Interpreting spatial layouts of nursing homes based on partitioning theory

ABSTRACT With the aging world and the improvement of people’s living standards, the quality of elderly life is gaining more attention. The nursing home is an important carrier for the life of the elderly, the reasonable layout of its spaces can provide a better wayfinding experience for residents. Based on the survey of 168 Japanese nursing homes, we found that some spatial layouts are more widely used in the design of nursing homes than others. The intention of this paper is to attempt to explore the characteristics of these spatial layouts within the theoretical framework of space syntax. The expectation is to discover certain design principles that are followed when the spaces in the nursing home are arranged, seeking to contribute to a better understanding of nursing home morphology. Setting out from the spatial models established by partitioning theory research, the paper examines the impacts of the different numbers of corridors, different corridor combinations, and different spatial distribution on their subjects. The result shows that designers will consider a combination of influencing factors when designing the spatial layout of the nursing home. In terms of the number of corridor and corridor combinations, the spatial layout which facilitates the residents’ wayfinding is adopt in more nursing home plans, but in terms of spatial distribution, the plan layout of nursing homes is more focused on a balanced design. Finally, this paper analyses the design principles of spatial layout for nursing homes by using spatial syntax theory and method based on experience. It suggests that the theory and method can provide designers with a new way of thinking about understanding architectural space.


Introduction
With the aging world, the quality of daily life for older people is gaining more attention. The older people's health status and mobility decrease as they age. The elderly people are more willing to migrate to the nursing home with better accessible amenities from their own home environment (Buurman et al. 2014;De Jong et al. 1995). The built environment is the most primary, enveloping, the largest and the most socially significant artefact that humans create (Hanson 1998). In addition to the meaning of a physical shelter, a built environment is a meaningful formation of information that expresses the psychological need, social premise, and lifestyle of the inhabitant (Tuan 1977;Tao and Ding 2015). The nursing home is a social welfare-built environment that specially provides comprehensive assistance in daily activities, complex care, and nursing needs for the elderly (Duncombe, Robbins, and Wolf 2003). Normally, in addition to providing living arrangements, nursing homes are equipped with some basic care facilities. Therefore, it is a topic worth exploring how to arrange the various functional spaces in a nursing home to provide a better living experience for the elderly.
Although the research to date on nursing homes has covered many disciplines, most have focused on care and social environment. In terms of the built environment, researchers have been working on the physical environment of nursing homes in the hope of improving the quality of life of residents. For instance, Burke et al. attempted a grounded theory approach to enabling articulation of new conceptual-spatial relationships between residents and physical environment of the nursing home based on residents' lived experiences and aspirations (Burke and Alejandro 2021); Davis et al. hoped to focus thinking for the design of the built environment from the perspective of the occupant's living experience to address and create a more occupant-friendly physical environment of the nursing home (Davis et al. 2009). Eijkelenbooma's team researched the impact of architectural factors on the sense of home for elderly people living in nursing homes and developed design guidelines to improve the experience of nursing home residents (Eijkelenbooma et al. 2017); Burton et al. studied the effect of a particular spatial environment on the elderly (Burton and Sheehan 2010); Rodiek et al. studied the preference of elderly residents for specific environmental characteristics of nursing homes (Rodiek and Fried 2005) ; Parker, et, al. studied the individual space's design objects such as security or privacy through the assessment of the physical environment (Parker et al. 2004); Shuang and Masataka's team studied the facility configurations of different types of Japanese nursing homes (Shuang and Chiori 2008;Masataka et al. 2016). However, research on the built environment of nursing homes mainly focused on the studies of individual spaces and the configuration of facilities, but little is known about the spatial layout of nursing homes.
The spatial layout is not only an important part in the architectural design process but also a key attribute of the building's expression of its own function. More importantly, studies suggested that the spatial layout can predict people's wayfinding behaviour in the building and can affect people work behaviour and productivity (Saif and Craig 2003;Lesley, Robert, and Christian 2021). So, how are the spaces laid out in the built environment of vulnerable people? This paper investigated the 168 existing commercial nursing homes in Japan to address this issue. The survey revealed a certain similarity among the spatial layouts of these nursing homes. In particular, some similar solutions regarding spatial layout, which include the number of corridors, the corridor combination, and spatial distribution, are widely used. Normally, the design of a building influenced by many factors, such as local requirements, client's request and preferences, building regs, designer's personality. Why did these spatial layouts end up being selected and widely used under a multitude of factors? Is it because these spatial layouts provide a better living experience for the residents, such as better wayfinding? Or is the spatial distribution in them more rational? Regarding these questions, this paper attempts to explore these spatial layouts within a theoretical framework to seek a better understanding of the spatial layout of nursing homes. This combined framework is built through a series of case studies of the nursing home and seeks to develop a theoretical consensus based on experience. Finally, the paper examines the impact of the possible forms of different spatial layouts on their subjects in the light of the spatial models established by space syntax studies. The expectation is to discover the design principles that designers follow when laying out spaces for nursing homes, generating enlightening information and guidance for designing nursing homes. The research framework is shown in Figure 1.

The theory of space syntax: total depth
The methods used for the study of the physical environment of buildings are diverse. In the study of the physical environment of nursing homes, SCEAM (Sheffield Care Environment Assessment Matrix) and EAT (Environmental Audit Tool) are two of the more commonly used methods. SCEAM is a method for recording the characteristics and use of a nursing home, often used in combination with other methods, and preferring to be adopted for research on aspects of care (Torrington 2007). EAT works by creating design-related questions that incorporate the lived experiences of the residents of the nursing home, then using questionnaires to obtain data to assess the design quality of the building, and finally clarifying the spatial relationship between the residents and their physical environment (Fleming and Kirsty 2015;Quirke 2019). In assessing the spatial arrangement of nursing homes, EAT focuses more on the design details and relies on the sense of the residents. Space syntax techniques for the analysis of spatial layouts were the first to demonstrate, in a numerical way, clear and systematic relations between spatial design and observed functioning across a range of building and urban types (Hillier 1996(Hillier , 1984. The main interest of space syntax is the relation between human beings and their inhabited spaces. Space syntax research aims to develop strategies of description for configuring inhabited spaces in such a way that underlying social meaning can be enunciated (Dursun 2007). In addition, space syntax attempts to constitute a configurational theory in architecture by generating a theoretical understanding of how people make and use spatial configurations. It leads to the study of architecture intuition with its rational and more rigorous creation, then makes the deployment of non-discursive intuition more rational and therefore more discursive (Hillier and Hanson 1997). In the study of the built physical environment, spatial syntax is often used to study people's activity and movement in space. For instance, Saif et al. investigated the topological knowledge of hospital layouts and obtained the result that spatial layout can predict people's wayfinding behaviour in buildings (Saif and Craig 2003); Kali studied museum building design and exhibition layout and demonstrated that museums can convey their intended messages and meanings through the interplay of spatial layout and exhibit display (Tzortzi 2007); Byun et al. examined the social logic of the activities of Korean family members using spatial syntax (Byun and Choi 2016). Lee interpreted the sustainability of three buildings from the degree to which the space attracts people together by using spatial syntax theory (Lee 2020). Therefore, space syntax is advantageous for the study of spatial layout from the perspective of people's movement paths through spaces.
The method for analyzing the interior spaces of a building by space syntax is based on using variables to represent how deep or shallow a certain space might be. It provides a quantitative and numeric way of thinking to understand how spaces are arranged, and how the spatial formation works (Lee 2020;Hillier and Penn 1991). Total Depth, which is one of the variables, will be focused on in this paper. Its calculation formula is: where the numerator represents the Total Depth value of unit space i. n is the total number of nodes in the spatial system. dij represents the Depth of node i to j, namely the minimum number of connection steps from node i to j. The Total Depth value TDi is usually used to represent the sum of the minimum number of connecting steps from space i to all other spaces. The smaller the Total Depth of a space, the shallower or the higher the degree of convenience (accessibility) it is in that spatial system. A simple graphic illustration may serve to explain. Figure 2a shows a simple, rectangular buildings divided by a partition into two, cell A and cell B, with a doorway creating a relation of permeability between them, space C is on outside of the building but connected to the Cell A. In space syntax, the justified graph is an immediate, visual way of showing spatial configuration. Figure 2b shows the justified graphs, which are drawn from different spaces in the building. Finally, the process of calculating the total depth of each space is shown below each graph in Figure 2.

Partitioning theory: gaining or losing total depth
How does the change in the spatial formation affects the depth of the space? How does it manifests itself in the gain or loss of depth? Regarding these questions, Hiller suggests partitioning theory (Hillier 1996). Consider a 6 × 6 complex in which a total of 36 cells are evenly distributed within a square, and all cells are open to each other (Figure 3a). Figure 3(b) shows the connections between all the cells in the complex. Then, the total depth values of each cell can be obtained by the formula of calculation. Finally, adding up the total depth values of all the cells, this complex has 5040 depths as shown in Figure 3(c). How does depth gains or losses determine spatial formations? An explanation is creating a larger space in the complex resulting in a change in the total depth (Hillier 1996). For instance, in Figure 4(a, b), larger spaces in the complexes are created by eliminating the existing twothirds partitions in four adjacent cells. It can be noted that the larger spaces with different shapes and locations have implications for the total depth of the complex from the four complexes ( Figure 4 (a,b)). The total depth of the complex with a centrally placed open "square" is 102 depths less than that of the complex with a peripherally placed open "square" (Figure 4 ((a-1), (a-2))). Figure 4 (a-1,b-1) shows that a linear larger space will result in the complex losing more depth than a compact one. In the two examples in Figure 4(c), the larger space consisting of four open cells is divided into 2 two-cells spaces. By comparing 'c-

complexes'
with 'a-complexes', 'b-complexes', a comparable number of discrete spaces create more depth for the complexes than continuously joined spaces.

Classification and description of the main spatial layouts
In the early stage of the study, nearly 1000 commercial nursing homes are surveyed through email enquiries, web searches and reviews of relevant books and magazines. Finally, only 168 nursing homes provided us with clear and detailed floor plans. These 168 nursing homes are drawn from 34 cities in 8 regions of Japan, are distributed between 218.51 m 2 and 12,901.2 m 2 in size, have between 1 and 10 stories in height, and are opened between 1991 and 2020 years. And in terms of the number of beds, these nursing homes have between 5 and 200 beds, 85% of which are equipped with between 10 and 70 beds. Spaces for movement form an integral part of any building organization, and the overall configuration of the circulation paths indirectly describes the spatial layout of a building (Ching 1979). Within a nursing home, corridors link a series of individual rooms and act as circulation routes and transitional spaces for residents to move between spaces. Of 524 plans in the 168 nursing homes, 485 plans have at least one distinctly long corridor per unit. One plane even consists of up to nine corridors. Hence, the number of corridors and the combined form of corridors are crucial factors in forming the spatial layout of the nursing home and the flow of residents. Based on the survey of 168 Japanese nursing homes, four main spatial layouts are derived depending on the number of corridors the plan consists of. Table 1 shows the relevant sketch maps, cases, and numerical data for four spatial layouts. The descriptions of four spatial layouts (T1 to T4) are as follows: T1. A series of individual rooms in a nursing home plan is organized by a single corridor.
T2. Two corridors, which form in a T-shaped or L-shaped, organize a series of individual rooms in a nursing home plan.
T3. A series of individual rooms within a nursing home plan is organized by three corridors, which normally form in an H-shaped or C-shaped. T4. A ring circulation path, which consists of four corridors, organizes a series of individual rooms in a nursing home plan.
The survey results show that 97% of 168 nursing homes adopt at least one of four spatial layouts as the organization of interior space. It is rare that a plan of the nursing home utilizes over four corridors. This indicates that the use of the number of corridors in the design of nursing homes is limited by the effect of some factors. These factors perhaps stem from local building regs, the needs of the client, the character of the building and the personality of the architect, etc. In addition, the data reveals certain spatial layouts are more favored to be adopted in the plan design of the nursing home. Clearly, the highest usage rate is for the spatial layout of T1, which is used in approximately two-thirds of nursing homes and half of the plans. Subsequently, the spatial layout of T2 appears in 44.26% of the nursing homes and 33.21% of the plans. Nevertheless, the usage rate of the spatial layout of T3 and T4 is merely 11.9% and 2.97% of the nursing homes and 5.91% and 2.29% of the plans, respectively. These data demonstrate that a spatial layout constituted by comparatively fewer corridor are more widely used when a nursing home plan is designed. Finally, the spatial layouts in 480 plans adopt one of these four types, whereas the spatial layout in the rest of 5 plans consists of more than 3 corridor and in which the combined form of corridors does not belong to any of four types. It should be noted that the spatial layout in 12 plans consists of the ring-corridor type made up of more than 3 corridors, but in 10 plans the spatial layout is constituted by 4 corridors.
Further investigation, based on four spatial layouts, reveals that similar solutions are adopted for the spatial distribution of the nursing home, particularly in the arrangement of the number of rooms. Firstly, a pattern of spatial distribution that we can call homogeneous type is present in around 93% of the plans with the spatial layouts of T2. The pattern exhibits that the number of rooms on corridor 1 is approximated by that on corridor 2 as shown in Table 1 (b-1). In contrast, the non-homogeneous type accounts for a very small ratio. Secondly, Table 1 (c-1) depicts a dominant pattern of spatial distribution (homogeneous type) in the plans with the spatial layouts of T3. This pattern illustrates that the number of rooms on corridor 1 and corridor 3 are similar, meanwhile they exceed the number of rooms on corridor 2. Likewise, an explicit pattern of spatial distribution exists in the plans with the spatial layouts of T4. It is consistent with that the number of rooms on the opposite side corridor approximates each other as shown in Table 1 (d-1). In short, these domain patterns reflect the spatial distribution characteristics of relative symmetry in morphology.
Therefore, we can infer that there is a preference for the number of corridors, the combined form of corridors, and spatial distribution when a nursing home plan is designed based on and limited to the data collected and analyzed. The preference indicates that these spatial layouts are considered to be relatively superior and reasonable compared to their homologous counterparts. However, these spatial layouts here derive from a summary of dataset in a morphological generalization. In fact, the final spatial layout of a project may also be influenced by local requirement, clients request and preferences, building regs, overall budget of the building, constructive aspects, etc. Therefore, the preference only seems to tell us that in practice these spatial layouts are more favored to be used in terms of form. Finally, we attempt to adopt a theoretical approach to explore the properties of these spatial layouts to seek a better understanding of the spatial layout of nursing homes.

Application of partitioning theory
How is a nursing home plan diagram translated into a depth diagram? An example based on a nursing home plan can help us to understand. Figure 5a shows a nursing home plan in an "L-shape". The plan is turned to a diagram of the connected relation between convex spaces or function spaces (see Figure 5b) (a convex space is defined as a space in which all points can completely see each other, and it is represented as a rectangle in the paper.). The convex space diagram is then transformed into a justified graph, and the total depth of each space is calculated according to Equation 1. Figure 5c shows a justified graph drawn from space R1, and the process of calculating total depth of space R1. Finally, the total depth of the space is labelled into each convex space as shown in Figure 5d.
Within the partitioning theory, it has been clarified that the shape, location, and the number of larger spaces will affect the total depth of a complex. A corridor in a building plan acts like a large space in a complex of the partitioning theory. Similarly, it can be inferred that the number of corridors and the corridor combination have an impact on the total depth of a building plan. Then, the ideal models which resemble the grid complexes of the partitioning theory are created depending on the spatial layout within the nursing home. These models are shown in detail as follows: Firstly, suppose that there are 10 individual cells, without a direct connection, linked by the corridors within a complex. Secondly, four models are conceived according to the number of corridors they consist of as shown in Figure 6 (a-d). Another three models composed of identical number of corridors but with different combined forms of corridors are created as shown in Figure 6 (d-f). Thirdly, different spatial distributions within each model are represented by altering the number of cells on the corridor, such as complex (b-1) to complex (b-4) in Figure 6. Finally, six groups of models are formed. Considering that the change in the number of corridors gives rise to the deviation in the total number of spaces within the complex. The deviation will affect the gain or loss of the total depth of the complex. Therefore, here the sum of the total depths of the 10 cells are used to represent the total depth of the complex. Finally, the total depth of each cell is marked separately on each cell of each complex. The total depth of each complex is shown below each complex in Figure 6.
The first point can be drawn from Figure 6 is that the discrete corridors create more depth for a complex than a single corridor. This can be derived from the fact  that the total depth of the complex (a-1) is less than that of any other complex in Figure 6. The second point can be obtained is that the complex tends to gain more depth as the greater the number of corridors it is composed of. Complex (a-1), (b-1), (c-1) and (d-1) are taken as examples. The total depth increases from 190 in the complex (a-1) to 360 in the complex (d-1) as the number of corridors rises from 1 to 4. This can also be seen by comparing the average total depths of the complex in different models. As the number of corridors within the complex increasing from 1 to 4, the average of total depth of the complex changes from 190 in Model (a) to 253 in Model (b), 325.75 in Model (c), and 355 in Model (d). The third point is that the gain or loss of the total depth of a complex is related to the cell distribution. In each model, each complex, which represents a cell distribution, has a unique total depth of the complex. Particularly in four complexes of Model (b), the number of cells on corridor 1 declines from 5 to 2, and that on corridor 2 increases from 5 to 8. The result follows a sequential reduction that the total depth of the complex reduces from 260 to 258, 252, and 242. The last point concerns that the total depth of a complex will be affected by the combined form of corridors the complex consists of. This can be evidently derived by comparing the complex (d-1), (e-3) and (f-1) in Figure 6. The cell distribution where each corridor carries 3, 2, 2, and 3 cells in sequence, is consistent among the complex (d-1), (e-3) and (f-1). However, the three complexes have the total depth of 360, 382 and 406, respectively. In addition, the average total depths of the complex in Model (d), (e) and (f) are different, especially the average total depth in Model (d) which has 355 is lower than that with 398.5 in Model (e) and that with 395.5 in Model (f). Therefore, preliminary results suggest that the total depth of the complex will be affected by the number of corridors, the combined form of corridors and the cell distribution.
The further explorations will focus on two goals. The first one is to explore which factor has a greater impact on gain or loss of the total depth of the complex. For instance, the total depth increases 60 from the complex (b-1) to (c-3), while it gains 77 from the complex (b-1) to (c-1), an additional 17. The consequence is subject to two factors including the number of corridors and the cell distribution, but, in the instance, which factor is dominant cannot be determined intuitively. The second goal is to discover the correlation between the total depth and the cell distribution. When a complex is partitioned by the corridors, the total depth varies with the number of cells connected on the corridors. The first relatively apparent correlation is expressed between the total depth of the complex and the cell distribution in the complex. It can be elaborated as that the complex gradually loses the total depth as the cells are progressively clustering on a corridor. In other words, the complex has more potential to gain total depth when the cell distribution tends to be homogeneous. This correlation is clearly presented in the variation of the four complexes in Model (b). From the complex (b-1) to (b-4), the number of cells on corridor 1 decreases from 5 to 2, conversely, that on corridor 2 increases from 5 to 8, resulting in a decrease in the total depth of the complex from 260 to 242. This can be interpreted that as the difference between the number of cells on corridor 1 and 2 rises from 0 in complex (b-1) up to 6 in complex (b-4), which means the cell distribution tends to inhomogeneous, the total depth of the complex reduces 18. In addition, the variation between part of the complexes in Model (c) or (d) also reflects the correlation. For instance, the most cells with 5 on corridor 1 but the least with 2 on corridor 3, and the difference is 2 in the complex (c-4) with the total depth of 319. Yet in the complex (c-2) with the total depth of 327, the most cells with 4 on corridor 1, the least cells with 3 on corridor 2 or 3, a difference of 1. The cell distribution in the complex (c-4) is more inhomogeneous than that in the complex (c-2), the total depth of the complex in the former is less than that in the latter by 8. Similarly, the difference of the cells in the complex (d-4) is 4, and that in the complex (d-1) is 1, the complex (d-1) has 16 total depths of the complex more than the complex (d-4). Hence, it suggests that the shifts between these two complexes in Model (c) and (d) are consistent with the first correlation.
The second relatively apparent correlation is related to the relation between the cell distribution and the total depth of the individual cell. In detail, the total depth of the cell on a corridor decreases as the number of cells on that corridor increases, and vice versa. That is, the greater the difference between the total depths the cells on the different corridors have as the cell distribution becomes more inhomogeneous. A clearly regularity about the correlation is presented in the complexes of Model (a). As the difference between the number of cells on corridor 1 and 2 increases from 0 in the complex (b-1) to 6 in the complex (b-4), the difference between the total depth of the cells on corridor 1 and on corridor 2 expands gradually from 0 to 2, 4, and 6. In the complex (c-4) with 3 differences of the number of cells, the cells on corridor 3 have the highest total depth with 37, but that on corridor 2 have the least total depth with 30, a difference of 7. In the complex (c-2) or (c-3) which has only 1 difference of the number of cells, the difference between the total depths of the cells is 5. So, it suggests that the change in cell distribution from the complex (c-4) to (c-2) or (c-3) tends towards homogeneous, with a reduction in the difference between the total depths of the cell. Similarly, as the homogeneity of the cell distribution increases sequentially from the complex (d-4) with 5 differences of the number of cells to the complex (d-3) with 4 differences, and the complex (d-1) with 1 difference, the difference between the total depths of the cell decreases gradually from 8 to 4, and 0.
A relatively clear pattern of variation between the complexes in Model (b) reflects the above two correlations. However, the two correlations seem to be reflected just in part of complexes of Model (c) and (d). For instance, the cell distribution in the complex (c-1) is more inhomogeneous compared to the other three complexes in Model (c). However, the complex (c-1) gains the most total depth; meanwhile, where the difference between the total depths of individual cells is minimal. This seems to embody the exact opposite character to the two correlations. In addition, despite the differences of the number of cells are identical in the complex (c-2) and (c-3), the complex (c-3) has 7 more total depth than the complex (c-2). Within the Model (d), the shifts between the complex (d-4) and (d-3) are fully consistent with the two correlations. Nevertheless, the shifts between the complex (d-2) and (d-1) exhibit a unique correlation. In detail, although the cell distributions are different in complex (d-2) and (d-1), the total depths of the complexes are identical as well the total depth of each cell. Therefore, these suggest that the shifts between the complexes in Model (c) and Model (d) are possibly influenced by other rules in addition to the two correlations. However, with only four complexes of each model, the exploration here is limited, the results may be full of chance and are lack of conviction. Hence, the next study will further refine these two goals, and validate and complement the findings here from a more comprehensive range of cases.

Acquisition of cases
Assuming that there are "n" individual cells, without a direct connection, connected by the corridors within a complex, the number of cells on each corridor is labeled as a, b, c, and d, as shown in Table 2. Three cross-referenced experiments are designed, where the total number of cells in each experiment is 10, 20, and 30, respectively. Each experiment consists of six subexperiments (T1 to T6). The variables among the former four sub-experiments (T1 to T4) contain the number of corridors and the number of cells on each corridor. The latter two sub-experiments served as control groups for sub-experiment T4, where the variables include the combined form of corridors and the number of cells on each corridor. Then, each complex within each sub-experiment represents a kind of cell distribution. Meanwhile, the total depth of each complex and the total depth of each cell need to be recorded. Finally, the number of complexes in Table 2 shows all possible cell distributions in each subexperiment. These numbers have excluded duplicated cell distribution due to symmetrical distribution.

The extent to which the number of corridors, the combined form of corridors and the cell distribution effect on the total depth of the complex
Normally, each complex within the identical subexperiment obtains a unique total depth due to the different cell distribution. Thus, each sub-experiment generates a range about the total depth of the complex. Figure 7(a-c) shows the maximum, minimum, and average total depths of the complex within each subexperiment of three experiments. The maximum total depth means that within the identical sub-experiment the total depth of a complex is larger than that of any other complex. Whilst the definition of the minimum total depth is the opposite of the maximum total depth. The average total depth conveys the central tendency of the total depths of all the complexes in a sub-experiment. Finally, the range of total depth (maximum total depth -minimum total depth) describes the range and dispersion of the total depth of the complex. The distribution of the range of the total depth for all sub-experiments is displayed in Figure 7(d).
From Figure 7 (a-c), the first point can be noted that each sub-experiment possesses a unique range of total depth. It indicates that the variations of cell distribution generate different degrees of impact on the total depth of the complexes in different sub-experiments. Therefore, we cannot conclude which complex from the different subexperiments obtained more total depth. However, the average total depth within each experiment shows a clear upward trend from sub-experiment T1 to T4 or T5, or T6. The trend statistically reflects that the complex consisting of fewer corridors has a greater potential to gain less total depth. Particularly, the single corridor has the absolute advantage of keeping the complex with least total depth. In addition, the average total depth of sub-experiment T4 is apparently lower than the average total depth of T5 or of T6. It illustrates that a ring-corridor brings less total depth to the complex than other forms, which consist of the identical corridor with the former. Meanwhile, the rising trend of the average total depth attenuates on sub-experiment T4. These means that the ringcorridor can effectively weaken the total depth of the complex brought by the increased number of corridors.
Figure 7(d) shows the tendency of the range of the total depth among the different sub-experiments. First, vertically, the range of the total depth in each column noticeably expands as the number of cells in the complex rises from 10 to 30. Second, horizontally, the range of the total depth in each row tends to enlarge when the corridors within the complex increase from 1 to 4. Finally, the tendency of the range of the total depth is overall exhibited radially from the point of subexperiment T1. However, an exception is that the trend drops off at the point of sub-experimental T4. The range of the total depth in sub-experimental T4 is narrower than that in sub-experimental T5 or T6, even smaller than that in sub-experimental T3. This drop is greater as the number of cells in the complex increases. These trends statistically articulate that the variation of the total depth of the complex is less and more stable when the complex consists of fewer corridors, fewer cells, or a ring-corridor.

Correlations between the total depth and the cell distribution
The complex of sub-experiment T1 consists of one corridor, where the cell distribution and total depth are constant. Hence, the objects of the study focus on the complexes in sub-experiments T2, T3, and T4. The experiment n = 20 is selected as the case.
First, probing and verifying the regularity between the total depth of the complex and the cell distribution in the complex. Here, the difference in the number of cells represents the degree of homogeneity of the cell distribution. The difference derives from that the number of cells on a corridor minus that on another corridor. For instance, in a complex consisting of three corridors and 20 cells, the number of cells on each corridor is 2, 5, and 13, respectively. Then, 13 (max)-  2(min) = 11 represents the difference in the number of cells in this complex. When the difference is taken as the independent variable X, the total depth of the complex as the response variable Y, a graph of the relationship between them is obtained. The three graphs in Figure 8 represent the distribution of all the complexes in sub-experiments T2, T3, and T4, respectively. In Graph (a), the total depth of the complex shows a parabolic decrease as the difference in the number of cells becomes larger. Graph (b) and (c) overall show the trend like that in Graph (a). Partially, several exceptions which are against the trend exist in Graph (b) and (c). The reason is that the trend is influenced by additional factors in sub-experiment T3 and T4. In sub-experiment, the factor is that the variation in the number of cells on corridor 2 has a greater impact on the total depth than that on other corridors. As shown in Graph (b), the size of the dot indicates the magnitude of the number of cells on corridor 2, the bigger the dot, the more cells. Graph (b) shows that within the identical difference in the number of cells, the smaller the total depth of the complex as corridor 2 carries more cells. It suggests that when the cells are concentrated on corridor 2, the complex has more advantage in obtaining less total depth. That is, despite the difference in the number of cells increases, The total depth lost in the complex due to an increase in the difference in cell numbers perhaps be less than the total depth gained in the complex due to a decrease in the number of cells on corridor 2.This is reflected in Graph (b). As shown by the arrow, as the difference in the number of cells expands, a few complexes gain more total depth. In sub-experiment T4, the factor is related to the difference between the number of cells the mutual opposite side corridor has. Specifically, the difference refers to the difference in the number of cells on corridors 1 and 3 or that on corridors 2 and 4. Graph (c) takes the sum of the two differences as another independent variable in addition to the difference in the number of cells. The size of the dot indicates the magnitude of the sum of the differences, the bigger the dot, the greater the sum. Clearly, the complex with smaller sum gains more total depth, when the sum is 0, the complex obtains the most total depth. Besides, within the identical difference in the number of cells, the complex loses the total depth as its sum increases. These lead to that the complex where the difference in the number of cells increases does not necessarily lose total depth but has a global trend in losing the total depth. Second, the section will focus on discussing the regularities between the total depth of the individual cell and the cell distribution in the complex. Likewise, the difference in the number of cells is taken as the independent variable X, while the range in the total depth of cell as the response variable Y. The range in the total depth of cell represents the degree of homogeneity of the cell distribution in terms of the spatial depth conception. The range derives from the difference between the total depth a cell has most and that another cell has least within an identical complex. Graph (a), (b), and (c) in Figure 9 show the correlation between two variables in three sub-experiments. Graph (a) illustrates the standard positive correlation between the two variables. This means that within the complex of subexperiment T2, the inhomogeneity of the cell distribution in the number directly determines the degree of inhomogeneity in depth. Whilst the correlation in Graph (b) appears relatively ambiguous but is linear globally. Likewise, the size of the dot indicates the magnitude of the number of cells on corridor 2, the bigger the dot, the more cells. In general, the range in the total depth of cell expands with the increase of the difference in the number of cells in sub-experiment T3. However, the ranges in part of complexes decline as the difference rises within the region where the difference is less than 10. The range is smaller as fewer cells on corridor 2, as shown by the arrows. It illustrates that when the cell distribution within the complex is relatively homogeneous, keeping the smaller cells on corridor 2 enables a more uniform cell distribution in depth. Conversely, when the cell distribution within the complex is extremely heterogeneous, the unevenness of cell distribution in depth can be reduced when the cells are concentrated on corridor 2. Graph (c) demonstrates the apparent correlation between the variables. In the graph, the size of the dot indicates the magnitude of the sum of the differences, the bigger the dot, the greater the sum. When the difference and the sum are smaller, the cell distribution in the complex is more homogeneous in depth. It means that the homogeneity of the cell number distribution in the complex of sub-experiment T4 is reflected by the difference and the sum.
Finally, the three graphs in Figure 10 are supplemented to illustrate the relationship between the total depth of the complex and the range in the total depth of cell. An anti-correlation is observed from the graphs that the range in the total depth of the cell is smaller, the total depth of the complex is larger. This implies that the cells within the complex are more homogeneously distributed in terms of depth, but most of the cells are instead at deeper locations.

Discussion
This study objectively interprets the properties of the spatial layout being widely adopted in a quantitative and numerical way of thinking by using partitioning theory. The architecture educators, particularly those who are interested in the design of nursing homes, may benefit from this study. The design of a nursing home should be based on a human-centred philosophy, and aim to provide the best possible living experience for the residents. However, the design process of a building is certainly influenced by many factors such as local regulations, the requests and preferences of the residents, investors' requirements, building regulations, etc. The final building that is presented to us is created by the architects after balancing all these factors. Therefore, the design solutions that are currently widely used, including spatial layout, are to some extent relatively more rational and superior than others. There are many factors that affect the quality of life of the elderly in a nursing home, such as the care environment, the adequacy of facilities and the built environment. Wayfinding behaviour is inevitable in the daily life of elderly people in nursing homes. A good spatial layout can facilitate the wayfinding behaviour of elderly people with relatively fragile physical conditions. The total depth in the space syntax is a variable that measures the degree of spatial accessibility in a building, and in this paper is used to assess the overall accessibility of planes with different spatial layouts.
The results show that a complex consisting of fewer corridors has more potential to gain less total depth, leaving the space within the complex at a shallower position globally. Particularly, the spatial layout of a single corridor brings the least total depth to the complex, resulting in the spaces with a high degree of accessibility in the complex. It means that residents can find their destinations more easily in the plan consisting of fewer corridors. In practice, in the commercial market of nursing homes in Japan, the nursing home in which the spaces in the plan are organized by one or two corridors is also dominant. It is even rare that a nursing home plan consists of over four corridors. In terms of corridor combinations, the ringcorridor can effectively reduce the total depth of the complex brought by the increased number of corridors, thus increasing the accessibility of the interior space of the complex. In fact, when the number of corridors forming the nursing home plan increases, especially when there are more than three, the use of ring-corridors occupies a dominant position compared to the other corridor combinations, but in shape, the ring-corridor forms a closed circular route in a plan thus reducing the number of dead ends, and residents within that plan can find their destination from any point on the corridor. In this respect, the spatial layout of the ring-corridor seems to be more friendly to the movement of elderly people with dementia. However, the nursing home in which the plan is organized by the ring-corridor occupies only a relatively small part of the market. Therefore, these suggest that the design solution that provides better convenience and accessibility for residents is a very important consideration in the design of nursing homes.
In terms of spatial distribution, these spatial layouts that are widely adopted in practice have the similar characteristic of being relatively symmetrical in form and relatively homogeneous in depth. That is, the spaces in these spatial layouts are at locations with similar spatial depths and have relatively equal accessibility and convenience. It suggests that in the plans consisting of these spatial layouts, the spatial distribution is more balanced, avoiding some spaces being located far away from the subject, but these spatial layouts bring the complexes more total depth. This means that in the plans consisting of these spatial layouts, the space is generally at a relatively deeper location with relatively poor accessibility, and it is more difficult for residents to arrive at their destinations. The principle of adopting a spatial distribution seems to be the opposite to that of adopting the number of corridors and the corridor combinations. So, in spatial distribution, the design solution seems to focus more on maintaining a balanced spatial distribution.
In addition, the results also show that the number of corridors and the corridor combinations have a greater potential impact on gain and loss of total depth in the complex than the spatial distribution. This implies that changing the number of corridor or corridor combinations has a greater impact on wayfinding of the residents in the plan than changing the spatial distribution. Therefore, it can be inferred that a design principle for the nursing home is to improve accessibility of the spaces by minimizing the use of corridors within the plan or using ring-corridor when the corridors increase and then to maintain an even plan design by adjusting the spatial distribution.

Conclusion
This paper analyses the impact of different spatial layouts on the accessibility of spaces from the perspective of people's movement in the plan through the spatial models established by using partitioning theory. The results show that changing the number of corridor or corridor combinations has a greater impact on wayfinding of the residents in the plan than changing the spatial distribution. Therefore, in terms of the number of corridors and the corridor combinations, the spatial layout which facilitates the residents' wayfinding is adopt in more nursing home plans, but in terms of spatial distribution, the plan layout of nursing homes is more focused on a balanced design and less on the convenience and accessibility of spaces. This shows that it is often necessary to consider a combination of possible factors, rather than focusing on one indicator when a building is designed.
In The Ten Books on Architecture, Vitruvius (15 B.C./ 1914) suggested that knowledge in architecture is the product of both theory and practice (Vitruvius 1914). The spatial models in this paper are built up through a series of case studies of the nursing home and provide an empirical explanation of the different spatial layouts based on a theoretical approach. Through a combination of theoretical and practical data, this paper demonstrates that the spatial layout determined in practice by a series of possible factors (such as user requirements, rule constraints, design experience, common sense, etc.) can also be shown to be relatively reasonable and superior in a theoretical approach.
In conclusion, it should be noted that the spatial model is proposed as a way of thinking based on the study of space syntax to reading the spatial layout of the nursing home. In that sense, it might be suggested that the method could be a valuable contribution to the design of the nursing home. It provides designers with a better understanding of principles and some knowledge of the systemic consequences of strategic design decisions. In terms of the accessibility of spaces, designers can use this method to simulate and analyze the spatial layout of the building before designing a building, as well as to assess the spatial layout of existing building. More importantly perhaps, it can also inform the application of new ideas, and encourage new ways of approaching the relationships between spaces.
However, the approach in this study is deficient in some aspects. Space syntax is helpful in predicting people's wayfinding as it focuses more on people's movement through building when it is used to study architectural space. However, space syntax is based on relatively pure spatial topological relationships and studies architecture from a perspective of more rational and dialectical thinking, thus lacking consideration of the subjective consciousness and will of the residents. This perhaps leads to the research results being biased towards idealization. In addition, the analysis in this study is based on simplified models built up from summaries of practical cases. The individual cells in these models are all identified as one type of space, which ignores the distinction between spatial types in the nursing home plan and lacks consideration of the differences in the size of space. Finally, wayfinding is just one factor that may affect the living experience of the elderly in a nursing home. The overall improvement of the quality of life of elderly people in nursing homes requires exploration and breakthroughs in a variety of areas. Regarding these shortcomings, further improvements are needed in the future research, in the hope that a more comprehensive exploration and better study methods can help us to better understand the spatial layout of nursing homes and improve the quality of life of the elderly in nursing homes.

Disclosure statement
No potential conflict of interest was reported by the author(s).