Urban and Regional Planning Review Vol. 5, 2018 | 1
Urban Form that Minimizes the Total Travel Cost Assuming Multiple
Floors in a Three-Dimensional City with Two-Stage Hierarchical Bases
Takehiro KONDO* and Tohru YOSHIKAWA**
Abstract: This paper calculates the optimal urban form for a given total floor area with minimization of the total travel cost and studies the impact of changing various factors, by assuming the presence of multiple floors in three-dimensional cities having a hierarchical space structure. In this paper the optimal urban form is defined as the urban form offering the minimum travel cost, from an arbitrary point in a city to a specific city center site. For vertical transfers, a model is formulated by assuming the presence of multiple floors. This assumption makes the model more realistic and closer to actual cities than conventional models where multiple floors are disregarded. The results show that the optimal urban form changed; retaining similarity when the total floor area was changed, but without doing so when the vertical travel cost or travel cost by short-distance transportation changed. The paper also compares the city model and an actual city form to clarify the difference. The optimal urban form derived from the model calculation has a far smaller horizontal size than the actual city and hence includes many high-rise buildings, which differs significantly from the form of the actual city. Consequently, the optimal urban form with respect to the travel cost was that in which ultra high-rise buildings were concentrated in the city center.
Keywords: Compact City System, Floor Height, Multiple Floors, District Centers, Hierarchy, Tama City
1. Background and objectives
This paper obtains the optimal form of a three-dimensional city with a hierarchical space structure with ease of transfer in mind and clarifies how it differs from the actual urban form. Here, the optimal urban form is defined as the urban form offering the minimum travel cost, from an arbitrary point in a city to a specific city center site. For vertical transfers, a model is created assuming the presence of multiple floors. This is to make the model more realistic and closer to actual cities than conventional models, where multiple floors are disregarded. Many high-rise buildings have been constructed in Japan as well as in large cities worldwide and construction has accelerated in recent years1). Plans for ultra-large buildings incorporating a city function, such as hyper buildings, have also emerged2). However, it takes time to move vertically in a city full of high-rise buildings and moving from place to place is not always convenient. Japan and other nations have been planning and establishing models of cities to visualize compact cities3). Achieving such a compact city requires a quantitative analysis of the
* Mitsubishi Electric Information Systems Corporation ** Department of Architecture and Building Engineering, Tokyo Metropolitan University
(C) 2018 City Planning Institute of Japan http://dx.doi.org/10.14398/urpr.5.1 Urban and Regional Planning Review Vol. 5, 2018 | 2 effects of the city and is crucial for future implementation of city planning policy based on social agreement4). This paper provides suggestions for this purpose. The following studies on optimal urban form focus on minimizing travel cost. Many articles including that of Holden et al.5) analyzed the relation between compact city indicators such as population density and costs such as travel cost in real cities. There also exist many articles including that of Oliveira Panao et al.6) on the form and arrangement of urban buildings optimizing the energy efficiency in terms of building facility energy consumption. Morimoto7) studied the effect of making an actual city compact, but generalizing these effects is difficult, since it is strongly linked to city-specific situations. Conversely, studies on the optimal urban form of the model city would be useful to determine the generality of the effect of a compact city. Studies of this type include those by Koshizuka8) and Kurita9), who showed an optimal three-dimensional urban form assuming a rectangular parallel-piped urban model. Suzuki10) obtained an optimal urban form, assuming an urban model of stacked rectangular parallel-piped boxes, but such studies still face the following problems. In the former model, in which an urban form was approximated with a single rectangular parallel-piped structure, the difference in building height between the urban and surrounding areas, namely the difference in residential density, cannot be reflected. In the latter model, where an urban form was approximated with stacked rectangular parallel-piped boxes, the difference can be reflected to a certain extent. However, the number of rectangular parallel-piped boxes is restricted due to computational limitations, while the urban form is limited by the shape and size of the rectangular parallel-piped boxes. Moreover, given the uncertainty of what city component the rectangular parallel-piped box represents, it is difficult to discuss the obtained optimal urban form. In addition, these models assume either direct movement from any point to a single base or movement between any two points. For this reason, these models overlooked the impact of the hierarchical space structure observed in actual large cities on the urban form. Therefore, the influence of hierarchical bases on the optimal urban form has not been clarified. In response, Kondo et al.11) derived an optimal urban form by introducing hierarchical bases and means of transportation. The optimal inter-base distance was calculated using travel cost ratios of pedestrian, bus and elevator routes fixed to realistic values. Consequently, they revealed how creating a compact city with ultra high-rise buildings was not always efficient, while developing the city horizontally and exploiting certain means of transportation could be efficient in some cases. However, this ignored the fact that actual city spaces consist of the vertical division of a building into multiple floors with constant distance or floor height; instead treating buildings as horizontally and vertically continuous spaces. Namely they assumed a population distributed uniformly over any of the three dimensions, i.e. a uniform density of citizens floating within a three-dimensional city space. The key advantage of this assumption was the chance to simplify calculation because calculus, a very effective means of calculation, could be used to calculate the cost of travel both vertically and horizontally. However, the population distribution and the vertical travel cost in this model differ from the model in actual cities, which comprises buildings with many floors and hence the optimal urban form obtained could differ from the correct one. In the case of introducing multiple floors, each floor can be assumed to be a rectangular cuboid with the height of the floor as the vertical height. Based on this concept, cross-sections of both models with equal volumes are shown in Figure-1. The figure is based on the one by Kondo et al.12), though the location of the urban envelop when not considering the floors is different. It is because in the three-dimensional case the urban Urban and Regional Planning Review Vol. 5, 2018 | 3 models with the same volume should be compared, while in the two-dimensional case the urban models with the same length are compared. First, the urban model with floors and that without them differs as to the population distribution at the end of the city. When introducing multiple floors, the entire population concentrates on the floor, and the population density per unit floor is constant. In contrast, without considering multiple floors, the edge of the city has an inclined shape like an attic and the population density at the edge is lower, since the height of the city is regarded as the population density. Furthermore, the end of each floor of the model without considering multiple floors moves inward compared to the case considering the floors (Figure-1). This means that the urban model without considering multiple floors has a different population density at the edge of the urban area compared to actual cities and that the horizontal movement is overestimated. Also, in the case of considering multiple floors, the population distributed in the air between floors in the case without considering multiple floors is distributed on the nearest downward floor. Thus, the vertical travel distance is reduced by about half of the floor height as shown by the downward arrows in Figure-1. This shows that the urban model without considering multiple floors overestimates the travel cost in the vertical direction. These factors cause the difference in urban forms. This difference is expected to be conspicuous especially when the city size is small. Therefore, there is a danger that the optimum city shape obtained may differ from the actual one.
Figure-1. Difference in population distribution according to the existence of multiple floors
The above facts suggest that assuming multiple floors has the following advantages: Firstly, in an urban model comprising buildings with multiple floors, there would be less difference in the above-mentioned travel cost between the model and actual cities compared with conventional models. Secondly, there is scope to analyze in more detail how a change in various factors such as floor height, which has increased in recent years in Japan, will affect the urban form. Thirdly, not only does using a continuous function reduce the amount of computation, it is also expected to circumvent the difficulty in interpreting the travel cost and the calculated urban form arising from the above problem in preceding studies to a certain extent. This is because assuming multiple floors equates to describing city borders with a continuous rather than discrete function, given the minimum height in the actual cities as the height of stacked rectangular parallel-piped boxes in the previous study described above. To challenge this problem, Kondo et al.12) 13) 14) derived an optimal urban form, assuming the presence of multiple horizontal flat floors at certain heights in the vertical direction and a uniformly distributed population on each floor. The urban model used in their study was limited to a two-dimensional city, developed along a railway or a major street or a cross-section of a Urban and Regional Planning Review Vol. 5, 2018 | 4 three-dimensional city. Accordingly, there has been no study of the optimal urban form by assuming more realistic multiple floors of a three-dimensional city. The present study was designed to fill in this research gap.
2. Outline of the assumed urban model and the research approach
This paper defines the urban form that minimizes the total travel cost as the optimal urban form. The three-dimensional urban model has multiple floors with a fixed floor height in the vertical direction, and the people are assumed to be uniformly distributed on the floor. By these assumptions, as the optimum urban form, the base position minimizing the total travel cost and the height of each point are obtained (Figure-2). Also, we use the result obtained to compare the optimal urban form and an actual city with the same floor area to clarify the difference between the urban form of the optimal city and an actual city. The following assumption is made for the urban model above in this paper for the sake of simplicity. Namely the total travel distance is calculated only for traveling to the secondary base. The study assumes a hierarchical city structure with four primary bases and one secondary base to obtain the optimal urban form, which minimizes the total cost of travel from an arbitrary point to the secondary base. Here, in order to facilitate comparison with preceding studies, as in the preceding studies, the article adopts the following assumptions. The article assumes four primary bases from O1 to O4; lying with the distance d from the secondary base O5 on the origin, as shown in Figure-2-a. Define r as the distance from the secondary base O5 to the city edge. People are assumed to travel only in the directions of x-, y- and z-axes and the distance is measured as the three-dimensional Manhattan distance (Figure-2-c). The inverse of the average travel speed in each case,15)-20) namely the time required to travel a unit distance, is used as the travel cost and the ratio of each cost to the travel cost by bus is shown in Table-1. The same values are used in the table as those in Kondo et al. 11)-14) for the purpose of comparison. The cost of public transportation or base development and management is also excluded from consideration in this paper. An example of horizontal travel includes the assumption of travel to the primary base or the secondary base located close to the primary base on foot, i.e. at low speed and that travel from the primary to the secondary base is by bus, i.e. at intermediate speed.
For vertical travel, people are assumed to use elevators. In what follows, cd presents the vertical travel cost, ch1 the horizontal travel cost on foot and ch2 the horizontal travel cost by bus. For specific distance values, the unit is kilometers. The enveloping surface of the multiple floors is called the building outline, which represents the city form. However, in the portion higher than the highest floor, the building outline is defined as the surface that is the enveloping surface of each floor extended upward. The vertical distance from the ground surface to the building outline is called the height of the building. The height of the building at the location (x, y) is denoted by h(x, y). This function is not a non-negative integer but a non-negative real value and continuous at an arbitrary point. The floor space height is denoted by a, which is a constant, and the difference between the highest floor and the maximum building height is denoted by h’ ( '0 ah ).
Urban and Regional Planning Review Vol. 5, 2018 | 5
Table-1. Travel costs Speed Cost
Travel in the vertical direction by elevator C d 2.7km/h 10/27 on foot C 4.8km/h 5/24 Travel in the horizontal direction h1 by bus C h2 16.8km/h 5/84
a. Contour map b. Sectional view
c. Travel route Figure-2. Assumed model
3. Formulation of travel cost with multi-floor buildings
A point in the city is presented with coordinates x, y and z and the city is assumed to lie in the following area on the xy plane (Figure-2-a):
r x r and S N xYyxY . Here r is an arbitrary real value from zero to infinity. Since the city is symmetrical with respect to the x- and y-axes, only the first quadrant is considered in what follows. The number of stories in a building at (x, y) is given by: hyxh ', 1 a This is expressed as m 1 Let z (non-negative integer satisfying mz 10 ) be the floor number, S(z) be the floor area at z-th floor and D be the total floor area in the city. Then we have m1 D 21 1 kSmSSS . k1 4 The total floor area D on the right-hand side is divided by four because we only consider the first quadrant as described above. The floor is continuous in a city where the travel cost is minimized. Also, if it is assumed that every resident in the city needs a floor of the same area, we see that the total floor area D is equivalent to the city population.
The travel cost of a person at P(x,y,z) in Figure-2-c to move to the secondary base O5 is the sum of the vertical travel cost from P(x,y,z) to the ground and the horizontal travel cost from the Urban and Regional Planning Review Vol. 5, 2018 | 6 ground point to the secondary base. The person selects the path that minimizes the cost. The person goes to the primary base O1 only when traveling via O1 that minimizes the horizontal travel cost and vice versa. Namely, only the horizontal position x and y determines whether the person travels via O1, irrespective of the height z of P(x,y,z). Since the person travels to the secondary base at the lowest cost, he/she goes there via the primary base O1 only when it costs less than traveling directly to the secondary base O5. Accordingly, in this case, we have . h1 h1 h1 h 2 dcycdxcyxc or h1 h1 h1 h2 dcdycxcyxc Solving this (1), when we have in the first quadrant cd cd 1 h2 x or 1 h2 y, 2 c 2 c h1 h1 the person goes to the primary base O1 or to the secondary base via the primary base O2. Also, when cd cd h2 , h2 1 dx 1 dy , and h1 hhhh 1121 h2 dcdycxcdcycdxc , 2 ch1 2 ch1 namely when x y, the resident travels to the secondary base via the primary base O1. Conversely, when h1 hhhh 1121 h2 dcdycxcdcycdxc , namely when x y, the resident travels to the secondary base via the primary base O2.
Also, it is obvious that travels to the secondary base are via the primary base O1 when xd and yd , and via O2 when xd and yd . Since we use the Manhattan distance in this paper, the cost of traveling via O1 and that via O2 are the same when xd and yd . For the sake of simplicity, we assume that the person travels to the secondary base via the primary base O1 when
cd h2 cd h2 x y as is the case with 1 dx and 1 dy via O2 when x y (Figure-3). 2 ch1 2 ch1
Figure-3. Voronoi diagram
Accordingly, the travel cost from an arbitrary point P to the secondary base O5 is given by
cd h2 cd h2 11 dhh zcycxcz,y,xC for 0 x 1 , 0 y 1 , 2 ch1 2 ch1
cd h 2 ,, h1 1 hdh 2 dczcycdxczyxC for 1 x , x y, and 2 ch1
cd h2 ,, hh 11 hd 2 dczcdycxczyxC for 1 y, x y. 2 ch1 Here, in order to formulate the height of the building, consider the building outline in the cross-sectional view given by cutting the building outline in the x-axis direction or the y-axis Urban and Regional Planning Review Vol. 5, 2018 | 7 direction. As an example, it is assumed that the building outline is cut along line A-B in Figure-2-c. Since the Manhattan distance is adopted in this paper, the travel route from point P to the primary base O1 consists of the vertical travel P-A by elevator, the horizontal travel A-B on foot and horizontal travel B- O1 on foot. The horizontal travel B- O1 is outside the sectional view and is common among all floor ends. This means that the portion of the building outline in the cross-sectional view that passes through the same primary base O1 is identical to the optimal urban form of the two-dimensional model by Kondo et al. 12) when point B is regarded as the c primary base. Therefore, the slope of the building outline is the same value, h1 . Accordingly, cd in the cross section in the x-axis direction or the y-axis direction, the optimum urban form of the region passing through each base is a triangle with the summit as the highest point at the base c point and with a slope of h1 . This means that the building height H(x , y) at (x , y) is given by cd
ch1 ch1 cd d c y,xH x Cy for 0 x 1 h2 , 0 y 1 h2 , 1 1 2 c cd cd h1 2 ch1
ch1 ch1 cd h2 , 2 y,xH x Cy 2 for 1 dx , x y cd cd 2 ch1 c c h1 h1 , 3 y,xH x Cy 3 for xd , x y cd cd c c cd y,xH h1 x h1 Cy for 1 h2 dy , x y, and 4 4 2 c cd cd h1 c c h1 h1 yd . 5 y,xH x Cy 5 for , x y cd cd These have to satisfy the following equations:
3 ,sH 00 , 1 N xY,xH 0 , 2 N xY,xH 0 ,
3 N xY,xH 0, 4 N xY,xH 0 , 5 N xY,xH 0 cd cd cd cd H h2 ,1 Hy h2 ,1 y , xH , 1 h2 xH , 1 h2 , 1 2 1 4 2 ch1 2 ch1 2 ch1 2 ch1
2 3 ,, ydHydH , 4 5 ,, dxHdxH ,
x,xHx,xH 2 4 , 3 5 x,xHx,xH . So, we have
c c h1 h2 c h1 c h1 , C1 dr 1 CC 2 dr CC r c c 42 c 53 c d h1 , d , d
c d c h2 h2 d c , N drxxY 1 for 0 x 1 , h 2 c 2 c 0 y 1 h1 h1 2 c h1 d c 1 h2 dx N 2 drxxY for , x y, 2 ch1
N rxxY for xd , x y,
d c 1 h2 dy N 2 drxxY for , x y, 2 ch1
N rxxY for yd , x y. Urban and Regional Planning Review Vol. 5, 2018 | 8
If we write the highest height as H=h(0), we have c c h1 h2 hH 0 dr 1 cd ch1 and hence
cd ch2 . r dh 10 ch1 ch1 Where the diagrams shown by YN(x) can be classified into three patterns in Figure-4 according to the r magnitude relation between d and r. The left diagram in Figure-4 shows the case of d namely 2 d c d c r h2 h2 2 rd , the middle one for d and 2 dr 1 namely 3 2 dr and the 2 2 ch1 2 ch1
d ch2 d ch2 right one for 2 dr 1 namely r 3 . 2 ch1 2 ch1
d c d ch2 a. Case of 2 rd b. Case of 3 h 2 2 dr c. Case of r 3 2 c h1 2 ch1
Figure-4. Diagrams shown by YN(x)
Therefore, there is a discontinuous point in the urban form when 2dr . This is obviously inappropriate for the optimal urban form and only the case 2 rd is considered in the following. Since the urban form is symmetrical about yx, we limit our calculation to the region yx. The cross-sections along the x-axis and y=x are given in Figure-7, respectively.
a. Cross-section along x-axis b. Cross-section along y=x Figure-5. Cross-sections along x- and y-axes
c h1 3 d c d c c c For the cases of height z 2 dr , z h1 h 2 r h1 , z h1 dr and c d 2 c d 2 c d c d c d
ch1 ch2 z dr 1 , the number of stories is expressed as mA, mB, mC and mD, respectively and cd ch1 Urban and Regional Planning Review Vol. 5, 2018 | 9
S the floor area at height z as S(z). The floor area for the first to mA+1-th floors is written as 1m A , S S for mA+2-th to mB+1-th floors as mm BA , for mB+2-th to mC+1-th floors as mm CB and for mC+2-th S (2) to mD+1-th floors as mm DC . The total floor area D is given by the following equation with m being the number of stories: 1 mA 1 mB 1 mC 1 mD 1 SD S S S . 1mA mm BS mm CB mm DC 8 k1 mk A 2 mk B 2 mk C 2 The total floor area D on the left-hand side is divided by eight because we focus on the first quadrant and the region yx. Moreover, the population traveling to the secondary base via the primary base can be equivalent to the sum of the shaded floor area in Figure-6-a and is written as V. (3)
d c d ch2 a. Case of 2 rd b. Case of 3 h 2 2 dr c. Case of r 3 2 c h1 2 ch1 Figure-6. Population traveling via the primary base
From the above, we can calculate the total travel cost C:
d c c 1 mA 1 h 2 d 1 1 d 1 rak 2 ch1 ch1 2 cC h1 xdxx d c dxxdx dxdxx 0 1 h 2 d 8 k1 2 ch1
c d 1 sak cd ch1 x 1 dxdxrak c 1 d 1 rak ch1 ch1 2
m 1 1 cd 1 ch 2 d ch 2 B 1 drak 1 1 2 c 2 c 2 c cd ch2 h1 h1 xdxx h1 x 1 drak 1 xdx 0 1 cd 1 ch 2 1 drak 1 c c mk A 2 h1 h1 2 ch1 2 ch1
c d c d 1 rak c d c d d c x 21 dxxddrak h1 x 1 dxdxrak h 2 1 c d c 2 ch1 h1 h1
m 1 1 cd 1 ch 2 cd ch 2 C 1 drak 1 1 drak 1 2 c 2 c c c cd ch2 h1 h1 xdxx h1 h1 x 1 drak 1 xdx 0 1 cd 1 ch 2 1 drak 1 c c mk B 2 h1 h1 2 ch1 2 ch1
c d c d 1 rak c d c d c x 21 dxxddrak h1 x 1 dxdxrak d 21 drak c d c ch1 h1 h1
m 1 1 c 1 c c c D d 1 drak 1 h 2 d 1 drak 1 h 2 cd ch2 2 ch1 2 ch1 xdxx ch1 ch1 x 1 drak 1 xdx 0 1 cd 1 ch 2 1 drak 1 c c mk C 2 h1 h1 2 ch1 2 ch1 Urban and Regional Planning Review Vol. 5, 2018 | 10
m 1 c 1 A d 1 rak cd ch1 2 1 ydyyryak 0 c k 1 h1
m 1 c d c B d 1 ak 1 h 2 2 dr c c 2 c d h1 h1 1 ydyyryak 0 c mk A 2 h1
1 cd ch 2 1 drak 1 c c 2 c c d h 2 h1 h1 1 dryak 1 ydyy c d c d h 2 1 ak 1 2 dr ch1 c h1 ch1 2 ch1
c d 1 rdak cd cd ch1 1 ryak 1 2 ydydryak cd d ch 2 1 ak 1 2 dr ch1 ch1 ch1 2 ch1
m 1 1 c c C d 1 drak 1 h 2 cd ch 2 2 ch1 ch1 1 dryak 1 ydyy 0 mk 2 c c B h1 h1
c d 1 rdak c c c d d h1 1 ryak 1 2 ydydryak 0 c c h1 h1
m 1 1 cd ch 2 D 1 drak 1 c c 2 c c d h2 h1 h1 1 dryak 1 ydyy 0 mk 2 c c C h1 h1
2 mA 1 1 cd c 1 1 akrak d 4 c k1 h1
2 2 2 mB 1 2 2 2 d 2 12 dhh 2 1 dh h1 1 hhh 21 42 drdcccrczrccdccczc 1 ak 2 mk A 2 4 ch1 2 2 2 2 mC 1 5 210 10 5 rczrccdccczc d h1 2 dh 1 dh h1 2 mk 2 4c B h1
2 2 2 2 2 hh 21 10 h1 h2 2 hh 21 5 h1 dccccrdccc 2 1 ak 4 ch1
2 mD 1 1 1 cd ch2 drz 1 1 ak 2 2 c c mk C 2 h1 h1
h2 dVc . In the same way, the population traveling to the secondary base via the primary base (4), total floor area, and total travel cost are given by the following formulae in the case of
d ch2 3 2 dr . (Figure-6-b) 2 ch1
1 mB 1 mC 1 mD 1 SD S S mm BS mm CB mm DC 8 k 1 mk B 2 mk C 2 1 cC 8 h1
m 1 1 c 1 c d c B d 1 drak 1 h 2 1 h 2 cd ch2 2 ch1 2 ch1 xdxx 2 ch1 x 1 drak 1 xdx 1 c 1 c 0 d 1 drak 1 h 2 k 1 ch1 ch1 2 ch1 2 ch1 Urban and Regional Planning Review Vol. 5, 2018 | 11
c d c d 1 rak c d c d d c x 21 dxxddrak h1 x 1 dxdxrak h 2 1 c d c 2 ch1 h1 h1
m 1 1 c 1 c c c C d 1 drak 1 h 2 d 1 drak 1 h 2 cd ch2 2 ch1 2 ch1 xdxx ch1 ch1 x 1 drak 1 xdx 0 1 cd 1 ch 2 1 drak 1 c c mk B 2 h1 h1 2 ch1 2 ch1
c d c d 1 rak c d c d c x 21 dxxddrak h1 x 1 dxdxrak d 21 drak c d c ch1 h1 h1
m 1 1 c 1 c c c D d 1 drak 1 h 2 d 1 drak 1 h 2 cd ch2 2 ch1 2 ch1 xdxx ch1 ch1 x 1 drak 1 xdx 0 1 cd 1 ch 2 1 drak 1 c c mk C 2 h1 h1 2 ch1 2 ch1
m 1 c d c B d 1 ak 1 h 2 2 dr c 2 c cd h1 h1 1 ydyyryak 0 c k 1 h1
1 cd ch 2 1 drak 1 c c 2 c c d h 2 h1 h1 1 dryak 1 ydyy c d c d h 2 1 ak 1 2 dr c h1 c h1 ch1 2 ch1
c d 1 rdak cd cd ch1 1 ryak 1 2 ydydryak cd d ch 2 1 ak 1 2 dr ch1 ch1 ch1 2 ch1
m 1 1 c c C d 1 drak 1 h 2 cd ch 2 2 ch1 ch1 1 dryak 1 ydyy 0 mk 2 c c B h1 h1
c d 1 rdak cd cd ch1 1 ryak 1 2 ydydryak 0 c c h1 h1
m 1 1 c c D d 1 drak 1 h 2 cd ch 2 2 ch1 ch1 1 dryak 1 ydyy 0 mk 2 c c C h1 h1
cd
mB 1 2 2 2 2 2 2 d 2 12 dhh 2 1 dh h1 1 hhh 21 42 drdcccrczrccdccczc 2 1 ak k 1 4 ch1
2 2 2 2 mC 1 5 210 10 5 rczrccdccczc d h1 2 dh 1 dh h1 2 mk 2 4c B h1
2 2 2 2 2 hh 21 10 h1 h2 2 hh 21 5 h1 dccccrdccc 2 1 ak 4 ch1
2 mD 1 1 1 cd ch2 drz 1 1 ak 2 2 c c mk C 2 h1 h1
h2 dVc In the same way, the population traveling to the secondary base via the primary base (5), total floor area, and total travel cost are given by the following formulae in the case of Urban and Regional Planning Review Vol. 5, 2018 | 12
d c h2 r 3 . (Figure-6-c) 2 c h1 1 mC 1 mD 1 SD S mm CB mm DC 8 k 1 mk C 2 1 cC 8 h1
m 1 1 c 1 c c c C d 1 drak 1 h 2 d 1 drak 1 h 2 cd ch2 2 ch1 2 ch1 xdxx ch1 ch1 x 1 drak 1 xdx 1 c 1 c 0 d 1 drak 1 h 2 k 1 ch1 ch1 2 ch1 2 ch1
c d d 1 rak cd c cd c x 21 dxxddrak h1 x 1 dxdxrak d 21 drak c d c ch1 h1 h1
m 1 1 c 1 c c c D d 1 drak 1 h 2 d 1 drak 1 h 2 cd ch2 2 ch1 2 ch1 xdxx ch1 ch1 x 1 drak 1 xdx 0 1 cd 1 ch 2 1 drak 1 c c mk C 2 h1 h1 2 ch1 2 ch1
m 1 1 c c C d 1 drak 1 h 2 cd ch2 2 ch1 ch1 1 dryak 1 ydyy 0 c c k1 h1 h1
c d 1 rdak c c c d d h1 1 ryak 1 2 ydydryak 0 c c h1 h1
m 1 1 c c D d 1 drak 1 h 2 cd ch2 2 ch1 ch1 1 dryak 1 ydyy 0 mk 2 c c C h1 h1
2 2 mC 1 2 2 5 d h1 210 2 dh 10 1 dh 5 h1 rczrccdccczc cd 4c 2 k1 h1
2 2 2 2 2 hh 21 10 h1 h2 2 hh 21 5 h1 dccccrdccc 2 1 ak 4 ch1
2 mD 1 1 1 cd ch2 drz 1 1 ak 2 2 c c mk C 2 h1 h1
h2 dVc
Among the constraints in this model, the travel costs cd, ch1, ch2, the floor height a, and the total floor area D can be given as prerequisites of the model, and thus are treated as constants. Also, since the height h of the city and the distance r to the city end are determined by the inter-base distance d, finding the optimal urban form is synonymous with finding the optimum inter-base distance d. As described in Chapter 2, the unit of distance is km, the cost is the value of Table-1, the floor height 0.00285 km in consideration of general housing 21), and the floor area 1 km2. Based on this premise, the optimal values were obtained using the Solver function of Microsoft Excel. In the case of 2 rd , the inter-base distance d is 0.091 km, and total travel cost is
d ch2 0.0023. In the case of 3 2 dr , the optimal value is obtained when r = 2d, which is 2 ch1 Urban and Regional Planning Review Vol. 5, 2018 | 13
d c h2 the boundary value. In the case of r 3 , the optimal value is obtained when 2 c h1
d c h 2 , which is the boundary value. In the case of 2 rd , the optimal value is larger r 3 2 c h1 than the optimal value that is obtained in the case of 2 rd . Therefore, the optimal inter-base distance is 0.0091 km, and total travel cost is 0.0023.
4. Analysis of the characteristics of the three-dimensional optimal urban form 4.1 Comparison of urban models with or without consideration of floor height
Table-2 shows the optimal urban forms depending on the presence or absence of multiple floors. For the cases where the total floor area is 0.1 km2, 1 km2 or 10 km2, the optimum urban forms are given with the floor heights of 0.00285 km and 0.0035 km. Also, on the condition that the volume becomes the same as in the case of considering multiple floors, the optimum urban forms without considering multiple floors are given. In order to compare travel costs, the table shows the average travel costs per unit population. The average travel cost is given by dividing the total travel cost by the volume when not considering multiple floors, and by dividing the total travel cost by the floor area when considering the floor.
Table-2. Comparison of optimal urban forms (left: without floors; right: with floors)
Volume (km3) 0.000285 0.000350 0.00285 0.00350 0.0285 0.0350 Floor Area (km2) -0.1-0.1-1-1-10-10 Floor Height (km) - 0.00285 - 0.00350 - 0.00285 - 0.00350 - 0.00285 - 0.00350 Base Distance (km) 0.0424 0.0424 0.0454 0.0453 0.0913 0.0913 0.0978 0.0978 0.1968 0.1968 0.2107 0.2107 Urban area (km) 0.0995 0.0969 0.1065 0.1034 0.2143 0.2117 0.2295 0.2263 0.4616 0.4591 0.4944 0.4912 Height 0.0321 0.0307 0.0344 0.0327 0.0692 0.0677 0.0741 0.0723 0.1490 0.1476 0.1595 0.1578 at Primary Base (km) Height 0.0389 0.0375 0.0417 0.0399 0.0838 0.0824 0.0898 0.0880 0.1806 0.1792 0.1934 0.1917 at Secondary Base (km) Average Travel Cost 0.0108 0.0103 0.0116 0.0109 0.0233 0.0228 0.0249 0.0243 0.0502 0.0496 0.0537 0.0531 Percentage of the Base Distance 42.6% 43.7% 42.6% 43.8% 42.6% 43.1% 42.6% 43.2% 42.6% 42.9% 42.6% 42.9% to the Urban area Percentage of the Height of the Primary Base 32.3% 31.7% 32.3% 31.6% 32.3% 32.0% 32.3% 32.0% 32.3% 32.1% 32.3% 32.1% to the Urban Area Percentage of the Height of the Secondary Base 39.1% 38.7% 39.1% 38.6% 39.1% 38.9% 39.1% 38.9% 39.1% 39.0% 39.1% 39.0% to the Urban Area
Table-3. Ratios of specifications of the urban forms with floors to those without floors
Floor Area of the urban forms 0.1 1 10 with floors (km2) Floor Height (km) 0.00285 0.00350 0.00285 0.00350 0.00285 0.00350 Base Distance 100.0% 99.8% 100.0% 99.9% 100.0% 100.0% Urban area 97.5% 97.0% 98.8% 98.6% 99.4% 99.4% Height at Primary Base 95.6% 95.0% 98.0% 97.7% 99.0% 98.9% Height at Secondary Base 96.4% 95.9% 98.3% 98.1% 99.2% 99.1% Average Travel Cost 95.2% 94.5% 97.8% 97.4% 99.0% 98.8%
Compared with the model without multiple floors, the urban area becomes smaller and the primary base relatively moves outward in the model with multiple floors. Also, as the height of Urban and Regional Planning Review Vol. 5, 2018 | 14 the city decreases, the floor height increases, or the total area decreases, these phenomena become prominent. In the case of a large-scale city, the error is small, but even in cities with a total floor area of about 10 km2, the urban area, the height of the city, and the average travel cost differs by 1%. The population of a city with a total floor area of 10 km2 is approximately 150,000 in Japan, according to the example of Tama city, which is described later.
4.2 Analysis of features with different conditions for the urban model
Here, we calculate the inter-base distance and maximum height that minimize the travel cost using the urban model formulated in Chapter 3, while varying the constant values in Chapter 3. The optimal solution was numerically obtained using the Solver function of Microsoft Excel. Figure-7 shows the results for 20 different total floor areas from 5 to 100 km2 at intervals of 5 km2, while Figure-8 shows the results for 20 different floor space heights from 0.002 to 0.0039 km at intervals of 0.0001 km. Figure-9 shows the result for 24 different vertical travel costs from 0.2 to 4. Figure-10 shows the results for 24 different travel costs by bus from 0.02 to 0.208. These results were obtained by setting large parameter ranges, which could contain values for actual cities and for which the functional behaviors could be viewed. The total floor area is assumed to be 10 km2 and the height of a floor space is assumed to be 0.00285 km in Figures-8, 9 and 10. Based on this result, the optimal urban form minimizing the total travel cost is uniquely determined if the population, i.e. total floor area, is given. If a regression formula is applied to the relation between the inter-base distance d and maximum height H for this case, we obtain a relatively accurate relation as follows: (1) When the total floor area is changed, H 9138.0 d (2) When the floor space height is changed, H 9104.0 d (3) When the vertical travel cost is changed, H 0066.0 d 031.2 (4) When the travel cost by bus is changed, H d 2 d 4302.09022.11893.3
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Figure-7. Specifications when the floor area is changed
Figure-8. Specifications when the floor space height is changed
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Figure-9. Specifications when the vertical travel cost is changed
Figure-10. Specifications when the travel cost by bus is changed
When the total floor area is changed, the relation between the maximum height and the inter-base distance is linear. Here, therefore the optimal urban form changes while similarity is retained. Also, the relation between the population traveling via the primary base and the total floor area is linear. The maximum height, inter-base distance, total travel cost and urban area increase approximately in proportion to the cube root of the increase rate of the total floor area. When the floor space height is changed, the optimal urban form changes with similarity retained, as when changing the total floor area, while the maximum height, inter-base distance, total travel cost and urban area tend to increase in proportion to the cube root of the increase rate Urban and Regional Planning Review Vol. 5, 2018 | 17 of the floor space height. There is no obvious correlation between the population traveling via the primary base and the floor space height, because the increased vertical and horizontal distance due to the increase in floor space height have a complicated impact on determining the optimal urban form. When the vertical travel cost changes, the relation between maximum height and inter-base distance is expressed as a power math function and the optimal urban form changes without retaining similarity. This is because the change in vertical travel cost changes the line connecting the uppermost points of the city, namely the envelope curve. The inter-base distance increases in proportion to the cube root of the increase rate of vertical travel cost, while the maximum height increases in proportion to the -2/3 power of the increase rate of the vertical travel cost. As when changing the floor space height, there is no obvious correlation between the population traveling via the primary base and the vertical travel cost. When the travel cost by bus changes, the relation between maximum height and inter-base distance is expressed as a quadratic function, while the optimal urban form changes without retaining similarity. Unlike the other cases, it is not expressed by a simple formula. This is because the change in travel cost by bus changes the ratio between the population traveling directly to the secondary base and that traveling to the secondary base via the primary base. The population traveling via the primary base decreases with the growth in the travel cost by bus. A comparison of the optimal urban form of a three-dimensional city obtained in this paper with the optimal urban form of a two-dimensional city obtained in the preceding study12) shows the following. In the two-dimensional city, the maximum height and inter-base distance increase almost in proportion to the square root of the increase rate of the total floor area, while in the three-dimensional city, it increases in proportion to the cube root. Similar results were obtained for cases where the floor space height, inter-base distance, or vertical travel cost is changed. The maximum height of the two-dimensional city increases in proportion to the increase rate of vertical travel cost to the power of -1/2, while that of the three-dimensional city increases in proportion to the increase rate of vertical travel cost to the power of -2/3. The difference comes from the existence of one additional axis to consider in the three-dimensional city and the functional characteristics themselves are the same in both urban forms.
4.3 Comparison between the optimal urban form and the actual city
Here, we calculate the optimal urban form assuming a floor area equivalent to an actual city using the urban model with multiple floors and compare it with the actual city. We selected Tama City in suburban Tokyo, Japan, as an actual example because it is surrounded by greenery, separate from surrounding cities and includes many mid-rise buildings constructed as Tama New Town has developed. The optimal urban form of a three-dimensional city must also contain roads. In this paper, the road area ratio in the city relative to the total floor area was set to 20% and the roads were assumed to be uniformly distributed on each floor. Since the total floor area of Tama City is 7,474,635 m2, the optimal urban form of a city with a total floor area of 9,343,293.75 m2, including the road area (1,868,658.75 m2),22) is given in Table-4. The table also shows the optimal urban form of a three-dimensional city without the floors for which the volume is the same and the optimal urban form of a two-dimensional city for which the urban area width as a cross-section is identical.
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Table-4. Specifications of the optimal urban forms Values for two-dimensional city Values when floor height Values when floor height when floor height is NOT considered is considered is considered Floor Height (km) - 0.00285 0.00285 Floor Area (km2) - 9.3433 29.2063 Base Distance (km) 0.1923 0.1924 0.2179 Urban area (km) 0.4512 0.4488 0.4488 Height at Primary Base (km) 0.1457 0.1442 0.1299 Height at Secondary Base (km) 0.1765 0.1751 0.1649 Total Travel Cost 0.0490 0.4533 1.1841
The result shows that the inter-base distance is 192.4 m, the urban area size, i.e. half the entire urban area width, is 448.8 m, the maximum height is 175.1 m and the height at the primary base is 144.2 m. Namely the result indicates that a city with 100 m-high buildings continuously distributed over a horizontal distance of 200 m from the city center is optimal. Under the conditions that the urban area widths are the same, the optimal urban form of the two-dimensional model and the three-dimensional model is overlapped in Figure-11. The primary bases of the three-dimensional model are slightly closer to central secondary base, and the floor height is higher than those of the two-dimensional model. One possible reason for that is as follows. In the three-dimensional model, the y-axis was added, whereas the horizontal travel axis was only in the x-axis direction in the two-dimensional model, thus the spread of the plane in the suburbs increases and the share of the horizontal travel cost in the total travel cost increases.
Figure-11. Overlapping of the optimal urban forms and the actual city
Figure-11 also shows a cross-section of the optimal urban form of the urban model overlapped with a cross-section of Tama City along Kawasaki Street and with the center on Seiseki Sakuragaoka Station. One can see from the figure that the actual city has lower building height and a lower building density than the urban model. Also, the actual distance between bus stops is longer than the inter-base distance in the urban model. In Tama City, there is no residential building higher than 175 m, the maximum height in the optimal urban form. In fact, there are only a few residential buildings which exceed this height in Japan. A building of almost the same height as this one is, for example, Proud Tower Shinonome Canal Court in Koto Ward, Tokyo. There are no tall buildings within a distance of 50 m from the building. Namely the optimal urban form derived from the model calculation includes a far smaller horizontal size than the actual city and hence has many high-rise buildings, which differs significantly from the form of the actual city. The significant difference between the calculated urban model and actual city form is attributable to the following two reasons: Urban and Regional Planning Review Vol. 5, 2018 | 19
The first is the vertical travel cost value used in the city model. In this paper, the travel cost excludes the expense of constructing high-rise buildings, the expense of installing elevators, operational costs such as electricity expenses for elevators and waiting time for users, all of which mean the vertical travel could cost far more than assumed. To clarify the mathematical behavior of the model when the vertical travel cost changes, the optimal urban form is calculated for about ten times the value of the vertical travel cost, although the value is unrealistic and extremely large. The result shows that the optimal urban form spreads slightly horizontally but the ratio of height to width remains largely unchanged, as seen in Figure-11. When the vertical travel cost is 2, about ten times larger than the travel cost on foot, Figure-11 reveals that the maximum height of the city H is 57.2 m and the urban area size r is 796.2 m. Since the urban area size r is half the entire urban area width, the height is about 3.6% of the horizontal width of the entire urban area. When the travel cost by bus is 0.02, about one tenth of that on foot, Figure-12 reveals that the maximum height of the city is 155.2 m and the urban area size r is 499.4 m, meaning the height is about 15.5% of the horizontal width. In the actual city, the distance between Keio Nagayama Station and Keio Tama Center Station on the Keio Sagamihara Line in Tama City, Tokyo, is about 2.3 km. If we assume that the actual urban area is about 2.3 times larger than the inter-base distance as in the urban model, we have the urban area size r of 5.29 km. Even if the maximum height of the actual city is 175.1 m, the height is about 1.7% of the width of the urban area. Therefore, the horizontal width of the city model urban area is smaller than that of the actual city even when the vertical travel cost is assumed to be larger or the horizontal travel cost smaller. Figure-11 also indicates that the building height monotonically decreases with the increase in the vertical travel cost but the effect of the vertical travel cost gradually declines. Figure-12 also shows that even when the travel cost by bus declines significantly, the decrease in building height is limited, which indicates that the actual urban form is restricted by factors such as the nonlinear cost increase in building construction in a vertical direction. The second is the legal regulations governing the building form. For example, the city plan of Tama City designates the maximum height of buildings23) and does not allow successive construction of residential buildings of 100 m or higher. Therefore, the legal regulations prohibit the development of an urban form which minimizes the travel cost in actual cities.
5. Summary and future problems
In this paper, we assumed the presence of multiple floors in three-dimensional cities having a hierarchical space structure, calculated the optimal urban form for a given total floor area with minimization of the total travel cost and studied the impact of changing various factors. We also compared the city model and actual city form to clarify the difference. Consequently, we found that the optimal urban form changed; retaining similarity when the total floor area was changed but without doing so when the vertical travel cost or travel cost by bus changed. Also, the optimal urban form had a significantly smaller horizontal width than actual city forms. Accordingly, the present paper showed that the optimal urban form with respect to the travel cost was that in which ultra high-rise buildings were concentrated in the city center. In addition, the urban model without multiple floors has a smaller urban area compared to the urban model with consideration of multiple floors, and the travel distance in the vertical direction decreases. Therefore, the average travel cost becomes smaller. Even in the model with Urban and Regional Planning Review Vol. 5, 2018 | 20 a total floor area of 10 km2, by assuming multiple floors, the average travel cost decreases by 1%. For cities smaller than this, the range of decrease will increase, thus the model with multiple floors developed in this article will be useful in obtaining an average travel cost closer to the actual situation. Cities resembling the optimal urban form calculated in this paper, sometimes expressed as hyper buildings or archology, have been proposed by architects and companies. For example, Frank Lloyd Wright planned a gigantic 528-story building “Mile High Illinois” in his book A Testament.24) Toshio Ojima’s group at Waseda University proposed a 10-km high building “Tokyo Babel Tower” on a 110 km2 base area.25) In 1990, concepts of ultra high-rise buildings such as Aeropolis 2001 by Obayashi Corporation and Sky City 1000 by Takenaka Corporation were also proposed by various general contractors. Behind such proposals was the social need to prevent urban sprawl, the pursuit of technological potential and study on the feasibility of a future city including highly concentrated city functions. For example, a city model having the same base area as Tokyo Babel Tower is shown in Figure-15. In the figure, the base of Tokyo Babel Tower is assumed to be round. As described above, the city model in this paper concerns the “district” and “town” functions of a compact city system, which correspond to the lower part of Tokyo Babel Tower (i.e. residence, commercial floors, offices, hotels, etc.). In this regard, the optimal urban form obtained in this paper and the lower part of Tokyo Babel Tower are similar. Our result shows that Tokyo Babel Tower could be close to a rational urban form from the perspective of minimizing the total travel cost.
Figure-15. Comparison of Tokyo Babel Tower and the three-dimensional urban model
Most of these proposed cities are higher than the optimal urban form derived in this paper because the proposed cities form clusters in a vertical direction over several floors containing various city functions. The urban model used in the present paper could be modified to one with higher-rise buildings if we create bases vertically, like those horizontally and connect them with high-speed elevators. However, these proposed cities have not been realized. This could be because perspectives other than the travel costs studied in this paper, such as legal regulations and the costs of constructing and operating high-rise buildings, affect how a realistic form of cities is determined. Urban and Regional Planning Review Vol. 5, 2018 | 21
These perspectives should be taken into account when discussing a compact city. As future work, we need to study the occurrence of another primary base due to the horizontal development of a city and any change in the optimal urban form caused by such occurrence. In this paper, the urban form changes, while maintaining similarity in proportion to the total floor area. However, the horizontal distance to travel on foot is limited, as is the area a single primary base can cover. Accordingly, an additional primary base would be necessary when the total floor area is extended. In this paper, on a trial basis, we used the mathematically easiest model to calculate the case where only a single primary base exists on a horizontal line extending from a secondary base, meaning a city scale far smaller than the actual figure. Moreover, actual cities feature primary bases located almost continuously; with a shorter inter-base distance, lower maximum height as moving away from the secondary base. This underlines the need to create an optimal urban model by taking account of not only a single urban area but also connection between urban areas. If the costs of developing and operating the bases and of installing and operating the elevators and others are taken into account based on this future work to devise a more realistic urban model, a compact city system could be evaluated more precisely. It is also necessary to develop an urban model taking account of building gaps, given the gaps between ultra high-rise buildings in actual cities and since Kurita26) calculated an optimal layout of passageways connecting two buildings, assuming continuous vertical population distribution.
Acknowledgements Part of this study was supported by JSPS KAKENHI Grant Numbers JP24560752 and JP15K06370.
Notes (1) If dx 0 , we have
h1 h1 h1 h2 dcycdxcyxc and hence
0 hh 12 dcc Since we assume in this paper that the cost of traveling on foot exceeds the travel cost by bus, this is always true. Therefore if dx 0 , people always travel via the primary base. If dx 0 , we have
h1 h1 h1 h2 dcycdxcyxc and hence
d ch2 x 1 2 ch1
d ch2 So, if xd 1 , people always travel via the primary base. 2 ch1
ch1 (2) If we set 1 amz , the floor area for the first to mA+1-th floors when0 z 2 dr is cd given as follows: 2 1 c 1 c 1 c S d rz d rz d rz 1mA 2 ch1 2 ch1 4 ch1
ch1 3d ch1 d ch2 ch1 When 2 zdr r , namely for mA+2-th to mB+1-th floors, the cd cd 22 cd cd floor area is given as follows: Urban and Regional Planning Review Vol. 5, 2018 | 22
2 1 1 c c d h2 S mm drz 1 BA 2 2 c c h1 h1 1 c cd 1 c c cd 1 c c d rz 3 h2 d drz 1 h2 1 h2 d drz 1 h2 2 c 2 c 2 c c 2 c 2 c c h1 h1 h1 h1 h1 h1 h1 2 1 c c cd cd 1 c d drz d rz 3 h2 d 1 h2 d z dr 2 c c 2 c 2 c 2 c h1 h1 h1 h1 h1
2 2 2 2 2 2 d 2 12 dhh 2 1 dh h1 1 hhh 21 42 drdcccrczrccdccczc 2 4 ch1
3d ch1 d ch2 ch1 ch1 When r z dr , namely for mB+2-th to mC+1-th floors, the cd 22 cd cd cd floor area is given as follows: 2 1 1 c c 1 c c S d drz 1 h2 d drz 2 d drz mm CB 2 2 ch1 ch1 2 ch1 ch1
5 2 2 210 10 5 2 rczrccdccczc 2 d h1 2 dh 1 dh h1 4 c 2 h1
2 2 2 2 2 hh 21 10 h1 h2 2 hh 21 5 h1 dccccrdccc 2 4 ch1
c c c h1 h1 h2 When zdr dr 1 , namely for mC+2-th to mD+1-th floors, the floor area is cd cd ch1 given as follows: 2 1 1 c c S d drz 1 h2 mm DC 2 2 ch1 ch1 (3) In the case of 2 rd , the first quadrant and y x , the population traveling to the secondary base via the primary base (V) is given as follows: 2 2 m A 1 1 c 1 mC 1 1 c V d 1 rak d 1 drak 2 c 2 2 c k 1 h1 mk A 2 h1
m A 11 c 1 d c c 1 d c d 1 rak 1 h2 d 1 rak 1 h2 2 c 22 c c 22 c k 1 h1 h1 h1 h1
mB 1 1 c c d c d c d d h2 h2 1 drak 1 rak 3 d 1 mk 2 2 c c 2 c 2 c A h1 h1 h1 h1 2 mC 1 1 c d 1 drak 2 c mk B 2 h1 c d h2 y x (4) In the case of 3 2 dr , the first quadrant and , the population traveling to 2 ch1 the secondary base via the primary base (V) is given as follows: 2 mC 1 1 cd V 1 drak k 1 2 ch1
mB 1 1 c c cd cd d 1 drak d 1 rak 3 h2 d 1 h2 2 c c 2 c 2 c k 1 h1 h1 h1 h1 2 mC 1 1 c d 1 drak 2 c mk B 2 h1 d c h 2 y x (5) In the case of r 3 , the first quadrant and , the population traveling to the 2 c h1 secondary base via the primary base (V) is given as follows: Urban and Regional Planning Review Vol. 5, 2018 | 23
2 2 mC 1 mC 1 1 cd 1 cd V 1 drak 1 drak k1 2 ch1 k1 2 ch1
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