections are the deemed the plan's vulnerable points. In particular, we can see how Borobudur's has avoided placing any structures directly over the "diamond" intersections formed by the main diagonal lines of the plan.
During the last reconstruction efforts, archaeologists noticed that the walls of the first gallery had an inclination of more than 1 degree. Assuming this to be an architectural defect, they attempted to re-set the walls vertically but found that this would not allow the corners to meet. As this small inclination is now regarded to have been intentional we can suggest a plausible reason for its presence in the overall design. By the same token, the architect's avoidance of the plan's "points of vulnerability" at the lower levels of the monument may explain why walls have been "rotated" from true north/south or east/west alignments.
Up on the summit of the monument, the architect was unable to avoid the placement of stupas over the intersections of the plan's lines of breath, but in this case it did not matter. The perforation holes in the 72 stupas provide the means whereby the vastupurusha mandala's lines of prana retain their unimpeded access to the remainder of the monument. The stupa perforations are equivalent to the perforation holes in the bricks that the builder of the Vedic fire altar lays down over the image of the "Golden Man." According to the Satapatha Brahmana, these naturally perforated bricks are the breaths of the Purusha whose image is buried at the very bottom of the entire structure.
Figure 4: Borobudur's North/South axis in the horizontal domain conforms to the same 4:6:9 ratio used in the monument's layout in the vertical domain.
Figure 5: Borobudur's corresponding layout in terms of a 19 x 19 grid of squares.
The proportions that Borobudur's architect used in laying out the head, body and foot of his architectural plan also compare to proportion ratios that were later incorporated into the certain mandala traditions of Tibetan Buddhism, where the tripartite design of the kalachakra mandala, for example, represents the body, speech and mind of the Buddha. This does not necessarily mean that Borobudur's foot, body and head were also meant to represent the "three jewels" of the Buddha or the derivatives called the Buddha's three mysteries in certain Vajrayana Buddhist texts. However, the illustration presented below does bring to mind a saying of the Buddha Vairaja from the Lankavatara Sutra:
"Within the body, measuring one vyana...the measure of two extended arms...there is a world; the cause of its rising, the attaining of cessation, and the path (pratipad)--this I teach to sons of the Victor (i.e., to the Bodhisattva sons)."
Figure 6A: Borobudur's mandala with an overlay representing the proportion system that was incorporated into the later mandala traditions of Tibet.
It is also entirely possible that Borobudur's 4:6:9 ratio formula and its resulting 19 x 19 grid of squares was based on mystic mathematical diagrams that were well known in ancient China, where they were used in the design of buildings and even entire cities. They also were used in India as part of tantric practices involving geometrical magic diagrams called yantras. In the first set of examples presented below, the sum of each of the columns, rows and diagonals three squares equals 15. Each of the four sides of the square also represents one of the four elements of earth, air, water and fire.
Figure 6B, 6C and 6D: The rhythmic expansion of the nine (3x3) squares of the vastupurusha mandala occupied by Borobudur's main stupa (3^2 = 9; 9^2 = 81; 15^2 = 225; 19^2 = 361). (For a detailed description of the vastupurusha mandala and its parallels on Borobudur's summit, see the article Essence of Tathagatahood, Part Three.)
The ruling cipher of the first set of examples presented above is the numeral 5 at the center of each diagram, which enters into each and every calculation. Here the combined sum of the three rows and columns is 90, or 120 when the diagonals are included. With regards to Borobudur, the 3x expansion of the central 9 squares of the diagrams presented above, produces the sum of 3 x 120 or 360--which is also the total number grid squares that surround the central square of the entire diagram.
Moving on to the second set of the examples presented below, the ruling number is 6 and the combined sum of the three rows and three columns is 108 (18 x 6), or 144 when the two diagonals are included. At Borobudur, the 3x expansion of the central 9 squares of the second diagram presented below, produces the sum of 3 x 144 or 432--which is also the number of Buddha statues that were installed beween the summit and Borobudur's outermost walled perimeter.
There are many other possibilities inherent in the two sets of examples provided. I will leave it up to each reader to delve as deeply as he or she wishes.
Even if the mathematical yantras cited above were not expressly intended to play a numerical role in the monument's design, the rhythmic expansion of the brahmastana--the nine central squares of the vastupurusha mandala at the summit--is reflected in the major divisions to be found within the overall architectural plan.
Figure 6E and 6F: In ancient China, the Lo-shu number diagram centered around the number 5 was intended to represent the divisions of the earth. Its celestial counterpart, in which the number six is the ruling cipher that enters into every calculation, is presented in the diagrams below. (For further details on how these diagrams were incorporated into building designs in China, see Architecture, Time and Eternity, Volume II by Adrian Snodgrass.)
Borobudur’s summit is crowned by a stone platform that supports the large stupa located over the monument’s central axis. This structure is the only round terrace platform that presents a truly circular form. The other three terrace platforms at the summit have two major axes in the intermediate directions as well as two minor axes that are aligned with the cardinal points.
Photo 4: A statue of the Buddha Vairocana with Borobudur's main stupa in the background.
To derive the diameter of the main stupa platform, the author first measured the structure's circumference and then divided the results by pi. The circumference of the main stupa platform as measured at its mid-point in the vertical domain equals 50.83m or 222.0t. The equivalent diameter was calculated by divided the main stupa platform's circumference by pi, which equals 16.18m or 70.66t. What is particularly significant here is that the diameter of the platform also equals the averaged dimension of the monument's architectural "body" in the vertical domain (See Figure 1 above) according to Professor Atmadi's pioneering study of Borobudur's defining ratio formula.
The total length of Borobudur's horizontal axis in the North/South direction (between the tips of the makara snouts on the ground-level north and south staircases) equals 123.13m. When this dimension is divided by the diameter of the main stupa platform, the ratio of 7.61:1 is produced, which comes very close to equaling the decimal equivalent of Borobudur's southern latitude of 7 degrees, 36 minutes, 29 seconds (7.608).
The length of Borobudur's axis in the East/West direction-- i.e., between the tips of the makara snouts on the ground-level east and west staircases--is considerably shorter, equaling just 122.3m. At the location of 7.608 degrees south latitude, the circumference of the Earth is indeed slightly shorter than the polar circumference of the globe. In this respect, Borobudur conforms precisely with its geographical location on the sphere of the Earth.
We should not be surprised if Borobudur's architect intended for all this information to be embedded into his architectural representation of the universe in microcosm. The sages of India had known that the Earth was a sphere long before Borobudur was constructed. In addition, the ancient Hindu astronomy textbooks taught the means whereby the latitude of any location could be determined through calculations based on the length of the shadow cast by a vertical stake or "gnomon."
In Angkor Wat: Space, Time and Kingship, Eleanor Mannikka explains how the architect of Angkor Wat employed a similar method for indicating the temple's location in terms of latitude.
"The latitude unit was first derived from the central sanctuary of Angkor Wat. The sanctuary has many coordinates that place it in relation to the earth, conceived as a globe floating in space. One of these coordinates is the north-south axis between doorways: 13.43 cubits. Since latitude is one's position on a north-south meridian, the north-south measurement is appropriate in orientation as well. If this unit were confined to the central sanctuary it would be interesting in its own right. But it turns out that 13.43 cubits was a construction module in the prieu cruciform and second gallery as well. It is close enough to 13.41 to be acceptable as the temple's latitude, and so I have called it the latitude unit." (24)
When the distance between the apex of Borobudur's summit and Candi Pawon is divided by the diameter of the main stupa platform (1750 m/16.18 m) the ratio of 108:1 is produced. This particular dimension has also been embedded into Borobudur itself, where it equals the distance of the six circuits of worshippers (1750.06m) around Borobudur's second, third and fourth galleries (x2 each), a prerequisite for "reading" all of the reliefs in this particular part of the monument in their correct order.
According to a study by Jacques Dumarcay, Borobudur's three terrace platforms at the summit were probably added during the third stage of the monument's construction, at which time the architect also elected to expand the monument's rectangular terrace levels through the addition of the walled perimeter, rounded cornice and broad base that surrounds the entire monument. (25) Based on the author's own measurements, Borobudur's exterior walls extend outward from the first gallery at the gateway points by 4.03m to the North, 3.99m to the South, 4.15m to the East, and 4.02m to the West, or 16.19m in all, which once again suggests the possibility of a connection with both the monument's "body" in the vertical domain (16.18m) as well as the diameter of the main stupa platform. In addition, the length of the eastern staircase, from the staircase at promenade level to the base of the first round terrace platform at the summit, equals 1/2 of the main stupa platform's circumference (25.44 m or 111 tala). In addition, the total length of Borobudur's eastern staircase--from the very first step up to the promenade level to the base of the first round terrace platform at the summit--equals 111 t or 1/2 of the main stupa platform circumference (25.44 m).
However, it must be pointed out that this writer's decision to apply the main stupa platform's circumference at its vertical mid-point is entirely arbitrary for there are no signs in the platform's design to indicate a favoring of vertical mid-point over vertical top (16.16m/70.57t) or bottom (16.225m/70.85t). At various junctures later on in this article series, it shall be demonstrated how the use of the bottom circumference dimension can be employed to provide potentially significant results).
Figure 7: The dimensions of the monument's summit form the "cross of the four seasons" of the solar year in the cardinal directions (364 talas) as well as present a total of 505 talas in the intermediate directions, which may bear a relationship to the 504 Buddha images that were incorporated into Borobudur's original plan, together with the entity represented by the main stupa itself.
From the bottom edge of the main stupa platform, the author measured outward along the four axes of the cardinal directions until each of the measurements terminated at the edge of the inside-facing wall that surrounds the summit area. The sum of these four dimensions, which were presented above in Figure 7, equals 83.41m or 364.24 tala.
At Borobudur, the four spokes of the summit area in the cardinal directions suggest the four quadrants of a 365.24-day solar year, with perhaps the main stupa providing the final unit required for completing the calendrical symbolism. Alternatively, the architect may have intended for the four quadrants to represent the duration of one sidereal year, according to the observational practices of astronomy.
The sidereal day, which consists of the time interval between two successive passages of a star over the same meridian of longitude, is nearly four minutes (3 minutes 56.555 seconds) shorter than the apparent solar day--that is, the amount of time between two successive transits of the Sun over the same meridian. Over the course of the solar year, this apparently insignificant time difference accumulates to just over 1,460 minutes, equivalent to 24.3333 hours or 1.0138 days. For this reason, the four seasonal axes that radiate outward from Borobudur's main stupa platform could also represent the solar year in terms of sidereal solar time as opposed to the apparent solar time. In addition, there is a possible relationship with the cycle of Jupiter, which according to the Suryasiddhanta completes 364,221 revolutions in a Great Age or "kalpa" of 4,320,000 terrestrial years.
The sum of the round terrace cross-axes in the intermediate directions, which terminate at the four corners of the summit area, equals 505.28t (See Figure 7 above). Borobudur’s architectural plan originally called for the placement of 368 Buddha statues in the niches of the lower galleries, together with the 64 statues in niches of the fourth gallery and the 72 statues within the 72 stupas that grace the monument’s summit, or 504 statues in all.
The dimensions of the summit’s three round terrace platforms in the intermediate directions are also significant in several other respects (See Table 1 below). For example, the sum of all six round terrace axes equals 1,079.2t, yet another possible correlation with the Hindu sacred number 1,080, which the architect also used to define the fourth gallery perimeter. In addition, multiples of 364 are generated when the axial dimensions of the summit’s three round terrace platforms are multiplied together. For example, the product of the four NE/SW cross axes are approximately 364 million talas, while the product of the four NW/SE cross axes is almost 364.5 million talas (See Table 2 below).
Table 1: Sum of Round Terrace Axes (Intermediate Directions)
Round platform Cross-Axis (m/tala)
1st RT - NW/SE:
54.73 m 239.00 t
Subtotal NW/SE: 123.63 m = 539.9 t
1st RT - NE/SW:
54.60 m 238.42 t
Subtotal NE/SW 123.51 m = 539.35 t
TOTALS 247.14 m = 1,079.21 t
Table 2: Product of the Cross-Axes of the Round Terrace Platforms (Intermediate Directions)
NE/SW Axes: (in m/tala) (product)
MS RT 16.18 m 70.66 t ----
3rd RT - 27.11
m 118.38 t = 8364.73
NW/SE Axes: (in m/tala) (product)
RT 16.18 m 70.66 t ----
The average of the two axes equals 364,255,074.8
Atmadi’s estimate of the monument’s original height above ground level is echoed in the dimensions of the second round terrace platform in the intermediate directions: 182.53t from NE/SW and 182.79t from NW to SE, or 365.32t in total. In addition, the monument's pivotal mid-point in the vertical domain coincides with the vertical mid-point of the second round terrace platform. The coordinates appear to delineate the boundaries for an invisible geometric sphere located at the very core of the monument that is defined by dimensions that are related to the duration of a solar year. (26)
Figure 8: Round Terrace Platforms, North-South Direction
Figure 9: Round Terrace Platforms, East/West Direction
Figure 10: Intermediate Direction NE/SW
Figure 11: Intermediate Direction: NW/SE
Figure 11: A complete measurement survey of Borobudur's summit.
(3) Ibid., p. 31.
(10) “Eight is an important number to the Hindu-Balinese, representing the four cardinal directions and the four intercardinal directions. One god is assigned to each of these eight, plus Siwa in the center, completing the so-called nawa sanga, the compass rose symbolism.” Eiseman (1990):196 [vol. II].
(13) Atmade, Parmono. Some Architectural Design Principles of Temples of Java. Yogyakarta: Gadjah Mada University Press (1988):182-183. The Javanese may also have associated the tala with the sole of a human foot, a distinction of particular significance with respect to the ritual procedures still employed by the Balinese architect. According to an Old Javanese religious text called the Bhuvana-Samksepa, “Tala is the sole of the foot.” See Chandra, Lokesh. Cultural Horizons of India. New Delhi: Aditya Prakashan (1997):142 [vol. V].
(15) Bernet Kempers, A.J. Ageless Borobudur. Servire: Wassenaar (1976):43. The slight variations that exist between the figures presented by Kempers and those provided by the author in Figure 1 below can easily be accounted for. The vertical walls of the main stupa platform curve inward slightly. If the figures provided by Bernet Kempers had been derived by measuring the circumference of the top of the main stupa platform rather than at the platform's bottom, this would generate a difference of nearly 7 cm in all subsequent platform diameter measurements, which appears to be just what has happened. The figures also appear to have been rounded off to the nearest whole measurement unit. In addition, Bernet Kempers' statement does not take into account the slight differences in dimensions between the N/S and E/W axes of the three round terrace platforms.
(16) Ibid., p. 59.
(17) Dumarcay, Jacques. "Report on Measurements and Dimensions: Remarks on the Drawing of the After Reconstruction Situation of Borobudur" Borobudur Reconstruction Committee document Pelita CC/III7 (1974):225 - 232.
(18) According to a text on Hindu temple architecture called the Manasara, a class of one-story buildings called Vikalpa should be defined in terms of half-cubit measurement units (see Admadi 1988:324-325). The Vikalpa is a sub-category of the South Indian Jati Vimana (from "vi ma," which literally means "to measure out") class of temples, "...which represent each a collection of various classes, namely the storeyed temples, on the head of which is placed a small shrine plus a rampart of chapels surrounding each story" (see Kramrisch 1946:292-293).
(20) Kramrisch 1946:74.
(22) In addition, 182.576 talas comes very close to equaling the square root of 33,333 (182.573).
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