Abstract
Geometric construction in Gothic architecture has been a popular subject of research for centuries. The use and extent of quadrature and triangulation grids has long been debated, as the few architectural drawings that survived from the era do not indicate the construction process. Modern surveying methods, however, allow us to inspect the exact geometry of a building with all its irregularities – in the case of this article we use 3D point clouds created by laser scanning, from which accurate measurements can be taken.
In our study we analyse four churches from mediaeval Hungary: the Franciscan church of Szeged-Alsóváros, the fortified church of Cincu (RO, Nagysink/Großschenk), the Franciscan church of GyöngyösAlsóváros and the fortified church of Mediaș (RO, Medgyes/Mediasch). Each of these are built with elongated choirs and approximately hexagonal apses. We explore the possible construction systems of these buildings and compare them to each other. We aim to present the similarities and differences between them and to offer likely explanations of their irregularities. Moreover, we propose the significance of construction circles used for creating a triangulation grid.
1 INTRODUCTION
The utilisation of geometric patterns in architecture in the gothic period is a long-researched subject that has captivated the interests of architects and historians for centuries. In this paper we present the potential of using terrestrial laser scanning (TLS) as a sufficiently accurate survey method to investigate possible construction processes. This technique allows us to explore the exact shape and layout of structures – along with their flaws and irregularities.
Our goal is to recreate probable geometric construction processes of churches with approximately hexagonal apses – a less commonly occurring shape than the octagonal layout. We examined four gothic churches of this type from the territory of mediaeval Hungary: the Franciscan church of Szeged-Alsóváros, the fortified church of Cincu (RO, Nagysink/Großschenk), the Franciscan church of Gyöngyös-Alsóváros and the fortified church of Mediaș (RO, Medgyes/ Mediasch). These examples were specifically chosen for the study based on the geometric similarities between their choir apses. We aim to determine the connection between the hexagonal shape of the apse and the general layout of the choir, or the entire church respectively. Thus, we can examine whether the construction method was based on the same logic for similarly shaped structures in different regions within the country and of different time periods.
We assume that discovering the geometric construction of either a part or the entirety of a building may in some cases imply its period in itself or in relation to other parts or structures around it, which would be especially beneficial in cases where very few historical records are available. To find regularly used methods could lead to researchers being able to date a building that we know little about or even link it to other buildings that were built using the same logic.
1.1 Building history of the Franciscan Church of Szeged-Alsóváros
Szeged is the county-seat of Csongrád-Csanád County in southern Hungary. The Franciscan church (Fig. 1) stands in the quarter called Alsóváros, its monastery is connected from the north. The entrance of the church on the west end leads to a small narthex, which opens to the single nave of the building. A triumphal arch on the east side of the nave leads to the choir with a polygonal apse. The tower connects to the choir from the north, through which we may access the sacristy. The nave, choir and sacristy all have different types of ribbed net vaults, the tower has a stellar vault without ribs.
The choir (a) and the nave (b, c) of Szeged, 2023
Citation: Építés – Építészettudomány 52, 3-4; 10.1556/096.2024.00119
A church near the site of the monastery (but not below the current church) dedicated to Saint Peter was first mentioned in 1459 in a certificate by King Matthias I, and last mentioned in 1497. The current church, the beginning of the building works of which is debated but could be placed around the second half of the 15th century1 functionally replaced the first church upon its completion. The consecration is known to have happened in 1503 and thus presumably the building was also ready for use.2 After the consecration work continued on the tower3 and the vaults4 until the Ottoman invasion that reached Szeged in 1543.
Exact dates of certain parts of the building are unclear due to the scarcity of contemporary records. Nevertheless, through archeological research some details became clear: Zsuzsa Lukács discovered a change of brick size in the choir and nave,5 István Harsányi noted a line of detachment at the triumphal arch along the entire height of the structure.6 These findings indicate a difference in building period. The shape of the apse was altered midway through the building process, the walls of the hexagonal layout were built up to about 4.7 m, where it was formed to be closer to an octagon.
1.2 Building history of the fortified church of Cincu
Cincu (RO, Nagysink, Großschenk) is a village in Brașov County, Transylvania, Romania. The fortified Lutheran church (Fig. 2) of the village in its present form begins on its West end with a massive tower. Towards the East the building continues with a narthex and a western gallery above that. From the narthex we may step into the nave, with aisles on both the north and south sides. The east side of the nave ends in a triumphal arch, which separates it slightly from the elongated choir with a polygonal apse. Both the nave and choir have eye-catching ribbed net vaults, while the aisles have simple cross-vaults without ribs.
The nave, triumphal arch and choir (left), and western gallery (right, photo by Ábel Hajas) of Cincu, 2023 (The terms of the CC BY 4.0 License do not apply to the photo on the right by Ábel Hajas.)
Citation: Építés – Építészettudomány 52, 3-4; 10.1556/096.2024.00119
The church was originally built in the first half of the 13th century,7 this Romanesque structure was severely damaged in the Ottoman invasion.8 During the 15th century the attacks became frequent in the southern parts of the region. The denizens of market towns and villages found their churches the safest places to stay and carry through in these harsh times, thus the fortification of these buildings began. This phenomenon was most prevalent in Saxon communities.9
The church was rebuilt around the turn of the 15th and 16th centuries in gothic style. The earliest date indicated on the structure is 1494, carved near the entrance of the north-eastern tower,10 which might mark the beginning of the renovation. The old semi-circular apse was demolished, the remains of it can only be discovered in the foundation.11 The new walls were erected in a polygonal shape with buttresses at their corners. A new stellar net vault was raised in both the nave and the choir, on the latter the year 1522 was carved and painted (no longer visible) which likely indicates its completion.12
In the later centuries the building had undergone a handful of changes during which the layout remained unaltered. Originally the church was built as a basilica, which arrangement was modified in 1693, when galleries were added above the aisles.13 In 1769 the vault of the aisles was altered as well and in the same year the windows were replaced here.14
1.3 Building history of the Franciscan Church of Gyöngyös-Alsóváros
The town of Gyöngyös is in Heves County in northern Hungary. The Franciscan church (Fig. 3) in the quarter called Alsóváros stands today with the monastery connected to its northern side. From the western entrance we may step into a baroque narthex with a western gallery above, joining these from the south is a baroque tower. Continuing eastward from the narthex we get to the nave. The church has a single aisle on the northern side which is a later addition. The nave ends in a triumphal arch on the east, which leads to the choir that ends in a polygonal apse. Following the Franciscan mediaeval tradition, a tower is located at the shoulder of the choir. The ground floor of the mediaeval tower, through which we can access the sacristy, is connected to the choir from the north. The nave and aisle have baroque lunette vaults, the choir and sacristy have ribbed cross-vaults.
The choir (left), the nave (middle) and the western gallery (right) of Gyöngyös, 2023
Citation: Építés – Építészettudomány 52, 3-4; 10.1556/096.2024.00119
Along with many other Franciscan churches, the one in Gyöngyös-Alsóváros was founded sometime in the middle of the 14th century.15 Some parts of the current building are from this period: the sacristy and the corridor next to it along with the eastern tower, the choir, and the nave.16
The first written mention of the building is in the so-called Bakócz-codex, which implies the year of completion to be 1494.17 The cross-vault and the windows of the choir were presumably built before this completion date – the double-lancet windows sometime in the first half of the 15th century, the triple-lancet windows, and the vault around 1460–1470. The windows of the nave are thought to be built around 1500 due to their similarity in shape to the windows of the reformed church of Nyírbátor, which are known to have been built at this time.18
The Ottoman invasion left its mark here also, as only minor repairs were made for the next few centuries despite the desperate need for large-scale refurbishments. These, however, were only able to happen after the Treaty of Szatmár, closing the war of independence led by Ferenc Rákóczi in 1711. In 1718 in the first phase of building construction from March to October a new triumphal arch and nave vault were built, along with a completely new aisle on the northern side of the nave. In the next phase from about 1740 to 1746, the nave was lengthened towards the West with a new narthex and the western tower was erected. Above the narthex a western gallery was built of stone to functionally substitute the old wooden one in a new place. The last section when significant changes were made lasted from about 1758 to 1766, during which the western gallery needed to be rebuilt due to structural problems with the one built in the previous decades.19
1.4 Building history of the fortified church of Mediaș
Mediaş (RO, Medgyes/Mediasch) is a town in Sibiu County, Transylvania, Romania. The fortification of the now Lutheran church (Fig. 4) is still partially visible today. A tower makes up the north-western corner of the church. Immediately to the south of it is the main entrance that leads to the nave, which connects to an aisle from both the north and south sides. A triumphal arch separates the nave from the elongated choir with a polygonal apse. The sacristy connects to the choir from the north. The nave, apse and sacristy all have different types of ribbed net vaults, the aisles have ribbed cross-vaults.
The choir (left) and the nave (middle, right) of Mediaş, 2022
Citation: Építés – Építészettudomány 52, 3-4; 10.1556/096.2024.00119
The building period of the first – then Benedictine20 – church started in the second half of the 13th century. This basilica with an aisle on each side of the nave had presumably never been fully completed.21 The second church was built in the 14th century, the polygonal apse of the first one was replaced with a longer choir with a similar apse, most likely due to the lack of available space for the congregation. The foundations of the apses of these first and second churches were found under the current building during archaeological excavations in 1971–1972.22
The third building period began at the end of the 14th century: the north wall of the second church was demolished and a north aisle was built. At the beginning of the 15th century the aisle was vaulted. Around 1440 the works on the current choir started, which carried over to the second half of the century along with the rebuilding of the nave and building of the south aisle. The year 1488, according to Historia Transilvaniae by Georg Soterius, marks the completion of the church. However, the heightening of the towers took place several decades later, in 1550– 1551.23
2 METHODOLOGY
Original drawings of most gothic buildings – if they ever existed – have not survived. A handful of examples24 are known, nonetheless, they are quite neat and tidy, the steps of their construction are rarely decipherable. Moreover, in some churches large-scale carvings have been found,25 which suggest that elements, including plans were drawn on site.26
In our study we test possible geometric construction methods of the churches described above. As a start we compare notable measurements to check for correlations and possibly identify the measurement unit used.27 Gothic constructions on the scale of plans and sections are generally thought to have utilised grids of 45-, 60-, or 90-degree patterns. These were presumably built on the famous principles of ad triangulum and ad quadratum, both of which had been known since antiquity and in use in the Middle Ages.28 In our analysis we test different iterations of these grid patterns and attempt to determine the most probable method.
In his book29 Robert Bork emphasises the importance of looking for an internal logic of a building which the construction was based upon. When seeking a possible geometric construction method to a building, many approaches might give similar end results that may be applicable to a certain structure. Nevertheless, if a similar system is discovered to be fitting for multiple parts (of the same period) of the building, it is safer to assume that the given theory is correct.
This type of geometric analysis of mediaeval, especially gothic structures has been a popular topic of study for centuries, in which quadrature and triangulation patterns were drawn on ground plans, sections, elevations and smaller details as well.30 It is difficult, however, to analyse large structures without remarkably accurate surveys which need to contain their imprecisions. Without today’s technology, surveying a building with traditional methods accurately enough for proper analysis required vastly more time and effort. If a survey drawing does not contain the exact geometry, the results of analyses might be misleading.
Our research relies on surveying by terrestrial laser scanner. We used a BLK360 scanner by Leica, that has a nominal accuracy of 4 mm at 10 m distance, which far surpasses that of the mediaeval builders’ and is thus adequate for our examinations. We chose multiple standpoints around each site for scanning, the data of these were then processed through Leica 360 Cyclone Register, which subsequently generated 3-dimensional point clouds. The clouds were next analysed in Graphisoft Archicad. For the analysis, thin sections of the point clouds are cropped, in the case of plans – unless made impossible by circumstances – close to the floor, so the builders’ intentions are clearer, and the vertical imperfections of walls do not need to be considered.
3 RESULTS
3.1 Analysis of Szeged
The analysis of Szeged was described in a paper in the process of publication at the time of writing the present study by Jobbik Eszter and Fanni Budaházi.31 As seen on the ground plan (Fig. 5), the layout is quite accurately constructed and built, which precision made it possible to confidently assume its construction technique.
Plan of the fortified church of Szeged with a 60-degree construction grid (Jobbik–Budaházi 2024)
Citation: Építés – Építészettudomány 52, 3-4; 10.1556/096.2024.00119
The two main approaches when seeking a possible grid system on the ground plan of all our analysed churches were a 90-degree grid made up of squares, based on quadrature – method one – and a 60-degree grid of equilateral triangles and their construction circles, based on triangulation – method two. Both systems are relatively easy to construct even with rudimentary tools and full scale on the ground. These grids of varying node densities were digitally drawn on each plan that was acquired from the point cloud.
The plan in Szeged was discovered to have been constructed with method two fit separately on the choir and the nave, as seen on Fig. 5.32 Even though the layout of the apse was built rather precisely, the corners were closer to the constructing circles than the grid nodes. The vertical elements did not align with the system of the ground plan.
The discoveries made in Szeged launched the continuation of the analysis. We searched for churches with similarly shaped apses to investigate the prevalence of the construction method deduced here. Thus, the analyses of the rest of the buildings were built on the same principle as here.
3.2 Analysis of Cincu
The analysis of the fortified church of Cincu started with its ground plan. After reviewing the dimensions of the building, we could not undoubtedly link them to any known mediaeval measuring unit. The clear ratios between measurements of the church are: the width of the choir can fit 1.5 times into the width of the nave and 3 times into the width of the aisles and nave; the length of the nave is 3.05 (approximately 3) times the size of its width. It needs to be noted, however, that the building is generally rather irregular in terms of geometry: the angles between the apse walls are a few degrees larger than that of a hexagon, the choir width is smaller at the western side than at the eastern, and right angles are rarely present.
In the case of Cincu, to get the layout the cloud was sliced horizontally 0.75 m above the main floor of the choir, since at the apse the floor was elevated by about 0.7 m compared to the rest of the choir floor. Method one yielded no convincing results. Interestingly, in the case when the size of a grid square was the sixth of the choir width, 11 squares could fairly accurately fit in its length from the apse to the outer side of the triumphal arch. The apse corners, however, could not have been defined by the nodes of this grid. Moreover, the system did not match any measurements of the nave and aisles. Since the method only partially matched a part of the building, it could not be identified as the internal logic and was thus ruled out as a possibility.
Method two, on the other hand, seemed to be more successful. In Fig. 6 the plan is illustrated with the 60-degree grid along with the construction circles which would have been used to easily draw the grid on site as well. The walls which were not visible at the level the plan was taken are shown with dashed lines. The radius of the circles is 3.04m. This system marks out some notable points and measurements with good accuracy: the corners of the apse33 and the width of the nave and aisles. The eastern and western walls of the nave somewhat align, though the angles of the structure in these parts makes it difficult to pinpoint a good match.
Plan of the fortified church of Cincu with a 60-degree construction grid (method two)
Citation: Építés – Építészettudomány 52, 3-4; 10.1556/096.2024.00119
This system is also in line with the vertical elements of the building. The same size construction circle was drawn on the sections in Fig. 7. In the choir two circles were drawn, their centres the length of their radius apart, this way the bottom of the lower circles touches the floor and the points of the upper circle fit on the cross-section of the vault – all in all, the ratio between the maximum height of the choir and its width is 3:2. In the nave the grid is rotated 90° compared to the choir. The construction of the height here is also shown in Fig. 7: the lowest circle much like in the choir touches the floor,34 the uppermost circle touches the highest point of the vault. This correlation in both spaces suggests that the current floor level is the same as, or very close to how it was at the time of building the vaults. The vaults of the aisles were built during the baroque period and thus not considered in this evaluation.
Section of the choir (C1 – on the left) and the nave and aisles (C2 – on the right) of the fortified church of Cincu. On C1 the light grey horizontal dashed line indicates the elevated floor level of the apse
Citation: Építés – Építészettudomány 52, 3-4; 10.1556/096.2024.00119
3.3 Analysis of Gyöngyös
Just as in Cincu, we could not certainly identify a known measurement unit in the church of Gyöngyös. The only simple ratio that could be admissible on the plan is between the widths of the choir and nave, which is roughly 2:3.35 The aisle was not included in the analysis due to its baroque building period.
The western ending of the mediaeval church was in about the same position as the eastern side of the narthex today. The traces of this wall can still be seen at the top of the northern and southern walls above the baroque vault in the attic (Fig. 8).
Photographs from the attic in Gyöngyös showing the signs of the mediaeval western wall on the northern wall (left), southern wall (middle), and a mason’s mark on a stone of unidentified purpose (right)
Citation: Építés – Építészettudomány 52, 3-4; 10.1556/096.2024.00119
The construction methods we considered were the same here as in Cincu, and the analysis came to a similar result. To examine the plan of Gyöngyös, a thin horizontal slice of the point cloud was taken 10 cm above the current floor level of the choir to avoid misinterpretation caused by the vertical irregularity of the walls.36 Method one consisted of right-angle grids, after several attempts with different densities, however, we could not find any system that would convincingly match with both the nave and the choir. Since the building period of the choir and nave are supposedly the same, it is unlikely that the builders would have applied different methods in their design.
Method two (Fig. 9) involved a 60-degree grid with its construction circles. The diameter of the circles applied in the choir was chosen to be the average width of the structure, as some irregularities can be found in the layout. The grid drawn this way fits the corners of the apse37 and choir length with good precision. In the nave a same-sized grid as opposed to marking out the inside of the western wall, it seems to mark out the outside. It is worth mentioning, though, that only the position of the top part of the mediaeval western wall was possible to measure, as the wall itself was demolished in the baroque period and the base could have been in a somewhat different position. In the case of the choir, method two certainly looks to be the correct one. Even though in the case of the nave both methods can provide a plausible result, considering the integral logic of the building, method two looks more likely.
Plan of the Franciscan church of Gyöngyös-Alsóváros with a 60-degree construction grid (method two). The mediaeval western wall of the nave is marked with dark blue
Citation: Építés – Építészettudomány 52, 3-4; 10.1556/096.2024.00119
When inspecting the vertical elements in Gyöngyös we were left without success. The current nave vault was built during the baroque period and even though the imposts of the gothic vault can be located in the attic, the shape and height of the vault is unknown. Although the choir still retains its mediaeval vault, no correlation was found between that and the construction system of the plan. Either there never was a connection to begin with or the floor level was altered enough for it not to be recognisable.
3.4 Analysis of Mediaș
The analysis of Mediaș also began with the plan. We could not link the dimensions of the church to any mediaeval measurement unit with certainty, furthermore, no simple correlation could be found between the lengths and widths of the current spaces.
The choir and nave are known to be built during different periods. Therefore, it is not assumed that their construction systems were built on the same logic. 90- and 60-degree grids were nonetheless tested on the layout – separately on the choir and on the nave and aisles. In the case of the nave, we could not find a convincing match, therefore we chose not to present the tried methods in drawing.38 Method one, the 90-degree grid, did not fit the choir too well in any iteration.
The 60-degree grid of method two (Fig. 10) was made using construction circles with the radius of 4.32 m in the choir. Despite the significant imprecision of the walls, this method fits its layout, the corners of the apse – as emphasised in later chapters – align more to the construction circles than the triangular grid.
Plan of the fortified church of Mediaș with a 60-degree construction grid on the choir (method two)
Citation: Építés – Építészettudomány 52, 3-4; 10.1556/096.2024.00119
During our analysis of the vertical elements of Mediaș we discovered the construction circles of method two were relevant on the section as well (Fig. 11). In the choir the bottom of the construction circles are somewhat close to the floor level (they extend 12 cm below it) and the top of the system reaches the maximum height of the vault. This indicates a possible correlation in case the floor level was altered since the completion of the vault. Moreover, the top circle – disregarding the irregularities – marks out the curvature of the vault, too.
Section of the choir (M) of the fortified church of Mediaș
Citation: Építés – Építészettudomány 52, 3-4; 10.1556/096.2024.00119
4 DISCUSSION
4.1 The utilisation of grids
In all the cases described above, the most viable geometric construction method of the choirs was the 60-degree grid drawn with circles, with the diameter of the choir width. Also, we discovered that the corners of the apse polygon in each case matched the construction circles at different points than where the nodes were. Below are the close-up drawings of the choirs with their corners (and other matching points) highlighted, and tables containing the errors with which the actual corners differ from the construction circles and grid nodes.
The choir in Szeged (Fig. 12) is the largest, yet the most accurately constructed out of the examples. The difference of errors is less significant, but still relevant. At W2 and W3 in Table 1, the errors from the nodes are an order of magnitude larger than the errors from the circles. This smaller discrepancy39 still fits the pattern of construction of the other three churches, while the general preciseness helped us to conceptualise our hypothesis.
Choir plan of Szeged (Jobbik–Budaházi 2024)
Citation: Építés – Építészettudomány 52, 3-4; 10.1556/096.2024.00119
Error of choir corners in Mediaș depicted on Fig. 12
| Corner | Error (cm) | |
|---|---|---|
| From circle | From node | |
| W1 | 7.6 | 7.7 |
| W2 | 2.6 | 23.5 |
| W3 | 3.0 | 29.0 |
| W4 | 2.9 | 4.1 |
| W5 | 2.9 | 3.8 |
| W6 | 1.2 | 4.5 |
In Cincu the construction system in the choir ends at the western side of the triumphal arch, marked with a dotted line in Fig. 13. On the construction circle right of the line a quarter point of the circle (X1) fits on the line with remarkable accuracy, as shown in Table 2. The apse corners (X2–X5) – except for one (X3) – align to the circle with errors an order of magnitude less than they do to the grid nodes.
Error of choir corners in Cincu depicted on Fig. 13
| Point | Error (cm) | |
|---|---|---|
| From circle | From node | |
| X1 | 0.5 | – |
| X2 | 2.8 | 15.2 |
| X3 | 4.5 | 4.5 |
| X4 | 1.5 | 20.0 |
| X5 | 2.4 | 53.4 |
At first glance the choir of Gyöngyös seems to be more regular, however, in Table 3 it is evident that the phenomenon observed in Cincu occurs here, too. The corners (Y1–Y6) marked in Fig. 14 align with the construction circles quite well, while the error of the fit of the grid is an order of magnitude larger in almost all cases (Y2–Y6).
Error of choir corners in Gyöngyös depicted on Fig. 14
| Corner | Error (cm) | |
|---|---|---|
| From circle | From node | |
| Y1 | 6.9 | 6.9 |
| Y2 | 2.4 | 47.1 |
| Y3 | 1.7 | 24.1 |
| Y4 | 1.1 | 31.7 |
| Y5 | 4.7 | 45.1 |
| Y6 | 6.4 | 12.5 |
Choir plan of Gyöngyös
Citation: Építés – Építészettudomány 52, 3-4; 10.1556/096.2024.00119
The choir of Mediaș (Fig. 15) is the least accurately constructed out of the four analysed examples. This, however, made another interesting observation possible that may be a hint for the possible construction process in more detail. At the apse corners (Z2–Z5) in Table 4, the divergence of the grid and the corners is by far the largest of all examples, the corner at Z2 is almost 1 metre away from the corresponding grid node. The circles meanwhile fit relatively well, much like in the other churches.
Error of choir corners in Mediaș depicted on Fig. 15
| Corner | Error (cm) | |
|---|---|---|
| From circle | From node | |
| Z1 | – | 24.8 |
| Z2 | 2.4 | 91.8 |
| Z3 | 4.0 | 49.0 |
| Z4 | 3.1 | 3.6 |
| Z5 | 2.9 | 37.2 |
| Z6 | – | 21.5 |
Taking a closer look at the corner Z2 in Mediaș (Fig. 16), the northern wall itself curves: from the closest grid point to the corner the construction circle fits on the line of the wall quite well. The explanation for this rather unusual feature is unclear, but it appears to suggest that the construction circle mentioned had indeed been used in this position.
Northern corner (Z2) of the apse in Mediaș
Citation: Építés – Építészettudomány 52, 3-4; 10.1556/096.2024.00119
Our analysis above clearly shows that the corners of the inspected choirs align considerably better with the construction circles of the 60-degree triangular grid, than the grid itself. Triangulation, as mentioned, was a known method of geometric construction in the Middle Ages. However, it is not well understood to what extent it was used. In the case of floor plan construction, it seems likely that the process did not start by drawing a web of equilateral triangles, but by marking partial circles on the floor. One explanation for the large errors from the grid nodes at the apse could be that after drawing the circle that the corners should align with, the builders started to mark them out starting from one side, using ropes and stakes with a moderate attention to accuracy. Then, after designating the first one, they continued to mark out the next from the first corner, and so on, increasing the error as they went. This appears to be a feasible way in which the choirs of Cincu, Mediaș and Szeged were constructed.
Our findings further show the significance of circles in geometric construction predominantly through the sections. Floor levels and vault heights are determined mostly by the lowest and highest points of circles of the construction system, respectively, as seen in Cincu, Mediaș and Szeged. This seems to indicate that the circles were just as crucial in these systems as their points of intersection. Interestingly, in Cincu this also seems to apply to the floor plan, as seen on Fig. 13.
4.2 Comparison of construction systems
The four analysed churches largely vary in their dimensions. Nevertheless, all of them – at least partially – appear to have been constructed using the same base principle. The proportions of their spaces differ as well, the exceptions to this are the choirs of Gyöngyös and Szeged which share the same length to width ratios. Evidently, despite some similarities, this method of construction did not carry a fixed set of proportions, instead, the manner of application more likely depended on required and available space and resources.
The exact starting point where the construction began is difficult to determine, though in the choirs we attempted to narrow down the options. If the first circle to be drawn had been the easternmost that defined the apse, the system would be substantially altered in the cases where the construction was notably inaccurate (e.g. in Gyöngyös and Mediaș), since the centre point of the circles drawn next would have been the northern and southern apse corners. As demonstrated in Fig. 17, this would result in the collapse of the system (on the left), or it would become remarkably askew (in the middle). However, if the construction started on the West, the main inaccuracies were left to the end of the process and the grid fit well on the entire layout (on the right). As seen in chapter 4.1., the grids of the choirs analysed fit well on the plan, much like the last version described. Thus, we find it feasible that in at least some cases40 their geometric construction started on their western parts, with the easternmost circle assigning the position of the edges of the apse polygon.
Theoretical inaccurate choir plan: the first construction circles drawn with red long dashed lines, the circles of the next step drawn with red short dashed lines, the apse corners highlighted as on Figs 12–16
Citation: Építés – Építészettudomány 52, 3-4; 10.1556/096.2024.00119
Regarding Gyöngyös, we found an interesting connection between the corners of the choir (Fig. 18). We assumed the beginning of its construction to be as described above, the centre of the first circle positioned on the western edge of the triumphal arch. However, our methods did not explain the exact placement of the apse corners. While the reasoning behind the deviation from a regular half-hexagon is unknown, we discovered a curious link between the northern-most and southernmost corners and the centre (point C) of the construction circle of the apse polygon. Arcs drawn from points Y1 and Y6 through C intersect with the northern and southern walls close to Y2 and Y5, 41 respectively.
Choir plan of Gyöngyös with possible correlation between corners and construction grid
Citation: Építés – Építészettudomány 52, 3-4; 10.1556/096.2024.00119
5 CONCLUSION
In our present article we analysed the geometry of four churches: the Lutheran church of Cincu, the Franciscan church of Gyöngyös-Alsóváros, the Lutheran church of Mediaș and the Franciscan church of Szeged-Alsóváros. Case studies were chosen based on the geometric similarity of the apse polygon, closely related to a regular hexagon. Our analysis was based on the point clouds we produced by laser-scanning these buildings, through which we could inspect the exact geometry of the structures. To determine their large-scale possible geometric construction systems, our primary focus throughout was on the ground plans and sections.
In regard to the overall ground plans of the churches, we gave possible explanations for their construction methods. Most of these used a 60-degree grid, or triangulation and its construction circles. In Cincu the choir, nave and aisles appear to be connected, using the same logic and same size circles. In Gyöngyös, the construction system of the choir continues without a break in the nave, with a slight difference in the size of the circles. In Mediaș the choir and the rest of the building were built in separate periods, their systems do not match, the nave and aisles might have been constructed with the principle of quadrature. We compared these geometric constructions with the church in Szeged, which was analysed previously.42
We have examined the possible utilisation of rectangular and 60-degree triangular grids. Rectangular grids did not seem to match with the geometric layout of the choirs, but our results show plausible evidence that a triangulation grid similar to the one the construction of the church of Szeged-Alsóváros had probably been based upon is traceable in the case of at least the choir construction of the case studies. It looks likely though that the intersecting circles that trace out the system of regular triangles play a more important role in the construction than the edges of the triangles themselves. This may throw new light upon the utilisation of triangulation grids in late mediaeval Hungarian architecture. It is also important to mention, however, that the irregularities detectable in the geometry of these ground plans suggest that master builders used geometrical grids with relatively moderate consistency.
ACKNOWLEDGEMENTS
We would like to thank Eszter Jobbik for her collaboration with surveying and Ábel Hajas for lending us his photographs. We are also thankful to the congregations of the Franciscan church of Szeged-Alsóváros, the Lutheran church of Cincu, the Franciscan church of GyöngyösAlsóváros and the Lutheran church of Mediaș for welcoming us and to the organisers of Szászbogács Nyári Egyetem (Băgaciu Summer School) for making two of these visits possible.
The research was supported by the National Research, Development, and Innovation Fund of Hungary under Grant TKP2021-BME-NVA.
Vitéz 1944. 92, Lukács 1999. 3, Harsányi 2001. 297, Levárdy 1980. 7.
The supposed date of consecration is visible on stone plates placed thereafter on the northern and western walls of the nave – Lukács 2000. 143–144.
Lukács 2000. 146.
Harsányi 2001. 297.
Lukács 1994. 453
Harsányi 2001. 303.
Császár 1990. 169–170.
Oprescu 1961. 49.
Fabini 1999. 247.
Fabini 1999. 246.
Entz 1996. 188–189 – the year carved into the nave vault was noted also by Erhard 1982, who already mentioned it as covered up.
Fabini 1999. 247.
Fabini 1999. 248.
According to oral history the church was founded around 1400 by the Báthory family, however, there are no certificates to back up this claim. – Dezséri 1944. 59. See also: Bártfai–Csemegi 1937. 3–4.
Bártfai–Csemegi 1937. 3–4.
Baranyai 1978. 158.
Baranyai 1978. 159.
Oprescu 1961. 34.
Fabini 1999. 284 – we could not acquire information on the exact shape of the previous choir plans from published works, however, an approximate drawing is found on the referenced page of the book.
Fabini 1999. 285.
For example, many mediaeval drawings were collected and published by Böker 2005.
Most notably in York Minster and Wells Cathedral, described for example by Harvey 1997.
Pacey 2007. 62.
Some measurement units are known from this era, however, these were not always used for building construction. The size of a building was determined by a base measurement, which could have been arbitrary – Hoppe 1994. 6.
They are visibly used in Villard de Honnecourt’s drawings – Barnes 2009. folio 20.
This type of research was also prevalent in Hungary, the examples include: Császár 2001, Csemegi 1936, 1937 and 1953, Szakál 1978 and 2007, etc.
It needs to be mentioned, however, that method one is also applicable in the case of the nave, but not the choir.
Which align more with the construction circles than the grid itself.
The original level of the floor is estimated to be the same in the nave as it is in the aisles. The nave currently has a rather uneven and unstable flooring of wooden planks.
The ground plan of the nave is somewhat irregular, the width on the eastern side is 10.09m, on the western side it is 10.35 m. The choir width multiplied by 1.5 is 10.43 m, which is interestingly much closer to the western side than the eastern.
Considering the accuracy of mediaeval master builders, we assume that the junction of the foundation and the wall is where the actual floor plan is closest to the desired geometry.
Just as in Cincu, the corners align with the construction circles rather than the triangular grid.
Regarding the nave, however, it is worth mentioning that there may be a possibility that the length was altered at some point. A strangely angled buttress is visible on the wall of the southern aisle (on Fig. 10), the second one from the West is built like the ones on the corners. On the northern side of the church in this position is the eastern wall of the western tower, which currently does not align with the buttress in question. Nonetheless, as mentioned in 1.4., the towers were heightened after the church was completed. To account for the added load, we can assume that their walls were also widened for reinforcement. We do not know the exact former thickness of the tower walls, but they might have been thin enough that the eastern wall of the western tower aligned with the buttress in question better. And if that buttress was indeed on a corner at some point, a former western wall might have been the continuation of the former eastern wall of the tower. This means that the nave and southern aisle might have been lengthened, which would explain the odd position of the tower and the length of the westernmost bays. This way the width–length ratio of the nave and aisles would have been 2:3.
It is important to mention that the difference between the width of the choir on the eastern and western end is about 7 cm, however, the intention was obviously to build two parallel walls. Therefore, we can consider this error as the “general” margin of error of building construction of the time.
This order may have been applied in Szeged, Gyöngyös and Mediaș, while it seems unlikely in Cincu – here the width is larger at the eastern side of the choir than it is on the western, if the construction had started on the West, the circles that fit the structure would have been smaller.
The error of the intersection points from Y2 is 3.2 cm, from Y5 is 3.6 cm.
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