AN INVESTIGATION INTO GEOMETRIC RATIOS OF THE SUNKEN COURTYARDS IN TRADITIONAL IRANIAN HOUSES (A FIELD STUDY ON YAZD AND KASHAN)

. In Iranian architecture, the emphasis on the use of geometric ratios such as human scale and modularization, caused beauty and harmony. Unfortunately, in contemporary architecture, the use of these scales has been forgotten, and their presence has been diminished. Therefore, the main goal of the present study was to analyze and evaluate the geometry and proportions used in six remaining traditional Iranian sunken courtyards in Yazd and Kashan. For each house, the length, width, height and the ratio between these dimensions were measured for the sunken courtyard, courtyard and earth of the case studies. Then, to find out which kind of proportions were used in these sunken courtyards, we proposed some statistical tests to compare our measurements with the traditional proportions used. In the end, the results showed that the proportions used in the design of the sunken courtyard, courtyard, and earth of the case studies are related and the traditional Iranian sunken courtyards have been designed mostly based on the use of Gereh (a unit of measurement), which was the most appropriate and most widely used scale in housing architecture.


Introduction
In the ancient world, two civilizations such as Babylon and Egypt made the foundation of geometric knowledge more than other civilizations. The first scientists of geometry never wanted to know geometry but turned to geometry for their daily needs. Probably the ancient Egyptians, Sumerians, and Babylonians were the first to discover the principles of geometry (Najafgholi Pour Kalantari et al., 2017, p. 481). In addition, in Islamic art, all universes can be understood by geometry, numbers and the alphabet. The development of Islamic geometric patterns began with the translation of Greek Sanskrit 1 texts (Akkach, 2012, p. 18). Of course, this happened about three centuries after the advent of Islam, and we can say that there is a gap in the growth and development of geometry of buildings from the beginning of the 7th century to the end of the 9th century. At that time, the first examples of the geometric layout of buildings in Islamic countries were created (Abdullahi & Embi, 2013, p. 244).
A quantitative approach to evaluating architectural design proposals for historic settings needs to supplement existing qualitative design guidance (Hu et al., 2017, p. 93). Using the geometry and proportions in design as an art to create shapes, patterns and proportions recalls the role of God as the architect of the world. In other words, geometry and proportions are considered as a key element in establishing the relationship between components of design which is expressed by its constructor and it is perceived by the audience and reference to his being (Nasrabadi et al., 2016, p. 42). Various built-environment attributes related to physical form are posited "to create a safe, inviting, and well-used public realm with visual interest" (U. S. Green Building Council, 2009, p. 17;Oreskovic et al., 2014, p. 220). In Iran, the main emphasis of architecture and urbanism has been based on a kind of aesthetic worldview. Over the centuries, the Iranian people have always had a great value for beauty and the science of geometry, and the proportions were considered as a powerful tool for creating balance, harmony, beauty, and order in the world of creation (Nadimi, 1999, p. 32). The most popular proportions used both in Iranian architecture and in European architecture are golden ratio with a value of 1.618, 1.41 ratios (especially in palaces of Persepolis and Apadana), 1.73 ratios (Kasra palace in Ctesiphon), 1.118 ratio (Sarvestan and Ctesiphon palaces) (NoghrehKar, 2011, p. 189).
Geometry is a basic element for establishing unity in Iranian architecture and is the important subject for architects for its discipline and rule (Danaeinia & Azad, 2019, p. 21). In the past architecture of Iran, the design and implementation of building forms and sizes was done using the geometry knowledge. The use of the Hanjar triangle method was one of the methods in which the architect of the building performed all the large and small divisions of the building simply and accurately (Mehdizadeh et al., 2011, p. 15); Also from the Timurid period, the use of checkered plates based on geometry to draw a plan of the buildings has been common (Najiboghlo, 1999). Aboalghasemi emphasized that the design of the courtyard based on the Iranian golden ratio, provides a healthy and pleasant environment for residents throughout the year and provides the best lighting for the rooms and spaces around the courtyard (Aboalghasemi, 2005).
Paying attention to the geometry and proportions of buildings has always been of interest to architects and researches in this field, and the discovery of the elements of Iran's climatic architecture is of great importance (Mahdavi Nejad et al., 2013;Silvayeh et al., 2013;Nabavi & Ahmad, 2016;Sajjadzadeh et al., 2015;Abdi et al., 2016).
A house serves as a shelter and, ideally, matches its natural surroundings by being designed according to the exposure, views, daylight, sunlight, shade, greenery, and other conditions of the site (Bjerregaard, 2005;Forshed & Nylander, 2003;Wågø et al., 2016, p. 330). Considering the different-climate zones in Iran, the architects of building in each region have always sought solutions to create satisfactory conditions for life. To this end, architects in hot, dry and, desert climate of Iran, have been looking for a way to adaptation the burning heat of the desert and create climate-friendly space as well as access to flume water. The sunken courtyard is an architectural space that is an appropriate answer to the design of the building in this climate and to achieve the goals set by the architects of the buildings. In fact, the sunken courtyard has sunk into the deep soil to create a temperature balance and favorable climatic conditions (Guia et al., 2013, p. 91). There is little research on the proportions of this traditional architecture space in Iran.
More research on the sunken courtyard has been carried out into the history, definitions and the role of this architectural element. Hence, the lack of studies in the field of geometry and proportions of sunken courtyards in Iranian houses has led to this study. Furthermore, the answer to this question is very important, which kind of ratio has been used between the length and width of this architectural space?
The rest of this paper has been presented as following. The Section 1 named "materials and methods". Section 2 inclodes research literature. The results are presented in Section 3 and finally in the last section we show the conclusion.

Materials and methods
For such research, there is a need to use statistical analysis. For that, in this study, the measure of the length and the width of sunken courtyards have been calculated and analyzed via appropriate statistical methods and tools. Statistical tools are often used as a means of reporting results. When there is a discussion about which means to use for different metrics, the impact of the underlying assumptions involved in reporting results in the architecture community has been largely unexplored. In the current work, we plan to investigate the validity of assumptions such as the normality of the measured data and the use of significance tests in the 95% and 99% confidence level. The main assumptions for the statistical parametric test are to have: Normality: Data have a normal distribution (or at least is symmetric), Homogeneity of variances: Data from multiple groups have the same variance, Linearity: Data have a linear relationship, Independence: Data are independent. Several distribution tests have been suggested to check the normality of data and whether the data set is well-modeled by a normal distribution. We use the Shapiro-Wilk test because it is recommended in the NIST Handbook and the Kolmogorov-Smirnow test. The statistical null hypothesis is H0: Distribution is Normal and for p-values greater than 0.05 it works.
Finally, after confirmation of the normality of the data, by t-test and descriptive statistics, the superiority of each of the hypotheses is compared with others.

Geometry, proportions and module
In Persian Dehkhoda Dictionary, geometry is referred to as a particular subfield of mathematics, in which the space, shapes, and imaginable objects are studied and discussed (Bemanian et al., 2010). Geometry is the fundamental science of forms and their order. Geometric Figures, forms, and transformations build the material of architectural design. In the history of architecture geometric rules, based on the ideas of proportions and symmetries formed, fixed tools for architectural design. Proportions were analyzed in nature and found as the general aesthetic categories across the nature and art (Leopold, 2006, p. 1). Geometry and geometrical patterns in Iranian architecture are such a critical issue that, without paying attention to their structural role, the spatial-structural analysis of this type of architecture will be essentially descriptive (Kiani & Amiriparyan, 2016).
The system of proportion exists everywhere in nature, such as the human body, shapes, artwork, music, painting, and even in the cosmic design of the universe. In architecture, the proportion is defi ned as a relationship between one part of the design to others or to the whole design that may create harmonious and aesthetical (Gangwar, 2017, p. 172). Th e proportional systems are mental, not visual constructs. Th us, you cannot see the usage of numerical ratios, which are expressed in terms of the local unit of measure, when a particular proportional system was designed (Cohen, 2014, p. 3). Proportion plays an important role in all three parts, as it provides guidelines for laying out useful spaces, for designing structural systems, and for creating an aesthetically pleasing environment. Th e proportions of a space can dramatically change how visitors feel in it, and the proportions of a facade design can aff ect whether a building appears welcoming, threatening, or impressive (Josephine, 2017, p. 266). An example of the proportions used in the design and construction of Iranian architectural buildings is the proportions of 1.118, 1.618, 1.41 and 1.73, each of these is briefl y explained in the following.

Root rectangles
First, we draw the square whose side is 1 unit and then draw an arc as long as its diameter. Th e larger side of the rectangle obtained is equal to the square diameter (2) and the diameter of this rectangle is equal to 3. With a rectangle diameter of 3, we can draw a rectangle 4 and this process continues (see Figure 1). Th ese rectangles are called dynamic rectangles (Jormouzi & Salehi, 2014).

Golden ratio
Th e concept of the golden section is theorized by Greek Philosopher Plato 2 . He has established the relationship between the two-unequal segment of the line, as per his theory that if the ratio between the sum of two segments of the line to larger part is same as the ratio between large segment to small segment, the two-part is in the golden ration (Gangwar, 2017, p. 172) (see Figure 6). Th e Golden Ratio (also called the Golden Proportion, Golden Section, Golden Mean, Divine Ratio, and Divine Proportion) (Markowsky, 1992) is a transcendent ratio found in funda-mental forms: plants, fl owers, viruses, DNA, shells, planets, and galaxies. Although the Golden Ratio is primarily regarded as a proportion, not a number; as a numerical quantity, it is defi ned to be 1.618 (Hejazi, 2005). To draw a golden rectangle, divide the square with the side length of one unit into two parts, then draw the rectangle diameter and we draw the arc as much as the rectangle diameter. Th e resulting point represents the location of the golden rectangle formation (see Figure 2).

Gaz as an Iranian Module
Specifi c units were used for the majority of the parts in a traditional building. An example of this is the usage of the specifi c brick size. Th is is done for the purpose of harmonizing various buildings put together. Th e measurement unit in Iran is called Gaz 3 (see Table 4). All elements, especially openings used to be built based on this unit and its proportion (Nabavi & Ahmad, 2016, p. 146) (see Table 1).

Research area
People in the warm and dry regions of Iran have always been looking for ways to cope with the climate conditions of the region (Tavassoli, 2003, p. 649). In these areas, such as Yazd and Kshan, buildings are built in an introverted way, in the form of a central courtyard and providing a climate of comfort for the inhabitants of the building (Shateryan, 2008, p. 309). Using this method, the people of this climate were able to reduce the demand for energy compared to buildings on the ground using the soil insulation characteristics and land cover (Imani chat ghaye & Heidari, 2018, p. 90).

Central courtyard
The central courtyard is a kind of courtyard that has at least one side and a maximum four sides of it surrounded by a building and is the main space for providing lighting and ventilation and an internal connection to the spaces of the houses (Samarghandi, 1964, p. 59).

Sunken courtyard
The sunken courtyard is a kind of central courtyard whose height is lower than the level of the house and the passageway. In some cases, the sunken courtyard was part of the central courtyard which made the central courtyard have two levels (Soltan Zadeh, 2011, p. 82) (see Figure 4). In general, it can be said that the sunken courtyard is not related to a certain period, and during the time when the climate of the desert areas became hot and dry, the building was mostly dipped in a hole until they were formed in the sunken courtyard (Guia et al., 2013, p. 94). In Figure 5, the process of building a sunken courtyard has been shown schematically.

Investigation and analysis of case studies
In this study, the case studies have been selected based on the following criteria: 1. The first criterion is that the building was built during the Qajar period. 2. Availability and access to architectural documents of the building and the possibility of field visits and measurements. 3. Having historical value and being located in the historical context of the city.
Therefore, based on the mentioned criteria, Among the residential buildings of the hot and dry climate of Iran, three houses from Yazd and three houses from Kashan have been selected which were made during the Qajar era (see Figures 6, 7, 8). In Table 2 the brief introduction of these buildings and the position of the sunken courtyard on the plan and section of each house have been shown.

Analysis the proportion of case studies
To analyze the proportions of case studies, the length, width and height of the sunken courtyard, courtyard and  earth of the case studies have been measured and documented. Since the mentioned cases have different length, width and height, the average of them has been considered. These dimensions and differences between their ratios and the other proportions introduced in the previous sections are shown in Table 3. In order to analyze whether Iranian architects in designing and constructing sunken courtyard, courtyard and the earth of the mansions have used Gereh as a traditional measurement method, it is necessary to convert the length, width and height of the mentioned cases to Gereh. Therefore, if we convert the numbers obtained in Table 2 to centimeters and divided by 6.66, then the result will be in Gereh (see Table 4). Table 5 shows the results of the tests, both p-values are greater than 0.05 which confirms that our measured data modeled by the Normal distribution. The decision is to fail to reject the null hypothesis. In consequence, the data in this study follow the normal distribution. In the next step, the significance test is used to determine if the mean of measured data significantly differs from the Iranian architecture standard measures in Table 3 (1.41, 1.73, 1.118, and 1.618) in 95% and 99% confidence level. For each case, we do the one-sample ttest. Based on the results in Table 6; the measured data is not significantly differing from the standard measure (p-values are greater than 0.01 or 0.05).

Statistical analysis
In Table 6, we have extended the confidence interval from 95% to 99% to have more chances to reject the inequality of the measurements. There is no significant difference between our data and the standard measure. According to data in Table 4, another way to analyze the significance of results is to use the independent student's t-test to check the difference in Gereh and its units. The null hypothesis can be stated as H0: the means of the measured Gereh is equal to the same means for the units with significance level α = 5%, So if the p-value is less than 5%, the null hypothesis is rejected, and the means have a 5% significant difference. The independent sample student's t-test is a good fit for this case. In the first part of the Table 7, there is a significant statistical test named Leven's test for equality of variances. The p-value of the test 0.719 is greater than 0.05 and it means the equality of variance in both groups. About the differences in the mean of group, with the p-value is 0.993, there is no significant differences between the two means. Table 8 includes the descriptive statistic of the standard measures. The measure of the Coefficient of Variation (CV) is the main part of the table in which, we can easily inference that how much our measured data differed from the standard one. The smaller measurements of CV guarantee the smaller variation of our data from standard measures. In among all the CV's, Gereh has the smallest one and it could be chosen as a most appropriate measure in Iranian architecture.

Conclusions
The results obtained from the researches that have been done so far on the geometric ratios used in traditional Iranian buildings, indicate that the architects of this buildings sought to bring the dimensions of the spaces as close as possible to some geometric ratios, for this purpose, proportional adjustment systems such as (1.118, 1.618, 1.41, 1.73, Gereh) heve been used. This method, in addition to organizing and spatial arrangement of these buildings, provides maximum use of sunlight for the sunken courtyard and surrounding spaces for the comfort of the building's residents in different season. Therefore, it can be said in this research, based on the statistical analysis of all measurements and their coefficient of variation in Table 8, Gereh, has the better agreement to be an appropriate scale matching to the real scales of that architects. It means, the proportions used in the design of the sunken courtyard, courtyard, and earth of the case studies are related and the Gereh as a proportional adjustment system has been used in designing of sunken courtyards in Yazd and Kashan mansions.