Geometry in Gothic Architecture (2024)

“The Gothic cathedral is a blossoming in stone, subdued by the insatiable demand of harmony in man. The mountain of granite blooms into an eternal flower, with the lightness and delicate finish as well as the aerial proportions and perspective of vegetable beauty.” — Emerson

Geometry has always been an integral part of architectural design. From antiquity to the 21st century, geometry has brought to life extraordinary shapes and constructions of absolute geniality, ultimately portraying the highest level of human intelligence. If we go through the millennial of architectural development, there are some milestones to be highlighted — one of which is Gothic architecture. The Gothic style is seen as a deviation from the conformity and standards of classical Greek and Roman architecture. Nevertheless, it gave birth to audacious designs and complex configurations originating from fairly simple and organic geometry, composed of mainly circles and straight lines and limited operative systems.

The growing impact of the Christian church after the fall of Rome influenced a new type of architecture — that of building churches and cathedrals. Lead by the Romanesque architecture of the Mediterranean Europe, architects of the northern part of the continent sought a new style — one that allowed more light to enter high buildings, illuminating their interiors even more. This eventually created a new way of conceptualizing churches: architects would use light as the tool of achieving greater interior heights with as much uninterrupted vertical space as possible. The concept of soaring vertically combined with simple geometrical configurations produced a both visually dramatic and elegant building structure. But how did it all start?

The Arch

Gothic church architecture is one of soaring vastness towards heavens, and its revolutionary design was initiated with the rise of the equilateral arch. This type of arch was first seen at the Abbey of St. Denis in the 12th century. In his supplement on Gothic geometry, Joe Chiffriller describes that the geometric construction of this arch is based on the intersection of two circles, which form two arcs, whose endpoints lie on the horizontal baseline that hold the arch itself (Chiffriller, 4). As seen through abbot Suger’s cathedral of St. Denis, the equilateral arch has an upward pointed shape, which created a new style that eventually spread all over Europe in church architecture. When looking at St. Denis’ equilateral arches, there is a particular technological trait that needs to be emphasized: When comparing them to round arches, pointed arches are more effective in distributing the force of heavier ceilings, and support more weight than the latter (Stanley, 337). In the way that they are constructed, pointed arches allow for more flexibility of wideness by maintaining the same height, as seen below (Haveman, Fellner, 2004).

The Vault

The pointed arch eventually gave birth to more sophisticated and risky constructions. A genuine achievement of such a methodology was the vaulted ceiling. These ceilings could directly consist of a pointed arch or could be “groined, ribbed, and filled with tracery in the most costly manner,” as John Hopkins distinguishes in his “Essay on Gothic Architecture” (Chapter 1). When analyzing the construction of these ceilings, one can notice striking kaleidoscopic patterns in them. Fractal-like shapes, unconventional proportions of rib-like forms created by the intersecting arches extract, what the well-known photographer David Stephenson would call, “heavenly vaults” — a system of sublime symmetries that are complex, yet beautifully articulate in their geometric representations. This symmetric structure is formed by two barrel vaults intersecting at right angles. The result is four curving surfaces that draw the attention to the center. The areas between the vaults create ribs that emphasize both the vault’s strength and aesthetic appeal. The application of these ribs creates a very systematic design based on simple shape configurations, such as triangles, which allow for a more stable surface and usage of less complicated supportive structures. To take an example, Durham Cathedral in England, is the first building in the country that claims the fame of incorporating the engineering of vaulted ceilings. First intended only for the choir, the ribbed vaults were eventually added to the aisle and the rest of the church, creating an “iconographic” structure, which opened a new trend in Europe afterwards (Stalley, Briefing).

The Flying Buttress

Another main characteristic and innovative trait of Gothic architecture is the flying buttress. Essentially, this type of buttress, used at the exterior of the church, supplements the strength of the entire structure by pushing the weight of the roof away from the walls and putting it in the exterior surrounding the edifice. A typical flying buttress is composed of “an arch capped by a flat stone band,” and its mass is distributed accordingly, depending on the location of its centroid (midpoint), inclination and thickness. According to calculated studies, “the thicker the flyer (flying buttress), the better it is,” as it allows for a bigger thrust that keeps the building structure more stable (Nikolinakou, Tallon, 2351). The structural behavior of these buttresses, although simple in their geometric configuration, makes it able for us to understand the entire architectural and engineering mechanism used behind the heavy cathedral walls.

Gothic Windows

Gothic architecture gave birth to a new way of allowing more light to enter buildings. Gothic style windows were first documented in the 12th century and were popular until the 16th century. They have long been admired and highly sought after for their simply-structured ornamental beauty and elegant combination of geometrical patterns. To examine the level of sophistication used in these windows one should first realize that Gothic ornamentation is a morphology of various shapes, such as triangles, polygons, circles and rings of circles intersecting each other. A typical design example is the trefoil, a foliation found in the upper part of Gothic windows, which consists of a three-sided leaf formed from partially overlapping circles circ*mscribed in an equilateral triangle (Chiffriller, 11). In the same manner, the foils can be added up to six of them — forming a hexafoil, in which the six circles are drawn around a regular hexagon. Alternatively, the circles can also be transformed in pointed arches, producing a pointed hexafoil, which is generated by displacing the radius of the circles and scaling them to create the arch-like forms (Havemann, Fellner, 4).

Bringing it all together

So far, we have been discussing about linear and two-dimensional shapes as the basis of Gothic architecture — lines, triangles, circles etc. How does all this come together to create these great cathedral structures that we inherit nowadays? According to Havemann and Fellner, the usage of “profiles” gives Gothic architecture “the fascinating and impressive three-dimensionality” of structure that we see in the buildings. These profiles are created by rotating polygon shapes into specific angles and multiplying the effect so that it encompasses a designated space. The most typical shape that undergoes this process is the square, but this process can be analyzed using other shapes such as hexagons, triangles, circles etc. A component of Gothic churches which obeys the rule above is the chevet, the eastern end of the church in which a curved ambulatory is added behind, constructed of a series of apses and small chapels radiating from it. In the example of the Chartres Cathedral, the space between the ambulatory walls and the chapel is filled with buttress walls, which form a massive structure similar to the construction of a rose window. Meanwhile in St-Denis, the depths of the chapels vary and the buttresses holding them “are not aligned with the radii of the chevet” (Bork, 56). According to many researchers, this irregularity at St-Denis is due to the usage of a particular type of geometry called epicyclic, or it may have some cosmological sense (Bork, 56–57). The structure of the chevet, as with many classical cases of geometrical usage of organic shapes, is mostly composed of squares inscribed in circles, rotated around a triangular configuration, which gives the chevet a half circular representation and which is aligned with framing buttresses.

Church Interiors

Geometry has played a great role in designing the interior of gothic churches too. The wide usage of polygon configurations in designing the main parts of the interior of the churches eventually became a cult that had many historical, political and religious implications. In the example of the Palatine Chapel in Aachen, we can see political thinking merging with the architectural and geometrical vision of the cathedral itself. The octagonal shape of the chapel, for instance, reflects the political and religious will of Emperor Charles (Sharp, 185). The entire structure has a spiritual meaning as well — the circle represents the divine and transcendent world; the square represents the rational world, and the octagon stands in between, representing both worlds (Hausmann). This type of imperial architecture transmits a new form of message embedded in the heritage of Gothic architecture nowadays.

Ornamentation

A final interesting segment of the interior design of Gothic architecture worth mentioning is the implementation of geometry in its ornamentation. The aesthetic qualities of the ornamentation of Gothic cathedrals are mostly found in the intersection between sculpture and columns. Gothic sculpture of the 12th century was predominantly architectural in character. The sculptures used in cathedrals were of a big scale and mostly attached to the entrance doors. Because they were connected to the colonnettes by which they were supported, they were called “statue-columns” (Hinkle). What Gothic architecture introduced to the way the entrances and exterior facades of the cathedral looked like, was a trait called the “flamboyant style.” Originating in France during the 1380s, this ornamental structure of flame-like configurations brought in many new geometrically elongated motifs (originating from the classical pointed arch), in which ellipses, ribs or rose-patterns combined to create a rather lavish and extravagant look of Gothic churches.

Gothic architecture brought to life extremely beautiful, elegant and innovative engineering techniques which are seen in the surviving cathedrals around the world. The pointed arches, vaulted ceilings, rose windows and ornamental forms were revolutionary in the architecture world, as well as in the perception of the rising power of the Christian church. As the famous French author Victor Hugo would write in the Hunchback of Notre Dame: “In the days of architecture, thought had turned into a mountain and taken powerful hold of a century and of a place” — the power of architecture was so vast, that it was able to keep the order of societies, generate systems directly related to its development and serve as a reference point of human achievement. Gothic architecture, indeed, created a beautiful system of mathematical precision and historical importance that it now remains a world heritage and has romantically been sought by many.

Citations:

Axel Hausmann, “Kreis, Quadrat und Oktogon Struktur und Symbolik der Aachner Kaiserplatz”,( Meyer & Meyer Verlag, Aachen — 1994)

Hopkins, John Henry. “Essay on Gothic Architecture, with Various Plans and Drawings for Churches: Designed Chiefly for the Use of the Clergy” Burlington: Smith and Harrison.

Chiffriller, Joe. “New York Carver: Tips and Tricks to Gothic Geometry.” 2002.

Nikolinakou, Maria A. and Tallon, Andrew J. “New Research in Early Gothic Flying Buttresses.” Massachusetts Institute of Technology. Web. <http://www.arct.cam.ac.uk/Downloads/ichs/vol-3-2347-2362-nikolinakou.pdf>

Dennis Sharp, ed. The Illustrated Encyclopedia of Architects and Architecture. New York: Whitney Library of Design, an imprint of Watson-Guptil Publications, 1991. ISBN 0–8230–2539-X. interior photo, p185.

Hinkle, William M. “Gothic Art and Architecture.” Online Encyclopedia 2000. Web. <http://autocww.colorado.edu/~flc/E64ContentFiles/PeriodsAndStyles/Gothic.html>

Bork, Robert. “Ground Plan Geometries in Suger’s St-Denis. A Prototype for Altenberg.” Web. <http://www.art.uiowa.edu/files/art/bork_article.pdf>

Schneider, Michael S. “Geometry of the North Rose Window of Chartres Cathedral.” Constructing the Universe Activity Books. Web. 2009. <http://www.constructingtheuniverse.com/Chartres%20Window.html>

Bork, Robert and Nußbaum Norbert. Gotischer Baubetrieb am Aachener Münsterchor. In: “Sie glänzte wie ein kostbarer Edelstein, wie ein kristallklarer Jaspis.” 600 Jahre Aachener Chorhalle, (Schriftenreihe des Karlsverein — Dombauverein, Band16) Aachen 2014, pp. 22–42.

Stalley, Roger. “Briefing: The Structural System. Durham Cathedral and the Anglo-Norman Romanesque.” Web. <http://www.learn.columbia.edu/ma/htm/sr/ma_sr_discuss_dc_structure.htm>

Betz, William; Webb, Harrison Emmett; Smith, Percey Franklyn (1912), Plane Geometry, Mathematical texts for school. Ginn. page 321.

Havemann, Sven. and Fellner, Dieter W. “Generative Parametric Design of Gothic Window Tracery.” The 5th International Symposium on Virtual Reality, Archeology and Cultural Heritage. Web. 2004. <http://www.generative-modeling.org/GenerativeModeling/Documents/window-tracery-vast04-final.pdf>