Round Heaven and Square Earth, the Unity of the Pagoda and Statues—A Study on the Geometric Proportions of the Architectural Space, Statues, and Murals in Ying **an Fogong Si Shijia Ta 應縣佛宮寺釋迦塔 (Sakyamuni Pagoda of Fogong Temple in Ying County) ^†

Abstract

In Ying ** of the architecture and statuary (including the murals on the first floor) combined with geometric design analysis, this paper proposes the following: First, there exists a clear geometric proportion among the interior space and statues on each floor of the pagoda. Second, clear proportional relationships also exist among the statues on each floor, and each of the 26 statues has ‘classical’ proportion rules. Third, the height of the giant Buddha statue on the first floor is the module for not only the height of the statues on each floor but also the construction of the whole pagoda such that the height of the statue on the first floor is 1/6 of the total height of the pagoda (excluding the base). And the ratio of the pagoda’s total height to the first floor’s diameter, the ratio of the total height to the top-story height under the column capital, and all the other geometric proportions are closely related to the architectural modeling. And finally, in the construction of the pagoda, statues, and murals, the scale is deduced to be 1 chi 尺 (Chinese foot) = 29.5 cm. These values give clear scale logics not only to the construction but also to the details of the statues. Accordingly, the most frequently used proportions in the architectural space, statues, and murals of the Pagoda of Fogong Temple are $\sqrt{2}$, 3:2, 5:3 (or 8:5), and 9:5, which are imbued with cultural messages, like Zhou Bi Suan **g 周髀算經 (The Mathematical classic of the Zhou shadow-gauging instrument), Ying Zao Fa Shi 營造法式 (Treatise on Architectural Methods or State Building Standards), the ancient Chinese world view—tian yuan di fang 天圜地方 (the dome-shaped heaven and the flat, square earth) reflected from “ yuan fang tu 圓方圖 (rounded-square map)” and “fang yuan tu 方圓圖 (squared-circle map)”, ancient Chinese ideas that ”san tian liang di er yi shu 參天兩地而倚數 (‘three’ is the number of the heaven and ‘two’ is the number of the earth, and all numbers are based on them)” and “jiu wu zhi zun 九五之尊 (nine and five are the numbers of the honorable central position)”, and most probably related to the “mandala” of Esoteric Buddhism and to the Western “Golden Ratio”, which all need further research in depth.

1. Introduction

The Sakyamuni Pagoda of Fogong Temple in Ying County, Shanxi Province, also recognized as the Timber Pagoda in Ying County, was constructed in the second year of Qingning during the Liao Dynasty (1056). This remarkable structure stands as the sole surviving ancient pagoda with a pure timber framework in China, holding the distinction of being the world’s largest wooden pagoda. Its architectural significance is unparalleled in its historical context.

Notably commendable is the pagoda’s unique architectural feature known as the five “ming ceng 明層 (bright layers)”. These layers, interspersed with “an ceng 暗層 (hidden layers)” give the structure the form of “wai wu nei jiu 外五內九 (outside five inside nine)”.1 Each of these layers, referred to as “ nei cao 內槽” (inner troughs), houses meticulously preserved statues, totaling 26 in number. This arrangement imparts a semblance of a three-dimensional Buddhist temple, enabling devotees to ascend the pagoda’s steps, worshi** Buddha floor by floor. Consequently, the pagoda unfolds as a vertically layered, three-dimensional Buddhist temple.

Liang Sicheng, in his inaugural paper “Wo Men Suo Zhi Dao De Tang Dai Fo Si Yu Gong Dian 我們所知道的唐代佛寺與宮殿 (Buddhist Temples and Palaces in the Tang Dynasty Known to Us)”, published in Zhongguo Yingzao Xueshe Huikan 中國營造學社彙刊 (the Bulletin of the Society for Research in Chinese Architecture) (1932), pointed out that “the Buddhist Hall is the Buddha’s residence…” (Liang 1932, No. 1). Following Liang’s point, enshrining and worshi** Buddha is the core function of the Pagoda of Fogong Temple. This aligns with the pagoda’s original name, “shijia ta 釋迦塔 (Sakyamuni Pagoda)”, emphasizing its purpose as a vertical Buddhist hall. The intricate interior space is tailored for the statues on each floor, creating a visual and spatial experience that resonates solemnity and harmony.

Building upon prior study on the Pagoda of Fogong Temple, this paper meticulously measures the architecture and statues (including the mural on the first floor),2 conducting a comprehensive surveying and map** of the structure. The findings reveal clear geometric proportions within the interior space and statue complex on each floor. These proportions are particularly evident in the relationships between the height of stories and statues and between the giant Buddha statue and the murals on the first floor. The geometric relationships also extend to each statue’s and mural’ s proportions. The statue on the first floor is assumed to have a pivotal role, serving as the module for both subsequent statues and the overall architectural design. The construction scale of the pagoda, statues, and murals is speculated as 1 chi = 29.5 cm. From this, the proportional relationships among the height of the pagoda, the base, each story, the pagoda-top finial, and each statue and mural can be inferred. The construction of the pagoda as a whole also has its “classical” ratio control.

The subsequent sections will provide a brief overview of the existing literature, followed by a detailed analysis of the geometric proportions inherent in the pagoda’s interior space and statue complex and the deduction of the scale of construction. This paper will conclude by summarizing various classical geometric proportions adopted in the construction of the architectural space, the statues, and the murals, including $\sqrt{2}$, 3:2, 5:3, 8:5, and 9:5, underscoring their cultural significance such as “tian yuan di fang 天圜地方 (a dome-shaped heaven and a flat, square earth)”, “san tian liang di 參天兩地 (three as the heavenly number and two as the earthly number)”, “jiu wu zhi zun 九五之尊 (the numbers nine and five as denoting an honorable position)”, and even concepts like the “Golden Ratio” and “Esoteric Buddhist mandala”.

2. Literature Review

The pioneering investigation of the Pagoda of Fogong Temple and the statues was undertaken by Liang Sicheng 梁思成 and Mo Zongjiang 莫宗江. Their collaborative efforts encompassed a comprehensive examination, yielding a series of pagoda surveys and official sections, visually representing both architectural features and statues (see Figure 1 and Figure 2). Liang Sicheng’s paper, “Shanxi Yingxian Fogong Si Liao Shijia Muta 山西應縣佛宮寺遼釋迦木塔 (Liao Sakya Timber Pagoda of Fogong Temple in Ying County, Shanxi)” (completed in 1936), provides detailed insights. In this work, Liang highlighted the measured values, such as the approximately 12.30 m height of the Buddha on the first floor, the 4 m height of the two-story Buddha and its base, and the approximately 4.85 m height of the four-story Buddha.

The surveying and map** drafts (1933) by Liang and Mo, collected in Tsinghua University Archive, have measurements of all statues except the giant Buddha statue on the first floor, which was too big to measure at that time (Chen 2001, p. 113; 2002a, pp. 551–53).3 Beyond the Sakyamuni Pagoda of Fogong Temple, Liang Sicheng extended his survey and cartographic endeavors to other Buddhist structures, including the Guanyin Pavilion of Dule Monastery in Ji County, the Three Bodhisattvas Hall of Guangji Monastery in Baodi, the Hall of the Three Sages, Samantabhadra (Puxian) Pavilion, and Mahāvīra Treasure Hall of Shanhua Monastery in Datong, the Bhagavat Sutra Hall of Huayan Lower Monastery in Datong, and the Great East Hall of Foguang Monastery in Mount Wutai. The survey reports authored by Liang often incorporated a dedicated section addressing the aspect of “statues”.

Building upon the preceding research, Chen Mingda 陳明達 conducted a comprehensive examination of the geometric proportions of the Pagoda of Fogong Temple, analyzing the construction from a holistic and detailed perspective, creating a paradigm in the study of the design methodology employed in ancient Chinese timber structures. In his seminal work, Ying **)” concept in various locations, such as Cave 004 of the Grottoes of Mount Maiji, the Great Hall of Nanchan Monastery, the Great East Hall of Foguang Monastery, the Pagoda of Fogong Temple, and the Guanyin Pavilion of Dule Monastery. The findings indicate that the elevation angle of the line of sight between the second to fifth floors of the pagoda typically ranges between 23 and 25°.5 (Figure 4).

In the paper Yixian Fengguo Si Daxiong Dian Diaocha Baogao 義縣奉國寺大雄殿調查報告 (Investigation Report on the Mahāvīra Hall of Fengguo Monastery in Yi County, Liaoning), Ding Yao 丁垚 and Cheng Li 成麗 conducted a preliminary examination of the scale of seven statues through 3D laser scanning and map** of the statue complex within the Mahāvīra Hall (Ding and Cheng 2008, pp. 25–42). Similarly, in 2012 Shanxi **yao Zhenguo Si Wanfo Dian yu Tianwang Dian **gxi Cehui 山西**遙鎮國寺萬佛殿與天王殿精細測繪報告 (Report on Fine Surveying and Map** of the 10,000 Buddhas Hall and the Celestial Kings Hall of Zhenguo Monastery in **yao, Shanxi), Liu Chang 劉暢 et al. employed 3D laser scanning and color map** techniques to analyze the standing and seated Attendant Bodhisattva statues situated on the east side of the 10,000 Buddhas Hall and the Celestial Kings Hall of Zhenguo Monastery. The study also briefly discusses the geometric proportions observed among the myriad Buddhas depicted in the mural (Liu et al. 2012, pp. 155–72).

The author conducts an investigation of the geometric proportions linking architecture, interior space, and statues together. The study also encompasses diverse examples, including the Great East Hall of Foguang Monastery in Mount Wutai, the Guanyin Pavilion, and the Mountain Gate of Dule Monastery in Ji County, the Mahāvīra Hall of Fengguo Monastery in Yi County, the Mahāvīra Treasure Hall, and the Haihui Hall of Shanhua Monastery in Datong, the Bhagavat Sutra Hall of Huayan Lower Monastery in Datong, and the Dasheng Pavilion of Puning Monastery in Chengde. Notably, the examination focuses on ratios related to $\sqrt{2}$ and $\sqrt{3}$/2 derived from the squared-circle map. This paper will introduce and elaborate on the design methodology termed “duo xiang gou wu 度像構屋”, which means designing the interior space of Buddhist architecture—such as halls, pavilions, and pagodas—in accordance with the principal statues (N. Wang 2017, pp. 29–36; 2018a, pp. 103–25; 2018b, pp. 216, 256; 2018c) (Figure 5).

In the article Duo Mian Wei Fan Die Fo Hua Ta—Fogong Si Shijia Ta Yi Ti Hua She Ji Mo Shi ji Ying Zao Li Nian Tan ** of extant Buddhist architecture and the statues inside is an indispensable prerequisite for further study. Fu ** techniques to deeply explore the proportional relationships in the construction of Buddhist structures, by analyzing their architectural space, with a particular emphasis on their interior space, statues, and murals located on each floor.

3. Geometric Proportion of the Interior Space and Statue Complex of the Pagoda of Fogong Temple

In the pagoda, there are presently 26 statues distributed across its floors, including 1 on the first floor, 5 on the second floor, 4 on the third floor, 7 on the fourth floor, and 9 on the fifth floor (see Figure 6 and Figure 7). Historical documentation from 1933, when Liang Sicheng and Mo Zongjiang conducted a survey and map** of the pagoda, indicates that there were 11 statues on the fourth floor at that time, suggesting a total of 30 statues within the entire pagoda (Liang 2007, vol. 10, pp. 113–17).7 Liang Sicheng posits that while the statues underwent modifications in subsequent periods, they generally adhere to the original form. The most recent restoration occurred in the winter of the 18th year of the Republic of China (Liang 2007, vol. 10, p. 114). Fu ** data, from 2021, the dimensions of the six statues and murals on the first floor are as follows (see

Table 1. Dimensions of the giant Buddha on the first floor and the six Buddhas in the murals in the central chamber of the Pagoda of Fogong Temple (unit: m).

Name	Top Height	Total Height	Total Width of the Base	Clear Height	Clear Width	Height of Head
Giant Buddha	10.32	10.32	6.955	8.478	5.537	2.625
Southwest Wall Buddha	7.366	6.374	4.691	5.07	3.597	1.473
Western Wall Buddha	7.421	6.429	4.37	5.126	3.644	1.491
Northwest Wall Buddha	7.4	6.408	4.728	5.17	3.682	1.493
Northeast Wall Buddha	7.076	6.11	4.731	4.714	3.326	1.311
Eastern Wall Buddha	7.169	6.203	4.246	4.668	3.323	1.296
Southeast Wall Buddha	7.247	6.281	4.619	4.894	3.747	1.344
Average of the Six Buddha Murals	7.28	6.301	4.564	4.94	3.553	1.401

Within the scope of this article, the term “top height” delineates the vertical measurement from the apex of the statue or mural to the ground. The concept of “total height” encompasses the aggregate of the statue’s height and its base. “clear height” specifically pertains to the statue’s height, excluding the dimensions of its base. The term “total width of the base” denotes the broadest section of the base, typically determined by the maximum inner diameter of the lotus situated atop the Sumeru seat. Lastly, “clear width” denotes the horizontal span measured between the knees of a cross-legged seated Buddha/Bodhisattva or between the elbows of a stood Buddha/Bodhisattva.

).

The following conclusions can be drawn through the geometric map** and data analysis of the detailed surveying and map** of the pagoda’s first-floor space, statues, and murals.

3.1.1. Analysis of the Proportional Relationships between the Space, the Statues, and the Murals

(1) Height of the First Story (14.57 m)8/Total Height of the Giant Buddha (10.32 m) = 1.412 ≈$\sqrt{2}$ (99.8% coincidence).

(2) Total Height of the Giant Buddha (10.32 m)/Top Height of the Six Buddhas in the Mural (average 7.28 m) = 1.418 ≈$\sqrt{2}$ (99.7% coincidence). Total Height of the Giant Buddha (10.32 m)/Total Height of the Six Buddhas in the Mural (average 6.301 m) = 1.638 ≈ 5:3 (98.3% coincidence).

(3) Height of the First Story (14.57 m)/Top Height of the Six Buddhas in the Mural (average 7.28 m) = 2.001 ≈ 2 (99.9% coincidence).

(4) Total Height of the Giant Buddha (10.32 m)/Inner Diameter of the Central Chamber (10.244 m at the foundation of the wall)9 = 1.007 ≈ 1 (99.3% coincidence).

(5) The Distance from the Top of the Octagonal Caisson Ceiling to the Ground (12.825 m)/Total Height of the Giant Buddha (10.32 m) = 1.243 ≈ 5:4 (99.4% coincidence).

To sum up,

The Height of the First Story/Total Height of the Giant Buddha (equal to the inner diameter of the Central Chamber)/Top Height of the Six Buddhas in the Mural is ≈ 2: $\sqrt{2}$:1. Therefore, the space on the first floor (especially the central chamber), the giant Buddha statue, and the six Buddha murals are designed as a whole and have a perfect geometric proportion (Figure 9 and Figure 10).

3.1.2. Proportional Analysis of the Statues and the Murals

The following can be seen from

Table 2. Proportional analysis of the giant Buddha on the first floor of the pagoda and the six Buddhas in murals.

Name	Total Height/Total Width of the Base Measured Value Ideal Value (Coincidence Degree)	Clear Height/Clear Width Measured Value Ideal Value (Coincidence Degree)	Total Height/Clear Height Measured Value Ideal Value (Coincidence Degree)	Clear Height/Height of Head Measured Value Ideal Value (Coincidence Degree)
Giant Buddha	1.484 3:2 (98.9%)	1.531 3:2 (98%)	1.217 6:5 (98.6%)	3.23
Average of the Six Buddha Murals	1.381 7:5 (98.6%)	1.39 7:5 (99.3%)	1.276 9:7 (99.2%)	3.526 3.5 (99.3%)
Name	Total Height: Total Width of the Base	Clear Height: Clear Width	Total Height: Clear Height	Clear Height: Height of Head
Southwest Wall Buddha	1.359	1.41	1.257	3.442
Western Wall Buddha	1.471	1.407	1.254	3.438
Northwest Wall Buddha	1.355	1.404	1.239	3.463
Northeast Wall Buddha	1.291	1.417	1.296	3.596
Eastern Wall Buddha	1.461	1.405	1.329	3.602
Southeast Wall Buddha	1.36	1.306	1.283	3.641

:

(1) The clear height-to-width ratio of the giant Buddha statue is about equal to the total height-to-width ratio (that is, the ratio of the total height to the total width of the base, the same below), both of which are about 3:2, forming a partial and overall harmony of the sculpture of the Giant Buddha shape (Figure 11).

(2) Except for the Southeast Wall Buddha, the clear height-to-width ratio of the other five Buddhas is close to 7:5, that is, “fang wu xie qi 方五斜七 (square five, oblique seven)” (close to $\sqrt{2}$);10 (Figure 12). The average value of the total height-to-width ratio of the six Buddhas is close to 7:5, but there is a large difference among them; in contrast, the clear-height-to-width ratios are more unified.11

(3) The head-to-body ratio of the six Buddhas in the murals is close to 1:3.5, which conforms to the craftsman’s formula of “li qi zuo wu pan san ban 立七坐五盤三半 (seven-heads tall when standing, five-heads tall when sitting, three and a half heads tall when sitting crossed-legged)”. The Six Buddhas’ clear height/height below the head ≈ 3.5:2.5 = 7:5, that is, “fang wu xie qi 方五斜七 (square five, oblique seven).” (S. Wang 2001)12

(4) The proportion of the head of the giant Buddha statue is significantly higher than that of the six Buddhas in the mural.

3.2. Analysis of the Second-Floor Space and Statues

A square wooden Buddha altar is placed at the center of the second floor. On the altar, the Sakyamuni is sitting cross-legged at the center to the north, with two Attendant Bodhisattvas standing on both sides. The two Bodhisattvas sitting cross-legged on the two sides of the altar are Manjusri and Samantabhadra, respectively (which can be distinguished by the lion and elephant sculptures at the waist of the Sumeru seats) (Figure 13).

According to the 2021 surveying and map** data, the dimensions of the statues on the second floor are as follows (see

Table 3. Dimensions of the statues on the second floor of the Pagoda of Fogong Temple (unit: m).

Name	Top Height	Total Height	Total Width of the Base	Clear Height	Clear Width	Height of Head
Sakyamun	4.42	4.052	2.667	3.126	2.051	0.876
Attendant Bodhisattva in the northwest	3.26	2.892	0.856	2.726 2.598 (Bun not included)	0.793	0.499 0.37 (Bun not included)
Attendant Bodhisattva in the northeast	3.348	2.98	0.81	2.809 2.558 (Bun not included)	0.799	0.603 0.364 (Bun not included)
Bodhisattva Manjusri	3.007	2.639	1.595	1.838 1.67 (Bun not included)	1.135	0.602 0.421 (Bun not included)
Bodhisattva Samantabhadra	2.985	2.617	1.607	1.802 1.671 (Bun not included)	1.066	0.56 0.399 (Bun not included)

).

The following results can be obtained through the geometric map** and data analysis of the measured drawings of the space and statues on the second floor.

3.2.1. Analysis of the Proportional Relationship between the Space and the Statue Complex

(1) The Height of the Second Story (8.82 m)/the Top Height of the Buddha (4.42 m) = 1.995 ≈ 2 (99.8% coincidence).

(2) The Height of the Second Story (8.82 m)13/the Width of the Buddha Altar (5.29 m) = 1.667 ≈ 5:3 (100% coincidence).

(3) The Width of the Buddha Altar (5.29 m)/the Width of Each Façade of the Inner Trough (average 5.305 m)14 = 0.997 ≈ 1 (99.7% coincidence).

To sum up,

The second-floor height/the width of the Buddha altar (that is, the width of every side of the inner trough)/the height of the Buddha ≈ 10:6:5. Therefore, the space of the inner trough, the Buddha altar, and the statues on the second floor form an overall design, with obvious and basic proportions (Figure 14).

3.2.2. Proportional Analysis of Statue Complex

(1) The Total Height of Buddha (4.052 m)/the Total Height of Attendant Bodhisattva (average 2.936 m) = 1.38 ≈ 7:5 (98.6% coincidence, that is, “fang wu xie qi 方五斜七 (square five, oblique seven)”).

(2) The Total Height of the Attendant Bodhisattva (average 2.936 m)/the Total Height of the Manjusri Bodhisattva (average 2.628 m) = 1.117 ≈ 10:9 (99.5% coincidence).

To sum up,

The total height of Buddha/the total height of the Attendant Bodhisattva/the total height of the Manjusri Bodhisattva ≈ 14:10:9. Therefore, the statue complex on the second floor has a clear proportional relationship in elevation design (Figure 15).

3.2.3. Proportional Analysis of the Statues

As can be seen from

Table 4. Proportional analysis of the statues on the second floor of the Pagoda of Fogong Temple.

Name	Total Height/Total Width of the Base Measured Value Ideal Value (Coincidence Degree)	Clear Height/Clear Width Measured Value Ideal Value (Coincidence Degree)	Total Height/Clear Height Measured Value Ideal Value (Coincidence Degree)	Clear Height/Height of Head Measured Value Ideal Value (Coincidence Degree)	Clear Height (Bun Not Included)/Height of Head (Bun Not Included) Measured Value Ideal Value (Coincidence Degree)
Sakyamun	1.519 3:2 (98.7%)	1.524 3:2 (98.4%)	1.296 13:10 (99.7%)	3.568 3.5 (98%)
Attendant Bodhisattva in the northwest	1.655 5:3 (99.3%)	1.619 8:5 (98.8%)	1.436 10:7 (99.5%)	3.053 3 (98.2%)	3.967 4 (99.2%)
Attendant Bodhisattva in the northeast	1.629 8:5 (98.2%)	1.69 5:3 (98.6%)	1.452 10:7 (98.3%)	3.218	4.053 4 (98.7%)
Bodhisattva Manjusri	3.379	3.438 3.5 (98.2%)	1.061	5.463 5.5 (99.3%)	7.022 7 (99.7%)
Bodhisattva Samantabhadra	3.679	3.516 3.5 (99.4%)	1.061	4.658	7.027 7 (99.6%)

,

A: Buddha Statue

(1) The height-to-width ratio (including the clear height-to-width ratio and the total height-to-width ratio) of the Buddha on the second floor is about 3:2, which is the same as the ratio of the Buddha on the first floor.

(2) The head-to-body ratio of the Buddha is about 1:3.5, which is in line with “li qi zuo wu pan san ban 立七坐五盤三半 (seven-heads tall when standing, five-heads tall when sitting, three and a half heads tall when sitting cross-legged)”. The clear height of the Buddha statue/the height below the head ≈ 3.5:2.5 = 7:5, that is, “fang wu xie qi 方五斜七 (square five, oblique seven)”. All the ratios above are the same as the statues on the first floor (Figure 16 and Figure 17).

B: Manjusri, Samantabhadra

(1) The clear height-to-width ratio and the total height-to-width ratio of Manjusri and Samantabhadra are about 8:5 or 5:3, both of which are close to the Western “Golden Ratio” (about 1.618).

(2) The ratio of Manjusri’s and Samantabhadra’s total height to their total height is about 10:7, that is, the craftsman’s formula “fang qi xie shi 方七斜十 (square seven, oblique ten)”, which is also the approximate value of $\sqrt{2}$.

(3) The head-to-body ratio (excluding the bun) of Manjusri and Samantabhadra is close to 1:4; Manjusri’s head-to-body ratio (including the bun) is about 1:3. By contrast, Samantabhadra’s head-to-body ratio is slightly smaller (Figure 18, Figure 19, Figure 20 and Figure 21).

C: Two Attendant Bodhisattvas

(1) The clear height to clear width ratio of two Attendant Bodhisattvas (the clear width is from elbow to elbow) is close to 3.5 (that is, 7:2).

(2) The head-to-body ratio of two Attendant Bodhisattvas (excluding the bun) is about 1:7 (that is, the so-called seven-heads tall), which also conforms to the “li qi zuo wu pan san ban 立七坐五盤三半 (seven-heads tall when standing, five-heads tall when sitting, three and a half heads tall when sitting cross-legged)”. And the golden section of the two Bodhisattvas’ clear height is located near the elbow when the arm is naturally sagging (that is, close to the navel), indicating the proficiency of the craftsman in human body proportions at that time (Figure 22 and Figure 23).

3.3. Analysis of the Third-Floor Space and Statues

The central Buddha altar on the third floor of the pagoda is octagonal, and four seated Buddhas are set on the altar, facing east, south, west, and north, respectively, generally considered to represent the Four Guardian Warriors (namely, the Eastern Akshobhya Buddha, the Southern Ratnasambhava Buddha, the Western Amitabha Buddha, and the Northern Amoghasiddhi Buddha).

Although the size of the four Buddha statues is roughly the same, their mudras are different, and the Sumeru seats under the statues on the north and the south are different from the Sumeru seats under the statues on the east and the west. Under the south and north Buddhas, the seat bottoms are two-layer-corbelled octagons, above which should have been the waist. However, the south Sumeru seat has an eight-horse sculpture (representing the steed of Ratnasambhava) instead of the waist, while the north Sumeru seat has an eight-bird sculpture (representing the steed of Amoghasiddhi: Garuda, the King Golden-winged Birds). Above each sculpture is a four-layer lotus throne supporting the Buddha, and the front petal is covered by the Buddha’s clothes. The east and the west Sumeru seats have two-layer-corbelled octagons at both the top and the bottom. The waist part of the east Sumeru seat is an eight-elephant sculpture (representing the steed of Akshobhya), and that of the west one is an eight-bird sculpture (representing the steed of Amitabha: the peacock). Above each sculpture, a lotus throne stands on the top corbelled octagon, supporting the Buddha (not covered by the Buddha’s clothes) (Figure 24).

According to the 2021 surveying and map** data, the dimensions of the statues on the third floor are as follows (see

Table 5. Dimensions of the statues on the third floor of the Pagoda of Fogong Temple (unit: m).

Name	Top Height	Total Height	Total Width of the Base	Clear Height	Clear Width	Height of Head
The Southern Ratnasambhava Buddha	3.313	2.746	1.699	1.969	1.304	0.591
The Western Amitabha Buddha	3.279	2.712	1.842	1.945	1.332	0.533
The Eastern Akshobhya Buddha	3.348	2.781	1.789	1.98	1.372	0.599
The Northern Amoghasiddhi Buddha	3.245	2.678	1.84	1.954	1.345	0.576
The average value of Buddhas	3.296	2.729	1.793	1.962	1.338	0.575

).

3.3.1. Analysis of the Proportional Relationship between the Space and Statue Complex

(1) The Height of the Third Story15 (8.59 m)/the Total Height of Each Buddha (average 2.729 m) = 3.148 ≈ π (99.8% coincidence).

(2) The Total Height of Each Buddha (average 2.729 m)/the side Length of the Altar (average 2.756 m)16 = 0.99 ≈ 1 (99% coincidence).

To sum up,

The height of the third story and the total height of the four Buddhas (equal to the length of the octagonal Buddha altar) are in a unified design. In particular, the height of the third story/the sum of the total height of the four Buddhas ≈ π:4 (that is, the ratio between the perimeters of a circle and its circumscribing square). Drawing a square with the total height of the four Buddhas (which is also equal to the side length of the octagonal Buddha altar) as the side length, the height of the third floor is exactly equal to the circumference of its inscribed circle. This composition is very similar to the “mandala” of Esoteric Buddhism, corresponding to the statues’ layout on the third floor of the Four Guardian Warriors of Esoteric Buddhism (Figure 25).

3.3.2. Proportional Analysis of the Statues

As can be seen from

Table 6. Proportional analysis of the statues on the third floor of the Pagoda of Fogong Temple.

Name	Total Height/Total Width of the Base Measured Value Ideal Value (Coincidence Degree)	Clear Height/Clear Width Measured Value Ideal Value (Coincidence Degree)	Total Height/Clear Height Measured Value Ideal Value (Coincidence Degree)	Clear Height/Height of Head Measured Value Ideal Value (Coincidence Degree)
The average value of Buddhas	1.522 3:2 (98.5%)	1.466 3:2 (97.8%)	1.391 7:5 (99.4%)	3.412 3.5 (97.5%)
Name	Total Height/Total Width of the Base	Clear Height/Clear Width	Total Height/Clear Height	Clear Height/Height of Head
The Southern Ratnasambhava Buddha	1.616	1.51	1.395	3.332
The Western Amitabha Buddha	1.472	1.46	1.394	3.649
The Eastern Akshobhya Buddha	1.554	1.443	1.405	3.306
The Northern Amoghasiddhi Buddha	1.455	1.453	1.371	3.392

, by taking the average value of the data of the four Buddhas, we can obtain the following:

(1) The height-to-width ratio of each Buddha is close to 3:2, equal to the ratio of the statues on the first and the second floors.

(2) The ratio of the total height to the clear height of each Buddha is about 7:5, that is, the “fang wu xie qi 方五斜七 (square five, oblique seven)”, which is approximately $\sqrt{2}$.

(3) The head-to-body ratio of each Buddha is close to 1:3.5, equal to the ratio of the statues on the second floor (Figure 26 and Figure 27).

3.4. Analysis of the Fourth-Floor Space and Statues

The layout of the statues on the fourth floor, according to Liang Sicheng, is “the most dramatic one in the pagoda”—the sitting Buddha in the center, with two disciples on either side, Kayah and Ananda. In front of each disciple are, respectively, Manjusri and Samantabhadra Bodhisattvas. Their lion and elephant are standing statues, walking and carrying the Bodhisattvas on lotus thrones, and the Bodhisattvas are all shaped with one foot droo** and one foot sitting cross-legged. In the 1930s, there were also four sculptures of kumara and two sculptures of foreigners on the altar, all in dance poses. In the 1960s, they no longer existed and were replaced by new sculptures of a lion slave and an elephant slave, which survived to the present. From Chen Mingda’s book Ying ** foot of the Samantabhadra was also remolded (Figure 28).

According to the 2021 surveying and map** data, the dimensions of the statues on the second floor are as follows (see

Table 7. Dimensions of the statues on the fourth floor of the Pagoda of Fogong Temple (unit: m).

Name	Top Height	Total Height	Total Width of the Base	Clear Height (Bun Not Included)	Clear Width	Height of Head (Bun Not Included)	Remarks
Buddha	4.71	4.274	2.798	3.235	2.115	0.935
The disciple on the northwest	2.754	2.318	0.732	2.136	0.689	0.299
The disciple on the northeast	2.738	2.302	0.703	2.12	0.676	0.309
Bodhisattva Manjusri	3.248	2.812	0.77	1.435/1.329		0.375/0.27	The head was remolded later
Bodhisattva Samantabhadra	3.278	2.842	0.788	1.438/1.34		0.372/0.273	The feet were remolded later
Lion	1.499	1.063		1.063		0.184	Remolded in Modern times
Elephant	1.522	1.086		1.086		0.192	Remolded in Modern times

):

3.4.1. Analysis of the Proportional Relationship between the Space and the Statue Complex

(1) The Height of the Fourth Story (7.75 m)17/the Top Height of the Buddha (4.71 m) = 1.645 ≈ 5:3 (98.7% coincidence).

(2) The Top Height of the Buddha (4.71 m)/the Width of the Buddha Altar (5.646 m)18 = 0.834 ≈ 5:6 (99.9% coincidence).

In summary, the height of the fourth story, the Buddha altar, and the Buddha are in a holistic design.19 (Figure 29).

3.4.2. Proportional Analysis of the Statue Complex

(1) The Total Height of Buddha (4.274 m)/the Total Height of Samantabhadra (2.842 m) = 1.504 ≈ 3:2 (99.7%). The Total Height of Buddha (4.274 m)/the Total Height of Manjusri (2.812 m) = 1.52 ≈ 3:2 (98.7% coincidence).

(2) The Total Height of Buddha (4.274 m)/the Clear Height of the Two Disciples (average, 2.128 m) = 2.008 ≈ 2 (99.6% coincidence).

(3) The Total Height of Buddha (4.274 m)/the Total Height of the Lion and Elephant (average, 1.075 m) = 3.976 ≈ 4 (99.4% coincidence).

To sum up,

The total height of Buddha/the total height of Bodhisattva/the clear height of the two disciples/the total height of the lion and elephant ≈ 12:8:6:3. Although Manjusri’s head and the lion and elephant are restored or reshaped, there is still a clear proportional relationship between the statues on the fourth floor (Figure 30).

3.4.3. Proportional Analysis of the Statues

As can be seen from

Table 8. Proportional analysis of the statues on the fourth floor of the Pagoda of Fogong Temple.

Name	Total Height/Total Width of the Base Measured Value Ideal Value (Coincidence Degree)	Clear Height/Clear Width Measured Value Ideal Value (Coincidence Degree)	Total Height/Clear Height Measured Value Ideal Value (Coincidence Degree)	Clear Height/Height of Head Measured Value Ideal Value (Coincidence Degree)
Buddha	1.528 3:2 (98.2%)	1.53 3:2 (98%)	1.321 4:3 (99.1%)	3.46 3.5 (98.9%)
The disciple on the northwest	3.167	3.1	1.085	7.144 7 (97.9%)
The disciple on the northeast	3.275	3.136	1.086	6.861 7 (98%)
Bodhisattva Manjusri	3.652		1.96 2 (98%)	4.922 (Bun not included) 5 (98.4%)
Bodhisattva Samantabhadra	3.607		1.976 2 (98.8%)	4.908 (Bun not included) 5 (98.2%)

,

(1) The height-to-width ratio (3:2) and head-to-body ratio (1:3.5) of the Buddha statue on the fourth floor are the same as those of the Buddha statues on the third floor.

(2) Each of the two disciples is about seven heads tall, and each of the two Bodhisattvas is about five heads tall (excluding the bun). If the bun is included, the height of Manjusri and Samantabhadra above the base is about three times the height of the head, which is the same as the layout of Manjusri Bodhisattva sitting cross-legged on the second floor. The two disciples standing on the fourth floor, the two Bodhisattvas, Manjusri and Samantabhadra sitting on one foot, and the giant Buddha sitting cross-legged, exactly constitute the proportion of “li qi zuo wu pan san ban 立七坐五盤三半 (seven-heads tall when standing, five-heads tall when sitting, three and a half heads tall when sitting cross-legged)”, which is consistent with the crafter’s formula. In addition, the lion and elephant remolded in modern times have a moving posture about six heads tall.

(3) The clear height of the Manjusri and Samantabhadra statues is about half of the total height (Figure 31).

3.5. Analysis of the Fifth-Floor Space and Statues

The layout of the statues on the fifth floor is the most solemn and clear. The Buddha altar is nearly square, and at the center is the seated statue of the Dari Tathagata Buddha (also known as the Piluchana Buddha). Liang Sicheng believes that “the head of this Buddha statue seems to be more ancient than statues on the lower floors.” The remaining four sides and four corners set up a total of eight Bodhisattvas’ sitting statues, in an octagonal shape around the central main Buddha. Many scholars believe that this layout reflects the “Foding Zunsheng Mantuoluo 佛頂尊聖曼陀羅” (or “nine-mandala”, “eight Bodhisattvas Mandala”, etc.), that is, according to the “foding zunsheng tuoluoni niansong yigui fa 佛頂尊聖陀羅尼唸誦儀軌法” (translated into Chinese by Bukong不空 in the Tang Dynasty) (Liang 2007, vol. 10, pp. 20–21),20 the Piluchana Buddha and the eight Bodhisattvas formed a “nine-orientation” statues system (Luo 2010, pp. 189–216; Cheng 2017, pp. 81–109; Guo 2007, pp. 45–63; ** data, the dimensions of the statues on the fifth floor are as follows (see

Table 9. Dimensions of the statues on the fifth floor of the Pagoda of Fogong Temple (unit: m).

Name	Top Height	Total Height	Total Width of the Base	Clear Height	Clear Width	Height of Head
Piluchana Buddha	3.97	3.414	2.242	2.398	1.59	0.784
(South) Sarvanivāraṇaviṣkambhin	2.508	1.952	1.162	1.305	0.844	0.416
(Southwest) Ksitigarbha	2.514	1.958	1.188	1.294	0.828	0.431
(West) Avalokitesvara	2.588	2.032	1.173	1.379	0.891	0.463
(Northwest) Maitreya	2.574	2.018	1.2	1.372	0.837	0.442
(North) Ākāśagarbha	2.602	2.046	1.18	1.412	0.855	0.483
(Northeast) Samantabhadra	2.563	2.007	1.193	1.372	0.849	0.438
(East)Vajrapani	2.557	2.001	1.194	1.348	0.850	0.439
(Southeast) Manjusri	2.607	2.051	1.162	1.371	0.861	0.463
Average of eight bodhisattvas	2.564	2.008	1.182	1.357	0.852	0.447

):

3.5.1. Analysis of the Proportional Relationship between the Space and the Statue Complex

(1) The Height of the Fifth Story (from caisson to the ground, 7.109 m)/the Top Height of the Buddha (3.97 m) = 1.791 ≈ 9:5 (99.5% coincidence).

(2) The Height of the Fifth Story (7.109 m)/the Width of the Lower Buddha Altar (7.111 m)21 = 0.9997 ≈ 1 (100% coincidence).

(3) The Height of the Fifth Story (7.109 m)/the Height of the Ceiling (4.938 m) = 1.44 ≈$\sqrt{2}$ (98.2% coincidence).

(4) The Height of the Fifth Story (7.109 m)/the Top Height of Each Bodhisattva (average 2.564 m) = 2.773 ≈ 2$\sqrt{2}$ (98% coincidence).

In summary, it can be seen that the foundation, the ceiling, the statue complex, and the Buddha altar are all unified in design, and the proportion is perfect.22 (Figure 33).

3.5.2. Proportional Analysis of the Statue Complex

The Total Height of the Buddha (3.414 m)/the Total Height of the Bodhisattva (average, 2.008 m) = 1.7 ≈ 5:3 (98% coincidence).

3.5.3. Proportional Analysis of the Statues

The following can be seen from

Table 10. Proportional analysis of the statues on the fifth floor of the Pagoda of Fogong Temple.

Name	Total Height/Total Width of the Base Measured Value Ideal Value (Coincidence Degree)	Clear Height/Clear Width Measured Value Ideal Value (Coincidence Degree)	Total Height/Clear Height Measured Value Ideal Value (Coincidence Degree)	Clear Height/Height of Head Measured Value Ideal Value (Coincidence Degree)
Piluchana Buddha	1.523 3:2 (98.5%)	1.508 3:2 (99.5%)	1.424 $\sqrt{2}$:1 (99.3%)	3.059 3:1 (98%)
Average of eight Bodhisattvas	1.699 5:3 (98.1%)	1.593 8:5 (99.5%)	1.48 3:2 (98.6%)	3.036 3:1 (98.8%)
Name	Total Height: Total Width of the Base	Clear Height: Clear Width	Total Height: Clear Height	Clear Height: Height of Head
(South) Sarvanivāraṇaviṣkambhin	1.68	1.546	1.496	3.137
(Southwest) Ksitigarbha	1.648	1.563	1.513	3.002
(West) Avalokitesvara	1.732	1.548	1.474	2.978
(Northwest) Maitreya	1.682	1.639	1.471	3.104
(North) Ākāśagarbha	1.734	1.651	1.449	2.923
(Northeast) Samantabhadra	1.682	1.616	1.463	3.132
(East)Vajrapani	1.676	1.586	1.484	3.071
(Southeast) Manjusri	1.765	1.592	1.496	2.961

:

(1) The height-to-width ratio (3:2) of the Buddha statues on the fifth floor is the same as that on the first to fourth floors; the ratio of the total height to the clear height ($\sqrt{2}$) is the same as that of the Buddha statues on the third floor (Figure 34 and Figure 35).

(2) The height-to-width ratio of each Bodhisattva (including the clear height-to-width ratio and the total height-to-width ratio, taking the average value) is between 8:5 and 5:3 (close to the Golden Ratio); the head-to-body ratio (including bun) is about 1:3, and the head-to-body ratio without bun is about 1:4. All aspects above are close to the geometric proportion of Manjusri and Samantabhadra on the second floor (Figure 36 and Figure 37).

3.6. Proportional Relationship between the Buddha Statues on Each Floor

In addition to the proportional relationship between the interior space of each floor and the statue complex mentioned above, there are also proportional relationships among the Buddha statues (the main statues) on each floor of the pagoda.

3.6.1. Comparison of the Geometric Proportion of Statues on Each Floor

First of all, there are many similar techniques in the composition ratio of the statues on each floor of the pagoda, as shown in

Table 11. Proportional analysis of the Buddha statues on each floor of the Pagoda of Fogong Temple (A).

Name	Total Height/Total Width of the Base Measured Value Ideal Value (Coincidence Degree)	Clear Height/Clear Width Measured Value Ideal Value (Coincidence Degree)	Total Height/Clear Height Measured Value Ideal Value (Coincidence Degree)	Clear Height/Height of Head Measured Value Ideal Value (Coincidence Degree)
The giant Buddha on the first floor	1.484 3:2 (98.9%)	1.531 3:2 (98%)	1.217 6:5 (98.6%)	3.23 10:3 (96.9%)
The Buddha on the second floor	1.519 3:2 (98.7%)	1.524 3:2 (98.4%)	1.296 13:10 (99.7%)	3.568 3.5 (98%)
Four Buddhas on the third floor (average)	1.522 3:2 (98.5%)	1.466 3:2 (97.8%)	1.391 $\sqrt{2}$:1 (98.4%)	3.412 3.5 (97.5%)
The Buddha on the fourth floor	1.528 3:2 (98.2%)	1.53 3:2 (98%)	1.321 4:3 (99.1%)	3.46 3.5 (98.9%)
The Buddha on the fifth floor	1.523 3:2 (98.5%)	1.508 3:2 (99.5%)	1.424 $\sqrt{2}$:1 (99.3%)	3.059 3:1 (98%)

:

The following can be seen from the above table:

(1) The height-to-width ratio (including the clear height-to-width ratio and the total height-to-width ratio) of the Buddha statues on each floor of the pagoda is close to 3:2 (the coincidence degree of most data is higher than 98%, and only the coincidence degree of the clear height-to-width ratio of the Buddha statue on the third floor [taking the average value of the four Buddhas] is 97.8%). The ratio 3:2 is the most common modeling technique for the Buddha statues in the Pagoda of Fogong Temple.

(2) The ratio of the total height to clear height of the four Buddhas on the third floor and the Buddha on the fifth floor is $\sqrt{2}$:1 (this ratio also exists on the two seated Bodhisattvas on the second floor, namely Manjusri and Samantabhadra). The ratio of the total height to the clear height of the Buddha statues on the first, second, and fourth floors is also a relatively simple integer ratio.

(3) The head–body ratio of the Buddha statues on the second, third, and fifth floor is close to 1:3.5 (that is, “pan san ban 盤三半 (three and a half heads tall when sitting cross-legged) “, and the six Buddhas in the mural on the first floor also adopt this ratio). The head–body ratio of the Buddhas on the first and the fifth floor is larger, and that of the Buddha on the fifth floor is the largest, with the head occupying one-third of the clear height.

3.6.2. The Total Height Ratio of the Buddha Statues on Each Floor

Secondly, there is a subtler proportion arrangement between the Buddhas on each floor, and it is most likely designed with the giant Buddha on the first floor as the basic module (that is, the modulus). See

Table 12. Proportional analysis of the Buddha statues on each floor of the Pagoda of Fogong Temple (B).

Name	Total Height (unit: m)	Percentage of the Total Height of the Giant Buddha on the First Floor
The giant Buddha on the first floor	10.32	100%
The Buddha on the second floor	4.052	39.26%
Four Buddhas on the third floor (average)	2.729	26.44%
The Buddha on the fourth floor	4.274	41.41%
The Buddha on the fifth floor	3.414	33.08%
The sum of Buddhas on the third and the fifth floor	6.143	59.53% (≈60%, coincidence degree 99.2%)
The sum of Buddhas on the second and the fourth floor	8.326	80.68% (≈80%, coincidence degree 98.9%)
The sum of four Buddhas on the second to fifth floor	14.469	140.20% (≈140%, coincidence degree 99.9%)

:

As can be seen from the above table, if the total height of the Buddha on the first floor is 100, the sum of the total height of the Buddhas on the second and the fourth floors is about 80, and the sum of total height of Buddhas on the third and the fifth floors is about 60, the proportion of the above three is 5:4:3, that is, “ gou 3 gu 4 xian 5 勾三股四弦五, the Pythagorean theorem”.

Is this ratio a coincidence? If we carefully consider the Buddhist doctrine and discipline behind the layout of statues on each floor of the pagoda, we will find that this proportional relationship is probably the result of careful arrangement: the themes of the second and the fourth floors of the pagoda belong to Exoteric Buddhism (the second floor is possibly Huayan Sect). The third floor is dedicated to the Four Guardian Warriors, and the fifth floor is dedicated to the so-called “fo ding zun sheng man tuo luo 佛頂尊聖曼陀羅”, all belonging to the Esoteric Buddhism. Each lotus petal on the top of the giant Buddha’s Sumeru seat on the first floor is painted with a small Buddha, which is similar to the Sumeru seat of the Vairocana Buddha in Fengxian Temple of Luoyang Longmen Grottoes and the Tōdai-ji Temple in Nara, Japan (Nara National Museum et al. 2002, pp. 22–26)23 and may also belong to the Huayan Sect. The layout of statues on each floor together constitutes a religious system, which combined Exoteric Buddhism and Esoteric Buddhism, and some scholars have pointed out that this was an important feature of Buddhism in the Liao Dynasty (You 2013, pp. 27–32). Therefore, the statues on the second and the fourth floors (Exoteric Buddhism) are regarded as a group, and their total height is 4/5 of that of the giant Buddha on the first floor; the statues on the third and fifth floors (Esoteric Buddhism) are regarded as a group, and their total height is 3/5 of that of the giant Buddha on the first floor. On the one hand, this layout conforms to the corresponding Buddhist doctrine and discipline of each floor; on the other hand, it presents the proportional relationships between the Buddha statues on each floor clearly.

Further, the sum of the total height of the Buddha statues on the second, the third, the fourth, and the fifth floors (taking the average of the four Buddhas on the third floor) is just 1.4 times the total height of the giant Buddha on the first floor, approximately $\sqrt{2}$, and almost equal to the height of the first story of the pagoda (the sum of the total height of the Buddha statues is 14.469 m, and the height of the first story is 14.57 m, in which the coincidence is 99.3%). If we imagine that the Buddha statues on the second to fifth floors are stacked together, the total height is exactly equal to the height of the first story (and is equal to twice the distance between the head of the Buddha on the first floor’s mural); this should also be the result of careful design (Figure 38).

3.6.3. Ratio of the Height of the Pagoda to the Total Height of the Giant Buddha on the First Floor

Finally, it is particularly necessary to point out that the total height of the giant Buddha on the first floor is likely not only the module of the total height of the Buddha statues on each floor but also one of the important modules of the pagoda. According to our measured data in 2021, the total height of the pagoda (excluding the base) is 61.99 m. It follows that

The Total Height of the Pagoda (excluding the base, 61.99 m)/the Total Height of the Giant Buddha (10.32 m) = 6.007 ≈ 6 (99.9% coincidence). (N. Wang 2018b, pp. 216, 256; 2018c).24

According to the measured data of Bei**g University of Civil Engineering and Architecture in 1991, the total height of the pagoda is 65.88 m, and the height of the base (average) is 3.667 m, so the total height of the pagoda (excluding the base) is 62.213 m. The total height of the giant Buddha on the first floor is about 10.2 m. It follows that

The Total Height of the Pagoda (excluding the base, 62.213 m)/the Total Height of the Giant Buddha (10.2 m) = 6.099 ≈ 6 (98.3% coincidence).

According to the measured data of the China Academy of Cultural Heritage in 2011, the total height of the pagoda is 65.838 m, (Hou et al. 2016, p. 18) and the height of the base (the average value from the terrace to the top on each side) is 3.572 m,25 so the total height of the pagoda (excluding the base) is 62.266 m. The total height of the giant Buddha on the first floor is about 10.375 m. It follows that

The Total Height of the Pagoda (excluding the base, 62.266 m)/the Height of the Giant Buddha (10.375 m) = 6.002 ≈ 6 (the coincidence is close to 100%).

In summary, through the analysis of the measured data from surveying and map** in 1991, 2011, and 2021, there is a clear proportional relationship between the total height of the pagoda (above the base) and the total height of the giant Buddha on the first floor (6:1).26 (Figure 39). If the total height of the giant Buddha is H, then the total height of the pagoda (above the base) is 6H, and the sum of the total height of four Buddhas on the second to fifth floors is 1.4H, so the sum of the total height of Buddha statues on each floor (the average value of the four Buddhas on the third floor) is 2.4H. The ratio of the sum of the total height of the Buddha statues on each floor to the total height of the pagoda (6H) is 2:5.