A case study of the use of logical data analysis in the Workmen’s Village in Tell el-Amarna, Egypt

1 Introduction

The search for patterns and rules in archaeological contexts is ever-present in the study of ancient civilisations and their ways of life. Finding these patterns can help archaeologists understand how certain processes took place, how people organised themselves in terms of religion, culture and community, and how they supported themselves and often their families. The field of Egyptology is no exception, as the ancient Egyptian civilisation realised many processes that Egyptologists are trying to understand, including the process of food production.

Food production is one of the many aspects of interest to Egyptologists. How did the ancient Egyptians produce their food? What types of fruit, vegetables, wheat, meat and nuts did they have and grow? What did they import from other regions? What patterns can be found in the production of food, the tools used and their functions?

This paper presents one aspect of the possibilities of interdisciplinary work between mathematics and Egyptology, namely the case study of food production in an ancient Egyptian Workmen’s Village in Amarna during New Kingdom Egypt (1549–1069 BCE) and the use of mathematical logical data analysis to find patterns in the distribution of tools in the houses of the workmen and their families.

With an Activity Area Analysis, the understanding of the social and cultural behaviour of a society can be better understood. Activity Area Analyses are mostly used in studies of households and domestic architecture to reconstruct everyday activities in the private sphere, as is the case in this research.

First, an introduction to the Egyptological background is provided, which is necessary to understand this case study of the Workmen’s Village of Amarna. Secondly, the mathematical logical data analysis and its rules will be explained. The results will then be contextualised and interpreted, and finally prospects for future work will be given.

2 Background: Amarna and the Workmen’s Village

During the so-called Amarna Period (1347–1332 BCE [1]) the pharaoh Akhenaten founded what was then the new capital of Ancient Egypt: Akhetaten (engl. Horizon of the Aten [2]), today called Tell el-Amarna (arab. ) [3], p. 1]. It lies on the eastern side of the Nile, about 312 km south of Cairo [4], p. 932] in Middle Egypt, as seen in Figure 1. It was founded as the sacred ground for the sun cult, which revolved around the ancient Egyptian sun disc Aten, and was only briefly occupied [1].

Figure 1:

Map of ancient Egypt and Sudan, showing the location of Amarna and other places. Map: A. K. Hodgkinson, using data from OpenStreetMap.

Pharaoh Akhenaten’s wife was Nefertiti, now best known for her famous bust in the Egyptian Museum in Berlin [5], and they had six daughters, the first of whom, Meritaten, was born before the founding of Amarna. In addition, prior to the founding of Amarna, Akhenaten had a stela erected in the temple of Karnak at Thebes, modern-day Luxor in southern Egypt, on which he proclaimed that all other gods and goddesses except Aten no longer existed. At the same time, he ordered the destruction of the names and images of the traditional gods and goddesses on Egypt’s monuments, especially those of the Theban deities Amun, Mut and Chons [6], pp. 5–6]. The area on which the city was then built was chosen because of the absence of other deities and their sanctuaries. Akhenaten himself justified the choice of location by a divine inspiration from Aten, which he had inscribed on one of the 16 boundary stelae, which surrounded the area chosen for the city [7], p. 35, 8], pp. 32–33, 35]. These boundary stelae describe in detail the planned buildings of the city, which can be equated with the buildings that were discovered by archaeologists [9], pp. 156–157].

Tell el-Amarna was habitable for up to 30,000 people [1]. It was inhabited for about 12 years as the capital of Ancient Egypt [3], p. 1]. During this time, Nefertiti gave birth to five more daughters in addition to the first one mentioned above. More on Akhenaten’s family can be found in Lyn Green’s ‘The Royal Women of Amarna: Images of Beauty from Ancient Egypt’ [10].

A total of 16 boundary stelae surround not only the settlement area on the east bank of the Nile, but also large parts of the agricultural areas on the west side of the river [8], pp. 32–33]. Amarna stretches over 10 km from north to south, and almost 5 km from the Nile to the eastern mountains, as shown in Figure 2 [1].

Figure 2:

Map of Amarna with highlighting of the Workmen’s Village. Courtesy of the Amarna Project.

Akhenaten died during his seventeenth year of reign and was buried in the royal tomb in Amarna. His mummy was probably later moved to the tomb KV55 (KV = King’s Valley, Thebes) and his cause of death is unknown. After Akhenatens successor, Semenkhkare, Tutankhamun slowly returned Egypt to its original, pre-Amarna conditions [6], pp. 8–11].

With the main background knowledge for Amarna now established, the following section will focus on the Workmen’s Village of Amarna as the main focus of this article. What follows is an overview of the Village, its surroundings and the history of its excavation.

The Workmen’s Village is the largest of two settlements in the eastern part of Amarna and is located just over a kilometre from the city centre [3], p. 21, 11], p. 71]. It consisted of a walled residential area, the Walled Village, which will be discussed in more detail later, and was surrounded by numerous chapels, tombs, animal pens and small gardens, as seen in Figure 3. Tombs were located both on the headland and on the slopes. Brick chapels were built near them and especially on the slopes of the nearby hills. They were places of remembrance for the ancestors of the inhabitants and formed a ritual zone. The ritual zone overlaps with the waste area and the animal pens, which were primarily used for rearing pigs [8], p. 194]. The so-called zîr-area can be found on the road between the city and the Workmen’s Village. The term zîr is derived from the Arabic word for a large clay jug (Arabic: ) [12] and refers to the numerous shards of large clay jugs that were used to transport water to the settlements from the Main City, which was over 1 km away. The zîr-area is seen as a meeting place, where the main purpose was to provide the water for the settlement, its inhabitants, plants and animals [8], p. 194]. The shards of clay lead to a small building facing the town, which represents a checkpoint for the transport of goods [3], p. 24]. Besides water, the delivery of grain was also via the zîr-area. The settlement was supplied with the grain required for e.g. bread production, partly as payment for work [13], pp. 125, 128].

Figure 3:

The Workmen’s Village of Amarna, with the square Walled Village and the surrounding area with animal pens, chapels, gardens and more. Courtesy of the Amarna Project.

The Workmen’s Village was a planned state-settlement whose purpose was to house the workmen and builders of the Amarna tombs and their families. Around 300–400 people lived in the settlement [14], p. 162]. It is often compared to the Workmen’s Village of Deir el-Medina, Thebes-West, as both settlements are similar in size and structure and exist during the New Kingdom. It is assumed that the workmen were relocated from Deir el-Medina to Amarna. The absence of faience rings with royal names in Deir el-Medina, which are typical of the Amarna period, confirms this. It is therefore concluded that the Theban Workmens’ settlement was (partly) abandoned at this time [8], p. 191, 15], pp. 87–89].

The aforementioned Walled Village is surrounded by an 80-cm-thick square brick wall [3], p. 24] and forms the centre of the settlement. It measures approximately 69 × 69 m and consists of over 70 houses, all of which have a similar structure. A larger house, located in the south-eastern area, probably belonged to the local overseer. The houses were built on five parallel streets. The Walled Village has only one entrance; another entrance was walled up in ancient times [8], p. 191]. The houses are rectangular, built mainly of brick, measure an average of 5 × 10 m and are divided into three sections. Located at the houses entrance were the front rooms. The middle rooms often have benches along the walls and other installations for daily life, such as cooking areas. They are considered the main area. At the back of the houses were smaller rooms with ovens and other installations. The roof was used for many purposes, like drying of grain or other foods [8], pp. 192–193, 11], pp. 71/73].

The houses are categorized into two types, type A and type B. The decisive criterion is the location of the staircase leading to the roof or to a second floor. In most Type A houses the staircase winds around a central pillar in the rear room, while in Type B houses, a straight staircase leads up from the middle room along the wall to the front room [16], p. 56]. Not all houses were excavated and sometimes house types cannot be identified.

There is no precise evidence for the period during which the Workmen’s Village of Amarna was actually inhabited. From various sources we can conclude that Amarna was inhabited for about 15–20 years, even when the city was not the capital of the country [17], p. 4]. At the beginning of Tutankhamun’s reign, probably at the time when the capital was moved, the settlement was abandoned, but later, still during the reign of the same pharaoh, it was resettled for a short time. The newly settled community continued with organized pig farming. The settlement was finally abandoned under Haremhab, two Pharaohs after Tutankhamun [17], pp. 42–43]. There are 20–22 years of settlement concluded [18], p. 4].

The history of excavation of the Workmen’s Village at Amarna began in 1921 with the work of the British Egypt Exploration Society (EES). Large parts of the site were first excavated by Peet and Woolley [19], p. 60], [20]. In one of his publications [20], Woolley used the now common name Workmen’s Village, but titled his photographs “The Gravediggers’ Village”. Both used the Workmen’s Village of Sesostris II in Lahun, in the eastern Fayyum region in west Egypt, for an architectural comparison [16], p. 58].

A comprehensive publication of their excavations appeared in 1923 in the EES series ‘Excavation Memoirs’ under the title ‘The City of Akhenaten. Part I: Excavations of 1921 and 1922 at El-Amarna’ [16].

Peet and Woolley’s publications still form the basis of knowledge about the Workmen’s Village at Amarna, as the state documented by Peet and Woolley was no longer the same in later excavations. The subsequent excavations were also organized by the EES, from 1979 to 1986 under the direction of Barry J. Kemp. He had been excavation director at Amarna since 1977 [19], p. 60], until early 2024. In eight campaigns, the area outside the already extensively explored Walled Village was brought into focus, in addition to which some areas of the Walled Village were re-examined and four further houses excavated. Preliminary reports were published on the first four excavation campaigns, after which the increase in finds and the complexity of the investigations necessitated the publication of annual reports in a separate series of the EES [18], p. 21, 21], p. 121]. Between 1984 and 1995, Kemp published six volumes, the Amarna Reports, in which the excavations were documented, and the results and observations published [22]. The Amarna Project still works on site today.

3 Food processing in the Workmen’s Village

The analysis of food production in the Workmen’s Village of Amarna was the focus of the master’s thesis of Klasse [23], which this case study is based on. The aim of this study was to identify the activity areas of food production within the Workmen’s Village. The focus of the investigation was on archaeobotanical and archaeological data. A central component of the research is the application of methods of digital humanities, such as QGIS and R, to support an activity area analysis, to create an overview of the available data and to visualise the results obtained. This analysis was based on the interest in the self-sufficiency of Workmen’s Villages in Egypt, since they are mostly state run, but do have components, like bakeries or at least facilities and tools that suggest they needed to e.g. make their own bread and cultivate some, if not most, of their own food. Since Amarna was, as mentioned before, abandoned as an Ancient Egyptian capital and the Workmen’s Village was only briefly re-settled shortly after, the state in which this settlement was excavated in set a well-founded and fairly undisturbed basis for research. The focus was placed on botanical and animal remains, as well as tools commonly used for food productions, as they are thematically related and provide information about food and its production. Such tools include mortars, pestles, saddle querns, ovens and fireplaces or hearths, which are sometimes harder to distinguish between in the archaeological context.

Data sources for this research were the find data from the aforementioned relevant publications of the excavations, namely Peet and Woolley 1921/22 for botanical remains and Kemp 1979–1986 for botanical and animal remains as well as tools used for food processing. Those publications were supplemented by more recent work on the processing of the finds, which focussed on certain material like botanic remains. In addition, the photo archive of the EES, which contains the photos of the 1921/22 excavations, and the associated Excel spreadsheet on the image information were used. The ‘Small Finds Database’ was used as well and is available as an open access spreadsheet on the Amarna Project website, which is based on the object card of the finds in the EES Amarna Archive. This database contains over 7500 objects that were given a registration number during the excavations from 1921 to 1936. These include finds from the Workmen’s Village, of which mainly the botanical remain references were taken. The partly published PhD thesis of the Egyptologist Ian Shaw from 1987, which was made available for the author for this study, contains a statistical analysis of the artifacts from Amarna, including objects from the Workmen’s Village. The data from the PhD thesis was used, albeit rarely, if published material from Shaw was not available. The PhD thesis by Egyptologist Thais Rocha da Silva, which was submitted in 2019 and has not yet appeared in print, and which deals with domestic space and the private sphere in the Workmen’s Village, was also consulted, mainly for insight and to provide reference to certain themes, which were not focused on in the master thesis.

The following Figure 4 shows the Walled Village, the main focus of the analysis, which was made based on an unpublished artwork by Barry J. Kemp, who in turn based it mainly on Peet and Woolley’s publication in 1923 and the Amarna Project excavations. Here, the relevant publications were used to distribute the tools used for food processing correctly in the Walled Village as detailed as possible in the houses and their rooms. Those were the plan drawn by Delwen Samuel of the Walled Village, the plan of its central north part by Kate Spence and certain parts of it published by Kemp as well [8], pp. 196, 200, 13], p. 136, 24], p. 92]. Sometimes the reference images and text sources did not entirely match, and the most probable one had to be chosen by the author.

Figure 4:

The Walled Village of Amarna’s Workmen’s Village and the distribution of botanical remains and tools. QGIS map by Sarah M. Klasse, using a base map courtesy of Barry J. Kemp (Amarna project) and information compiled from the sources mentioned in the text above [8], pp. 196, 200, 13], p. 136, 24], p. 92].

The map was made in the geographic information system QGIS [25], with the unpublished map as the base for contours, like walls, installations and stairs. The tools and botanical remains were distributed by the author according to the data sources. Animal remains were not reliably localisable in the Walled Village. QGIS was used as well to provide labels and the north arrow.

The following presents a summary of the results of the analysis conducted in the master’s thesis, which are a needed background to understand the case study.

The Activity Area Analysis done in this research is largely based on two publications by Kent [26], [27]. The purpose of his type of analysis is to visualise the use of space by human activity based on the distribution of objects in a specific context. Human action is understood to include both everyday and individual activities. Individual and repetitive activities can be reconstructed if patterns in the use of objects can be recognised. These activities are carried out by a group or one person. Activity areas are specific places where actions can be tracked [28], p. 33]. The activities are not limited to a house or a household, but also exist on a religious or economic level. They reflect all aspects of human societies at the social level. Activity Area Analyses work best when the house structures and their contents are well preserved [28], pp. 29–30], as is the case in the Workmen’s Village of Amarna. The following presents the summary of the analysis done in the mentioned master’s thesis.

Activity areas can be found both outside and inside the Walled Village, the animal pens outside the Walled Village providing evidence for self-sufficiency in the form of animal farming, mostly pigs, sometimes goats and cattle, even though cattle meat was evidently mostly provided by the state [18], pp. 36–40, 29], pp. 132–36, 30], pp. 155–56]. The gardens outside the Village provided fruits, vegetables, herbs and the like, additionally to the grain provided by the state, coming from the main city [13], pp. 125, 128, 31].

The tools within the Walled Village are the only one of the three categories of finds used for which activity areas are detectable. These activity areas become clear in connection with the finds of botanical remains such as grain, especially emmer, in the storage facilities of the houses and next to the mortars [20], p. 59]. Bread production, which can be best understood and reconstructed, is the most suitable for analysing the activity zones. It became apparent that in the three-part houses, the front room, whether roofed or not, was used for flour production through activities at mortars and saddle querns. Both tools were primarily used for this purpose, but their multifunctionality should not be denied. The grain, which was supplied to the inhabitants by the state, was processed into flour in these front rooms with the help of mortars, pestles and saddle querns, as well as grinding stones. Once the flour had been made, the mixing, kneading, and moulding of the bread dough could take place anywhere in and around the house. However, it is plausible that the room with the oven would have improved the rising of the dough through the radiated heat. The ovens, unless they were located on the roof or on a second floor, which have collapsed over the course of time, can be found as fixed installations in the back rooms. Bread was baked in them, but perhaps other foods were also processed. Intermediate activities such as drying the grain or mixing the dough took place on the roofs [32], pp. 139–140, 167, 170]. The cooking area, which can be found in the middle room of all the houses, together with other architectural finds such as the benches, represents a kind of communal space in which other foods were prepared over the fire and eaten together [33], pp. 44–45].

This means that at least four activity areas can be identified in the houses: the front rooms for food processing, the middle rooms for the production of other foods that could be made without an oven, the back rooms for baking bread, and the roofs for various food production activities.

With this information of the activity areas, possible activities and different tools used in food production in the Workmen’s Village of Amarna, it is possible to use this data to conduct a Logical Data Analysis.

A logical data analysis, explained below, gives logical rules for a data set, which helps identify patterns. In this case study the intention was to use the logical data analysis to help make a possible model for an expected pattern of the distribution of tools used for food processing in the houses, since not every house in the Workmen’s Village has been excavated. This interdisciplinary idea was motivated and considered by the collaboration between Mathematics and Egyptology, as it had already been successfully applied in a bachelor’s thesis [34]. The intention is to see if there is a pattern to the distribution of tools in the excavated houses and to find the ‘typical’ distribution of tools used for food processing, especially bread production. Architecturally the difference between the house types A and B, depending on the placement of the stairs, affects the placement of the oven, either in the backroom left or right, but the hope was to have a better idea where exactly the oven is to be expected. The unexcavated houses of the Workmen’s Village were the main focus of this analysis, as with patterns and rules one could create a model house, which shows the expectation of the analysed tools. New excavations must be well considered and justified. And if no further excavations are desirable or feasible, an expectation of what has not yet been excavated could be created based on this available data set. On the other side, this could be used as part of a justification of a new excavation as well.

The dataset was prepared as an Excel spreadsheet, in which for every excavated house the criteria of a certain tool in a certain room was created. The Supplementary Table was filled with “0” (not found) and “1” (found) for each house in each column. An example would be ‘Mortar in front room’, ‘Mortar in middle room’, and so on for each tool and rooms, including the space in front of the house, where some mortars where found. While answering in each of those columns it was irrelevant if it was only one tool or more, the point was if it was found there or not. This resulted in 41 small data sets, i.e. the 41 houses excavated, and a total of 17 variables (i.e. the criteria above). The columns which were only answered with a “0” were not counted in the logical data analysis, since they cannot be processed mathematically, as the rule would not be in the ‘if-then’ form. But they were looked at together with the results, as it is important to see what did not occur at all. The excel spreadsheet was then entered into the code, of which the background is explained below.

4 Logical data analysis and its rules

To achieve the detection of patterns in the distribution of tools in the houses of the workmen and their families, logical rules which can show said patterns need to be found with the Logical Data Analysis.

To understand, how logical rules are extracted from the data, you need to imagine that you want to select certain house of the Workmen’s Village by providing a “selection expression”. In a selection expression, there are criteria A, B, C, … that can be answered either “yes” or “no” for every house and room therein. Let us give a simple non-Egyptological example of such criteria. Let A be the answer to the question “Is it raining?”, then the variable A can be either 1 or 0 (“yes” or “no”). Another simple example is B: “Is the ground getting wet?” here too we can answer with 1 or 0. Now consider the logical rule: “If it rains, the ground gets wet.” This logical rule corresponds to an equation of selection expressions: the selection of all situations in which it rains is equal to the selection of all situations in which it rains and in which the ground becomes wet. This results in an equation for the two variables A and B that looks like this:

Multiplication “·” corresponds to the logical operation “AND”. An introductory textbook about Algebraic Logic [35] provides the theoretical background of what is discussed in the followings. Logical rules are formulated as mathematical equations: Two ways of selecting observations (or selecting a house) leads to the same subset.

You cannot formulate all possible selection expressions if you only carry out restrictions, i.e., by only using the logical operation AND. To be able to select all possible subsets of tools, houses and rooms, only one further operation is required, namely the XOR operation. A XOR B means that eXclusively A OR B is 1. In this context, A + ^[1] B selects all observations in which A = 0 and B = 1 or in which A = 1 and B = 0.

With these two operations you can do algebra – with Boolean rings. More precisely: operations in Boolean rings. These operations can be used to convert mathematical equations. The following transformation of an equation of selection expressions is important: Add a term to both sides of the equation that corresponds to one side of the equation. Take the equation (1) as an example and add A to both sides. An equivalent equation is derived:

Look at the left-hand side. What are the selected observations in which exclusively A or A is valid? This is impossible, there are no observations which fit to this selection expression, thus:

This equation only holds, if A => B holds. If B is a criterion that is independent of criterion A, then B should be false although A is correct, thus A + AB = 1 + 1 · 0 = 1. This is the above-mentioned example “If it rains, the ground gets wet”.

Mathematically speaking, A is the additive inverse of A in Boolean rings. By the last operation one can see: Every logical rule corresponds to a selection expression that – applied to all possible observations – selects the empty set. In our simple example, the selection expression A + A · B selects an empty set. The transformations that can be made with the two logical operations + and · are very similar to our ideas about “plus” and “times”. On the right-hand side of the last equation, we can apply factorization which equivalently provides:

In the followings we will usually omit the multiplication sign and write “A(B + 1)” instead of “A · (B + 1)”. The right-hand side of the last equation means: “Select all observations in which it is raining (represented by A) AND the ground does NOT get wet (represented by B + 1)”. The last equation shows that according to the given logical rule, an observation of that type does not exist.

It has been described before that logical rules correspond to selection expressions, which do not “find” tools in houses or rooms. Here are two examples from the data file constructed according to the archaeological data:

These are two different selection expressions, which select a subset of possible property patterns, but these patterns have not been observed. There has been no house with a pestle in the front room that does not have a saddle quern in the front room. This leads to the rule: Whenever there is a pestle found in the front room, then also a saddle quern has been found there. The other selection statement in (2) says that there has not been found a house in which there is a saddle quern in the front room and either a saddle quern in the left backroom or a Mortar in front of the house. This rule means that if we have found a saddle quern in the front room, then QBL = MfH. Either both are true, or both are false.

If we have found a selection expression that does not “find” a house, then restricting this statement also does not “find” a house and thus leads to a further logical rule. Mathematically speaking, if we multiply a suitable selection expression with any Boolean polynomial including our 17 variables, it leads to another suitable selection expression. For example,

is a restriction of PFR · (QFR+1) which leads to the rule: Whenever there is a pestle found in the front room and a saddle quern in the left backroom, then a saddle quern has been found in the front room as well.

If we have found two suitable selection expressions S1 and S2 which select an empty set of houses, then the selection expression S1 + S2 (meaning S1 XOR S2) also selects the empty set. In our example we can add the two expressions from (2) and arrive at

Also, this expression selects the empty set and provides a logical rule, but we will have to explain later how to derive rules from such expressions.

If I is the set of selection expressions which do not “find” houses, then the discussed properties mean that for any statement S1 ∈ I and any Boolean polynomial π, we have S1 · π ∈ I. We also discussed that for two expressions S1, S2 ∈ I we have S1 + S2 ∈ I. These two findings simply mean that I is an ideal inside the set of Boolean polynomials. This strong algebraic structure makes it easier to handle the big number of elements of I. The idea to use Boolean rings for the analysis of non-mathematical data sources has been developed in a project together with humanities [36], [37]. It is an exercise in Commutative Algebra textbooks (e.g., [38]) to show that every finitely generated ideal in a Boolean ring is a principal ideal. The ideal I is a principal ideal which means that it is generated by just one selection expression. In our context, this mathematical finding is also explainable in a different way, because if we have found the selection expression σ ∈ I which exactly selects all property patterns which are not observed in the houses, then the ideal I just consists of all restrictions π · σ of this selection expression, where π is a Boolean polynomial of our 17 variables. σ generates I. The advantage of knowing the algebraic structure of I: We can easily construct the selection expression σ. If S1 selects one existing/observed pattern and S2 selects one different observed pattern, then S1 + S2 selects exactly the union set of these two patterns. Thus, the sum of all different Boolean polynomials constructed according to the given {0,1}-assignments provides the selection expression σ which exactly selects all existing/observed patterns. Therefore, the complement σ = 1 + σ is our searched for generator of the ideal I. Thus, the ideal I is constructable in computer algebra systems [39]. The disadvantage of σ: If we would write down the generator σ of the Workmen’s Village data set, it would be a polynomial which fills many pages (we tried it, the “normal form of the polynomial” filled 9 pages).

σ of the last section is a basis of the ideal I. In contrast to vector spaces, different bases of an ideal can have different numbers of elements. We call the set of Boolean polynomials {b1, …, bn} a generator of I if every element f ∈ I can be written as a sum of restrictions of these elements, i.e., as a linear combination f = π1 · b1 + … + πn · bn, where π1, …, πn are Boolean polynomials. We call the generator {b1, …, bn} a basis of I, if any real subset of these elements is not a generator of I. In this sense, a basis is a minimal generator of I.

Imagine that two selection expressions S1 and S2 (which are Boolean polynomials) form a basis of the ideal I. In this case, the linear combinations of S1 and S2 include all set theoretical combinations of the pattern subsets selected by S1 and S2: The intersection of the subsets is selected by S2 · S1 + 0 · S2. The union is selected by (1 + S2) · S1 + 1 · S2. The difference S1 without S2 is selected by: (1 + S2) · S1 + 0 · S2.

Thus, we are searching for a basis of our special ideal I. This basis should be different from the 9-pages statement σ. As an example: The (expanded) selection expression σ “starts” with the leading monomial

whereas the expansion of PFR · (QFR+1) in (2) is QFR · PFR + PFR which “starts” with the leading monomial QFR · PFR. Roughly speaking, we are looking for a basis of I with polynomials having leading monomials that are as “small” as possible. QFR · PFR is “smaller” than the leading monomial of σ and there exists a basis of I which includes the polynomial PFR · (QFR+1). Another example: The selection expression of (3) is QFR · QBL · PFR + QBL · PFR. This is a restriction, i.e., a multiple of PFR · (QFR+1). The leading monomial QFR · PFR of PFR · (QFR+1) divides the leading monomial QFR · QBL · PFR of QFR · QBL · PFR + QBL · PFR. The leading monomial QFR · QBL · PFR is “bigger” than QFR · PFR. Having these examples in mind, we can formulate a possible criterion for the choice of a basis of the ideal I: If a selection expression f ∈ I is element of the ideal I, then there should be one element in the chosen basis of I having a leading monomial which is a divider of the leading monomial of f (like QFR · PFR is a divider of QFR · QBL · PFR). This excludes σ as a candidate for such a basis, because its leading monomial is not a divider of QFR · PFR which is the leading monomial of the polynomial PFR · (QFR+1) ∈ I. A basis which meets these “divider requirements” is denoted as Gröbner Basis (or standard basis) in algebra [40], [41]. We are looking for a Gröbner Basis of the ideal I. Such a basis can be computed using the Buchberger Algorithm applied to the ideal generated by σ [40], [42]. Generating a Gröbner Basis of the ideal I does not have a unique solution. The Gröbner Basis depends on weighting the variables (parameters) QFR, …, MfH. The weight of a monomial in this case is the sum of the weights of the variables which occur in this monomial. The leading monomial is the monomial with the highest weight. If there exists a “simple but general” rule about tools in the houses, then the leading monomial of the corresponding selection expression S1 ∈ I is small compared to the restrictions (multiples) of this selection expression. For every selection expression S1 ∈ I there is an element of the Gröbner Basis which has a leading monomial that is a divider of the leading monomial of S1. This means that “general” rules are preferred in the Gröbner Basis, if the leading monomial has a small weight. In our case we just want selection expressions with as few as possible variables in it. Thus, we choose every variable to have the same weight 1. Thereby, monomials with less variables have a smaller weight. Using the Buchberger algorithm together with these weights, we receive a Gröbner Basis consisting of 108 elements (in the software implementation of this algorithm, we have to add 17 further structural basis elements which only account for the Boolean ring structure).

Now it has to be described how to derive logical rules about the distribution of tools in the houses out of the elements of the Gröbner Basis of I. We will extract rules, which account for a special property, e.g. QFR. The selection expressions which include the variable QFR provide such logical rules. The general algebraic form of such selection expressions is S1 · QFR + S2, where S1 and S2 are suitable selection expressions (not necessarily elements of I). To give an example: One element of the Gröbner Basis can be written in this form S1 · QFR + S2:

Case 1 (positive). Imagine S2 is not in I, i.e., S2 selects an existing subset of houses. The requirement is S1 · QFR + S2 ∈ I. We can conclude that the selection expression S1 · QFR must select the same subset of houses like S2, because it has to “cancel out with S2”. This means that the houses in this special subset S2 must have QFR. Houses with an oven in the main room and a mortar in the right backroom must have a saddle quern in the front room.

Case 2 (negative). Imagine S1 · (1 + S2) is not in I, i.e., the intersection of the set of houses selected by S1 and the complement of the set selected by S2 is not empty. In this case, S1 · QFR + S2 can only select an empty set of houses, if the houses selected by S1 · (1 + S2) do not have QFR. In the special example of an element of the Gröbner basis this leads to S1(1 + S2) = MBR(OBR+1) and therefore to the following rule. A house with a mortar in the right backroom and without an oven in the main room does not have a saddle quern in the main room.

5 Results and interpretation

It may seem as if this method is trying to crack a nut with a sledgehammer, as the Gröbner basis is a powerful tool and the dataset small. The reason for this is that when we specify logical rules here, they look as if they ‘simply’ follow directly from the Supplementary Table. However, the completeness of the ideal and the reduction of the rules in the Gröbner basis ensure that when we look at the Supplementary Table or Figure 4, we do not see what we (unconsciously) want to see, but that we are presented with logical rules that we may not have thought of before. Logical data analysis makes the rules contained in the figure and in the Supplementary Table more comprehensible and accessible and removes the personal bias from the observation.

Using the above explained mathematics and code, many logical rules were achieved, of which some of course were very specific ones, but others were shorter and could be applied well.

The rules were given each for several times they apply, which can mean that there are exemptions, which are rare, so they are not considered.

Considered were the following and some more rules, of which one example on how to read them will be given:

• Neg: MFH

On the above example, the variable ‘MFR’ (Mortar in the front room) was the focus of this special code, with the above rule being “The mortar is not in the front room, if it is in front of the house”. The same type of reading can be applied to other rules, of which the following are the main selection: “The oven is not in the left backroom if the saddle quern is in the left backroom”, “The saddle quern is not in the left backroom if the oven is in the left backroom”, “The saddle quern is not in the left backroom if the oven is in the right backroom” and “The saddle quern is in the front room if the pestle is also in the front room”.

We also have to consider the rules, which are not seen by the Logical Data Analysis, because the columns in the data set are all “0”. Like the fact, which showed in the data set, that there is no saddle quern in the middle room or in front of the house or that there is no pestle in the middle rooms, the left back room or in front of the house. There is no fireplace in any other room than the middle one and the there are no ovens in the sometimes build adjacent rooms or in front of a house. But mortars are found in every room even in front of the house.

With these rules it was possible to create an expectation of the distribution of the tools used for food processing in a house of the Workmen’s Village in Amarna, seen in Figures 5 and 6.

Figure 5:

One possible model of the expected distribution of food processing tools. Made by Sarah M. Klasse.

Figure 6:

Another possible model of the expected distribution of food processing tools. Made by Sarah M. Klasse.

Figure 5 shows one of the possible models of the expected distribution of the tools. The mortar in the front room, while it is not to be ruled out that the mortar can be in front of the house as well. The fireplace or hearth is always found in the middle room, while a saddle quern cannot be expected in this room. The oven is either to be expected in the left or right backroom, depending on the house type and thus the placement of the stairs. The placement of a saddle quern, depending on the oven placement, can be either in the right backroom in this model, or in the front room if a pestle is also found in the front room.

Figure 6 shows another one of the possible models of the expected distribution of the tools. The mortar is in the front room again and the fireplace or hearth is in the middle room, as it can only be found in this room type. A saddle quern can be found in the front room and the oven is again either to be expected in the left or right backroom, depending on the house type and thus the placement of the stairs.

Of course, it cannot be emphasised enough, that those patterns and rules are modern and do not represent the thoughts of the ancient Egyptian people. Also, the unexcavated houses could discard and reshape these rules and it must also be noted, that not in every house every possible tool was recovered. For example, in 41 houses, there were only 18 ovens localised for the created map.

In conclusion, it is quite possible to create an expectation of the distribution of tools for the unexcavated houses and a possible excavation of the houses could make it possible to consolidate or redefine these rules. It must be noted that this model only applies to this settlement but could be applicable to other similar settlements with equally similar houses. Without such an excavation, however, the rules presented here apply for the time being. This pattern shows the typical previously mentioned bread production sequence well and comparison to the distribution of botanical remains in the Workmen’s Village, one could expect grain near the mortar in the front room. To find more possible models of such a tool distribution, e.g. for each type of house, the logical data analysis must be read and interpreted more times.

6 Prospects for future work

The application of logical data analysis can be considered a success in this case study, as it shows the possibility of the analysis of archaeological data sets to find desired rules and patterns. This could be compared to other state planned and partly self-sufficient Workmen’s Villages in Egypt to seek a comparable pattern in all or some of them.