TWO AND THREE DIMENSIONS

Historically, the use of numbers in the ordering of space in two or three dimensions is governed by the distinctive ecology of Japan. The country, in its natural state, is mountainous, with a dense forest cover.

Wood therefore is the obvious construction material. At the same time, rich volcanic soil and a warm temperate climate with reliable rainfall make the country well-suited for wet rice cultivation. The material implications of these factors still govern the use of numbers in two or three dimensions.

The terraced cultivation of rice is to be seen almost everywhere in Japan in the familiar dandanbatake on the hillside above the villages.

This is numerically significant for dividing the continuum into a finite number of discrete intervals. It would be convenient to see this as the model basis for the steps cut into the side of sacred mountains, but then there is the problem that the characteristic fan layout of the terraces is not linear. In practice the terraces belonging to any one village were named, but not numbered, and the names were sufficient to identify them with their respective owners. If the geography of the terraces seldom lent itself to systematic numeration, the same was true also of the distribution of the houses in the village. Although, in deciding upon

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the orientation and location of a house, the principles of geomancy could not be neglected, the natural constraints of the terrain often left little room for choice.

The form of the house itself was largely determined by the fact that it was built of wood. The essential constraint imposed by wood is that the ground-plan must be based upon a rectangular grid.³¹ This has been characteristic of the standard Japanese family house, or minka, from the time of the very earliest records. The traditional Japanese building always had but one main floor (Masai 1987:67), and the floor-plan was based either on an interpost-span module or a tatami module (Itoh 1972:111). In the simplest case, such as that of the fourteenth-century Rin’ami house (of which the floor-plan is now kept in the archives of the Toji temple in Kyoto), the house was a simple rectangle, so that the Rin’ami house had four bays along one side and six along the other (ibid.: 138).

The familiar tatami mat was already at this early stage the key to the ground-plan.³² Although the size of the tatami has never been entirely standardised, they are always about 6 feet long, and 3 feet wide.

Because in every case the length is exactly twice the width, they provide the modular basis for any room whose dimensions are based on fixed multiples of the length of the shorter side,³³ as illustrated in figure 8.1. This is now standard in Japanese domestic architecture, although there are still small variations in the size of tatami between different regions.

The modular principle established by tatami is essentially two dimensional. The construction of the traditional Japanese building does not generally allow its application to be extended into three dimensions, but there are occasionally instances of this which occurred, historically, by force of circumstance. One of the best instances is provided by the platform supporting the main buildings of the well-known temple complex of Kiyomizu in Kyoto, which are located on the side of a steep hill. The support for this platform consists of a three-dimensional complex of posts and beams, whose design is clear to any visitor. This is in the form of a three-dimensional modular grid, in which every module has the same length, width and height. The number of modules in each dimension seems to have been determined purely by engineering principles, and no special numerical significance is claimed. Indeed, many of the beams and posts are greatly foreshortened to take into account the profile of the hill on which the temple is located (and which explains the necessity for the whole structure in the first place).

In Japan the rectangular modular plan for buildings has seldom been

Figure 8.1 Japanese floor-plans based on interpost-span module (top) and tatami model (bottom)

Notes: a=1) length of one ken, 2) distance from centre of one post to another, 3) standard unit of measurement. Tatami for any given room are all of the same size, but the size may vary from room to room.

b=1) length of one tatami, 2) standard unit of measurement. Size of tatami is uniform throughout the plan.

=Buddhist altar.

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used for planning on the scale of a whole town. It is standard for any number of shrine and temple complexes, although in such cases it may arise from the necessity to achieve the correct north-south orientation of the buildings.³⁴ The use of such a plan for cities is far from standard, but there are two exceptional cases, one modern and the other ancient.

The former is Sapporo, the capital of the prefecture of Hokkaido, which embraces the whole of the most northern of the four main Japanese islands. The colonisation and development of Hokkaido is part of the history of the present century, which explains the circumstances in which it was possible to lay out the capital city on a rectangular grid. The model could hardly be simpler. The streets in both directions are numbered according to their distance from two central axes. The model is that of a graph, with x- and y-axes, and Cartesian co-ordinates. The point is significant for the Japanese numerical tradition, since the application of the basic ideas and notation of algebra (conceived of as a general, abstract model of numerical equivalences) to geometry, which by definition is based on configurations in two or more dimensions, was only achieved by Descartes at the beginning of the seventeenth century (Williams 1978:16). This type of analytical geometry only became known in Japan when the country became fully acquainted with western science in the later nineteenth century.

Traditional Japanese culture was if anything hostile to any such mechanical approach to town planning, and the results of this attitude are still to be seen in the chaotic street plans of almost any Japanese city, starting with Tokyo. Any sentimental attachment to a ‘home town’, however large or small, was based on the concept of kokoro no furusato, literally ‘the old country of one’s heart’, or by implication the village of one’s ancestors. From this perspective form (katachi) is less important than sentiment (kokoro), so that the spiritual domain is seen as hidden, profound, and essentially mukei, that is without form, where the material domain is y¯ kei, or has form. According to this way of thinking, the contrast to be made in looking at Tokyo is between the slopes of the Yama no te literally ‘hand of the mountain’, and the low-lying Shitamachi, but such a view imposes no numerical order on town-planning. Only in modern times is there some semblance of order in the forms of addresses, so that the whole of Japan is divided into nearly a thousand postal districts, each with its own three-figure code, whereas at local level a specific building will be identified with a two- or three-part numerical code, such as 2–182, for a house in the Kyoto ch) of F*jiyama.³⁵ This process is taken to it utmost extreme with telephone numbers, where the model of the system represented by the circuit

diagrams used by the engineers is no more than a remote abstract from the actual topography.

The Japanese distaste for Cartesian systems of town-planning had no effect on the layout of the ancient capital city of Kyoto. This is the second of the two exceptional cases already mentioned. The town-plan of Kyoto is a near perfect rectangular grid, and the streets running east-west are numbered from south to north, using the suffix -j) as a counter.³⁶ The model was borrowed from China, as was also the rule according to which the ancient capital was divided into wards. The basis is the mandala so that the centre, Nakagy), is surrounded by the ku³⁷ of Kita (N), Higashi(E)yama,³⁸ Minami (S) and Nishi (W).

Looking at the city from the north provides the orientation for the remaining four wards of Uky) (right), Saky) (left), Shimogy) (lower) and Kamigy) (upper).³⁹ The first two names look at the city from the traditional perspective of the emperor, who sees one of the two ku immediately below him on his right hand, and the other on his left.⁴⁰ The second two names, meaning ‘upper’ and ‘lower’ may relate back, through Chinese Buddhism, to the egg of Brahma in traditional Hinduism, with the visible, light, upper half being separated from the invisible, dark, lower half, by a flat plane representing the earth (Crump 1990:141). In the Buddhist tradition, the visible, top half came to represent Mount Sumeru, which once again brings us back to the mandala, which is a binary but not a Cartesian model.

The conclusion must be that the use of numbers to organise space on a large scale is marginal in the Japanese use and understanding of numbers. Kyoto as a type of town-planning may be a model useful for general application (as in Sapporo), but its numerical base is rooted in a tradition which has been largely superseded. The Cartesian idea is reflected in certain modern situations, such as the cadaster of rice-fields in the extensive flood plain in Niigata prefecture, or the complex of oyster beds to be seen off the coast of the Kii peninsular, but instances such as these are purely practical, and hardly part of the numbers game.

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