Korean Traditional Patterns: Friezeand Wallpaper

Hyunyong Shin, Shilla Sheen, Hyeyoun Kwon, and Taeseon Mun

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Frieze Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Wallpaper Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Some Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Abstract

All frieze and wallpaper patterns in Korean tradition are presented with theirlocations. They can be recognized audibly as well as visually through analyzingsymmetries of patterns. Two Korean traditional music instruments play the sevenfrieze patterns. A piano music is also introduced. Music can be accessed throughQR-codes. A cover of umbrella of regular 17-gon is proposed by the 24 patterns.

H. Shin (�)Korea National University of Education, Cheongju-si, South Koreae-mail: shin@knue.ac.kr

S. SheenGraduate School, Korea National University of Education, Cheongju-si, South Koreae-mail: mathesign@naver.com

H. KwonGyeongsangnam-do Office of Education, Changwon, South Koreae-mail: jihoosung@korea.kr

T. MunSeoul Metropolitan Office of Education, Seoul, South Koreae-mail: mathti@hanmail.net

© Springer International Publishing AG, part of Springer Nature 2018B. Sriraman (ed.), Handbook of the Mathematics of the Arts and Sciences,https://doi.org/10.1007/978-3-319-70658-0_17-1

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KeywordsFrieze · Wallpaper · Symmetry · IUC notation · Orbifold notation

Introduction

From a geometric point of view, there are 7 types of frieze patterns and 17 types ofwallpaper patterns (Conway and Huson 2002). In this article, the Korean traditionalfrieze and wallpaper patterns for each type are presented, and their locations areshown in the map (Fig.1) by Roman characters. For example, the frieze pattern oftype p1(∞∞) can be found in Seoul which is coded as I.

Fig. 1 Locations

II

I

XI IX

X

IIIIV

V

VI

VIII

VII

JAPAN

NORTH KOREA

CHINA

RUSSIA

SOUTH KOREA

Korean Traditional Patterns: Frieze and Wallpaper 3

In presented patterns, the color has been ignored to make our classification simple(Grünbaum 2006). Most patterns are from South Korea, Republic of Korea, and theothers are from North Korea, Democratic People’s Republic of Korea.

The IUC notation(Macbeath 1966; Blanco and de Camargo 2011) as well as theorbifold notation(Conway and Huson 2002) for the types of pattern are used. Thesymbols of symmetries can be found in various literatures (Blanco and de Camargo(2011), for example), but they speak for themselves.

Frieze Patterns

The symmetries of each frieze pattern are quite obvious. The fundamental regionsshowing symmetries are omitted in this article.

p1, ∞∞Bo (Dish for ritual offering), National Palace Museum of Korea, Seoul, South

Korea (I)

p11m, ∞*Girder, Donghak Temple, Choongchungnamdo, South Korea (IV)

p1m1, *∞∞National Treasure of Korea No. 68, Gansong Museum, Seoul, South Korea (I)

p2, 22∞Brass Censer, Sukwang Temple, Kangwondo, North Korea (IX)

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p2mm, *22∞Girder, Ryongheung Temple, Hamkyungnamdo, North Korea (X)

p11g, ∞×Dodecagon Porcelain Dish (Editorial Board of Chosun Relics 2002), North

Korea (XI)

p2mg, 2*∞Girder, Ryongheung Temple, Hamkyungnamdo, North Korea (X)

Wallpaper Patterns

The fundamental region of each of 17 types of wallpaper pattern is tabulated inTable 1.

Now the examples of traditional wallpaper pattern of Korea for each type arepresented with fundamental region showing symmetries. Some patterns can befound in Lee (1995), LEEUM (2014), Lim (2004), Shin and Sheen (2014a,b) orShin et al. (2014).

Korean Traditional Patterns: Frieze and Wallpaper 5

Table 1 Types of Wallpaper

p1, oNational Treasure No. 60 of Korea, National Museum of Korea, Seoul, South

Korea (I)

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pm, **Flower Door, Soodug Temple, Choongchungnamdo, South Korea (II)

pg, ××Cotton Jacket, Sook Myung Womens University Museum, Seoul, South Korea (I)

Korean Traditional Patterns: Frieze and Wallpaper 7

cm, *×Porcelain Dish with Wave Pattern (Editorial Board of Chosun Relics 2002),

North Korea (XI)

p2, 2222Bamboo Basket, Korea Bamboo Museum, Chollanamdo, South Korea (VI)

In all the patterns in this article, the interlacing is considered. In fact, if it is notconsidered, the pattern may be of different type. On this issue, Grünbaum (2006) isrecommended.

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pmm, *2222Flower Door, Dae Seung Temple, Kyungsangbookdo, South Korea (VII)

pmg, 22*National Treasure of Korea No. 198, Gyeongju National Museum, Kyungsang-

bookdo, South Korea (VII)

pgg, 22×Bamboo Winnow, Korea Bamboo Museum, Chollanamdo, South Korea (VI)

Korean Traditional Patterns: Frieze and Wallpaper 9

cmm, 2*22Door, Magok Temple, Choongchungnamdo, South Korea (III)

p3, 333Korean Kite with Samtaegeuk Pattern, Private Possession

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As mentioned before, color is ignored in all patterns.

p3m1, *333Flower Door, Soodug Temple, Choongchungnamdo, South Korea (II)

p31m, 3*3Girder, Naeso Temple, Chollabookdo, South Korea (V)

p4, 442Dining Room, Changduk Palace, Seoul, South Korea (I)

Korean Traditional Patterns: Frieze and Wallpaper 11

p4m, *442National Treasure of Korea No.95, National Museum of Korea, Seoul, South

Korea (I)

p4g, 4*2Door, Changduk Palace, Seoul, South Korea (I)

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p6, 632Bamboo Wife, Korea Bamboo Museum, Chollanamdo, South Korea (VI)

p6m, *632Flower Door, Boolgook Temple, Kyungsangbookdo, South Korea (VII)

Korean Traditional Patterns: Frieze and Wallpaper 13

Some Designs

• Original colorsThe following design consists of seven concentric circles with seven frieze

patterns with the original colors as in this article. Through QR-code below, apiano music that presents the friezes can be heard.

• Obangsaek(Korean traditional colors)The following design consists of seven concentric circles with 7 frieze patterns

with Obangsaek, the Korean traditional colors. Through QR-code below, aKorean music that present the friezes can be heard. The music is played byProfessor Chung Tae Seok of Seoul National University and Ms. Chung YunSoo, a teacher of Korean National School of Korean Music.

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• UmbrellaPrime number “7” is special because 7 is the first prime number p such that

regular p-gon is not constructible by straight edge and compass. Prime number“17” is also somewhat remarkable because 17 is the first “nontrivial” primenumber p such that regular p-gon is constructible. It may be also worthwhileto remember that 7 and 17 are the first two full reptend primes.

Now we construct regular 17-gon and get 7 concentric circles with 7 friezepatterns in this article and get 17 wallpaper patterns also from this article at eachside of the polygon. This sheet may be used as a cover of an umbrella of regular17-gon.

Korean Traditional Patterns: Frieze and Wallpaper 15

Conclusion

Group theory classifies types of frieze and wallpaper patterns. This article presentsall types of them from traditions of Korea. Through mathematics visual patternscould be presented as music. The seven frieze patterns can be “heard” by Koreanmusical instruments as well as by piano.

References

Blanco MFB, de Camargo HALN (2011) Symmetry groups in the Alhambra. In: Visual mathemat-ics, vol 13. Mathematical Institute SASA, Beograd

Conway JH, Huson DH (2002) The orbifold notation for two-dimensional groups. Struct Chem13(3/4):247–257

Editorial Board of Chosun Relics (2002) Cultural assets of North Korea I, II, III, IV. Seoul NationalUniversity, Seoul

Grünbaum B (2006) What symmetry groups are present in the Alhambra? Notices of the AMS53:670–673

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Lee Y (1995) Relics of palaces (2). Daewonsa, SeoulLEEUM (2014) LEEUM collections of Korean old arts. LEEUM, SeoulLim Y (2004) Traditional patterns of Korea. Daewonsa, SeoulMacbeath AM (1966) The classification of non-Euclidean plane crystallographic groups. Can J

Math 19:1192–1205Shin H, Sheen S (2014a) A mathematical approach to pattern: classification of Korean traditional

frieze patterns according to group theory. Arch Des Res 27(3):295–311Shin H, Sheen S (2014b) Korean traditional frieze/wallpaper patterns according to mathematical

classification. Presented at ICM2014, SeoulShin H, Sheen S, Mun T, Kwon H, Lee Y (2014) A study on development of teaching/learning

materials based on wallpaper patterns. J Korean Soc Math Educ Ser A: Math Educ 53(3):433–445