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ABSTRACT
Octagonal buildings in ancient Korea were mostly constructed in the likeness of Buddhist pagodas that emerged during the Goguryeo era. The ancient Chinese books of Jiuzhang Suanshu and Yingzao Fashi confirm that, in constructing Fogongsi’s Yingxian Wooden Pagoda, the ancients could not design a full-form regular octagonal plan in the pre-seventeenth century before the introduction of Western mathematics. Eight-cornered monuments in Japan and Korea faced the same challenge. Thus, this study examines the acceptance and limitations of adapting octagonal drawing methods in Korean architecture from Chinese and Western mathematics. It conducts comparative studies of architectural and mathematical history in pre-modern and modern times. It reveals that field carpenters applied Yingzao Fashi’s formula in constructing an octagon via a square’s diagonal ratio. Further, the Western method of constructing regular octagons was introduced to the Korean Peninsula in the eighteenth century but was not utilized at work sites. Ultimately, the concepts in mathematics texts had a certain influence on the formative beauty of wooden constructions.
Disclosure statement
No potential conflict of interest was reported by the author(s).
Notes
1 In East Asia, there is a belief that the Sinhalese “dagoba” gradually became “pagoda,” though this derivation is subject to debate. Liang Sicheng contends that the term originates from the “ba jiao ta” 八角塔 (eight-cornered tower) because “ba jiao ta” might have been read as “pa go da” during the Tang Dynasty. The term “pagoda” became the accepted name for such monuments in European languages. In the southern region of China, in particular, the pronunciation of “ba jiao ta” was “pa-chiao-t’a” or “pa-go-ta” (Liang Citation1984). However, Liang was unaware that many eight-cornered towers or pagodas already existed in ritual shrines and have since been excavated in the Jilin Hwando 丸都 (Wandu, ch.) Mountain Fortresses, Gyeongju Najeong Well 蘿井, and Mingtang 明堂 (luminous hall) of Empress Wu from the Goguryeo and Silla period. Alternatively, the Dravidian term pagoda/pagavadi was derived from the Sanskrit bhagavadi (goddess, especially in reference to Kali) or the Persian butkada (temple) (Kim Citation2011, 116).
2 Under Japanese rule, Yoneda Miyoji analyzed the octagonal schematics of the Seokguram Grotto constructed in the eighth century and the ground plans of Goguryeo octagonal wooden pagodas from the fifth century. He also began examining Goguryeo’s octagonal building sites and noticed eight additional octagonal buildings for other purposes.
3 The mathematical books include Jiuzhang Suanshu Lizhu, annotated by Guo Shuchun, and Zhoubi Suanjing Lizhu, annotated by Chen Zhenyi and Wen Renjun.
4 The mathematical treatise comprises 13 books attributed to the ancient Greek mathematician Euclid.
5 Given Euclidean principles, it is impossible to draw a regular full-formed octagon with a non-scaled ruler and a compass.
6 In addition to the wooden construction, China also has a budo (small stone grave for housing the sarira or relics of a senior monk), a pagoda, and a building made of painted stone, judged to be of an early period. For example, regarding eight-sided votive pagodas in the northern Liang period (397–439) or the Dunhuang Grotto, there are eight-angled buildings with a polygonal plan within the Amitabha’s Paradise tableau (Amituo) in the north wall in Mogao Cave 107 and above a preaching scene of the Utmost-Superior-Dharani-of-the Buddha-Topknot’s tableau (Fuding Zunsheng Tuoluoni) centered on the south wall in Mogao Cave 217.
7 It is known that Matteo Ricci (1552–1610), along with Xu Guangqi (1562 ~ 1633) published the Jihe yuanben 幾何原本 (Elements of Geometry 1607) by translating Euclid’s Elements, a mathematical treatise consisting of thirteenth books attributed to the ancient Greek mathematician Euclid.
8 The construction of the Foxiangge Pavilion, inspired by the Huanghelou 黃鶴樓 (Yellow Crane Tower) under Qianlong (r.1735–1796), is located at the center of the front hill of Wanshoushan 萬壽山 (Longevity Mountain). It was rebuilt by Cixi 慈禧太后 (1835–1908) in 1891 after it was destroyed by Anglo–French allied forces during the Second Opium War in 1860. It was reconstructed again by Cixi in 1903 after it was devastated a second time by European forces during the Boxer Rebellion in 1900 (Steinhardt Citation2019, 305–306). In 1987, a building survey of the Foxiangge Pavilion found that most of the wooden structures were seriously damaged, including missing decorative parts and roof tiles on the first floor. The brick joints also appeared to have wide cracks. In July 1988, large-scale repair work started after the Summer Palace Management Office 頤和園管理處reported a comprehensive repair plan approved by the Municipal Garden Bureau 市園林局, which was finished on September 13, 1989. (http://www.bjmacp.gov.cn/cn/spec/pastdiscovered/viewinfo.aspx?tabid=300907&iid=90&categoryid=100007, accessed on January 10, 2021)
9 A recent study arithmetically proves that the octagonal drawing method of the Yingzao Fashi applies the ratio of 5:12.07:13.07; thus, it is not considered a perfect regular octagon (Zhang, Yang, and Xiao Citation2018, 98). Even in the case of the Yingxian Wooden Pagoda, the calculations of the measured drawings show that it does not have an equilateral octagonal plan (Chen Citation1981, 72–80).
10 In addition to Yumedomo Hall, the Octagonal Hall at Eizan-ji Temple 栄山寺 八角堂 (763–764) has a length of 10.78 chi (3266 mm, 1 chi = 30.3 cm) on one side of each octagon and is defined as a building with an equilateral octagonal plan because all sides have the same length.
11 The archaeological remains at Toseongri and Sangori Villages (fifth century) have no cornerstones or internal rows in the foundation. These octagonal shapes are more inaccurate than other building sites because most cornerstones are lost; however, a few remain. Such anonymous sites were excluded from this study [ and ].
12 Yoneda analyzed the octagonal building site in Cheongamri Village and identified two ways of making an octagonal plan. These methods were derived from the ceiling structure of Goguryeo burial mounds. The first method was to initiate a gradual reduction by rounding off the edges forming one half of a square. The second was to divide one side into three parts in a square, reducing it to eight angles. Thus, Yoneda considered the construction of an octagonal building on the temple site from a square-based octagon and, consequently, considered the main building at the Cheongamri site as based on a regular square-based octagonal plan. It is very reasonable to draw the octagon by making two points on one side of a square. However, considering the inscribed circle of the square, the length of the two sides of the isosceles triangle at each corner, excluding the diagonal, is too large to produce a regular octagon [].
13 Assumedly, it has an octagonal plan on a platform; only the foundation remains are visible on the inner platform, while the rows of cornerstones are invisible. Further, the inner (outer) angle of the octagonal surface on the outermost rain gutter side is 135° (45°).
14 Chinese reports and recent research papers define octagonal buildings as square octagonal plans.
15 Although no column bases remain on the building site, the foundation remains, and the traces of platform siding have been maintained in relatively good condition.
16 In the actual measurement report, it was not described as an octagon since there are no cornerstones; only the approximate distance between the columns was described.
17 This temple site was named Heungnyunsa Temple when the Japanese surveyed the Silla era temples in Gyeongju in the 1910s. In 1976, however, tiles with the characters “xxx廟之寺” pressed into them were found there, and questions about the location of Heungnyunsa and Yeongmyosa were raised as soon as roof tiles with the name of Yeongmyosa engraved on them were found. Consequently, the name of the temple site was restored to Yeongmyosa Monastery 靈廟寺, not Heungnyunsa, which was built when Silla was under the reign of Queen Seondeok (r. 632–647). The Yeongmyosa Monastery site was investigated in test excavations in 1972 and 1977 and between 1978 and 1981, following the conclusion of the tests. Heungnyunsa 興輪寺 was the first Buddhist monastery ever established in Silla.
18 The first construction layout consisted of one pagoda and two halls arranged to the north and south of the pagoda respectively. In time, the temple fell into ruin and was then wholly reconstructed during the Unified Silla period (668–935 CE).
19 As per the Precision Measurement Survey Report on the Hwangudan, there are various dimensions between each column; however, an equilateral octagonal plan is considered by default.
20 Kim states that the method of dividing a regular quadrilateral in the octagonal drawing method is currently used by carpenters in the field. The same method is used when constructing a round purlin. In his writings, the term “regular octagon” is not used.
21 Regarding Korea, the Jiuzhang Suanshu and Zhoubi suanjing were recorded in Samguk Sagi 三國史記 (History of the Three Kingdoms) during the Unified Silla period in the middle of the seventh century. However, given the previous system, laws and ordinances, and astronomical systems, they may have been transmitted to the Baekje and Goguryeo dynasties in the fourth to fifth centuries. Moreover, in Japan, an arithmetic system similar to that in China was recorded in the Nihon Shoji in the sixth century.
22 They refer to the classical mathematical books from the Chinese Han Dynasty to the early Qing Dynasty as the official mathematical texts for imperial examinations in mathematics, as follows: Zhoubi suanjing (Zhou Shadow Mathematical Classic), Jiuzhang Suanshu 九章算術 (The Nine Chapters on the Mathematical Art), Haidao Suanjing 海島算經 (The Sea Island Mathematical Classic), Sunzi suanjing 孙子算經 (The Mathematical Classic of Sun Zi), Zhang Qiujian suanjing 張邱建算經 (The Mathematical Classic of Zhang Qiujian), Wucao suanjing 五曹算經 (Computational Canon of the Five Administrative Sections), Xiahou Yang suanjing 夏侯陽算經 (The Mathematical Classic of Yang Xiahou), Wujing suanshu 五經筭術 (Computational Prescriptions of the Five Classics), Jigu suanjing 緝古筭經 (Continuation of Ancient Mathematical Classic of Wang Xiaotong), and Zhui shu 綴述 (Method of Interpolation of Zu Chongzhi)
23 Xu likewise proved in Celiang yitong 測量異同 (Similarities and Differences in Measurement 1608) and Gougu yi 句股義 (Principle of Base and Altitude 1609) that Chinese gougu surveying bore theoretical, methodological, and instrumental similarities and differences with Western geometric-square surveying (Hashimoto and Jami Citation2001, 268).
24 Many studies have shown that the contents of the Jihe yuangben stem from lecture notes written by the French Jesuits Jean-François Gerbillon (1654–1707) and Joachim Bouvet (1656–1730). They do not refer to the fact that the first Chinese version of Euclid’s Elements bore the same Chinese title, the Jihe yuangben (1607). It was translated by Matteo Ricci (1552–1610) and Xu Guangqi 徐光啟 (1562–1633) (Ju et al. Citation2016, 111).
25 The Shuli jingyun describes the six-sided figure inscribed in a circle, the four-sided inscribed in a circle, the six-sided circumscribed in a circle, and the four-sided circumscribed in a circle.
26 There is evidence of proportional compasses provided by Fabrizio Mordente (1532–1608) and Thomas Hood (d. 1598) even earlier than Galileo, but these geometric instruments had functional operations inferior to the Galilean compass. (Pisano and Bussotti Citation2015, 213)
27 Fang met Adam Schall von Bell in 1659 in Beijing and satisfied his great desire of studying Western mathematics under the Polish Jesuit Johannes Nickolaus Smogulecki (1610–1656).
28 Korean scholars referred to the Xiyang xinfa lishu 西洋新法曆書 (Treatise on Calendrical Science 1645), later re-edited with the title, the Xinfa suanshu 新法算書 (New Methods in Mathematics 1666), and the Tianxue chuhan 天學初函 (First Collection of Writings on Heavenly Learning, 1626) (Jun Citation2006, 482–483).
29 Although Korean mathematics borrows from ancient China, it did not change with Chinese mathematics trend since the seventeenth century. Paradoxically, Korean mathematics developed, while Chinese mathematics declined during King Sejong’s regime (r. 1418–50).
30 According to the introductory (Qujingwei) section of the Yingzao Fashi by Li Jie (?–1110), the preliminary mathematical explanations end with an explicit quotation from Li Chunfeng’s commentary on the Jiuzhang suanjing “營造法式,取經圍: 九章算術及約斜長等密率修立下條. 諸徑圍斜長依下頂.”
31 Mathematical ideas in Korea mirror the basic ideas of ancient China, and it is necessary to refer to the systematic arithmetical ideas in the Lulizhi of Hanshu 漢書 to precisely ascertain them. Lulizhi 律歷志 (Treatise on Harmonics and Calendrics) says “The counting of numbers fits well with all things in the universe 萬物氣體之數, 天下之能事畢矣.” In this case, the basic propositions integrate the tune音律, the astronomical calendar 曆法, the art of divination 易數, and the weights and measures 度量衡. That is, the basic scales and mathematical systems of ancient architecture are indirectly included in this scope.
32 The Korean Mathematical History explains geometrical constructions in art crafts and architecture and the Yin-Yang and Five Element thought in the architectural plan, metrological system, and scale, revealing a close rapport with the logic of mathematics.