Examining Variance, Flexibility, and Centrality in the Spatial Configurations of Yazd Schools: A Longitudinal Analysis

Abstract

Past historical research on Iranian educational architecture has highlighted three socio-spatial characteristics: (i) the topological variance of spatial properties, (ii) the flexibility of spatial configuration, and (ii) the centrality of outdoor spaces, which remain under scrutiny. This paper aims to quantitatively examine the topological changes in the spatial configurations of Yazd schools over time. The study entails the application of a Justified Plan Graph (JPG) method to quantify and compare the morphological properties of three historical sets of 20 Yazd schools (traditional, transitional and modern, and contemporary). The results confirm the historical changes in the Yazd schools’ flexibility and their outdoor spaces’ centrality, although the variance of spatial properties over time is not clearly observed. In particular, due to the relatively higher centrality values of outdoor spaces, the traditional schools, regardless of their rigid spatial logics, developed more integrated and interactive spatial configurations. As a first historical study on the socio-spatial configuration of Yazd schools, this paper precisely demonstrates the JPG method for this longitudinal spatial analysis.

1. Introduction

Prior research has expounded upon historical changes in the spatial organization of Iranian educational architecture [1,2,3,4]. Notably, the rigid hierarchy exhibited by spatial structures of traditional courtyard schools has decreased over time [5,6]. The courtyard of traditional schools in Iran is reported to reflect both educational policies [3,7] as well as socio-cultural significance [4]. However, these socio-spatial features of Iranian educational buildings have elicited scant scholarly research. To bridge this knowledge gap, this paper aims to mathematically examine the longitudinal changes in the spatial configurations of Iranian schools. Specifically, this paper measures spatial variance, flexibility, and centrality—the degree to which the spatial properties of the schools have changed over time, the degree of configurational differentiation, and the significance of outdoor spaces in the spatial configurations of the schools, respectively—of traditional, transitional and modern, and contemporary schools.

Space syntax theory, a widely-accepted computational approach, has provided the analytical or speculative characteristics of architecture that are inherently topological and relational [8]. This theory highlights that the spatial configuration of built environments reflects the socio-cultural characteristics of societies along with the people that create them. In parallel, space can also shape a social pattern [9,10,11]. Space syntax is also considered a rigorous method of exploring the built environment’s historical characteristics and morphological qualities [10,12,13]. By utilizing conventional axial and convex maps, space syntax research provides insights into: (i) the historical background to a case study, (ii) syntactical growth of urban spaces, (iii) morphological histories, and (iv) spatial-locational histories [13]. Both local and global integration values are frequently adopted to analyse historical urban areas and their design interventions [14]. In this regard, historical axial maps in a series of time periods have also been widely developed [15,16]. As such, the use of space syntax in historical research is largely limited to urban morphologies and transformations.

In contrast, a “justified plan graph (JPG)”, “justified permeability graph”, or “j graph” method, precisely introduced in Hillier’s and Hanson’s books [12,17], has been applied to examine the spatial configurations of various historical building types. For instance, Franca and Greene [18] morphologically analyse the stages of growth in the Isla Negara house, identifying five spatial properties—shallowness, permeability and visibility toward the ocean, clearly defined levels of public and private spaces, a high degree of functionality, and three levels of integration—particular to the organisation of space over time. Jeong and Ban [19] also examine the spatial configurations of South Korean apartments built between 1972 and 2000. They reveal that spatial and functional patterns, quantified by the integration measures of functional spaces, are influenced by socio-economic changes as well as the evolving applications of traditional lifestyles [19]. In addition, the JPG analysis of vernacular architectural planning in the Malay Archipelago [20], Iranian residential layouts in Kerman [21], and Toon historical town courtyard houses [22] are explored in terms of their spatial configurations and morphological developments. Interestingly, the JPG analysis of Mexico’s social housing built between 1974 and 2019 reveals a historical trend in housing typology and hierarchy [23]. A few archaeological studies [24,25,26] also uses the JPG method to provide historical insight into built spaces and spatial modifications. In summary, compared to axial maps addressing movement or behaviour in a spatial setting, the JPG method is more useful to examine the arrangement of programmatic spaces and their hierarchical relationships [9]. Thus, this paper uses the JPG method to explore the topological and morphological properties of school buildings, measuring the syntactic properties of 20 Yazd schools over time.

The following section describes the historical background of Iranian educational architecture. This paper then presents its methodology involving case selection, the JPG application, and hypotheses. Afterward, the results derived from a series of JPG analyses are presented and discussed. Finally, this paper concludes with a discussion about its limitations and contributions.

2. Historical Background

In the traditional era (from the 7th century to the early 19th century), the education system was entirely dominated by seminaries, and mullahs were controlling the education process [27]. Iranian schools were regarded as religious institutions where the main goal was a better understanding of Islam [28]. The teaching method of the traditional period significantly relied on a discussion between participants sitting in a loop formation [29]. This type of educational practice was useful to both students and the public for delivering the religious content of the courses [30].

In the transitional era (from the mid-19th to the early 20th centuries), Iranians became gradually familiar with western civilization and their culture, and experienced extensive socio-political changes. Both mullahs and teachers who completed their education in the Western countries collaboratively ran the Iranian education system [27], and non-religious lessons were included in its curriculum [30]. In contrast, in the modern era (from the mid-20th century to the late 20th century), mullahs were completely excluded from the education system [27]. Religious lessons were also completely removed from the curriculum, separating “teaching” from “learning”. The major teaching method was transferring knowledge from a knowledgeable person (teacher) to an illiterate person (student). Commonly, students sat in front of the teacher, while active communication, participation, and discussion rarely happened between students and teachers [29]. This teaching approach, with minor adjustments in class content, is still being used in Iranian education [30].

A variety of open (courtyard), semi-open (porch), and closed (chambers, Madras, etc.) spaces can be seen in Iranian historical schools. Traditionally, as shown in Figure 1a, the courtyard is placed at the center of the school building with a close connection to the porches and chambers (classrooms) [2,7]. However, in transitional and modern or contemporary schools (see Figure 1b,c), the centrality of the courtyard is compact because buildings are usually surrounded by outdoor spaces, and semi-outdoor or indoor corridors [7]. Importantly, the centrality of the courtyard in the traditional school responds to discussion-based teaching and a strong bond between education and religion. In contrast, the transitional or modern education system replaces this with “passive learning”, and its spatial organizations have been changed accordingly [3,31]. This paper examines these historical changes in the socio-spatial configurations of Iranian historical schools.

3. Methodology

3.1. Case Selection

This paper analysed three syntactic properties, variance, flexibility, and centrality, of Yazd schools over time. The historic city of Yazd, located in central Iran, is recognized as a World Heritage Site by UNESCO. The city represents the unique Persian architecture, including windcatchers (Badgir), cisterns (Ab Anbar), underground channels (Qanat), coolers (Yakhchal), and courtyard buildings. Among them, this paper focuses on Yazd courtyard schools, registered as national heritage sites in Iran. Four main criteria were used to select Yazd schools’ cases, which were:

(i)

Originally built as educational buildings,

(ii)

Largely standing in their original condition,

(iii)

Representative of each era (traditional, transitional, and modern, and contemporary), and

(iv)

Documented by experts and available for analysis.

5 traditional, 9 transitional and modern, and 6 Yazd contemporary schools were selected in this way (see

Table 1. Key characteristics of selected schools.

Era	School ID	Name	Year of Est.	No. of Stories	No. of Spaces	Timeline
Traditional	1	Mosalla	1551	2	99	Safavid (1501–1736)
	2	Shafieiye	1660	1	53
	3	Khan	1772	2	113	Zand–Afsharid (1736–1789)
	4	Imam Khomeini	1825	1	113	Qajar (early) (1789–1850)
	5	Mullah Esmaeil	1842	1	49
Transitional or modern	6	Keikhosravi	1889	1	37	Qajar (late) (1850–1925)
	7	Abolmaali	1900	1	36
	8	Badr	1900	1	35
	9	Khosravi	1906	1	19
	10	Dinyari	1916	1	15
	11	Ardeshiri	1925	1	20	Pahlavi (1925–1979)
	12	Iranshahr	1934	2	39
	13	Markar	1934	1	60
	14	Rostami	1936	1	18
Contemporary	15	Javad-Al-Aeme 1	1994	2	32	Islamic Republic (1979–present)
	16	Owlia	2000	2	25
	17	Javad-Al-Aeme 2	2010	1	57
	18	12 Day	2011	2	25
	19	Yadegar-E-Emam	2018	2	27
	20	Danesh	2019	2	38

). The traditional, transitional and modern schools are classified and documented by the Yazd Cultural Heritage Organisation and registered as National Heritage Sites in Iran, while the contemporary schools are documented by the Yazd School Renovation Organisation. The layouts of the latter have been frequently repeated in the Yazd contemporary schools. Interestingly, the number of functional spaces gradually decreased over time, although some schools did not follow this trend. The third and fourth cases (the Khan and Imam Khomeini schools) had the most complex program (113 spaces), while the Dinyari school (School ID: 10) had only 15 functional spaces. In summary, Iranian schools selected in this paper have been categorized into three main eras, commonly suggested by Iranian architectural historians [3,32]:

• Traditional era (the 13th century to the mid-Qajar or mid-19th century),
• Transitional and modern era (the mid-19th century to the late 20th century), and
• Contemporary era (the late 20th century to the present).

3.2. A Justified Plan Graph (JPG) Method

This paper measured the topological properties of plan layouts through a Justified Plan Graph (JPG) method [8,22,23,26]. The JPG method primarily involves identifying the chief, functionally defined spaces in a plan and how these spaces are interconnected. As shown in Figure 2, a JPG was a set of nodes and edges, where every functional space was first converted to a node, and an edge (or link) was used as a connection between two nodes. The spatial properties of each node were then measured by the relative depth of the nodes from the “carrier” node that was the starting point [33]. In this paper, the courtyard was selected as the carrier. Nine space syntax properties—Total Depth (TD), Mean Depth (MD), Relative Asymmetry (RA) and Integration (i_RA), Real Relative Asymmetry (RRA) and Integration (i_RRA), Control Value (CV), Difference Factor (H) and Relativized Difference Factor (H*)—were calculated by formulas and examples from previous research [33,34,35].

The RA value indicates how segregated a node is within the graph, and its inverse is i_RA, which is the level of integration of a node. Integration is a measure of the centrality of space within the graph [36]. Simply, more integrated spaces are more public, and people will be more likely to meet one another in these spaces. Conversely, less integrated spaces are more private or have a reduced likelihood of being sites of random encounters [37]. Since the spatial values derived from a graph are sensitive to the size (numbers of nodes) of the graph, RRA was used to construct a comparison between different-sized buildings [38]. RRA normalizes the values relative to an idealized diamond-graph D, for K number of spaces [17]. A normalized integration (i_RRA) measure can also be developed from RRA. CV reflects the degree to which one node influences other nodes in a network [39]. Spaces with high values of CV tend to be “sites of attraction” or have “pulling potential or capacity” [33]. Integration values can determine the global centrality, while CV is conceptually like betweenness centrality [34,40].

The difference Factor (H) is the entropy of integrated results in a graph, based on Shannon’s H [41], while Relative Difference Factor (H*) presents the normalized H value. Hanson (1999) describes that “the closer to 0 the difference factor, the more differentiated and structured the space; the closer to 1, the more homogenized the spaces or labels are, to a point where all have equal integration values and hence no configurational differences exist between them” (ibid., p. 30).

3.3. Research Hypotheses

Hypothesis 1.

(variance): There will be dramatic changes in the spatial properties of the Yazd schools over time. This hypothesis will be true if the SDs of syntactic measures (TD, MD, RA, RRA, i_RA, and i_RRA) are more than 25% of the mean in each measurement. That is, the variance is calculated by SD/μ. The artificial indicator (25%) can be used for this purpose [8].

Hypothesis 2.

(flexibility): The level of flexibility in the Yazd schools will have increased over time.

This hypothesis will be supported if the H or H* value (or flexibility) of traditional schools is closer to 0 (H or H* < 0.50), and the trendline for H or H* rises over time. As such, the traditional courtyard schools will exhibit a rigid hierarchy or differentiation, while the contemporary schools will be more flexible in their spatial configurations.

Hypothesis 3.

(centrality): Courtyards in traditional schools will be the most controlling and integrated spaces, while outdoor spaces in transitional and modern schools and contemporary schools will not hold such central positions. The centralities of outdoor spaces will also have decreased over time. The first part of this hypothesis will be supported if courtyards in traditional schools have the highest CV and i result (or centralities) of any space in the school plan, but outdoor spaces in the transitional and modern, and contemporary schools do not show such results. For the second part of Hypothesis 3 to be true, the longitudinal trendlines for the CV and the i value of outdoor spaces will fall over time.

Figure 3 illustrates a methodological framework consisting of these hypotheses as well as corresponding space syntax measures and graphical and statistical tests applied in this study.

4. Results

4.1. Variance

Table 2. Syntactic results of 20 Yazd schools and their variance values.

Era	School ID	TD	MD	RA	iRA	RRA	iRRA
Traditional	1	357.25	3.64	0.05	20.31	0.70	1.70
	2	188.75	3.62	0.10	10.68	0.76	1.20
	3	483.32	4.31	0.05	18.05	0.64	1.56
	4	480.35	4.28	0.05	18.27	0.64	1.56
	5	219.95	4.58	0.15	7.05	1.10	0.90
Transitional or Modern	6	109.40	3.03	0.11	9.45	0.68	1.45
	7	116.27	3.32	0.13	8.02	0.79	1.25
	8	114.62	3.37	0.14	7.66	0.84	1.18
	9	51.57	2.86	0.21	5.08	0.96	1.10
	10	34.40	2.45	0.22	4.96	0.84	1.17
	11	54.00	2.84	0.20	5.43	0.88	1.12
	12	126.56	3.33	0.12	8.50	0.77	1.29
	13	229.66	3.89	0.09	10.52	0.75	1.32
	14	48.55	2.85	0.23	4.75	0.94	1.03
Contemporary	15	99.75	3.21	0.14	7.27	0.80	1.24
	16	67.28	2.80	0.15	6.90	0.75	1.33
	17	258.35	4.61	0.13	7.99	0.95	0.94
	18	73.84	3.07	0.18	6.02	0.88	1.11
	19	82.00	3.15	0.17	6.11	0.88	1.12
	20	147.42	3.98	0.16	6.37	1.00	0.98
Mean (μ)		167.16	3.45	0.13	8.96	0.82	1.22
SD		135.47	0.63	0.05	4.60	0.12	0.21
Variance (SD/μ)		81.0%	18.3%	38.5%	51.3%	14.6%	17.2%

describes the results of syntactic values of 20 Yazd schools and their variance values (see also JPGs in Table S1 in the Supplementary Material). The two schools (the Khan and Imam Khomeini schools, school IDs: 3,4) that have the most complex program (113 functional spaces), develop the highest total depths (483.32 and 480.35, respectively), and have relatively higher syntactic values (MD, i_RA, and i_RRA) as shown in the table. In contrast, the Dinyari and Rostami schools (school IDs: 10, 14) that have the least complex program, develop the least total depth (34.40 and 48.55, respectively) and have relatively lower syntactical values (MD, i_RA, and i_RRA). However, the syntactic results were not only related to the number of functional spaces but also relied on the spatial relationships. Nonetheless, RA and i_RA were sometimes impacted by the number of nodes. Interestingly, the Mullah Esmaeil school (School ID:5) developed the highest RRA value (1.10), while the Rostami school (School ID:14) had the highest RA (0.23). These results may reflect the difference between two different normalizations. The variance values for RA and RRA were also quite different (39% and 14%, respectively). These results implied that RRA and i_RRA were more useful for a comparative analysis of syntactic values between buildings.

The Yazd schools in the same era develop similar syntactic values with some exceptions. For example, all the traditional schools develop higher TD and MD values than their averages (TD = 167.16, MD = 3.45). In contrast, the other groups of schools present lower TD and MD values than the averages, except for the Markar, Javad-Al-Aemeh 2, and Danesh schools (school IDs: 13, 17, 20). Likewise, all of the traditional schools, except for the Mullah Esmaeil school (school ID: 5), are more integrated than the average (i = 8.96), but the transitional and modern schools and contemporary schools tend to be less integrated than the average.

In the meantime, only half of the syntactic variables (TD, RA, and i_RA) confirm the high level of variance (>25%). SD/μ for MD is only18% and RRA and i_RRA also do not confirm the level of variance over time. Furthermore, it should be acknowledged that MD, RRA, and i_RRA are more normalized values than the other corresponding variables. Consequently, Hypothesis 1 based on the variance (SD/μ) is completely rejected. That is, there is no dramatic change in the spatial properties of the Yazd schools over time. In contrast, as shown in Figure 4, the MD results of the five traditional schools (μ = 4.09, SD = 0.43) compared to 15 Yazd schools in the other eras (transitional and modern and contemporary) (μ = 3.25, SD = 0.55) demonstrated significantly higher values: t (18) = 3.075, p < 0.01. However, there were no differences in the other syntactic variables, although the average i_RRA value of the traditional schools (1.38) is higher than the other groups (1.21 and 1.12, respectively). In summary, the complexity of a functional program may have decreased over time, while the degree of isolation or depth tends to increase over time.

4.2. Flexibility

Only three schools, the Mosalla, Imam Khomeini, and Badr schools (School IDs: 1, 4, 8, respectively), exhibited a rigid hierarchy or differentiation (H* < 0.5). In contrast, the H values of the Yazd schools examined in this research were higher than 0.8, indicating a high degree of homogeneity or decent flexibility. Such as other normalized syntactic values such as i_RRA, H* may be more useful for this longitudinal study. Nonetheless, the trendlines for H and H* values in Figure 5 slightly rose over time (R² values are 0.3918 and 0.3979, respectively), confirming that the plan layouts of the Yazd schools have become more flexible and homogeneous over time. Furthermore, the H* results of the traditional group of schools (μ = 0.55, SD = 0.13) compared to the other two groups (transitional and modern and contemporary) (μ = 0.68, SD = 0.10) demonstrated significantly lower values: t (18) = −2.322, p < 0.05. This result indicates that traditional courtyard schools have rigid spatial configurations.

4.3. Centrality

As expected, the courtyards of the traditional schools were the most controlling and had integrated spaces in their spatial structure. In contrast, these centrality positions were replaced with corridors in the other historical groups of schools, except for the Abolmaali, Badr, and Khosravi schools (School IDs: 7, 8, 9, respectively). The three schools in the late Qajar developed similar topological patterns to the traditional school ones, clearly exhibiting the centrality of courtyards (the highest CV and I values). In the traditional schools, the courtyards not only played a substantial role in providing overall accessibility but also worked for social engagement. In contrast, in the other groups of schools (from transitional to contemporary), interior or exterior corridors mainly controlled the accessibility in the spatial structure of schools and held central positions. This also meant that outdoor spaces were no longer used for attractions and random encounters.

On the other hand, the CV and i value (i_RA and i_RRA) of outdoor spaces in Figure 6 moderately decreased over time (R² values are 0.5152, 0.5821 and 0.5172, respectively), confirming the second part of Hypothesis 3. That is, the centralities of outdoor spaces in the Yazd school decreased over time. Since each case had more than one outdoor space, the figure illustrates the average centrality values of outdoor spaces in each case. This research identified 57 outdoor spaces, 9 for the traditional schools, 25 for the transitional and modern schools, and 23 for the contemporary schools, in the 20 selected cases.

Table 3. The results of one-way ANOVA with Duncan’s post hoc test for three centrality properties (T: Traditional era, TM: Transitional and modern era, C: Contemporary era).

Centrality Property	Group	Mean	SD	F	Sig.	Duncan a,b
CV	T	10.55	7.01	39.298	0.000	T > TM, C
	TM	1.61	1.26
	C	1.09	0.69
iRA	T	28.43	15.54	35.367	0.000	T > TM > C
	TM	11.15	4.23
	C	6.23	2.02
iRRA	T	3.33	2.22	16.263	0.000	T > TM > C
	TM	1.93	0.77
	C	1.07	0.36

^a Uses Harmonic Mean Sample Size = 15.417. ^b The group sizes are unequal. The harmonic mean of the group sizes is used. Type I error levels are not guaranteed.

shows the results of one-way ANOVA with Duncan’s post hoc test for the three centrality properties (CV, i_RA and i_RRA) of 57 outdoors where significant differences were identified among historical groups. The outdoor spaces of the traditional schools (T) had more influence than the other groups’ outdoor spaces (TM and C). In terms of integration (both i_RA and i_RRA), the outdoor spaces of the traditional schools (T) were more integrated than the transitional and modern schools (TM), while the TM’s outdoors were more integrated than the contemporary schools (C). Thus, the results of ANOVA for the three properties also confirmed that there were significant differences in ‘centrality’ between the three historical groups of the Yazd schools, and that the centralities of outdoor spaces decreased over time.

5. Discussions

The digitalization of cultural heritage is a very important issue in architecture and urban design [42,43]. Likewise, this computational data developed from the syntactic analysis in this paper can be fundamental knowledge for this purpose. This paper has conducted a series of syntactic analyses, using the JPG method, on rigorously selected cases, quantitatively capturing historical changes in the socio-spatial properties of the Yazd schools. Compared to previous historical research on the spatial organization of Iranian schools [1,6,31], this paper has developed a comprehensive, longitudinal analysis to examine three hypotheses with 20 cases over time. However, it should be acknowledged that there are some unavoidable methodological limitations. Firstly, there were a lot of contemporary schools in Yazd. However, many were excluded from this analysis because they did not meet all the criteria. For example, many Yazd contemporary schools either had been extensively modified in recent years, or their architectural plans were not available. In addition, due to limited historical resources, the upper or lower floors of schools were not included for spatial analysis in this paper, although the additions might reveal augmented insights into the schools’ planning and configuration. If there are directional and/or weighted relationships, centrality measures may be useful to the dynamic but precise relationships in the built environment [34,40]. Lastly, the space syntax method, intensively discussed in past research, can also be limited to revealing the topological, co-present characteristics of school buildings, disregarding some historical implications or other dimensions such as socio-cultural, environmental, and cognitive factors [9,10,44,45]. That is, the JPG method in this research was based on the connectivity patterns in the two-dimensional (2D) architectural plan layouts of the schools, but there were other cross-sectional variations in architecture.

Interestingly, despite the weak growth of flexibility values in Figure 4, the centrality values of outdoor spaces in Figure 5 moderately decreased over time. As such, contemporary schools have become more private or spatially isolated. In Iranian traditional schools, the public, along with students, were welcomed not only to join discussion loops but also to attend congregational prayers and religious ceremonies [32,46]. To provide such social and religious functions, the architectural plan layouts of the traditional schools were inherently integrated [4,47,48]. In contrast, in modern and contemporary schools, none other than students had been allowed to enter the schools [7]. The combination of flexibility and centrality measures might be useful to reveal these hidden socio-spatial phenomena.

Two indicators for the variance and flexibility, 25% and 0.5, respectively, have been suggested by the literature, but they have never been fully tested in any research. Thus, the interpretation of both properties might be weak and need further clarification. For example, Hypothesis 1 was completely rejected by the variance results, but comparing means between three groups of schools partly supported their differences. The variance results also contradict the fundamental assumption that the spatial configurations of Iranian traditional schools have changed notably over time [6,30,49]. Thus, the two indicators should be addressed in future research. Although most Yazd schools had flexible spatial configurations, H* results for the two traditional schools, Mosalla (H* = 0.36) and Imam Khomeini (H* = 0.48) schools, indicated the spatial configuration of both schools was relatively differentiated and rigid. There were also three exceptional school layouts (the Abolmaali, Badr, and Khosravi schools) in the transitional and modern eras. Nonetheless, the syntactic results supported Hypotheses 2 and 3. Significantly, this paper confirms that courtyards are the heart of Iranian educational architecture, providing considerable socio-spatial functions such as accessibility and socialization [5,6,50,51]. However, this result contradicts the conventional proposition that Iranian traditional schools possess a high level of spatial differentiation or deliberation [5,6,52]. Nonetheless, such a finding, high flexibility of the traditional schools, would make more sense for a building type that serves various functions, such as educational, residential and social practices [48].

6. Conclusions

Although courtyard buildings are known to be one of the architectural characteristics responsible for Yazd’s UNESCO World Heritage status, little research has been conducted on their socio-spatial properties. A case study has addressed 20 Yazd schools built by five Iranian dynasties, from 1502 to the present. The first two schools in this paper were built in the Safavid dynasty, one of the greatest Iranian empires, while the third case was the only Zand–Afsharid school. Seven (school IDs: 4–10) were built in the Qajar era and four (school IDs: 11–14) in the Pahlavi dynasty, the last Iranian royal dynasty. The remainder were Islamic Republic schools. By the JPG method, this paper has statistically examined three important topological characteristics, variance, flexibility, and centrality, of the Yazd schools over time. Furthermore, this study not only demonstrates the application of the JPG method to longitudinal spatial analysis but also contributes to its methodological development for future historical research in architecture. As explained in the discussion section, in the first hypothesis (variance), the syntactic analysis was completely rejected by several syntactic properties, while the second hypothesis (flexibility) was marginally confirmed by H and H* values. In contrast, the third hypothesis (centrality) was well confirmed by the Control (CV) and integration (i) values of outdoor spaces over time.

In summary, the results indicate that the plan layouts of the Yazd schools have become more flexible, but their outdoor spaces have become isolated over time. Although the variance measures have not worked well for this purpose, the combination of flexibility and centrality measures have captured interesting socio-spatial phenomena. Specifically, due to the relatively higher centrality values of outdoor areas, functional spaces in the traditional schools have been remarkably integrated. Acknowledging some inherent methodological limitations, the findings of this paper contribute to syntactically interpreting historical changes and trends in the spatial properties of the Yazd schools. Furthermore, the variance, flexibility, and centrality measures presented in this paper would be useful to examine how the socio-spatial characteristics of architecture have changed over time.