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Spire-Polyhedra

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Spire-Polyhedra Journal for Geometry and Graphics Volume 11 (2007), No. 1, 111{126. Spire-polyhedra L¶aszl¶o Strommer Dept. of Architectural Representation, Budapest University of Technology & Economics Mue} gyetem rkp. 3, H-1111 Budapest, Hungary email: [email protected] Abstract. The shapes of the spires of Western European medieval churches show almost as high variability as their interiors, but while inside | after the ¯rst Millennium | the builders began to use curved surfaces (i.e., vaults), the spires mostly kept their polyhedral (or conical) shapes. Architecture | due to its necessities and restrictions | used only a limited portion of the in¯nite set of potentially possible polyhedral shapes | such a small subset, that it seems conceivable to categorise them, or at least most of them. This paper suggests a method of classi¯cation of spire shapes, postulating that the more complex forms can be produced as compounds (either intersections or unions) of some basic shapes. Key Words: spire, geometry, classi¯cation of spire shapes MSC: 51N05 1. Basics Strictly speaking, the subject of this paper is to describe a speci¯c subset of polyhedra, and in this sense, its pure geometry. At the same time, the goals and means of architecture play such an important role in de¯ning the domain and codomain of the design space, that it seemed not only reasonable, but | for the sake of brevity and clarity | almost inevitable to use some architectural terms and de¯nitions. 1.1. Base concepts, de¯nitions By de¯nition, a spire is a steeply pointed termination to a tower. In this paper I will use this term in a somewhat wider sense: not only for the most common pyramidal or conical shapes, but for any shape a roof of a tower can have. In the ¯gures below the top parts of the towers are also depicted, but they do not belong to the spires proper: the spire ends at the horizontal plane where its bottommost sloping plane ends. In other words the incidental gables are part of the geometry of the spire (in addition to the sloping planes of the roof proper of course), but the tower walls are not, even if they are in the same vertical plane. ISSN 1433-8157/$ 2.50 c 2007 Heldermann Verlag 112 L. Strommer: Spire-polyhedra 1. Spire apex 2. Ridge 3. Valley 4. Gable ridge 5. Gable apex 6. Verge 7. Bottom plane of the spire 8. Gable (gable wall) Figure 1: Parts of a spire A gable is a vertical plane (a wall, actually) whose existence is inevitable whenever the bottom edges of the sloping surfaces of the roof proper are not horizontal. A verge is the sloping outer edge of a gable, a gable apex is the highest point of a verge. A spire apex, however, is a point located over the centre of the base, usually the highest point of the whole shape. A valley is a concave break between adjacent surfaces, which therefore collects the water from them; while a ridge is a convex break, which consequently diverts, not collects water. Finally, a gable ridge is a ridge starting from the gable apex, mostly (but not always) connecting it with the spire apex. 1.2. Base principles The most obvious examples of architectural restrictions determining the applicability of shapes are the ones that are consequences of the limitations of the building materials, like the size and steepness restrictions in case of wood, stone or brick spires. Another consideration is the rationality of the plan: e.g., the claim for freer plan arrangement causes the dominancy of rectangular shapes. Finally, a third aspect is aesthetics, which | through the use of symmetry | e®ectively excludes every ad hoc plan and shape. Even if we take the above restrictions into account, the set of possible shapes is still so enormous, that it takes some abstraction to produce usable categories; however, in my opinion this does not decreases the applicability of the system. 1.3. Base plans The archetype of the medieval tower can be described as a building or part of a greater building (mostly a church or castle), whose height is considerably bigger then the dimensions of its base | which is usually a square, a polygon or a circle, or, in rare cases, a rectangle or an ellipse (arranged in order of decreasing frequency). In this paper I focus on spires with square or regular octagonal base, partially because conclusions derived from their examinations can be easily generalized, but mainly because these are (prominently) the most frequent shapes | so much so, that sometimes we ¯nd polygonal spires even on circular towers (e.g., in Maria Laach/Germany). Furthermore, the spires of circular medieval towers are usually simple surfaces of revolution | mostly cones. L. Strommer: Spire-polyhedra 113 2. Basic shapes First let us deal with the simplest shapes, as according to my hypothesis all architecturally relevant spire shapes can be generated from them. Note here that in case of these basic shapes the change of the steepness of roof planes will not produce a topologically di®erent spire shape, therefore a whole set of shapes can be derived from the ones being shown below using only a±ne transformations. 2.1. Regular n-gonal pyramid Perhaps the simplest way to cover a regular n-gonal base is to use congruent isosceles triangles. The a4 and a8 shapes of Fig. 2 are depicting pyramids of equal height, covering regular polygonal bases which can be inscribed in circles having equal radii1. By duplicating the number of the faces of a pyramid we soon get a shape which | at least in architectural practice |, can be classi¯ed as a cone. Perhaps this kinship is the reason why all three shapes appear on the church of Maria Laach. a4 ¡! a8 ¡! a0 Figure 2: Regular n-gonal pyramids 2.2. Convex 2n-gonal base-truncated pyramid If we grab the midpoints of the base of a regular n-gonal pyramidal spire described in 2.1, and move them slightly upward, the sloping triangular surfaces break, and because of these new ridges (connecting the spire apex with the apexes of the newly formed gables) the horizontal section of the spire becomes a 2n-gonal polygon. Since the spire apex is closer to the midpoints of the base then to its corners, if the gables are not too high, the new gable ridges are steeper then the ridges over the diagonals of the ¡ base (b4 ), but when the height of the gables exceed a certain limit (see 2.7), the diagonal + ridges become the steeper of the two sets (b4 ). The idiosyncrasy of shapes b4 and b8 of Fig. 3 is that all of their ridges have equal slope, and this way their horizontal sections (over the level of their gable apexes) are regular polygons. Therefore, these shapes can be described as base-truncations of regular 2n-gonal pyramids. I call a shape \base-truncated pyramid", when a regular pyramid is truncated by a rotational symmetric set of planes which are perpendicular to the base of the pyramid. For example, if we take an a8 pyramid, connect every second vertex of its base, and then keep 1For better comparison all shapes in Section 2 have equal spire height | and therefore equal slope over the diagonals of their base polygons. 114 L. Strommer: Spire-polyhedra ¡ + b4 b4 b4 ¡! b8 Figure 3: Convex 2n-gonal base-truncated pyramids only that part of the original volume which lies above this new square base, we get a shape similar (a±ne) to b4. + The most well-known examples of this shape (or more exactly its b4 subtype) are the spires of the Speyer Cathedral in Germany | but the regular b4 form also appears, for example on the towers of the Marienkirche in LubÄ eck/Germany. 2.3. Rotated n-gonal base-truncated pyramid If we raise the gable apexes higher, until the diagonal ridges are embedded into the roof planes (see 2.7), we get a rotated base-truncated n-gonal pyramid | or in other words, the part of a pyramid which lies over the n-gonal base we get by connecting the midpoints of its original base. c4 ¡! c8 Figure 4: Rotated n-gonal base-truncated pyramids Over a square base, the c4 shape of Fig. 4 appears when the height of the gable apexes is exactly half of that of the spire apex. Since in this case the verges and the gable ridges are parallel, the sloping planes of the spire are rhombuses, and the shape of the roof proper can be described as a translational surface. The c8 shape is obviously similar to the previous c4 one as it also lacks the diagonal breaks, but | just like shape b4 in 2.2 | it also can be described as a (di®erent) base-truncation of a similar a8 pyramid. The sloping planes of this shape are deltoids. The c4 form can be found for example on the western tower of the cloister church of Maria Laach, the octagonal c8 form on St. Martin MunsterÄ in Bonn/Germany. L. Strommer: Spire-polyhedra 115 2.4. Concave 2n-gonal base-truncated pyramid The obvious next step is to raise the gable apexes even higher, which again produces diagonal breaks in the roof surfaces | this time valleys. ¡ + ¤ d4 d4 d4 ¡! d8 Figure 5: Concave 2n-gonal base-truncated pyramid ¡ + It is true again that a small lowering (d4 ) or raising (d4 ) of the gable apexes does not change the basic attributes of this shape | and that we can ¯nd an equilibrium (d4, see 2.7), when the slopes of the verges and the diagonal valleys are equal.
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