The Geometrical Relationship Between Ancient Hindu Technical Treatises and the Planning and Organization of Angkor Wat

Abstract

Angkor Wat is the supreme masterpiece of Khmer architecture, built by King Sūryavarman II during the 12th century A.D. Jane Przyluski hypothesized that Angkor Wat was the tomb of King Sūryavarman II. On the other hand, George Cœdès thought that Angkor Wat complex was habitation in the form of a celestial palace. According to Henri Parmentier, though the buildings and constructions in Angkor Wat temple complex are majestic, they are geometrically out of place. The temple complex is non-symmetrical, as the complex’s center is left-aligned. The above controversial opinions inspire a deep examination of the geometric system of the architectural and structural design of Angkor Wat. This research investigates the architectural planning and frame structures of Angkor Wat stone temple complex using a Hindu grid system. The study was based on field survey data of the temple complex and Hindu ancient texts, specifically the Vāstu Śāstra. PhotoModeler Pro5 and Polycam for iOS-4.0.5 were utilized to render three-dimensional (3D) images of the entire temple complex. The analysis finds the geometric code (suitable module) used in the planning of 2.75 m × 2.75 m in the metric system (1 Phyeam 1 Hat 1 Thnob in) the local Cambodian measuring system). The geometric code (2.75 m × 2.75 m) highlights the design diagram and construction of the temple complex. The research also unveiled the use of a center-shifting technique where the vertical axis running through the center is deliberately left-aligned, to avoid numerical fractions occurring in the grid modules. The technique gives rise to the asymmetry of the temple complex. The findings led to understanding the symbolic meaning of spatial organization of the layout and plan of Angkor Wat design, which was meant to be a suitable residence for the god on earth, the king, and his citizens. Moreover, it also means the final abode of King Sūryavarman II after his death, represented by the image of Lord Viṣṇu.

1. Introduction

Angkor Wat is an ancient temple complex situated in Cambodia’s Siem Reap province covering an area of 162.6 hectares [1]. Angkor Wat was constructed in the 12th CE by King Sūryavarman II [2] of the ancient Khmer Empire. It is regarded as the supreme masterpiece of Khmer architecture [3,4].

The layout of Angkor Wat stone temple complex consists of seven layers of enclosure enclosing the central tower. The seven enclosures include inner, middle, outer galleries, and lower raised ground (1st, 2nd, 3rd, and 4th enclosures) as in the aerial view in Figure 1 and Figure 2, stone wall (fifth enclosure), inner bank moat (sixth enclosure) and outer bank moat (7th enclosure). The first square enclosure is made of stone bearing wall and on the highest raised base enclosing the central tower. The top of the stone bearing walls of the innermost (1st) enclosure is covered by stone corbeled arch, with an entrance on each side of the wall and a smaller tower at each corner.

The second rectangular enclosure (middle gallery), surrounding the first enclosure, is on a raised floor of lower height, with an entrance on each side of the wall and a smaller tower at each corner. The third rectangular enclosure (outer enclosure), surrounding the second enclosure, is on a still lower raised ground, with four sides of the wall featuring bas-reliefs. The fourth raised space is an enclosed rectangle with steps on each side. The fifth enclosure is the outermost rectangular stone wall with a Gopura (ornate entrance) on each side. The last enclosure is the rectangular surrounding moat.

According to various assumptions of many scholars that the conceptual design of Angkor Wat was the tomb of King Sūryavarman II, the complex was habitation in the form of celestial palace etc. Though the buildings and constructions in the Angkor Wat temple complex are majestic; they are geometrically out of place. The temple complex is non-symmetrical, as the complex’s center is left-aligned. But no one used the related Vāstu Śāstra, as an integral system in the study process of planning and designing the Angkor Wat temple complex.

To gain deep insight into the science of designing and constructing of the Angkor Wat complex, this research investigates the planning and frame structures of Angkor Wat using a geometrical grid cell system based on field surveys of the temple complex together with the Vāstu Śāstra, Hindu ancient technological treatises, the Mānasāra, and the Mayamatam: Treatise of Housing, Architecture, and Iconography, in particular.

This research, to the best of our knowledge, is the first of its kind to utilize a geometrical grid cell system in its analysis, because not only is the knowledge of mathematics and geometry very important in Vāstu Śāstra but also plays a pivotal role in architectural and structural design nowadays. The research findings offer a deeper understanding of the perfect relationship between the design concept, function, and frame structures of the Angkor Wat complex, on which scholars have had different opinions for more than 100 years.

2. Literature Review

For the past century, research studies on the design concept of Angkor Wat have involved numerous aspects by many scholars in various fields, both in the Western and Eastern spheres. This research, which is related to the geometrical relationship between ancient Hindu technical treatises and the planning organization of Angkor Wat, is classified into three categories:

2.1. Building Type

According to Przyluski, [5] Angkor Wat was supposedly constructed for use as the burial site of King Sūryavarman II because the buildings were oriented westward and the sequence of the stories on the bas-reliefs around the outer gallery proceeded in the customary direction of a funeral procession (i.e., counterclockwise circumambulation).

Cœdès, [6] an eminent French scholar and author of The Indianized States of Southeast Asia, who had spent three and a half decades in Indochina, wrote “Angkor Wat complex is a tomb, in the sense of resting place of a king after death, but precisely because a Khmer king went to heaven at his death, this was his habitation in the form of the celestial palace in the center of which was placed an image of the god with whom the king was identified”, and he wrote further that in “any attempt to understand the culture of Southeast Asia, and Cambodia in particular, the Indian influence must be taken into account”.

McNeill, [7] wrote in “Beyond Borders Angkor Wat” that Dravidian temples of South India are open and symmetrical and display perfect geometric shapes, such as circles and squares. This is the model on which Angkor Wat’s temples are based, and indeed the architects may have been from Pala India, the powerful Buddhist-supporting dynasty that ruled Bihar and Bengal from the 8th to the 12th century. The style, however, was indigenized by local artists. Whilst the region’s early Hindu temples were built on the same scale as their Indian models, with stepped, square terraces and narrative reliefs along the terraces, later buildings such as Java’s Borobudur temple represented a dramatic change in architectural character. Increasingly vast in scale and monumental in design, temples became works of power as kings sought to accumulate religious merit and prestige by constructing ever more magnificent buildings.

Dalrymple, [4] wrote about Angkor Wat thus: During the time when Angkor Wat was built by King Sūryavarman II there were no temples that existed in India, the center of Hindu world, were of the same scale. Sūryavarman II devoted to Viṣṇu, whose statue was dedicated on July 1131, his 33rd birthday and it was installed in the temple which was started to be built by him in 1122 CE. Ankor Wat is roughly contemporary to the great South Indian Chola temples of Tanjore and Chidambaram, and the Khmers and Cholas, the two great powers of Indian Ocean world, were firm allies and in close diplomatic dialogue. Angkor is not just the most spectacular of all Indic temples; it is the largest religious structure built anywhere in the ancient or medieval world. Sūryavarman II created the largest empire in the history of Southeast Asia. His empire covered the areas of almost the entire mainland Southeast Asia.

Dokras, [8] in his article “Design concepts of the Angkor wat temple” describes an architectural concept as consisting of an idea, notion, opinion, abstraction, philosophy, belief, inspiration, and it should primarily be generated from the areas such as intention, theory, image, plan, hypothesis etc.

Angkor Wat combines two basic plans of Khmer temple architecture:

• The temple-mountain and the later galleried temple. It is designed to represent Mount Meru, home of the devas in Hindu and Buddhist cosmology.
• Mount Meru, the concept of a holy mountain surrounded by various circles, wasincorporated into ancient Hindu temple architecture with a Shikhara (Śikhara) a Sanskrit word translating literally to “mountain peak”. Early examples of this style can be found at the Harshat Mata Temple and Harshnath Temple from the 8th century CE in Rajasthan, Western India. This concept also continued outside India, such as in Bali, where temples feature Meru towers.

2.2. Astronomy

In Ancient Angkor, Freeman & Jacques, [9] Angkor Wat is a completely realized microcosm of the Hindu universe, culminating in the five peaks of Mount Meru, an Architectural masterpiece in fine proportions and rich in detail: the apogee of classical Khmer construction.

• The temple proper combines two major features of Khmer architecture: a pyramid and concentric galleries. Pyramids, which in most cases were created by means of stepped terraces, were the Khmer method of symbolizing the center of the Hindu universe, Mount Meru, in the form of a temple-mountain. To put it as simply as possible, Angkor Wat is a pyramid of three levels, each one enclosed by a well-developed gallery with four gopuras and conner towers. The summit is crowned with five towers in quincunx.
• The importance of orientation to the West depends very much on how unusual it is and is considered to be untraditional.

Eleanor Mannikka, who had received her master’s degree in Chinese and Asian Art, as well as her Ph.D. in Southeast Asian art from the University of Michigan and who was a professor of Art History with a specialty in Asian arts, won a Choice Award for her 1996 thesis: Angkor Wat: Time, Space, and Kingship, one of the outstanding academic books of the year [10]. Her work deals with the history of Angkor Wat, the Khmer’s astronomy, political system, cosmology and the Devaraja cult. Mannikka’s research work, which pertains to architectural design, states that

• No standard cubit existed at Angkor Wat or anywhere else in Asia. She also speculated that a standard cubit in Cambodia would range roughly between 0.40–0.50 m, and thereby, by trial and error, used this range to divide axes and circumferences of Angkor Wat. She later asserted that a very precise unit of 0.43545 m yielded the most consistent results.
• She estimated that between steps, the axis should be 12 cubits long, reasoning that 11 cubits would make the cubit length too long, while 13 cubits was considered too short. She stipulated that the number 12 is of high significance at Angkor Wat because the figure perfectly corresponds to the number of staircases (12) that lead up to the central sanctuary at this level. The 12 stairways are thus excellent symbols of the yearly solar deity, one of the 12 gods of the solar months. She further applied her cubit system to forming several other assumptions.
• Angkor Wat may derive a good portion of its symbolism from the Vāstupuruṣamaṇḍala. In addition, she ascertained that the number 49 in Figure 3 is recurrent at Angkor Wat, whereas neither 64 nor 81 are part of the temple’s system of measurements. The 64-and-81-square designs are found in the literature on mandalas, and the 49-square grid appears to be a Khmer invention. Therefore, she has suggested a possible arrangement for the deities in the 49-unit grid, based on logic but not on any known text.

2.3. Geometry

In Angkor Vat: Description Graphique du Temple, Guy Nafilyan [11], a French architect, and his team, compiled architectural drawings for conservation and restoration work. His team was made up of Alex Turletti, MeyThan, Dy Proeung, and Vong Von. The book contains two volumes in French. The first volume is concerned with architectural aspects of the entire building cluster presented in more than a hundred architectural drawings, while the second contains photos and detailed descriptions of the various elements of the building. Those technical drawings of the building clusters are the best to have been published to date, and they are used as the basis for this research.

Henri Parmentier [12], Chef honoraire du Service Archéologique, Correspondant de l’Ecole Française d’Extrême-Orient, criticized the Khmer architects who built Angkor Wat as being underrated, and said thus: there were repeated geometric mistakes, that this was unlikely to be due to negligence or religious motivation and that they used their eyes to judge without using standard. Parmentier could not explain why the Khmer architects used an asymmetrical technique at Angkor Wat.

René Dumont [13], an assistant curator of the Angkor monuments from 1955 to 1958 and professor at the University of Fine Arts in Phnom Penh from 1965 to 1975 began to seek and found the geometric schemas that presumably controlled plans of Khmer monuments in 1970. In his book Angor Vat par la règle et le compass, published in 1996, he analyzes the plan of this temple by means of elementary geometry (Figure 4). He proposes strategies for geometric proportionate planning of Angkor Wat, whereby the basic geometry construction methods, such as the fish method, square method, and cyclic quadrilateral method, are utilized.

René Dumont gave light to the idea of the geometry planning of Angkor Wat, but did not include the greatest common divisor in the grid diagram arrangement of space. On the other hand, Henri Parmentier saw the temple complex as being non-symmetrical, as the complex’s center is left-aligned.

In 1999, the paper On the Measurement Planning of Angkor Wat: A Study on the Planning Method of Khmer Architecture by Tsuchiya Takeshi from the Department of Applied Physics of Waseda University, Tokyo and Nakagawa Takeshi, a research associate at Institute for Molecular Science of Kyoto University, explained the theory of a grid system in the planning of Angkor Wat, using the 1d (daṇḍa) = 484.92 mm diameter of the north library columns, which were located in the corner of the third-level (outermost) cloister as the basic unit of the square grid system in analyzing the floor plan, with the original drawing by Naflyan, 1969 (Figure 5a,b). In addition, the central axis of the castle’s inner hall often did not overlap with the axis of the master plan when the castle faced east-west. The axis line of the inner hall usually deviates to the south of the central axis line for a distance of approx. “three times the distance of 484.92 mm”. This distance is usually more pronounced for north–south wall measurements than the opposite direction [14].

In Cambodian Architecture Eighth to Thirteenth Centuries, Jacques Dumarçay and Pascal Royère explained that the concept of spatial organization of the Angkor Wat building complex includes the axonometric perspective and Naflyan’s architectural drawings, and that Angkor Wat was a genuine anticipation of the completed work and a profound change in perspective effects; these were no longer based on proportional reductions, but on breaks in the lines of sight created by the different enclosures which formed screens separating one part of the structure from another. These obstacles could only be overcome by taking designated passages through the entrance pavilions constructed on the axes of the structure [15].

Vacharasin [16] in her research titled Geometrical Relationship of Khmer Architectural Design in Thailand, with financial sponsorship from King Mongkut’s Institute of Technology Ladkrabang (KMITL), studied the spatial organization in architectural planning and form of the Khmer architectural design in Thailand of seven religious sites or sanctuaries. These Khmer sanctuaries, which predated Angkor Wat, are situated either on the hilltops or in the plains of Thailand near the Cambodian borders and along the historic royal route connecting Prasat Phimai in Thailand with Angkor Wat in Cambodia. through the geometrical grid system in the Mayamatam treatise on housing, architecture and iconography, which is the significant foundation of spatial organization of the Hindu Temple with respect to the suitable diagrams in architectural design, the study brought to light some new knowledge. Such a study had never been conducted in Thailand in the past one hundred years.

The main findings in the spatial organization in architectural planning and form of the two sanctuaries in the plain area at Prasat Phimai and on the hilltops at Prasat Phanom rung, both with one gallery enclosing the main building, in the research are the following:

• The discovery of the creative design concept of sacred space for communal worship which is expressed by the extraordinary symbolic proportions of the assigned spaces of the building plan. The basic module used to design suitable spaces of the Prasat Phimai and Prasat Phanom rung plans is based on the width of the gallery (i.e., raised covered passageway); the proportion of the total plan is 7 × 8, as shown in Figure 6 and Figure 7. The significance of the idea of suitable spaces for the statue of the most revered God and the “Devarāja” (God–King) is interpreted geometrically in accordance with the ancient Hindu text on architectural arts, which is the most important basic guidance in spatial organization of Hindu religious buildings. These elements, together with the planning of the structural system, are harmoniously incorporated in the layout design.

• The discovery of the subtle application of geometrical systems in balancing the centralized spatial arrangement of the inner building plan. Typically, the space on the left side of the plan should equal that on the right side; however, on the plan of Prasat Phimai and Prasat Phanom rung, as illustrated in Figure 8 and Figure 9, the space on the right is less than that on the left (when facing inward). This is somewhat similar to the Angkor Wat plan, which was observed with apprehension by the French expert architect, Henri Parmentier, head of the École française d’ Extrême-Orient (EFEO).

Prasat Phimai (Figure 8): the proportion of the total plan (7 × 3 BM): (8 × 3 BM) is 7:8. In the front side (southeast) of the outer gallery: the number of openings (orange rectangle) on the left and right sides are 9:8.

Prasat Phanom rung (Figure 9): the proportion of the total plan (7 × 3 BM): (8 × 3 BM) is 7:8. The front side (east) of the outer gallery: the number of openings (orange rectangle) on the left and right sides are 3:2.

The Indian Temple mirror of the world [17]. Dr. Sachchidanand Josthi, Member Secretary, the Executive and Academic Head of Indira Gandhi National Centre for the Arts (IGNCA), New Delhi, notes that Indian temples contain information related to the ideas behind the building of temples not only in India, but also across the continent and, indeed, the globe. This has been analytically interpreted here according to Indian theology and has been defined in the treatise on architecture.

Bruno Dagens, who translated the Mayamatam: Treatise of Housing, Architecture and Iconography [18] has elaborated thus:

In South India and Cambodia, the arrangement is far more articulated, even if it results in various forms and follows various modes. Thus, the division may be marked either by the setting of intermediary enclosures, or by the arrangement of a stepped pyramid topped by the temple itself, or else by combining both of these options.

In Cambodia, two types of arrangements are used, one with a stepped pyramid and another with enclosures that are concentric but provided with walls and entrance pavilions, both of more or less identical size.

For example, the difference between South India and Cambodia is that in South India the enclosures are without steps. However, Angkor Wat is a four-enclosure composition—three on the stepped pyramid and the fourth one twinned with a moat to make the border of the sanctuary—projected at the start and built at one time.

In “Angkor Wat as synthesizing the Khmer temple mountain tradition” [19]

• Analysis of site plan Angkor Wat with 9:9 pada Paramaśāyika maṇḍala (Figure 10).
• The article gave light to the idea of a site of eighty-one plots 9 × 9 pada Paramaśāyika maṇḍala on the 1st-floor terrace with enclosure surrounding the planning of Angkor Wat, although the gallery plan is a rectangle space.
• Analysis of asymmetrical of the westward and the eastward by a modular unit with a remainder (Figure 11).The article gave light to the idea of a modular unit with a remainder, a central feature of both Khmer and Indian, although not only the modular unit which divides from the temple’s length and width are not completely but also unequal to the unit of 9 × 9 pada Paramaśāyika.
• Analysis of site plan for Angkor Wat with diagonal alignment (Figure 12).

The article analysis was based on assumption, without scale drawing, as a result, the article did not give the source of building distance dimensions used for analysis.

3. Research Methodology

The actual measurement of the building is the foundation of the research. The architectural drawings that Guy Nafilyan and his team made for the conservation and restoration work of Ankor Wat are kept by the APSARA National Authority. The researchers and team were permitted access to the drawings in the year 2012. The PhotoModeler Pro 6.5 and Polycam application are used in the analysis of architectural drawings and the actual measurement.

Guy Nafilyan’s excellent data set, the PhotoModeler Pro 6.5, Polycam data set and the Vāstu Śāstra ancient Hindu science of designing and constructing buildings are used in the analysis of this paper.

3.1. Documentary Evidence

Ancient Khmer script originated from the Ancient Brāhmī script that evolved into the Brāhmī used during the Palava dynasty of South India. The oldest Khmer inscription that uses the script of the Pallavas is the one that dates back to 611 CE, found at Angkor Borei district, Takeo province, Cambodia [20]. This epigraphical evidence indicates that India and Cambodia had contact as early as the early 7th century.

Khmer’s religious buildings have roots in Brahmanism and ancient Khmer traditions. An amalgamation of Hindu architectural principles and Khmer architectural practices produced unique creations and buildings different from Indian architecture. Archaeological and artistic findings revealed the dominance of Hindu traditions over the religious practices and architecture of the Khmers.

Architectural remains revealed the ubiquitous use of geometry in Angkor Wat’s building structure and decorative patterns on the ceilings, beams, and columns. In fact, geometry was widely used in ancient India [21], suggesting the prominence of geometry in architectural design and structural design.

To gain deep insight into the architecture and construction of Angkor Wat, this research referred to the Vāstu Śāstra ancient Hindu science of designing and constructing buildings, which finds its origin in the Vedas (knowledge). The Vedas are the most ancient sacred scriptures of India [22].

Architecture of Mānasāra [23], and Mayamatam: Treatise of Housing, Architecture and Iconography [18], are two of the principal texts of the Vāstu Śāstra which deal with proper orientation and right dimensions of all the facets of gods’ and men’s dwellings in construction. It was written in the 11th century CE. This research consulted the Mānasāra and the Mayamatam for meanings and significance of diagram arrangements of grid cells.

The Mayamatam gives the significant qualifications of a Sathapati (architect) as follows: he must be well versed in all sciences, learned in mathematics, well versed in all aspects of architecture, be able to draw, know how to establish buildings, know the whole country, etc.

There are 32 grid diagrams arrangement of space with different meanings and significance, e.g., 1:1 (sakala), 2:2 (pecaka), 3:3 (pīṭha), 4:4 (mahāpīṭha), 5:5 (upapīṭha) in (Figure 13a) through to 32:32 (indrakānta).

For example, the site suitable for Brahmins is square (1:1), the site suitable for the king’s palace is 3:3, and for casket foundation is 5:5.

It is said that rectangular sites too are suitable for gods and for Brahmins [18]. The shape of the site must be perfect, and it must rise towards the west or south. The site must be bordered by a water course flowing to the right [18]. The space suitable for Śūdras is one-fourth more than its width (3 × 4), for Vaiśyas is one-sixth more than its width (5 × 6), and for kings is one-eighth more than its width (7 × 8) which are rectangular (Figure 13b).

The sanctuary enclosures are intended to protect and embellish the temple, and contain the attendants’ shrines, as well. The main temple occupies only 1 square; the diagram for the first enclosure is 9 squares (pīṭha), that for the second is 49 squares (sthaṇḍila), that for the median one is 169 squares (ubhayacaṇḍita), that for the fourth is 441 squares (susaṃhita) and for the fifth it is īśvarakānta (961 squares).

South India ignores the pyramid arrangement and uses only the one where there are several enclosure walls and entrance pavilions with increasing sizes, as is theorized in the technical literature, each surrounding the preceding one and each bigger in size and provided with bigger entrance pavilions [17].

Nevertheless, in Cambodia, two types of arrangements are used, one with a stepped pyramid and another with enclosures that are concentric but provided with walls and entrance pavilions [17].

The measuring units: a contemporaneous inscription (1052 C.E.) giving evidence of the measurement system in use at the time is the Sdok Kok Thom inscription (K.235) [2]. On the fourth side of the stone, lines 104–105 describe the measurement units of land: slik, phlas and hat (cubit).

The important piece regarding meter regulations, the grid system, old Khmer units, and weights and measures, written in 1952 by a Cambodian named Mr. Poch Yankee, has made a comparison between the ancient Khmer measuring system and the metric system [24].

The comparison findings correspond with what is in the Encyclopedia of Scientific Units, Weights, and Measures [25]. The measuring units are detailed with respect to human fingers, and are still used by the present Thai descendants of the Cambodians living in the lower northeastern region, as follows:

$$\begin{array}{ll}\mathrm{Measuring}\mathrm{units}\mathrm{longer}\mathrm{than}\mathrm{hat}:& \mathrm{Measuring}\mathrm{units}\mathrm{shorter}\mathrm{than}\mathrm{hat}:\\ 1\mathrm{phyeam}=4\mathrm{hat}=2.00\mathrm{m}& 1\mathrm{jumarm}=1/2\mathrm{hat}=0.25\mathrm{m}\\ 1\mathrm{sen}=20\mathrm{phyeam}=40.00\mathrm{m}& 1\mathrm{thnob}=1/12\mathrm{jumarm}=0.0208\mathrm{m}\\ 1\mathrm{yoch}=20\mathrm{sen}=16.00\mathrm{km}& \end{array}$$

3.2. Fieldwork Study

The actual measurement of the building is the foundation of the research. The architectural drawings that Guy Nafilyan and his team made for the conservation and restoration work of Ankor Wat are kept by the APSARA National Authority. The authors were permitted access to the drawings in the year 2012 (Figure 14).

To avoid unnecessarily disturbing the cultural heritage stones, the authors collected images and applied the PhotoModeler Pro5 and Polycam for iOS-4.0.5 application as raw data to calibrate the three-dimensional representation with real measurement in comparison with Guy Nafilyan’s excellent data set, in the analysis of this paper.

The PhotoModeler is a Windows program that helps extract measurements and 3D models from photographs (Figure 15). By using a camera as an input device, PhotoModeler lets the user capture a lot of accurate detail in a very short time by organizing the model building process as tracing over photos on the screen.

The Polycam application is the leading 3D scanning platform, effortlessly creating high-quality 3D models from photos, generating precise LiDAR scans, and capturing immersive 360° images (Figure 16).

4. Research Results and Discussion

The research found the greatest common factor (GCF) of the best-fitted individual grid-cell (BM) in each layer. A trial-and-error fashion applied to the individual grid-cells on all four sides of the innermost, the middle and the outer enclosures. As a result, the basic module (BM) fits significantly when superimposed on the plan of each layer.

4.1. The Central Tower and the Innermost Enclosure

The best-fitted individual grid-cell was found to be of square shape with 2.75 m × 2.75 m in dimension (1 Phyeam 1 Hat 1 Thnob) for each side, which is meaningful in Khmer measuring system equal to 2 m, 0.5 m and 0.25 m, respectively.

For the sake of simplicity, this research has termed the 2.75 m × 2.75 m individual square grid-cell the basic module (BM). As illustrated in Figure 17, Figure 18, Figure 19, Figure 20, Figure 21 and Figure 22 collections of the 2.75 m × 2.75 m individual square grid-cells fit significantly when superimposed on the plan of the central tower 3 BM × 3 BM (3:3).

If the size of the basic module is slightly larger (2.7708 m × 2.7708 m) or smaller (2.7292 m × 2.7292 m). This grid system will not be applicable to the galleries’ column system.

Foundation Deposit (garbhanyāsa):

The Mayamatam considers the garbhanyāsa or the foundation deposit ceremony essential before the construction of the building. It is described thus: A foundation deposit with all the prescribed components is a source of success; an incomplete deposit or the absence of any component leads to failure; this is why the foundation deposit should be set in place with the utmost care and without error, space suitable for foundation deposit is 1:1 [18]. At the central tower of Angkor Wat (Figure 14; Long Section, Figure 17 and Figure 18), the pit for depositing the casket (mañjūṣa) containing precious objects should be beneath the base supporting the image of Viṣṇu representing King Sūryavarman II. Those precious objects were plundered when Angkor Wat was converted to Buddhism in the later period [6].

4.2. As Regards the Middle Enclosure Enclosing the Innermost One

The analysis revealed that the BM-based dimensions of the middle enclosure are 35 BM:42 BM (5 × 6) in Figure 23 and Figure 24. The center of the inner square enclosure, where the central tower is situated, is not aligned with that of the middle and outer enclosures. The space created by 35 BM:42 BM (5:6) according to the Mayamatam, symbolizes a suitable place for the inhabitation of the Vaiśyas [18].

4.3. As Regards the Third Enclosure Enclosing the Middle One

The analysis revealed that the BM-based dimensions of the third enclosure are 60 BM:72 BM (5:6), in Figure 25 and Figure 26. The finding also revealed the use of a center-shifting technique (non-fraction modules). The space created by 60 BM:72 BM (5:6), according to the Mayamatam, symbolizes a space suitable for the inhabitation of the Vaiśya.

4.4. Conclusion of the Center-Shifting Contributed to an Unequal Number of Openings (Windows)

In the middle and outer enclosure, where the left side contained more openings than the right one. The number of the windows on the west side of the outer enclosure; the left, and the right side, are 15 and 14 (blue rectangle). In addition, the respective second and third rectangle enclosures enclosing the middle galleries (orange rectangle) are 35 BM:42 BM, and 60 BM:72 BM spaces (Figure 27, Figure 28 and Figure 29).

4.5. The Proportion Analysis of Angkor Wat’s Enclosures: Juxtaposing the Mānasāra and Mayamatum Texts

The technique of increasing the faith and awe of the people visiting the temple through a lengthened walkway passing seven enclosures is a most ingenious technique. The field survey data analysis to find the proportional relationship underlying the aesthetics of the whole construction discloses a very high level of construction efficiency, according to the excellent planning. On the surface, the planning seems complicated and perplexing; it nonetheless is almost unbelievable that the proportion of the spatial organization of the whole plan is derived from the simple increase and reduction of the basic module of the geometric grid-system science of designing and constructing buildings.

The proportions of Angkor Wat’s enclosures are juxtaposed with those as appeared in the Mānasāra and Mayamatam text, by which the meaning and objective of the construction of Angkor Wat are derived. The finding reveals that the existence of Angkor Wat is beyond mere aesthetics. The original proportions and simple ratios of the central tower and the seven enclosures of the Angkor Wat layout are as follows:

• The ratio of the Foundation Deposit is 1 BM:1 BM (2.75 m:2.75 m).
• The ratio of the central tower plan is 3 BM:3 BM
• The ratio of the first square enclosure enclosing the central tower is 18 BM:18 BM or 9 BM:9 BM or 3 BM:3 BM.
• The ratio of the second rectangular enclosure enclosing the first square enclosure is 35 BM:42 BM or 5 BM:6 BM (0.8333).
• The proportion of the third rectangle enclosure surrounding the second enclosure is 60 BM:72 BM or 5 BM:6 BM (0.8333).
• The ratio of the rectangular base underneath the entire building cluster is 91 BM:117 BM (0.77) or approximately 3 BM:4 BM (0.75).
• The proportion of the rectangular wall enclosing the entire building complex and spaces of all the aforementioned is 290 BM:365 BM (0.7945) or approximately 5 BM:6 BM (0.8333).
• The proportion of the rectangular moat banks which surround and are next to the fifth enclosure or wall is 320 BM:395 BM (0.8101) or approximately 5 BM:6 BM (0.8333).
• The proportion of the banks which are farther from the wall of the rectangular moat is 450 BM:525 BM (0.8571) or approximately 7 BM:8 BM (0.875).

To pinpoint the genuine purposes of the architectural works of Angkor Wat, the researcher has superimposed, with Angkor Wat’s main Prāsād at the center, three pairs of the Mānasāra’s and the Mayamatum’s diagrams of 3:3, 4:4, 5:5, 6:6 and 7:7, 8:8 respectively, on the 3:4, 5:6 and 7:8 rectangular diagrams of the temple. The rectangular diagrams are converted into square grids with the simplest ratios, as shown in

Table 1. Comparison of the proportions of Angkor Wat enclosures with the Mānasāra and the Mayamatum’s diagrams.

Sequence	Angkor Wat’s Enclosure Proportion (in BM:BM)	Mānasāra and Mayamatum Diagram	Mānasāra and Mayamatum’s Symbolic Meaning
Foundation Deposit	1 BM:1 BM 2.7 m:2.75 m BM denotes Basic Module	1:1 (One square)	The sakala diagram. The basic diagram; it is suitable for sacrifices to Pitṛ and immortals, as well as for the worship of the guru. It should be bounded by four cords: the western cord is assigned to Varuṇa the god of water.
The first enclosure	18 BM:18 BM (=9 BM:9 BM) (=3 BM:3 BM)	3:3 (Nine squares)	The Pīṭha diagram. Used for the temple’s first enclosure of the king’s palace, and for casket foundation.
		9:9 (Eighty-one squares)	The Paramaśāyika diagram. The one most often used for rites and for building operations.
The second enclosure	35 BM:42 BM or 0.8333 (5 BM:6 BM)	5:6 (Thirty squares) (0.8333)	The site ratio reserved for Vaiśya, the third caste of Indian society, referring to laity.
		5:5 (Twenty squares) (inferred from 35:35)	The Upapīṭha diagram. The proportion which is intended as the casket foundation.
		7:7 (Forty-nine squares) (inferred from 35:35)	The Sthaṇḍila diagram. The second enclosure of a temple and the kings’ palace.
		6:6 (Thirty-six squares) (inferred from 42:42)	The Ugrapīṭha diagram. The establishment of a temple’s enclosure with sixteen attendants’ shrines.
The third enclosure	60 BM:72 BM or 0.8333 (5 BM:6 BM)	5:6	The site proportion suitable for Vaiśya.
		5:5 (inferred from 60:60)	The Upapīṭha diagram, the proportion was intended for casket foundation. (garbha in Sanskrit)
		6:6 (inferred from 72:72)	The ratio is suited to the establishment of a temple’s enclosure with sixteen attendants’ shrines.
The fourth enclosure	91 BM:117 BM or 0.77 ($\approx$3 BM:4 BM)	3:4 (Twelve squares) (0.75)	The ratio of the site is suitable for Śūdra, the fourth caste of Indian society.
The fifth enclosure	290 BM:365 BM or 0.7945 ($\approx$5 BM:6 BM)	5:6	The site proportion suitable for Vaiśya.
The sixth enclosure	320 BM:395 BM or 0.8101 ($\approx$5 BM:6 BM)	5:6	The site proportion suitable for Vaiśya.
The s eventh enclosure	450 BM:525 BM or 0.8571 $(\approx$7 BM:8 BM)	7:8 (Fifty-six squares) (0.875)	The length of the site suitable for kings, gods and brahmins. The shape of the site must be perfect, must rise towards the west or south and be bordered by a water course flowing to the right (flowing according to pradakşiņa, run from west to south).

and Figure 30. The procedure offers more meaningful and useful insights for a profound understanding of Angkor Wat’s existence.

Table 1: Comparison of the proportions of Angkor Wat enclosures and of the Mānasāra’s and the Mayamatum’s diagrams. Table 1 tabulates the proportions of seven enclosures of Angkor Wat and those in the Mayamatam text. The first enclosure is the inner square gallery enclosing the central tower (18 BM:18 BM). The second enclosure is the rectangular middle gallery enclosing the innermost square gallery (35 BM:42 BM). The third enclosure is the rectangular outer gallery enclosing the middle gallery (60 BM:72 BM), and the fourth enclosure is the base supporting the outer galleries, the middle, and the innermost enclosures (91 BM:117 BM). The fifth enclosure is the rectangular walls enclosing the base supporting the inner space (290 BM:365 BM). The sixth enclosure is the inner bank of the city moat outside of the walls (320 BM:395 BM), and the seventh enclosure is the outer bank of the moat (450 BM:525 BM).

In Table 1, the diagram for Brahmins (ancient India’s highest caste) is of a square shape of any size. The diagram for the king’s residence and for the casket foundation is of square shape. This is consistent with the drawing showing long section in Figure 14 mentioned–above. The diagram for a place of the Vaiśya (ancient India’s third caste) and for the Śūdras (ancient India’s fourth caste) are of rectangular shapes.

In Figure 30 and Figure 31, analysis of Angkor Wat’s layout using the BM system revealed the proportions of seven sections (square and rectangular shape) enclosing the central tower, as tabulated in Table 1. Moreover, the analysis of field survey data unveiled high levels of construction efficiency, whereby the proportions of the spatial organization of Angkor Wat are achieved by varying the number of basic modules (BMs).

5. Conclusions

The paper studies the employment of a geometrical grid system in the planning of the Angkor Wat stone sanctuary. The study has found the geometric code, or each suitable module of one of the Basic Module (BM) used in the planning of 2.75 × 2.75 m in the metric system or 5 Hat 1 Thnob in the local Cambodian system, and the method of shifting of the center.

The diagrammatic arrangement of the grid cell suitable for both immortals and mortals in the Mānasāra and the Mayamatam, two of the ancient treatises of Hindu architecture, is achieved by skillful arrangement of the building plan and layout, using a concentric arrangement. In accordance with the treatises, Angkor Wat consists of stepped pyramids and several enclosures with surrounding walls, a cloister-like gallery, and the surrounding moat.

The finding of the analysis of the proportions of the central tower plan and layout of Angkor Wat enclosures by the basic grid module 2.75 BM:2.75 BM revealed a deeper understanding of the meaning in spatial organization of the layout and plan of the Angkor Wat design which expressed the concepts of the supposed suitable residence for the god on earth, the king, and his citizens, and, in addition, the final celestial abode of King Sūryavarman II on Earth after his death:

The diagram 1:1 (2.75 × 2.75 m), which is in the middle of the central tower of Angkor Wat, is suitable for the foundation deposit (Garbha) and according to the Mānasāra and Mayamatam.

The diagram 3:3 inferred from the central tower plan symbolizes the celestial residence of King Sūryavarman II on earth after his death and becoming one with Lord Viṣṇu.

The diagram 3:3 inferred from the in enclosure symbolizes the celestial abode of King Sūryavarman II after his death, represented by the image of Viṣṇu which may be the image of Viṣṇu, presently erected in the western copura. The space is also suitable for Brahmins who served the king in the form of the image of Viṣṇu when it was erected in the central tower [4,9].

The diagram 5:6 inferred from the middle enclosure enclosing the central enclosure and enclosure enclosing the middle enclosure, symbolizes the habitation of the King’s citizens who were the Vaiśyas [18].

The diagram 3:4 inferred from the rectangular base underneath the entire building cluster, symbolizes the habitation of the King’s citizens who were the Śūdras [18].

The diagram 5:6 inferred from the rectangular wall enclosing the rectangular base underneath the entire building, symbolizes the habitation of the King’s citizens who were the Vaiśyas [18].

The diagram 5:6 inferred from the rectangular inner moat bank which surrounds the rectangular wall symbolizes the habitation of the King’s citizens who were the Vaiśyas, supposedly when they were alive [18].

The diagram 7:8 inferred from the outer moat bank which surrounds the rectangular outer moat bank symbolizes the palace of the King Sūryavarman II, supposedly, when he was alive [18].

Angkor Wat reached its peak as regards the architectural design concept and construction. The precise and harmonious geometry of the temple form and structure will be the subject of future study, demonstrating the culmination of the perfection of Angkor Wat’s architectural design and structural engineering.