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    Faculty of Systems and Information Engineering, University of Tsukuba

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  • ル・コルビュジエが提示したモデュロールによる羽目板遊びの探求
  • ル ・ コルビュジエ ガ テイジ シタ モデュロール ニ ヨル ハメイタ アソビ ノ タンキュウ

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<p> In this paper, we seek the possibility of Modulor in the digital era by demonstrating computational methods to solve “The Panel Exercises” that Le Corbusier presented and confirming their “combinatorial explosions.”</p><p> The first Panel Exercise proposed by Le Corbusier is a game of dividing an initial square (2260 * 2260) recursively under Modulor (Fig. 1). According to the characteristics of the golden ratio, the method can be formulated such that one of the four division rules is applied in the case of red series, and one of the five rules is applied in the case of blue series, about either the horizontal (x) or vertical (y) direction of the intended rectangle. Fig. 2 illustrates the schematic diagram of these rules. Based on this analysis, a Python program was prepared in which four classes (BB, BR, RB, RR) are defined to represent these rectangle types. Each class has a method of bisection rules. The set of N panel combinations can be made from the set of N-1 panel combinations. Fig. 4 shows combinations generated by this program and selected at random. Eventually, an accurate total number of combinations can be counted as shown in Table2.</p><p> Le Corbusier proposed another Panel Exercise, dividing the initial square by means of the specified panels measured in Modulor. More specifically, the first batches of sixteen combinations were given with respect to a) 12 pieces of six different panels, b) six pieces of four different panels, and c) nine pieces of three different panels (Fig. 5). There is no known method of easily counting the total number of solutions for this kind of packing puzzle. The algorithm for this procedure is called a “depth-first search,” and the practical steps for solving this Panel Exercise are as shown in Fig. 6. However, it is noted that this kind of search tree sets off a “combinatorial explosion” according to an increase in the number of pieces. Efficient pruning of the search tree is required to avoid unnecessary calculations. The pruning of the search tree can be illustrated as shown in Fig. 7. Fig. 8 illustrates five combinations for each pattern, selected from all solutions explored by this program. The total number of solutions and calculation time are as shown in Table3.</p><p> Additionally, four panel exercises were presented in Modulor. The third panel exercise is evolved from the first exercise by changing the aspect ratio of the initial rectangle according to the Modulor order. Therefore, the total numbers of combinations for each initial rectangle can be calculated using the same Python program from the first exercise. The fourth through sixth panel exercises are evolved from the second exercise by augmenting series of five initial rectangles (A), as shown in Fig. 10. The difference is that the allocating pieces are not fixed and are collected, allowing duplication from five panels, two band widths (flexible length panels), and dotted panels for doors (B). Before enumerating, the two bandwidths (flexible length panels) have to be restricted to the lengths presented in other five panels, since these exercises have literally infinite combinations under permission of using bandwidths of any length. Then, the tentative collections in which the sum of the pieces’ area is equal to the initial rectangle’s area should be listed, whereas many of them have no solutions in the end of the depth-first search. According to Fig. 10, if the two bandwidths should be less than six and the door panels should be less than three, then the total number of collections and total number of combinations without residues are as shown in Table5.</p>


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