THE IMAGINARY LINE ACROSS YOGYAKARTA LAND AS AN ETHNOMATHEMATHICS STUDY

: In attempt to explore Yogyakarta, the heritage city of Indonesia, the well known landmark, Tugu Pal Putih and the Yogyakarta Imaginary Line is explored. The purpose of this paper is to describe that this cultural context may be used as a reference for geometry learning. The method used in this research is a descriptive qualitative involving two historian experts in interviews. The results are that this line can be explored through ethnomathematics studies based on points, straight lines, comparison of distances between points on the imaginary line and the extension of the imaginary line. This line can be illustrated by an imaginary point located at the Tugu Pal Putih as the starting point to focus on thinking of the mathematical object. This line not only divides Yogyakarta imaginatively but also according to natural boundaries such as rivers. The extension of the imaginary line forms an area called the fertility area. The description of the patterns formed on the Yogyakarta Imaginary Line through ethnomathematics is expected to be used further as materials in the mathematics learning process.


INTRODUCTION
Ethnomathematics is a broad view of mathematical knowledge in relation to its development in various cultures (D'Ambrosio, 2012). Ethnomathematics links mathematics practiced by members of cultural groups who have similar experiences and practices with mathematics in a unique form (Cimen, 2014). Several studies regarding Yogyakarta culture in ethnomathematics include: 1) research by Mauluah (2022) who studied mathematics around the Yogyakarta Palace, 2) research by Prahmana et al. (2021) who researched the pranatamangsa system and birth-death ceremonies in Yogyakarta, and 3) ethnomathematics studies on Javanese Primbon (Utami et al., 2019). However, research that reveals The Imaginary Line of Yogyakarta with ethnomatmatics studies does not yet exist.
The Special Region of Yogyakarta (Daerah Istimewa Yogyakarta, or DIY) is a province in Indonesia that has the status of a special region or special autonomy, which is a legacy before the independence of the Republic of Indonesia. The province consists of four regencies, i.e.: Sleman, Bantul, Kulonprogo, and Gunungkidul, and one municipality city, namely the city of Yogyakarta, with the well known tourist area namely Malioboro Street. DIY is a special region because it has a philosophical axis and is included in the palace, Kraton DIY Cultural Heritage Area based on the Decree of the Governor of DIY No. 75 of 2017. Malioboro Street, which is located in the city of Yogyakarta, is also part of the philosophical axis or commonly called the Imaginary Line, which is a destination and attraction of the city of Yogyakarta, which is rich in history, tourism, culture, and the center of the economy in Yogyakarta (Ashari, 2022), which is a semi-pedestrian area as a whole based on the Grand Design Master Plan for the Malioboro Area 2020 (Jovenski, 2022). Malioboro Street is laid out as an imaginary axis from north to south that connects the Yogyakarta Palace to Mount Merapi in the north and the southern sea as a supernatural symbol. This imaginary concept pattern was disrupted when the Dutch colonial built Vredeburg Fort (1790) located south of Malioboro Street (Ashari, 2022). Hence, it is very interesting to discuss the relationship between this imaginary line and Ethnhomathematics.
The Yogyakarta Imaginary Line is the line that connects Mount Merapi, the Yogyakarta Palace, and the South Coast. This line is called an imaginary line because there is no visualization of the line. The imaginary line stretching from north to south has various philosophical meanings. The imaginary lines symolize the journey of human life, i.e.: its origin and where it returns (Karsono & Wahid, 2008). This line is closely related to belief in the relationship between human life and the universe, as well as human life and God (Meliana & Caroline, 2020).
Apart from being filled with philosophical meaning, the Yogyakarta Imaginary Line plays an important role in the development of various historical buildings and areas in Yogyakarta. These various components are for example: Pathok Negoro Mosque (Setyowati et al., 2018;Meliana & Caroline, 2020) and Malioboro (Cahya et al., 2017).
Various buildings or areas that are built have a certain distance from the imaginary line or are on that line. This causes the formation of certain patterns on imaginary lines or new areas or buildings that can be measured with the imaginary line as a reference point.
The existence of patterns formed on the Yogyakarta Imaginary Line can be explored further. This exploratory process can be done through ethnomathematics.
The existence of exploration of the patterns formed on the Yogyakarta Imaginary Line through ethnomathematics can be used further as material in the process of learning mathematics. This can be done by relating it to the material of Geometry.
The use of Yogyakarta Imaginary Line through ethnomathematics in the Geometry subject plays a role in the contextual learning process. Contextual learning in mathematics helps in guiding students to master mathematical concepts (Hoogland et al., 2018;Salavera et al., 2019). This is because students are actively involved in choosing, compiling, organizing, touching, planning, investigating, questioning, and making decisions so that mathematical concepts are interconnected from one concept to another (Rochsun & Agustin, 2020).
Contextual learning of mathematics can make use of Indonesian culture. There are several studies that have tried to carry out contextual learning processes through Indonesian culture including: 1) the use of Kambuik Kuantan Singingi culture used in Geometry materials such as cylinders with a contextual learning process framework (Isroi et al., 2022), 2) further research explores the concept of division, congruence and similarity, triangular prism, and half sphere with Barongko Cake (Pathuddin et al., 2021), and 3) other research collaborates ethnomathematics-based contextual learning with Sundanese culture (Supriadi et al., 2014).
The purpose of this study is to explore the Yogyakarta Imaginary Line with mathematics through the study of ethnomathematics. This paper is expected to be a reference for the contextual materials in Geometry learning process. In this case, we explore Gometrical concepts of points, distances, areas, and their measurements and comparisons. We argue that using this context, students may not only be shown the example of mathematics in real life, but also be known of the existence of the Yogyakarta Imaginary Line and its underlying beauty.

Research Design
The method used in this research was descriptive qualitative. The research stages used were adopted from Lambert & Lambert (2012). The research design was adjusted to the objectives of this study. This research aimed to explore the Imaginary Line of Yogyakarta through ethnomathematics study. The specific research question was as follows: How can we explore the Yogyakarta Imaginary Line through ethnomathematics study?

Data Collection
The data collection process began with literature review. The search process was based on predetermined keywords. The source selection process in the article was carried out starting from reading the abstract then if it was deemed appropriate, reading the full text. The database used as a source of article searches came from Google Scholar. The keywords used in the search include: Yogyakarta Imaginary Line, ethnomathematics, Yogyakarta Culture, Yogyakarta Map, Geometry, and Mathematics. Data collection for map making was done through satellite image data accessed from usgs.gov. The Indonesian landform map data was accessed from the Geospatial Information Agency.
Further data collection was carried out using interview techniques. The resource persons in this research were selected using purposive sampling technique. This was done because sources who knew about the Yogyakarta Imaginary Line were selected. The interviewees in this study were one (1) lurah abdi dalem of the Yogyakarta Palace and one (1) lecturer in the History Education Department UNY. The interview instrument was made with two main questions, namely 1) the history of the formation of the Yogyakarta Imaginary Line and 2) the philosophical meaning of the Yogyakarta Imaginary Line.

Data Analysis
The analysis process was carried out based on various information that had been obtained at the data collection stage. The analysis was conducted on the philosophical meaning data obtained from the literature review process, interviews, and geometric analysis of the Yogyakarta map.
The analysis of philosophical meaning was carried out by mixing data obtained from literature review and interviews. The interview process served to clarify various information that has been obtained at the literature review stage. At this stage, codefication was carried out on the sources. The codefication includes the abdi dalem as "AD" and the expert lecturer as "IS".
Further analysis was carried out through the map results. Various data obtained were analyzed using ArcGIS 10.8.1 and Google EarthPro software to produce maps along with imaginary lines, which were then analyzed with ethnomathematics. The analysis was conducted through the distance between objects along the line, namely Mount Merapi, Tugu Pal Putih, Yogyakarta Palace, Krapyak Stage, and Parangkusumo Beach, as well as the distance of surrounding objects, namely vegetation and rivers with imaginary lines as reference points. The results of the analysis were then used as contextual content for Geometry material. The stages of analysis can be seen in Figure 1. The analysis flow in this study can be seen in Figure 1.

Figure 1. Analysis Flow
The results of the review were written in three parts. These sections include: 1) point and straight line, 2) comparison of the distance between points on the Imaginary Line, and 3) expansion of Yogyakarta Imaginary Line.

RESULTS AND DISCUSSION
The imaginary line stretches from Mount Merapi in the north to the South Coast.
Along the imaginary line there is also a philosophical line that connects Tugu Pal Putih, Keraton, and Panggung Krapyak (See Figure 2). There are several mathematical components that can be explored on the Imaginary Line of Yogyakarta, including: point and line.

Points and Straight Lines
A point has a position but has no size or magnitude, so it is said to be dimensionless (Pereira, Wijaya, Zhou, & Purnama, 2020). Meanwhile, a straight line is the shortest distance between two points (Webb, 1974). A line is uniquely determined by two points, and the line passing through points A and B is denoted by AB ⃡ . Similarly, the length of a finite line segment ending at these points can be denoted by AB ̅̅̅̅ . A line can also be denoted by a lowercase letter (Jurgensen et al., 1963). The line between A and B can be seen in Figure 3.  Table 1. process. This is shown by sincerely worshipping God Almighty accompanied by a determination towards the welfare of the people (golong-gilig) and based on a pure heart (white color) (Karsono & Wahid, 2008;DIY Cultural Office, 2016).

Comparison of the Distance between Points on the Imaginary Line
The distance between points on the imaginary line can be seen in Figure 4. Based on this information, the comparison of the distance between points along the imaginary line can be determined.

Figure 4. Distance between points along the Imaginary Line
The comparison can be seen in equations (1) and (2) (2) Based on these results, it can be seen that the comparison results of equations (1) and (2) are around the value of 1. This shows that the location of the Yogyakarta Palace is in the middle of the imaginary line and the philosophical axis. This is in line with the philosophical meaning of the Yogyakarta Palace as the centre of world activities ranging from power, wealth, and also the centre of glory. Yogyakarta Palace is also the centre of the source of life with the eternal light that never dies. Yogyakarta Palace is also a place for Sultan Hamengkubuwono I -X who controls and owns Yogyakarata. The palace is a place for the Sultan to calm himself in the Siti Hinggil room and look at Mount Merapi (Syamsuddin et al, 2021). The palace became the centre of community activities ranging from religion, tradition, culture, and also government politics.

Expansion of Yogyakarta Imaginary Line
Based on the comparison of distances between points that have been obtained in equations (1) and (2) Another meaning is that Tugu Golong-Gilig is the main point of view of the Sultan when carrying out meditation in Bangsal Manguntur Tangkil in North Sitihinggil. This is also corroborated by the following interview results: "The Sultan looks towards the monument to concentrate on deciding things"-IS This shows that it is possible for the imaginary point to be a source of unity and a source of calmness, hence making it possible to think with high concentration.

Figure 5. Expansion of the Yogyakarta Imaginary Line
Another extension is done by drawing other lines that begin from the same initial point (Mount Merapi) to the imaginary line. These lines are Line 1, Line 2, Line 3, and Line 4 (See Figure 5). An imaginary line from Mount Merapi to Parangkusumo Beach divides Yogyakarta Special Region into two parts with different environmental conditions (Adinugroho et al., 2016) (Rahmadi, Wiantoro, & Nugroho, 2018). Each region has a different level of geographical difficulty due to varying topography.
The level of geographical difficulty will be higher the higher the area, such as the Menoreh and Sewu Mountains (Adinugroho et al., 2016).  (Rahmadi, Wiantoro, & Nugroho, 2018). The apex between these two mountain ranges makes low areas more developed such as Bantul Regency and Yogyakarta City (Adinugroho et al., 2016).
Along the imaginary line, dense tree vegetation is found. Vegetation from north