Knowing the geometrical logic behind particular regular pentagon constructions, it seems obvious to ask the question as to how these could have been applied to the design of actual structures. It must be stressed that the final form could differ slightly from the original concept as a result of the inaccuracy of the execution. Later modifications and renovations can also add to this divergence. Accordingly, taking the small margin of accuracy between the particular methods into consideration, it is apparent that the construction used in the design can hardly be determined by simply analysing the accuracy of a pentagon represented in certain structures. Also, an accurate survey of relevant medieval architectural features is not always available for comparison.
Besides their mathematical features, the geometric character of the regular pentagon approximations, namely the initial data and the principle of the construction, can determine the type of architectural situation in which they could have been applied. Furthermore, in most cases, the author or the circle of the possible author and the period in which the drawing arose, can help ascertain the scope of relevant monuments to be associated with certain approximations. If available, the master’s original pentagonal plans can be regarded as the most direct application of their method. Therefore, these should be given special consideration during any analysis.
As Carl F. Barnes, the author of the latest facsimile of the portfolio of Villard de Honnecourt has claimed, ‘no building anywhere can be securely attributed to Villard’ (Barnes 2009: 218). In addition, it is also uncertain as to whom the pentagonal tower drawing (no. 1 in Table 1) can be attributed, considering that Hand IV has copied a formerly existing figure from folio 20 verso; however, the drawing was without doubt created in Northern France in the second half of the 13th century. Villard surely visited the construction of the Cathedral of Reims and other newly built Gothic cathedrals of the era, (Barnes 2009: 220–227) where several architectural appearances of the pentagon can be mentioned. For instance, in the Cathedral of Reims and Laon, there are tracery windows of five-petalled parts, and the choir of Reims has been designed with a half decagon. Nancy Wu has also identified a pentagon in her theoretical reconstruction for the geometrical layout of the plan of Reims Cathedral (Wu 2002: 165). In accordance with Barnes, no operation can be attributed to Villard in Reims; however, in the sketchbook, his drawing for the half-decagonal choir of the Cathedral of Meaux with the inscription ‘vesci les ligement de le glize de miax de saint estienne ‘(‘See here the plan of the church of Saint Etienne at Meaux’) (translated by Barnes 2009: 96) represents a direct example for the application of the pentagon (Fig. 9a).
The designers needed to inscribe five sides into the semi-circle of the apse. The initial data of the pentagonal approximation in folio 21 recto, technically, has been the side length, but the method based on the angles could have been easily adapted to a situation where the circumcircle was determined (Fig. 9b, c). The 1 to 3 ratio is entirely appropriate for tracing out the direction of the first side, and then, by rotating the angle, starting from both sides to meet in the middle, the complete half-decagon can be constructed. Jean Wirth suggested a parallel solution for the construction of this choir plan, using right triangles of sides of 3, 4 and 5 to approximate 36° (Wirth 2015: 127–128). A similar method has been described by Josep L. Ginovart and his colleagues concerning the heptagonal (half-tetradecagon) choir of the Cathedral of Tortosa in the Traça of Guarc. (Ginovart et al. 2013). Robert Bork recognised that in the case of Villard’s drawing of Cambrai Cathedral (folio 14 verso), the distortion of the half-decagon of the choir was larger on the left side than on the right side (Fig. 9d) Bork 2011: 32–33). Perhaps the reason for this inaccuracy is just the sketchy nature of the drawing; this could have resulted in the method of the construction. If the 1 to 3 ratio had been used from the right side of the hemicycle, the distortion of the half-decagon grew from right to left (Fig. 9e).
It is important to note that the half-decagonal apse was quite common in the major part of the Middle Ages (Feher et al. 2018a, b; Pentagons: 309–311). Lisa Schürenberg has considered this layout as a separate group within French Gothic architecture (Schürenberg 1934: 31–62; Kuthan et al. 2016: 77). In the Cathedral of Prague, the half-decagonal apse (Fig. 10d) was designed by Matthias of Arras, the previous master to Peter Parler, in the mid-14th century; he was also of Northern French origin and had worked in Avignon previously (Kuthan et al. 2016: 79). Without suggesting a direct link between Villard’s portfolio and the choir of the cathedral of Prague, it is obvious that several examples of similar choirs from all over Europe suggest the problems of constructing the half-decagon often emerged, and a method similar to the one presented in Villard’s portfolio could offer a useful solution. Jiří Kuthan and Jan Royt have considered this type as a modern appearance in Prague (Kuthan et al. 2016: 79), but the shape had already occurred centuries earlier not only in Villard’s age but also, for instance, in Kalocsa (Hungary) (Marosi 2000; Szakács 2019) or Southern France in the 12th century.
Concerning the pentagon construction published in the Geometria Deutsch (no. 2 in Table 1, Fig. 10 a), it is important to point out that both Mathes Roriczer and his probable fellow lodger, Hans Schmuttermayer, have emphasised in their books on pinnacles that their knowledge had been inherited from the ‘Junkers of Prague’ (‘dj iungkherrn von prage’ and ‘die Junckhern von prage’ Shelby 1977: 82, 126), that are generally identified as the Parlers. This century-long transfer of knowledge on pentagon drafting is of great significance. Lon R. Shelby has proposed that the link between the Prague masters and the Roriczer family could have been Wenzel Roriczer, Mathes’s grandfather, who had worked with the Parler lodge of the Saint Vitus Cathedral in the second decade of the 15th century (Shelby 1977: 7–8). There is no clear evidence, however, proving that the ‘Junkers’ could have been identified with the Parlers, rather than any other masters of the Prague lodge (Legner 1978: 7). What is certain, is that after the activity of Peter Parler, several members of the next Parler generation (for instance, Johann, Wenzel, Janco or Michael) worked in Prague or other parts of Bohemia, such as Kutna Hora or Kolín (Legner 1978:11; Marosi 1997: 149–152).
Whether the ‘Junkers’ were, in reality, the Parlers, it is sure that the examples, comparable to Roriczer’s pentagon construction, range from mid 14th-century Bohemia to late 15th-century Germany. Peter Parler probably studied in Cologne and previously worked on the Frauenkirche of Nuremberg before his arrival to Prague in 1356 (Kuthan et al. 2016: 87). He took over the operation of Saint Vitus Cathedral with the finished ambulatory of the half-decagonal choir designed by Matthias of Arras. Throughout the construction, several pentagonal tracery windows of five-petalled sections can be found, such as those of the eastern window of the southern transept (Kuthan et al. 2016: 127); the upper part of the pillars of the flying buttresses of the eastern choir; (Kuthan et al. 2016: 230) the western façade (Kuthan et al. 2016: 433) and the tracery composition above the portal of the southern transept vestibule built in 1367 (Fig. 10b, c) (Kuthan et al. 2016: 113, 133–134, 136–137, 238). In case of the southern facade of the ground floor of the Great Tower, it is significant that both the present form rebuilt by Josef Mocker in the 19th century and the original plans, conserved in the Vienna Collection of Prints and Drawings contain pentagonal parts (Kuthan et al. 2016: 133, 139, 142, 152, 242). While the half-decagonal apse was Arras’s design, an identical geometrical assignment is mentioned in Parler’s works in an entirely different architectural situation: the semi-circular lunette of the northern portal of the Saint Wenceslas Chapel decorated with five petals (Kuthan et al. 2016: 105).
With regard to the Roriczer family, the construction of the Cathedral of Nuremberg was directed by Konrad Roriczer with the assistance of his son Mathes, and it is also likely that a century earlier Peter Parler had been there. Mathes also worked in Regensburg, where pentagonal window traceries can be found; (Fig. 10e) although, neither the churches of Nuremberg nor those of Regensburg have been designed with half-decagonal apses.
The frequent application of the pentagon in tracery windows, however, can hardly be associated with the method Roriczer has published and dedicated to his Prague predecessors. While the initial data of this approximation is clearly the side length of the pentagon, for a five-petalled form, the circumcircle should be determined first, especially in a complex composition of traceries where the role of this particular form is subsidiary. Roriczer’s pentagon construction is inappropriate for such designs (Fig. 9). Hence, the question of the real application remains open; it also likely that both the Parlers and the Roriczers were aware of other pentagon approximations for the design of five-petalled forms.
The initial data of diagram A and B of the Vienna Collection of Prints and Drawings (no. 3 and 4 in Table 1) is the side of the pentagon. While diagram A of sheet no. 17079 is clearly a sketch for a pentagon approximation, the purpose of diagram B is much more uncertain. As the spine of the pentagon is far too high, it is doubtful that this diagram has been referred to as a pentagon. Concerning the architectural examples where diagram A (Fig. 11a) could have been applied, window traceries of Saint Stephan Cathedral of Vienna should be mentioned, but they raise the same contradiction as in the case of Roriczer’s method (Fig. 11c). However, the collection of prints and drawings contain some ideal plans embracing pentagons, providing the most direct examples of a probable application of the method represented by diagram A. On sheet no. 16889, in the ideal plan of a gothic baldachin, two lateral pentagonal bays are attached to a central hexagon (Fig. 11d) (Böker 2005: 200, Koepf 1969: Fig. 247). In this composition, it is entirely clear that the side of the hexagon has provided the initial data for the construction of the pentagons; thus, the method of diagram A could have been appropriate. The figures from sheets no. 16994 and no. 17002 (Böker 2005: 304, 314, Koepf 1969: Figs. 368, 378) are similar cases as far as the starting point of the construction is concerned. In the former, the pentagons of the small additional baldachins join to one side of a buttress of determined length, while in the latter, a system of rotated pentagons is presented in which the whole composition is based on a pentagon, so any data could have been fixed arbitrarily (Fig. 11d).
In the case of Diagram C of the Vienna Collection of Prints and Drawings (no. 5 in Table 1), where the crucial lines of Boulerice’s suspicions are missing from the original drawing, it is doubtful if it is a pentagon approximation. However, it is quite confusing that the resulting pentagon is astonishingly accurate. Moreover, as a practical application, this construction could have been extremely useful for medieval architects as this is the only one where the circumcircle can be constructed directly from the side length. If this series of almost ad hoc steps had been a pentagon construction, it could have been applied both in the plan design of pentagonal buildings and in tracery works; although, in the case of the latter, it is not logical that the circumcircle could be constructed indirectly (Fig. 10).
In case of the figure on folio 18 of the Frankfurt Lodge Book of Master WG (no. 7 in Table 1), it is clear that the author’s initial intention could hardly be a pentagon construction to be applied in architectural design. The mathematical analysis has also revealed that the inaccuracy is highly significant. The starting point of the evidence, as speculated by Hoppe (1995: 148–151) is a square with an inscribed octagon, but neither the side nor the radius of the inner circle or circumcircle of these figures has any connection to those of the eventual pentagon. The radius of the inner circle and a quasi-circumcircle of the pentagon appears in a rather accidental way. Beyond that, it is rather hard to imagine any architectural features where an octagonal form should be linked to a pentagonal one, which also suggests that the link between the two figures of folio 18 is undoubtedly questionable.
Hans Hammer’s pentagon construction (no. 6 in Table 1) is the only one among the approximations occurring in medieval sources that needs the circumcircle as the initial data and follows the logic of dividing the perimeter of the circle into five parts. Thus, this method would certainly be appropriate for the design of the rather common five-petalled forms in tracery windows. Hammer’s career, although he has noted some autobiographical data in his sketchbook, (Entz 1992; Fuchs 1992) is rather obscure, except for some personal information and the fact that he became the master of the Cathedral of Strasbourg in 1482. (‘1482 vff die Paffen Fasznacht da wart ich Pallier’ Fuchs 1992: 11, 13) The great western rose window of the cathedral contains pentagonal parts, but both the original plan conserved in the Vienna collection and the realised version can be attributed to Michael von Gmünd from the Parler family, (Marosi 1997: 151) and no other pentagonal building elements can be attributed to Hammer himself. The most direct application of his method can be found in his sketchbook, right next to the pentagonal figure, which also sheds light on the purpose of some blind lines of the drawing that have not yet been solved (Fig. 12a). The curves AD, BE, CF and CD are not related strictly to either the pentagon or the heptagon construction, as both can be traced with the radii of the circumcircles (Fig. 12b). The curves and the segment BE can be explained in the context of the figure next to the pentagon, representing the construction of the profile of a great portal jamb and a rib of the vault (Fig. 12a after http://diglib.hab.de/mss/114-1-extrav/start.htm?image=00041). The pentagon of the previous figure, with its blind lines, plays a role in the geometry of these profiles, (Fig. 12b) providing new information about the application of the pentagon and the design methods of medieval master masons. Hans Hammer’s figures confirm that pentagons were also used in the design process of profile elements such as ribs, window sills and traceries, expanding the scope of the applied form from individual pentagonal or decagonal buildings, church choirs, gothic traceries and floor plans. (In the theme of plan design see Hiscock 2000; Hiscock 2002: 108–116 and Shortell 2002).
Beyond the architectural features that can be linked to the certain approximative methods for the construction of a regular pentagon, the following example shows that other constructions must have been invented by master masons to solve more complicated geometrical problems. The late gothic tower of the Church of the Poor Clares in Bratislava, with a pentagonal plan, was added to the former 13th-century nave in the 15th century. Being an additional construction attached to an existing one, multiple data were determined for the geometrical layout of the pentagon (Fig. 13). Both the height and position of the perpendicular cord of the pentagon were determined, although the dimensions of the pentagon depended on the master’s decision. Finding the position of the centre of the circumcircle, or settling any points to result an appropriate pentagon, are all complicated operations demanding a high level of geometrical knowledge. The application of numerical ratios, such as 1:3 or 10:17 seems to be the simplest way to determine the angles of the sides. The construction can also be easily solved by drawing a pentagon of adequate dimensions on the planning table and positioning it to the angle of the existing walls; it is quite probable that a more complex construction could also be ascribed to the medieval master.