Geographical Review of Japan Vol. 57 (Ser. B), No. 1, 1•`21, 1984

The Codes of Japanese Landscapes -An Attempt to Topological Geography

-Ichiro SUIZU*

In the articulations forming a landscape on various levels, we can recognize the relation

ships of "material E morpheme c constituent," and "constituent •¼ part of a landscape." A "region" can be regarded as a unity of places where the articulations of landscapes serve

as a ground and figure for actions to be physically performed, and where the loci of these

performances are stratified to reform and reorganize the given articulations. Thus, we first notice that the existence of relations which are latent in a "region" corresponds to the structure of a language. Since ancient days, the Japanese have had the concept of "ma," a concept related to a

space-time continuum. A versatile concept, it further connotes even relative relationships among objects in the space-time realm. In other words, various performances have been

considered to manifest themselves in concrete forms in the realm of "ma." Accordingly, it is presumed that among the traditional codes (langue) which have determined performances

(parole) based on the background of various articulations of landscapes consisting of morphemes

peculiar to Japan, there are some codes which contribute to providing a unique structure for "ma." This paper aims at varifying the existence of these codes on the basis of the results of

studies on the historical geography in Japan and at clarifying their characteristics from the

viewpoint of comparative geography. Furthermore, this paper aims to elucidate the codes that have some aspects corresponding to the topological space beyond Euclidean space and

to give a concrete explanation to the morphogenesis of landscapes in conformity with the theory of bifurcation in topology. In addition, the author intends to clarify part of the

topological underlying structure of "regions" in general.

Actions which are unfolded with a land mum units which are meaningful to human scape (Landschaftsbild) as a background, life. We call these the "morphemes" of a produce hidden multidimensional loci of landscape. For instance, a house, which is a performances. Since no corporal actions are constituent of a landscape, has such morphemes performed independently of a three-dimentional as floors, pillars, doors, a roof and eaves, as landscape, a close reciprocity exists between well as furniture and various tools. And a a space for these concrete performances and parcel of farmland, another constituent, has the structure of landscape (NAKAMURA,1982). morphemes including furrows, ridges and Therefore, no topological spatial theory on arable soil. Here, let us presume that these corporal actions can be established abstractly morphemes are corresponding to monemes in without regard to the analysis of a landscape linguistics. Just as each moneme is a unity or its constituents. It is suggestive that even of phonemes, so each morpheme is composed in the field of topology, which is supposed to of many varied materials such as wood, stone, have little relation with visual images, R. metal and soil. Taking the form of Fliese or THOM systematized dynamic theories of mor Okotop in nature, these materials are artificially phogenesis (THOM, 1977). extracted from nature and then processed into In disassembling a landscape into smaller morphemes. pieces, we find that they are reduced into mini Only when landscapes constituents as struc

* Department of Geography , Kyoto University, Yoshida-hon-machi, Sakyo-ku, Kyoto 606, Japan. 2 I. Suizu tural units composed of morphemes are combin We can see the following corresponding ed into a small or large landscape or a part relationships among the units of a landscape, of it with articulations at different levels in psychology and language (See Table 1). accordance with specific codes, does a world Thus, preparations have been completed for meaningful to communal life manifest itself . systematically analysing the structure of a "region" as a unity of places for f As long as landscapes or their constituents unctions (space are constructed by individuals, they differ for performances), that is, a set of landscapes with individuals. However, as long as human with a group of functions integrated in it beings are social beings, the functions and (region X)1), with an analogy to language structures of landscapes, or their constituents (Suizu, 1982a). A landscape is a corporal belonging to each human group, must have a extension (MERLEAU-PONTY,1945). system common to them. These landscapes and their constituents can be compared to I. The codes of historical landscapes in Japan parole in linguistics, while the system corre sponds to langue (SAUSSURE,1916). The codes of landscapes are corresponding But, the establishment of a structure com to the grammar of langue, while each land parable to a phonological structure is far from scape is comparable to parole (Sulzu, 1980). easy in geography. A language itself has no In this thesis, I would first like to examine form and is composed of a one-dimensional the geographical langue by focusing on some chain parlee. Therefore, it may seem extreme examples of (1) landscape construction patterns ly different in nature from "regions" under recognized in sacred or non-daily structures lain by landscapes which are composed of such as Buddhist temples, Shinto shrines, and concrete shapes on a land surface. Never historic gardens and architecture, and of (2) theless, in order for a parole organized on a landscape representation patterns seen in chain parlee to actually fulfill the function of paintings and drawings. Then, I would like communication, there arises the necessity of to proceed to prove the existence of codes of a "place for communication" between the historical landscapes in Japan by verifying speaker and the listener, a place beyond a that these patterns have been also applied to one-dimensional realm. When various actions individual daily landscapes. This will be then are performed in one place, the place also followed by the clarification of corresponing serves as a "place for communication" which relationships between these codes and "mathe assists in performing the actions more smoothly matical reality." by the use of parole as a systematic whole of symbols. Therefore, "places for communi 1. Landscape construction patterns cation" may well be regarded as a "space (i) The pattern of marking off (kugiru) for performances" in a common phase between (ƒÒ00) •¿linking (tsunagu) (ƒÒ11) geography and language. A sacred realm and a secular realm: These

Table 1. Corresponding relationships among the units of a landscape, psychology and language Codes of Japanese Landscapes 3

two realms are symbolically marked off by such morphemes as a torii (Shinto shrine archway) 2. Landscape representation patterns , gate, moat and bridge, shimenawa (sacred Black-and-white landscape paintings called Shinto straw festoon stuck with white paper sansuiga, bird's-eye-view paintings, pictures which has been cut into pieces) and kekkai of scenic spots on trips (dochu zu) , and paint (wooden fence erected in the main building ings of the inside and outside of Kyoto-all of a temple to show a boundary between the these traditional Japanese paintings share some clergy and laymen), which are too permeable landscape representation patterns such as the to be definite boundaries. There are transi incorporation of scenic spots, the movement tional spaces. of the focus and successive changes of scenes (ii) The patterns of dominating (shita (YAMORI, 1982). These representation patterns gaeru) (ƒÒ21), setting off (hikitateru) and are often in reverse perspective and bear re complementing (tsumaninaru) (ƒÒ23) (rock semblance to the above-mentioned landscape garden arrangement or huseki pattern) construction patterns. Especially , the patterns The arrangement of rocks in Japanese-style of "moving," "fusing," "abbreviating" and "linking" are domi gardens: With the layout of the Japanese nant. garden grasped in its totality, principal stones All these construction and representation (trump stones) are placed and then supple patterns were completed under the influence mentary stones are arranged among them. of the natural and cultural climate of Japan The "space" between the stones is not mere in and after the medieval ages in close relation empty space but it conveys a tense atmosphere with such factors as the qualities of construc characterized by the patterns of "dominating," tion materials typified by timber and bamboo "setting off" and "complementing , ." Here is and the distinct changes of the seasons as a way of placement due to circumstances. well as Buddhism and the principle of ten-chi

(iii) The patterns of likening (mitateru) jin (a concept of aesthetic equilibrium marked (ƒÒ31), fusing (tokeau) (ƒÒ32) and incorporating by dissymmetry and dynamic equilibrium) (ikedoru) (ƒÒ33) which is incorporated not only into the arrange Gardens of the four seasons: The gardens ment of rock gardens but also into the sectors of the four seasons are those adapted to each of flower arrangement and tea ceremony. And season and laid out in miniature in scenic among these patterns, mutual infiltration is spots (shukkei) to induce man's immersion into recognized. and harmony with the landscape. The gardens ƒÒ00_??_ƒÒ11_??_ƒÒ21_??_ƒÒ31_??_ƒÒ41_??_ƒÒ51 were used to be likened to the Pure Land

(Sukhavati) in Buddhism and so on. However, the discussion of aesthetics of (iv) The patterns of moving (ugoku) (ƒÒ41), these non-daily patterns is not the purpose of changing (utsurou) (ƒÒ42) and curving this paper. Aesthetics should be studied in (mageru) (ƒÒ43) the fields of architectural history or art history. Landscape gardens in the go-round style: As What is required in geography is to verify is indicated by the expression "the visualiza that even in urban areas and rural villages tion of movement," which is used to describe daily functions {ƒÒ} are apt to be combined ex these gardens, they are characterized by the clusively with specific groups of morphemes change in the viewpoint and successive changes instead of others in conformity with the above of scenes. mentioned patterns, and thereby to prove that

(v) The patterns of irregularizing (kuzusu) these patterns have served as the codes of (ƒÒ51) and abbreviating (habuku) (ƒÒ52) historical landscapes in Japan. Tea ceremony houses and karesansui (dry As space is limited in this paper, I would Japanese gardens): These tea ceremony houses like to present only an outline of this veri and gardens are marked by simplification, fication by giving some examples, but the symbolization and irregularization. pattern of abbreviating, which has been re peatedly noted by many scholars, will not be included. 4 I. SUIZU

groove along them. In fields in Europe, where 3. The pattern of marking off •¿ linking irrigation grooves are not so much required, Open verandas (nure-en), gardens and gates: phenomena like this are seldom seen. Instead The inside and outside of a house or a room in Europe, fields (Zelgen) are often surrounded are not only marked off but also linked by by pasture fences in order to prevent livestock these morphemes. Curtain screens (kicho), rattan from going into the fields where crops are or bamboo blinds (sudare), folding screens being raised. Thus, fence landscapes (Hecken (byobu) and paper-sliding doors (shoji) also landschaften) are prevalent. It is suggestive that serve in the same way, though they are all the word " Zelgen " in "Dreizelgenwirtschaft" used indoors (SUIZU, 1982b). Space under the in the medieval ages is derived from the word "Zaun" meaning "fence eaves (nokishita) also functions as a joint space ." Such a confronta where the inside and outside of the house tional concept in Europe as is indicated by the interact with each other. fact that the words "Umfriedung" (enclosure)

Streets : Kyoto, where a grid pattern was and "Frieden" (peace) have the same etymology introduced from China in ancient days and is not applicable to Japan (BOLLIVOw,1963). its blocks were marked off by the streets, As was shown above, "marking off" and "linking" in Japan are closely related with began to show changes in communal formation in and around the 15th century: townsmen's each other. In this sense, the situation is houses standing on both sides of a street quite different from that in Europe, where a began to form a new communal unit called sharp distinction is recognized between these "machi ," or "town." Thus, so-called "ryo codes. This characteristic in Japan has much gawa-machi," or "two-sided towns" of towns to do with the existence of kaiwai (activity men came into existence (ASHIKAGA, 1971). space which has no distinct boundary between Mostly formed in the shape of a hexagon with a central part and its vicinities), mutual inte a north-south street as the axis, these units, gration between urban and rural areas (urban which were almost the same in size, made up rural continuum), and transfer of a central reticular networks. In castle towns in the area and its environs. In Japan, neither capital Edo Period (1603•`1867), too, streets fulfilled cities in ancient times nor castle towns in the the unique function of "marking off •¿ linking" Edo Period had a solid city wall as cities on the inside and outside of the mercantile the continent did (SUIZU, 1976b; TAKAHASHI, houses along them, with the houses having a 1982). In addition, as is shown by the common lattice door, komayose (a low fence to prevent expression "a rural area in the capital of horses and pedestrians from coming near the Kyoto (Kyo ni inaka an)" or "a mountain house), noren (sign curtain), doma or toriniwa cottage near the city (zahen no yamazato)", few cities were endowed with absolute social and (earth-floored corridor), and tsubo-niwa (court yard). It cannot be denied that these helped cultural values. Furthermore, it was not rare to delay the development of streets as a public that the central government office was trans zone. In contrast to the situation in Japan, ferred from the inside of the capital to an plans to make the streets wider and straight outside rural area. began to be carried out in Europe in and around It should be noted that Japan was considered the 15th century. And this later led to baroque in ancient times to be "the Central Country urban planning. With each street and the (Nakatsunokuni) of Abundant Reed Plains and facades of the buildings along it integrated Rice Fields," as is mentioned in the Kojiki into one, a harmonious exterior space was (The Record of Ancient Matters, a chronicle completed as a public zone. compiled in 712); but it came to be regarded Ridges along irrigation grooves: Ridges be in the medieval ages as "a small country in tween arable land usually serve as a bound the peripheries of the world (zokusan-hendo)," ary in terms of possession or management. which later led to attempts to give a positive But in Japan, it is often the case that the significance to the Buddhist idea of "onri-edo" owners of adjacent rice fields jointly form a (regarding this world as an impure one and small group (mizu gumi) to share an irrigation avoiding it) in relation to the pious faith in Codes of Japanese Landscapes 5 the realization of the Pure Land in this insular 5. The patterns of curving and irregularizing country (UCHIDA, 1971). These patterns plays an active role in the 4. The pattern of fusing, likening and avoidance of straight lines, which are seldom incorporating found in nature, and in the adoption of indented

The patterns of fusing, likening and in streets (sumichigai), dead-end paths (hukuro corporating may well be called an extension komichi), partial curvature of the axial lines of the above-mentioned pattern of "marking of roads, recessed space (kubomi), non-parallel off •¿ linking." They serve to re-link what have system (sumikake), and obstruction of distant been partitioned off. In other words, they views (tomishadan). These morphemes or have a unique function of connecting a place groups of them served as a factor to "curve" which is close to one, with the external world. or "irregularite" the continental gridion street The pattern of fusing: Among the ex pattern which had been introduced from China, amples of the fusion of nature with human and thereby contributed to adding variety to beings are a solitary pine tree on the outskirts urban and rural life in and after the medieval of a village which meets or sees off travelers ages. No geometrical uniformity could be

(demukae or miokuri no matsu), a pair of seen in the layout in the regrouping of former sloping roads leading to a temple or shrine, hamlets (Verdorfung) in the medieval ages, one with a rougher slope (manly slope= either. Instead, the formation of those settle otokozaka) and the other with a softer slope ments was characterized by a deviation from

(womanly slope=onna zaka), and a spot appro the ancient rectangular-pattern allotment of priate to enjoy a beautiful view of Mt. Fuji land, frequent curvature of the axial lines of (Fujimi-cho), where many accounts of the streets, and the irregularity of the location of intimacy between man and nature have been the centers of the communities (SUIZU, 1965; given. KINDA, 1971). Even the temple or shrine The pattern of likening: Remote areas be which functioned as a symbol of spiritual yond the sea and the boundaries of villages unity of each community was usually located or towns were often likened to the Pure Land at one side of a settlement. Layout patterns believed in Buddhism to exist far to the west, like this were inherited and applied to the and to the Children's Limbo (sainokawara) planning of castle towns later in the Edo respectively. And a group of places within Period, too (FUJIOKA, 1962; YAMORI, 1970). a small area were compared to fudasho These patterns have been heretofore treated (Buddhist temples where amulets are issued lightly in morphological analyses because they to pilgrims) on a noted pilgrimage route as a seemingly look "irregular." smaller version of its original counterpart. Now, it is necessary to prepare for new The idea of reducing a landscape to a smaller analytic methods which make it possible to scale (shakkei) is also found in a miniature study these irregular patterns in comparison landscape stone and a dwarf tree on an alcove. with the "mathematical reality" and thereby The pattern of incorporating: An external to surpass conventional morphological theories. distant landscape is "incorporated" into a For, as will be mentioned in detail later in view of a dwelling as the background (shakkei, this paper, some of these patterns are presumed or borrowing space). to be corresponding to mathematical bifurca These patterns can be considered to be re tion sets. For instance, some winding roads lated to our ancestors' traditional custom of are corresponding to the "butterfly set" in worshipping remote mountains from their catastrophe theory and some irimoya style villages where a secular life was led, regard roofs (roofs with two slanting surfaces on the ing them as sacred. Especially the idea of upper part and four on the lower part) to the "cusp catastrophe shakkei is suggestive of a primitive form of ." map which probably played a symbolic part When combined with other patterns, these in representing a sacred realm on ancient patterns of curving and irregularizing serve festive days. as codes peculiar to Japan. 6 I. SUIZU

village where the deities were believed to re 6. The patterns of dominating, setting off side (HORI, 1982). With this as a fundamental and complementing structure, the codes of "dominating," "setting These patterns are recognized in layouts in off" and "complementing" are employed in which the principal elements such as the res various forms in accordance with the principle idential quarter and streets in a settlement, of "the rock garden pattern" which allows and the castle, its main gate and the temples variations suitable to the situation. Thus, or shrines in a castle town were first laid out landscapes with both variety and flexibility and then the other elements were arranged were formed. in accordance with the situation. Similar to

patterns employed in process planning, these II. The geography of ma patterns are different from static ones which can be analyzed by the use of random distri Interacting with one another, all the codes bution, regular distribution or agglomerative mentioned above have served as "conventions" distribution. In layouts like this, in which conductive to the formation of historical land

principal elements are first arranged with the scapes peculiar to Japan. Judging from the whole geographical situation taken into account, fact that these "conventions" are recognized it is natural that less importance should be in the field of literature including haiku attached to the gridrron street pattern, and (seventeen-syllabled poems) and renga (linked thus the patterns of curving and irregularizing verse) and in the realm of theatrical perform come to the fore. And as a natural conse ances including No plays, they may well be

quence of the initial arrangement of the prin referred to as the patterns of Japanese culture cipal (dominating) elements, the elements itself. One could safely say that these codes which "set off" or "complement" them are are equivalent to the structure of the Japanese then arranged. Arrangements of this kind langue (frequent dropping of the subject, were closely related with and influenced by special function of particles, order of words "the idea of fusui (the wind and water) ." and so on).

Originally introduced from China, this topo Naturally there must be a divergence of

graphical idea served as more than a mere opinions on the interpretation of these codes folk belief (UNNO, 1982). This idea, which as being peculiar to Japanese landscapes be explains that a community with mountains cause Europe does have some irregular parcels behind it to the north, hills to the east and of land, and some plans using the linear allot west, and water before it to the south is geo ment of land were carried out in Japan, as is graphically ideal, was suitable almost to the typified by the linear ridges between rice natural environment of Japan including its paddies opened up in the Edo Period (shinden). topographical features. The idea of locating But, the use of these codes in Japan implies a community in the bosom of hills was a re not a mere orientation toward disorder but flex of experimental wisdom. The pattern of the creation of a world with positive meanings "marking off •¿ linking" is also recognized in through irregularity. Landscape patterns in this idea. A fundamental composition often Japan marked by the integration of such seen in the formation of villages in accordance factors as the changeability of marking off, with this notion is as follows. First, the res an active deviation from the gridiron street idential quarter is laid out at the pivot of an pattern, and a propencity for curved lines can alluvial fan and then the kadota (the fertile be considered to be the underlaying and unique rice paddies in front of the settlement) are attitude toward landscape formation and its arranged. Plots of land in the village, marked harmony with nature which is hardly found out with ridges and footpaths in conformity in Europe. In contrast, landscape formation with the contour lines, and with the fields against nature in Europe is characterized first expanding in a fan-shaped alluvial plain, are of all by geometrical patterns. For instance, gently slanting and tapering off toward the geometrical patterns were adopted by the imposing mountains (Oku-yama) behind the Roman Empire. They were inherited by the Codes of Japanese Landscapes 7

Frankish Empire in the form of orthogonal (b) limited time as in "hiruma" (day land allotment plans including the national time), "tsukanoma" (a short period of colonization plan (frankische Staatskolonisa time) tion). Furthermore, in and after the 12th (2) intermitted time as in "taema" (inter century, settlements with an open space in mission) the shape of a square or a circle became pre (3) required time as in "ma ga nai" (to have valent both in urban and rural areas. Espe no time) cially villages in accordance with a plan of (4) discontinuous time as in "harema" (a uniform allotment of land (Plangewanndorf, lull in the rain) green village) were conspicuous (HOVERMANN, (5) important time as in "ma ni au" (to be 1957). After this, the spirit of Euclidean geo in time) metry continued to be strengthened in the (6) adjusted time as in "ma ga au" (to be Renaissance and Baroque ages, too (SUIZU, in rhythm) 1976a). (7) chance or opportunity as in "ma ga ii" The Japanese have had the traditional con (to be high time) cept of "ma," a concept related to a space Thus, the word ma has a wide range of time continuum. Various patterns are con meanings and the concept of ma has taken sidered to manifest themselves concretely in root even in the realm of the aesthetic sense a variety of "ma" through "en" (relation, of the traditional arts of Japan (SHIRAISHI, destiny, karma) in accordance with the codes 1973). already mentioned. It is significant that the As is shown above, the concept of "ma" is word en means originally a margin and an related to both space and time. It can be re open veranda. ferred to as a primitive concept of the space In order to have a better understanding of time continuum. This notion of "ma" held the word "ma," I would like to give some in Japan, where this world was considered, as is examples of the meanings it signifies. above-mentioned, "to be onri-edo" or "shogyo Among the meanings it has in terms of mujo" (space where all earthly things come space are: and go), led even to the development of an (1) (a) space between objects as in "arima" artistic concept of "utsuroi," or "mutability (interval) and "iwa no ma" (space be and transitoriness" as typified by flowing tween rocks) clouds and water ("gyoun-ryusui"). In con (b) limited space as in "taema" (gap), trast, the English word "space" or its German "kyoma" (intercolumnar space) and equivalent "Raum" does not connote the con "tsuginoma" (next room) cept of time. In this respect, the geography (2) What is made up of similar objects as in of "ma," where and when the codes of Japa "nakama" (company or group) nese landscapes function, is too high in dimen (3) discontinuous space as in "kakima" (empty sion to be analyzed by the use of Euclidean space in a hedge) and "mabiku" (to thin space, which adheres to a system of orthogonal out) coordinates. (4) adjusted space as in "ma o kubaru" (to arrange objects with space between them) III. The No stage (5) important point as in "ma ga nukeru" (to lack a point or to be out of place) and In order to clarify the geography of "ma," "ma o miru" (to judge the situation) I would like to examine the system of creative (6) function or ability as in "noroma" (slow actions which are determined by various codes. in function, that is, dull-witted) and "ma As the initial step of the examination, I would nuke" (lack of ability, that is, stupidity) like to focus on the No stage, where the The meanings of the word "ma" in terms scenery (a group of morphemes) is eliminated of time include: to a great extent. No is a traditional art for (1) (a) time between events as in "aima" public entertainment which originated from a (interval) primitive festive event in rural villages. 8 I. SUIZU

Though the No stage is simple, it still has On a spherical surface, a great circle which some setting so as to create an almost ex is drawn when the shortest course is taken cessive sense of unity between the stage and between two points is equivalent to a straight the audience. Among the sets are four pillars line. An infinite number of straight lines can erected at the four corners of the square stage, intersect at each of the two points (xn, yn) a bridge-form passageway, No masks and a but none of them run parallel to each other . mirror room (kagami no "ma," a room furnish Let us examine this model with a movement ed with a mirror where the actors put on (performance) from xn to yn given the name their No masks). The unique significance of "f" and one from y n to xn named "g." In these morphemes will be mentioned later. What the case where a projective transformation is to be examined now is the mechanism of "the maintaining the relationship "g=f-1" exists tense relationship between the performances and the identification of xn with yn (xn=yn) by the actors on the stage as a sacred realm is established2), the movement f-1f is a crea and the audience who has participated in the tive performance. Space where the relation sacred realm from the secular everyday life ship xn=yn exists is a projective plane p2. by passing through the entrance into the When a number of performances in con theatre" (TOITA, 1982). This mechanism is formity with this projective transformation suggestive of the structure of space produced (G=F-1) are possible, the mechanism of X= by performances in general which are mean Y, which can be referred to as a "latent ingful in terms of creative values. dimension," will be established in three Therefore, I would like to examine in detail dimensional landscapes with different mor the transformation of a system consisting phologies, and space for creation will be only of relationships between performances produced. and places. Let it be supposed that the body The No theater mentioned at the beginning passes through places x1, x2, x3,•c toward of this chapter can be regarded as space for some place infinitely far away, living each a "duel" between the actors and the audience moment with the indifferent attitude of an which meets the above-mentioned qualification onlooker. The space (x1+x2+x3•c) which for creation. Through the medium of the can be likened to shadows projected by these stage setting, this space manifests itself as actions is nothing but a mere extension of the admirable p2 and assumes an unorientable place xi and is not relate to a projective mechanism. Thus, the creation of an art is transformation in mathematical terms. Crea recognized.

tion cannot be possibly expected from actions It should be realized once again that not like these. Then, what kind of space do cre only the code of "marking off •¿ linking" but ative actions in the definite world produce? also the codes of "fusing" and "likening" When a creative action is performed in the have an unorientable mechanism characteristic bounded world, the places xl, x2, •c, xn should of projective planes like p2. This is why be united into a whole with "duration (duree)" these codes have been able to serve as codes in which x1, x2, •c, xn seem to the performer contributive to space for unique cultural crea to interact with one another in the same way tion. In the case of No plays, these codes as "all the sounds of a musical composition must be hidden on the stage with four pillars seem to us to be integrated into one, each erected at its four corners with no walls be interacting with another." Only performances tween them. that produce a mechanism like this can take

part in "the creative evolution (l'evolution IV. In pursuit of symmetry creatrice)" mentioned by H. BERGSON (BERGSON,

1889). On an Euclidean plane, it is difficult A projective plane produces symmetrical

to find a model which can meet this quali (bisymmetrical) patterns in principle. If all fication. At least there arises the necessity creative performances are essentially supposed of considering sphere S2 (Riemann surface) as to produce projective planes, symmetrical a model. morphogenesis can be expected of landscapes Codes of Japanese Landscapes 9 or their constituents and morphemes which in early times. The same desire must have contain these performances. The word "sym been behind the totemic communal system. metry" originates from a Greek word mean Images reflected on the surface of water pro ing "equilibrium" or "harmony." It seems bably helped to provide concrete ideas for the that it is the word "symmetry" as a mathe materialization of this desire. Not only the matical term that is connected in some respects water surface of springs or ponds but mirrors with the "balance" or "equilibrium" which also made a direct contribution to enhancing has been vaguely expected of a "region" in the concept of bisymmetry and were regarded geography. as mysterious as early as the prehistoric age. Not a few individual objects with a sym In No plays, too, which tacitly reflect and metrical form are found in nature. Not only convey the feelings and sentiments of the animals and plants but also human beings are ancient Japanese, there are not a few scenes almost perfectly symmetrical. This phenom where an actor gazes at himself with the enon may be related to the dispute on the surface of water as the mirror. The mirror preservation or non-preservation of parity room is not a mere dressing room but is a hidden deep in nature (GARDNER, 1964). Apart place where actors experience transformation from this subject, it is presumed that the and rebirth (TOITA, 1982). mechanism of projective planes underlies the In various parts of the world, we can find system in which one organism sustains itself many varied animal masks including those of by digesting another organism or inanimate strange imaginary animals. Similar to totem objects as food, that is, a system in which one ism, this must be a concrete representation of organism can be part of the environment of the concept of regarding oneself in the same another (UEXKULL, 1940). light with the animal, the fowl or the arbor, Ever since early times, mankind has not a concept which was held by mankind who merely depicted individual objects with a sharply discerned his proper position in "the symmetrical form but also concretely repre structure of nature." In the Japanese language, the word meaning "mask" also signifies sented symmetrical space consisting of a group "face of individual objects in the form of painting ." The masks used in No plays emit even a mysterious feeling. Thus, such codes as or sculpture. According to H. WEYL, the "fusing" and "likening" come to the fore in ancient Sumerians already had a strong taste for strict bisymmetry (WEYL, 1952). Symmet the projective plane system. Behind this there rical patterns were not unusual in ancient lies a sharp discernment which is aptly de scribed by the excellent proposition that "one Japan and China, either. A symmetrical pattern can be obtained when is one of the others" (SAKABE, 1976). a figure is reflected in a mirror. When a Then, the idea of symmetry in one's life figure is turned 180 degrees on an axis on the sphere developed into the formation of images plane, a "symmetrical" pattern is obtained of the world. For instance, maps depicted on but this pattern is not symmetrical in a strict sand on festive days by tribes indigenous to sense, for the figure obtained after turning is America reflect their strong desire to compre hend the world as symmetrical about an axis placed downward. A bisymmetrical pattern in a real sense can be obtained only when you (HISATAKE,1979). Among the other examples get a reflected view of the other surface of are Buddhistic world maps in ancient India the figure obtained by turning. This is the and Christian world maps in the medieval same as a pattern produced by a projective ages, both of which are of bisymmetrical arrangement. Furthermore, the symmetrical plane. Mankind depicted symmetrical patterns for sacred places, in spite of the difficulty of juxtaposition of the celestial world and the getting the reflected view. This must have nether world or paradise and hell with this been underlain by a strong desire to incorporate secular world between them is surprisingly into communal life the mechanism of projective omnipresent. According to M. ELIADE, it was planes which mankind had perceived latent in not unusual in early days that life on the "the structure of nature" (UEXKULL, 1940) surface of the earth was regarded as a reflex 10 I. SUIZU

of that in the celestial world (ELIADE, 1952). istence by giving a wealth of examples Later people sought for perfection in place (CAILLOIS, 1973). of the celestial world: the ancient Chinese As long as life space contains many varied recognized perfection in a square shape and solid objects with different functions , they the ancient Greeks in a globular shape (Suizu, were soon to face the matter of dissymmetry 1969). This is probably because these shapes in landscape planning including urban and most clearly represent symmetry. In Europe, rural planning. Especially in the case where even the idea of "antipodes" was advanced a view of directions is closely related to an under the influence of the theory that the idea of deities (for instance, in Japan four earth was round. But there remains the ques deities were believed to be corresponding to tion of to what extent the European people the four cardinal directions), dissymmetry held a symmetrical image of antipodal persons appears as a natural consequence . In the or antipodal points. A Christrian theologian Buddhistic view of the world, too, the adoption in the medieval ages named Augustinus denied of the belief that the Pure Land lies in the west this idea saying, "I can not possibly believe immediately distorted the previous symmetrical that there are inhabitants called 'antipodes' map of the world. It should be noted, however, on the earth and that they dwell in an area that dissymmetry differs from chaotic asym where the sun rises when it sets in the area metry in that the former indicates a high level we live in and walk with their legs upside of development following the partial destruction down to our legs" (ODA, 1959). Since the of symmetry. Dissymmetry can display its days of ancient China and Rome, the creation value simply because it is comprehended in of symmetry has been witnessed concretely relation to symmetry. in civil engineering plans (landscape plans) Even in the case of the belief of the four deities carried out on the surface of the earth such corresponding to the four cardinal directions, it as national land planning and rural and can not be denied that behind the belief there urban planning. We can clearly recognize the must lie the concept of symmetry. The four concept of an axis of symmetry or a rotary pillars of the No stage are also said to have center behind efforts to locate constituents been originally corresponding to the four including temples and a ruler's palace in a dieties the communities believed in. This be square street network as a way to construct lief was underlain by the theory of "inyo as symmetrical a whole as possible. Not only gogyo," a Chinese view of the world which the urban planning of a gridiron street pattern consists of two principal ideas. One is the but also the jori system (asquare partition of idea of inyo (yin and yang), which maintains arable land) in ancient Japan and the centuria that all things in the universe come into ex system in the Roman Empire show the con istence and change through the interaction of sistent application of a syntax with symmetry pairs of conflicting factors such as right and as its principle to an ancient grid pattern as left, up and down, man and woman, birth and a langue (Suizu, 1969). death, and right and wrong. The other is the idea of gogyo, according to which all things V. The appearance of dissymmetry in the universe are composed of five elements (wood, fire, earth, metal and water). As is mentioned by R. CAILLOIS, only re In the course of history, the formation of flections in mirrors are in conformity with the landscapes as a communal environment branch mechanism of projective planes. He also points ed into two directions, one guided by the out, however, that real symmetry obtained like symmetry of the constituents and parts of a this sometimes serves as an obstacle prevent landscape and one with dissymmetry as its ing the unrestricted development of nature guidepost. The former served as a syntax of and communities. The unrestricted develop a landscape in Europe and the latter in China ment of nature and communities is found when and Japan. For example, the symmetrical this rigid symmetry has been broken. He arrangement of sacred objects was adopted in discusses dissymmetry thus brought into ex the construction of temples in ancient Greece Codes of Japanese Landscapes 11 and cathedrals in medieval Europe. A persist in the construction of "islands" in garden ent trait toward symmetrical spatial arrange ponds in the medieval ages: in addition to an ment has been also witnessed in Europe in island likened to the Taoistic land of deities geometrical construction including that of called Shinsen-to, another island representing urban areas since the days of ancient Rom. the Pure Land was constructed in some garden In contrast, a tendency to recognize the value ponds with lotus flowers in bloom. With the of a slight impairment of symmetry became passage of time, dissymmetry became more stronger in China and Japan. R. CAILLOIS conspicuous in the layout of ponds like these. points out that in comparison with classical European culture, Japanese culture seems to VI. The cloud-mist painting method have averted the harm caused by the fascina tion of symmetry, which often produces sterility. No attempt to examine the lower strata of To substantiate this statement, he showed some "regions" can be possibly made without facing examples. Among them is the Zen master the problems of projective planes and bisym IKKYU (1394-1481) having had a few fallen metry. But a projective plane is no more leaves intentionally scattered on a garden that than an Euclidean plane with the point at in had been cleaned. Another example is the finity added to it. This plane, composed of importance being attached to motion centering straight lines and curves of second order, has on one point which breaks the stillness. too many restrictions. As might be indicated The patterns of "curving" and "irregulariz by the concept of "ma" in and after the ing" which can be considered characteristic medieval ages, the geographical "region" X of the historical landscapes of Japan are repre seems to belong to topological space with a sentative codes with which to embody dissym plasticity which at least contains and exceeds metry. The arrangement of rock gardens in both the Euclidean plane and the projective the medieval ages is a faithful reflex of the plane. above-mentioned concept of ten-chi-jin (aesthet Paintings of the inside and outside of Kyoto ic equilibrium), which may well be referred in the 16th and 17th centuries admirably de to as a typical Japanese concept of dissym pict scenes and events in and around the metry. This concept was to be adopted as a capital with changes in the four seasons and principle in flower arrangement in the Edo annual events interwoven into them. These Period, which followed the medieval ages. paintings, which are generically called rakuchu Another example of dissymmetry was found rakugai-zu, are marked especially by the pur

Figure 1. A rough sketch of a painting of the inside and outside of Kyoto, and the critical points in the clouds. 12 I. SUIZU

poseful abbreviation and separation of scenes of the viewpoint floatint among the clouds by the depiction of clouds and mist on the and mist serving as "blanks." The pictures canvas. Each of the divided scenes is in of Kyoto and its vicinities give an impression dependent of the others in terms of both time that the environment of the area is "in an and viewpoint. For instance, one of these actual existence as though it were refusing paintings concurrently depicts a New Year to be given a fixed shape (SHIGA, 1979). Feast being held in a garden of the Imperial Therefore, the environment represented by Palace, floats of the Gion Festival in procession these pictures is closer to "actual space" in summer along a street lined by two rows (espace vecu) (MERLEAU-PONTY, 1945). In of townsmen's shingled-roofed two-storied order to reason that these pictures are truer houses, and farmers busy gathering the har to life, it would be necessary to clarify vari vest in autumn on farmland on the east bank ous matters, transcending the image of space of the Kamo River. Both the street running produced by modern paintings and maps and through the "two-sided town" and the river the established concept of "regions." The are curved. Pictures like this convey a com above-mentioned environment which can be prehensive atmosphere of Kyoto to those who called "changing space" is removed from the appreciate them, through the views changing principle of isotropy and homogeneity, a prin one after another. This method of depiction ciple of Euclidean space as artifical space. is completely different from the European Here as a temporary measure, I would like methods in which the viewpoint is always to briefly analyze it by comparing topological fixed at one point. manifold (which will be mentioned later in Bird's-eye views painted in Europe which detail) to "ma" recognized in the paintings are similar to photographs must be more of Kyoto and its vicinities. accurate in terms of the representation of a "It is possible to give a 'shape' to X by scene at a specific moment. But the method assuming topological manifold in Euclidean taken in the paintings of the inside and out space. But the shape is no more than a side of Kyoto is superior in terms of the 'provisional shape .' The abstract quality delineation of life space as a whole which which the original X has as topological space constantly changes in different places with is in actual existence behind the provisional the lapse of time. shape as though it were refusing to be given Known as "the cloud-mist painting method," a shape." (SHIGA, 1979). Let us first suppose this technique is not limited to the paintings that the singular point and the critical point of the inside and outside of Kyoto but is often of topological space are hidden in the "blanks" seen in old picture scrolls. Peculiar to Japan, which are represented by clouds and mist, this technique is characterized by the descrip and that with the transition of the viewpoint, tion of space with "blanks" or negative space the topological mapping of different scenes (yohaku), which are closely connected with change one after another. If X is represented the concept of "ma." Perhaps only deforma on torus as Xa, the critical points shown in tion might be perceptible to those who are the right part of Figure 1 can be presumed. accustomed to European paintings character ized by a fixed viewpoint and three-dimensional VII. The codes of "changing" and representation with clear contours or to modern "irregularizing" maps marked by the abstract symbolization of concrete objects. These European painings For the purpose of deeply penetrating the "region" X and modern maps differ from the pictures , I would like to discuss it by painted by using the cloud-mist technique in showing an example in Europe, where topo that the former stand face to face with one logical mathematics was born. who appreciates them while the latter allow Kaliningrad, located at the northeastern ex him to be successively involved in the different tremity of the Gulf of Gdansk facing the scenes, as it were, because of "ma" or space Baltic Sea, is a city which once had a shelter time continuum produced through the medium castle of ancient Prussia. The construction of Codes of Japanese Landscapes 13

the city along the banks of the Pregolya Riv pleasure in discussing the possibility of going er was commenced with a fortress built by a through the four quarters A, B, C and D by German equestrian corps in 1255 as the center. crossing each of the seven bridges only once Until 1945, when it was annexed to the Soviet before coming back to the starting point. Union, it had been called Konigsberg and had It was L. EULER (1701-1783), a mathematician developed as the principal city of East Prussia born in Switzerland, who clearly proved the in Germany. The Pregolya, running through impossibility of doing so. This is quite well the central part of the city, is joined by a known in the history of graph theories. At tributary. An island called Kneipfhof was the young age of 28, Euler, who had inherited formed in the river in 1327, and had, among G. LEISNITZ's geometry of site, presented his others, the city hall and a cathedral. In the admirable solution to the Russian Academy in 18th century, this island (A) in the river and Petersburg. In solving the problem, he made the three quarters (B, C and D) surrounding a point stand for each of the four quarters A, the river were linked with one another by B, C and D and a line linking two points seven bridges, fewer than today (See Figure 2). represent each bridge (See Figure 3). By It is said that until the early 18th century, examining the figure consisting of only ver when the urban area had not extended beyond texes and edges (1), he found that only when the city walls yet and citizens could fully a figure has even-points and no odd-points, is enjoy the atmosphere of a Hansa commercial it possible to cross all the bridges (lines) with city still ruling the whole city, they found out crossing any bridge twice. It is also

possible when a figure has two odd-points but you cannot return to the starting point. In the geography of today, the index ƒÊ to show the connectivity between two places is often employed. Though this index has some

problems to be solved, I would like to apply it to the figure under discussion now.

Figure 2. The central area of Konigsberg in the 18th century. Figure 3. The central area of Konigsberg I: university II: castle represented diagrammatically. III: municipal office (14th century) a: bridge crossed once b: bridge not crossed IV: Gothic cathedral (1297-1302) c: bridge crossed twice 14 I. SUIZU

The index ƒÊ=the number of edges-the number of vertexes+the number of subgraphs =7-4+1=4.

When citizens strolled through all of the

quarters A, B, C and D, some bridges were not crossed while others were crossed twice. Among the routes used to stroll through the four quarters were the routes (2), (3) and (4) shown in Figure 3. Whatever route the citi zens took, their itineraries left figures with even-points and no odd-points when they visited all the quarters. In the case of route

(2), no bridge is crossed twice but a bridge between A and B, one between A and C, and a stable mass like the marbles in Greece but the one connecting A and D are not crossed. rather to those of something changeable like Route (3) shows that the bridge between A the Japanese gardens and picture scrolls. In and D is crossed twice while one between A other words, they are marked by the codes and B and one between A and C are not of "changing," "moving" and "irregulariz crossed. On route (4), a bridge between A ing." and B, one between A and C, and the one When substituting (F-EULER'S index) for between A and D are each crossed twice. (E-V) in the formula showing the index ƒÊ Accordingly we can obtain the following re after obtaining (F-EULER'S index=E-V) from sults; EULER and POINCARE'S formula, you can obtain the following formula. the index ƒÊ of route (2)=4-4+1=1 the index ƒÊ of route (3)=6-4+1=3 the index ƒÊ=(F-EULER'S index) the index ƒÊ of route (4)=10-4+1=7 +the number of subgraphs

This means that the connectivity among the Since (F-EULER'S index) is affected by the four quarters of the old urban area of Konigs characteristics of surfaces, the index ƒÊ also berg changed according to the way the bridges changes, depending on the surface. Therefore, were crossed: the urban area was a pedestrian it would be almost meaningless to simply ex space with the index ƒÊ changing from 4 to 1, amine the index ƒÊ alone in disregard of these 3 or 7. This indicates that it would be too changes when clarifying a "region" consisting hasty to judge the actual connectivity between of multilayered performance spaces. The codes of "changing" and "irregulariz places on the basis of the index ƒÊ uncondition ally. ing," which are characteristic of the historical L. EULER continued to study the mathemtical landscapes of Japan, are far beyond the re characteristics of surfaces. And in 1752, he strictions of dissymmetry mentioned previously discovered a rule determining relationships and have served in close conformity to the among the number of vertexes (V), that of changeable mechanism of actual performance edges (E) and that of surfaces (F) of a poly spaces: they have been conductive to the hedron: V-E+F=2. formation of unique landscapes through the This index of L. EULER'S was then expanded combination of morphemes suitable to the by J. B. LISTING and H. POINCARE and it was mechanism. These codes must have had dif proved that the index changes, depending on ficulty in developing in landscapes in Europe the surface (L. EULER and H. POINCARE'S which were mostly composed of solid stone formula, shown below). buildings with almost permanent durability. The characteristics of space clarified by But even in these landscapes, too, the above topological mathematics which has shown a mentioned mechanism must have operated in conspicuous development since the 19th century the lower strata of performance spaces. have proved to be comparable not to those of Then, what kind of space structure was Codes of Japanese Landscapes 15 latent in the central part of Konigsberg, where given conditions. pedestrian space could be transformed, depend Let us analyze a central place and its sphere ing on the route taken? The space structure of influence as an example. It may be assumed was anything but that of the transcendental that the daily shopping areas for instance, are "world of form" advocated on the basis of in the shape of an almost regular hexagon theories of EUCLID and NEWTON by E. KANT, when drawn on the basis of distances covered who led a life of meditation in the university on foot. But it would be meaningless to discuss in this city. Instead, it is presumed that a the commuting radius, a radius on a higher plastic manifold Xb similar to Xa mentioned level, on the same basis. For the commuting in relation to "ma" was latent in the central radius is determined by many varied factors part of the city: the space structure of the including the means of commuting and the city center can be projected onto two-dimen conditions of location of a place of work. A sional open disks by using the mappings ƒÓ1, number of daily shopping areas which are "marked off" from one another are "linked" ƒÓ2, ƒÓ3 and ƒÓ4 in the respective shapes of the figures (1), (2), (3) and (4), with the latent co together when viewed from the standpoint of ordinate systems represented by (x1, y1), (x2, y2), commuting radiuses or areas for purchasing

(x3, y3) and (x4, y4) respectively. In other quality goods. In the "region" X, where words, in its lower strata, the city center also radiuses of different levels are concurrently had different coordinate systems veiled in existing and stratified, the center and peri invisible "clouds and mist." pheries are variable. Each place within X can The artists who painted the pictures of the be the center or a periphery, depending on inside and outside of Kyoto depicted a whole the conditions of the radius. As is revealed set of different localities each corresponding by Figure 4, behind the existence of these to different coordinate systems not by solving relationships there lies a fact that space pro the differential equation but through a sharp duced by the central places and their spheres of intuition resident in artists: they symbolically influence is at least a kind of dual space which integrated different coordinate systems into is beyond Euclidean space. The code of "marking off •¿ linking" in Japan is considered their paintings by the adoption of the cloud mist painting method. Topological "mathe to be a symbolical example of the early com matical reality" latent in the actual world is prehension of the duality of space. ingeniously incorporated into Japanese methods Accordingly, in order to clarify space pro of constructing or representing landscapes. duced by the central places and their spheres of influence, it is necessary to sweep away "the

VIII. The code of fear world of form," which can be likened to an empty bowl. For space produced by the central

According to K. SHIGA, a mathematician, k places and their spheres of influence is in posses dimensional space "does not spread quietly" sion of "the same multiformity as mathemati in "a mathematical background," or in "the cal space itself has." In other words, actions world of form" (SHIGA, 1979). In the same performed in the space are incorporated into way, space dealt with in geography "does not a strange manifold in which porous toruses spread quietly" behind relationships among may be piled up one on top of another. It is various objects. true that the central places of a specific level "Mathematical space" like this is completely and their spheres of influence can be described different from invariable space which has been on a plane graph. But the plane as space for regarded as a geographical model space ever performances is transformed into a projective since the days of J. H. VON THUNEN. Here is plane realizable in four-dimensional Euclidean where I would begin to examine the code of space. Furthermore, if consideration is given to "marking off •¿ linking" recognized on the all levels of the central places and their spheres earth's surface on the premise that geographi of influence, the description of them becomes cal space has a variable structure, that is, a extremely complicated. Here I want to apply structure which changes in accordance with the name Xc to t-dimensional manifold cor 16 I. SUIZU

As was previously explained, the superiority of a central area to its peripheries was not necessarily established in Japan, where the idea of urban-rural continuity was prevalent. The prevalence of the traditional topographical notion of regarding the recesses or depths (oku) as superior to the centrality seems to have been related with this. Therefore, when discussing "the theory of central place" by W. CHRISTALLER,it is necessary to regard Xc as the same topological space as Xa and Xb (mentioned before) beyond a two-dimensional Euclidean image (CHRISTALLER,1933). In Japan, which had its own geographical codes, every phenomenon was considered to take place, and assume a "provisional shape" through the medium of en in accordance with the codes. The comprehension of the plastic nature of X, which never stops changing, can be aptly exemplified by the common expression "all things transmigrate" or "Matter is void ." The idea that all things are reflected in vari ous shapes and values in ma through mapping "en" bears resemblance to the ideas of top ological space. By the way, Yi-fu TUAN, a scholar of phe nomenological geography, shows us a wealth of examples of "landscapes of fear" (TUAN, 1980). They range from ghosts drawn by children to crimes in large cities. The Japa nese, too, have aimed at comprehending the "space of fear ," that is, space where "all things change and nothing is permanent (sho gyo mujo)," as one of the inevitable aspects of "changeable space" like Xa, Xb and Xc. If Xa, Xb and Xc contain a part correspond ing to a differentiable manifold, it would be possible to apply the "theory of bifurcation" in a local dynamic system to it. The phe nomenon of bifurcation of manifold is really suitable to the "space of fear," especially to the climate of Japan, which can be colored by Figure 4. The spatial duality of central places constant mutation and perpetual motion. The and their spheres of influence. above-mentioned ideas of "onri edo" as well as "shogyo mujo," which regard this world r esponding to all levels of the central places as impure and mutable and thus avoiding it and their spheres of influence. It seems that must have been a reflex of the Japanese the mechanism of Xc is not so much different people's intuitive insight into the mecha from that of "changeable space" represented nism of bifurcation of differentiable manifold in the paintings of the inside and outside of latent in the actual X through their experi Kyoto. ences in the medieval ages preceding the days Codes of Japanese Landscapes 17

of topological mathematics. This idea can be nese have continued to make efforts to "liken" further considered to have served the "codes" a "region" to topological manifold by making peculiar to Japan which were systematized on use of such codes as "likening" and "in the basis of this perception. corporating." Supposing that this manifold is In Japan, topological recognition has existed an n-dimensional differentiable manifold as a "mathematical reality since the very remote ages. But it was not ," it is possible to de until the 13th century that the recognition scribe the state in the "region" by the use freed itself of the ancient grid pattern intro of differential equations. Here, we see a new duced from China and was developed and re integration of qualitative geography and qual fined in the form of the codes of landscapes itative mathematics. mentioned before. It is suggestive that it was The "region" X as part of land surface the 14th to the 15th centuries that the No can be considered to contain X which is cor play, derived from the Sarugaku farce in rural responding to a multidimensional differentiable villages, was completed by KAN'AMI and his manifold3). Sx, a vector field which can be son ZEAMI. Furthermore, in the 16th century, referred to as a mapping of X, is the whole the genre, generically called "the pictures of of the loci drawn by actions (the flow of hu the inside and outside of Kyoto" characterized man beings and objects and their functions) by the cloud-mist painting method, reached performed with the perceptible landscapes of completion. In addition, both what is called the region as the ground and figure. The the Verdorfung in the medieval ages and the vector field composed of the flows of individual construction of castle towns in the Edo Period performances contains "bifurcation sets (ƒ°)" with variegated plans woven into it were also as a system of instability which causes various closely connected with the completion of the catastrophes and forces X to undergo sudden topological codes of landscapes. local changes4) (THOM, 1977; WILSON, 1981). Naturally, these changes must lead to changes IX. "Likening" to "mathematical reality" in X. This process seems to have correspond ing relationships with the codes of "changing" Since olden days, the practice of "likening" and "irregularizing." Please see "SUIZU, has been recognized in Japan: rocks arranged 1982a" for details about the mathematical in gardens have been likened to the Taoistic analysis of these relationships. land of deities called Shinsen-to, spots in the The other codes of historical Japanese land precincts of a temple to the thirty-three tem scapes listed early in this paper also help to ples on the noted pilgrimage route in the provide unique patterns for everyday life, with western part of Japan, and the depths of a set of catastrophes sticking to each of the mountains to the other world. Thus, "liken morphemes such as open verandas, paper ing" is marked by reproducing objects with a sliding doors, ridges, indented streets and distinctive feature such as Shumisen (the lofty trump stones, and also to each combination of mountain believed in Buddhism to soar in the morphemes. Here, we recognize the "space center of the world), temples on a pilgrimage of situation" with a stable periodic orbit route, and the other world, and by regarding (YAMAGUCHI, 1975) and we are even reminded them as identical with the originals. In this of the "daily prism" mentioned by T. sense, the idea of likening is more similar to HAGERSTRAND (HAGERSTRAND, 1977). When topological transformation than to the congruity viewed from R. THOM's theory, a landscape or similarity in Euclidean geometry. Not only structure corresponds to a combination of sets the idea of abbreviating a landscape in essence of catastrophes, which determine the multi on a smaller scale but also that of incorporat dimensional grammar of the constant succes ing an external distant landscape into a view sive occurrences of the performances in a "region of a dwelling as the background can be thought ." This multidimensional grammar has of as a kind of local coordinate transformation even an aspect comparable to the "creode" of part of a landscape. mentioned by the biologist C. H. WADDINGTON

If this is the case, it means that the Japa (WADDINGTON, 1961). This is an inevitable 18 I. SUIZU

Figure 6. The creode of capturing. A: Regime 7 with 3 points of elliptic umbilic. B: When food is coming near, hyperbolic umbilic appears with a wave crest. C: Regime 3 is captured by regime 7. (p=food) D: Food is digested and elliptic umbilic appears again.

Figure 5. Bifurcation sets. "mathematical reality" that ancient people A: Cusp B: Profile of a cusp C: Swallow-tail D: Profile of a swallow tail E: Consecutive profiles of a wave crest became aware of the types and functions of (Hyperbolic umbilic bifurcation set) catastrophes. They may have grasped some of them in a metaphorical manner through the medium of, for instance, curvature seen path followed by two or more sets of Catastro at a profile of strata, a crevice in a cave far phes, as is recognized in the morphogenesis away from human habitations, a tree or bamboo of organisms. bent by a gale, and waves broken on the Needless to say, some of the factors con beach. Or they may have understood them trolling cultural landscapes which have been through a series of events from the birth of produced by mankind must be different in an organism to its growth and eventual death nature from those of the morphogenesis of (the creode of capturing). It should not be organisms. But as long as both landscapes overlooked either that a sharp-edged profile and organisms have morphologies created in of the "cusp" catastrophe was concretized as space of less than four dimensions, it can be a stone axe to cut things with as early as the safely said that there must be some points prehistoric age. The morphology of a cusp common to them. In this sense, D.W. was also applied to the gable roofs of Shinto THOMPSON's remarks to the effect that each shrines and Buddhist temples. In spite of his organism has functions suitable to its own unawareness of the mathematical theory of morphology abound with suggestions the "cusp" or "hyperbolic umbilic " catastro (THOMPSON, 1945). phe, the painter HOKUSAI in the Edo Period It is not through the understanding of admirably depicted the beauty of the lively Codes of Japanese Landscapes 19

motions of wave crests enclosing ("incorporat codes in and after the Meiji Period (1868-1912), ing") Mt. Fuji in the shape of a cusp in one especially because of the rapid changes in the of a series of his paintings entitled "the 1960s and afterward. At the same time, I Thirty-Six Views of Mt. Fuji." The codes realize the importance of the irreplaceable of "marking off •¿ linking," "incorporating" historical codes which are becoming extinct and "likening" are closely related with the and their immense significance to our con "hyperbolic umbilic" catastrophe or the creode temporaries. of capturing. Needless to say, it is far from appropriate Thus, the mechanism of 7 elementary catas to deny the merits of the traditional system trophes including "cusp" and "hyperbolic established by European geography in the umbilic " catastrophes was perceived and sys modern age, because objective local space pro tematized in the form of geographical codes duced by one's performances, especially space with profundity in the framework of the cli in terms of directions and distances, always mate and history of Japan. I can not help has a mechanism similar to that of three but admire this wisdom of our ancestors be dimensional Euclidean space, just as the ex fore the age of science. ternal surface of a landscape does. We are In the morphology of landscape, it has been only ready to investigate the codes of land usually the case that the so-called "regular scapes as space for varied creative perform ances and to roughly judge their merits and patterns" have been tacitly regarded as "general" in the analyses of morphological demerits now that we have reached the stage of topological geography which contains the patterns. But I think "regular patterns" are actually special. Therefore, I consider it im framework of Euclidean space as its special case. portant to start with the so-called "irregular Much is left to be studied in the future: it patterns" and examine corresponding relation ships between them and "mathematical re is necessary to substantiate corresponding ality." Only with this reorientation of the relationships between the codes of historical way of thinking, would it become possible to landscapes in Japan and X as topological space take initial steps toward the attainment of a in a more concrete manner, to reflect these fine command of decipherment of variegated codes in the mirror of "mathematical reality" more precisely, and to analyze corresponding picturesque patterns of cultural landscapes which were referred to by A. v. HUMBOLDT relationships between these codes and the in the 19th century as "Naturgemalde (natural structure of the Japanese language in detail. paintings)" (SUIZU, 1982a). It goes without saying that the field of codes other than the Japanese ones which will be objects of comparative study is practically an X. Concluding remarks unexplored land. It is found that the geography of "ma," which has traditional profundity based on the (Received October 1, 1983) (Accepted December 17, 1983) recognition that "all things change and noth ing is permanet (shogyo mujo)" and that "this Notes world is impure and therefore should be avoid ed (onri edo)," is a far better approach to the 1) X=LV/L=a set of landscapes, V=a unity of their functions, V={ƒÒ} clarification of the real state of topological X 2) f:xnyn|xn•¸X than the geography of Euclidean "space," g: ynxn|yn•¸Y which is in pursuit of the illusive inflexible g=f-1 image of X. What I realize in the analysis •ˆ x n=yn of dynamics in the vector field attached to X 3) X•½X is the necessity of clarifying the mechanism 4) XhS|x•¸X, h•¸S of great shocks and commotions the codes of historical landscapes in Japan have undergone K•¾•ahƒ°|K=catastrophe sets because of their encounter with European h-1(E) 20 I. SUIZU

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