Flat Tessellations in Search of a Dome Shape

Abstract

Based on the design solutions of the Orthodox Church and while maintaining the conventions of polyhedral surfaces, the author describes and analyzes new 3D geometric solids (structures) modeled on tessellations. The pretext for undertaking the research was the need to find a polyhedron shape that could replace the curvilinear shape of a burning candle flame that had been used for centuries in Orthodox church architecture. The innovative idea of designing polyhedral domes of Orthodox churches led the designer to interesting structures that are not derived from either regular or semiregular polyhedra. These objects, according to the author, constitute a kind of bridge between flat tessellations and three-dimensional semiregular or regular polyhedra. Their analysis unexpectedly leads to the source, which is precisely flat, semiregular tessellations.

1. Introduction

The author of the monograph [1] quotes the statement: “Whatever we perceive, we can understand it only by perceiving its structure and thinking through structural analogy and comparison (Cyril Stanley Smith) [2]”. This was also the idea that guided the author of this work in analyzing the geometric structure of the domes of the Church of the Resurrection in Białystok (Poland). The analysis of the geometry of the domes leads to the description of new structures and at the same time constitutes an important addition to the content of the monograph [1].

The domes of churches usually have the shape of a burning candle flame (Figure 1b–d) [3]. In ref. [4], the following shapes of curvilinear roof domes of churches are distinguished: ovoid, helmet-shaped, onion-shaped, pear-shaped, conical, and umbrella-shaped (Figure 1) [5,6,7,8]. The helmet, onion, and pear shapes refer directly to the shape of a candle flame, with only a subtle difference between these shapes. Depending on the geographical location, or rather cultural area, the shape of a burning candle flame can be found in Russia, Belarus, Ukraine, and partly in Poland. In Greece and the Balkans (e.g., Serbia, Cyprus, Bulgaria, Albania), domes are ovoid. In Georgia, domes take the shape of a cone’s surface, and in the USA, we find churches for which different domes on the same church are of all the types listed above [3]. In Białystok (Poland), we find domes of the types listed above except for the conical shape (Figure 2 and Figure 3). It is worth adding that the number of domes is defined theologically and can take the following values: 1, 2, 3, 5, 7, 9, 12, or 13 [3].

Given the prevalence of curvilinear domes, with the predominance of domes shaped like the flame of a burning candle, the designer’s idea is to design a church with a polyhedral structure; in particular, a church with polyhedral domes that resemble the flame of a burning candle.

The main problem of this work is as follows: How to create a polyhedron in the shape of a candle flame?

The solution can be found among Platonic polyhedra; after all, such studies are found in other sciences. Polyhedral surfaces, in fact, have special applications in crystallography, chemistry, and physics [9,10,11]. The polyhedral symbol is sometimes used in coordination chemistry to indicate the approximate geometry of atoms coordinating around a central atom. This is defined by so-called polyhedral symbols [12]. Usually, a model is constructed on the basis of Platonic polyhedra and then derived by truncation based on Archimedean polyhedra. Recent studies and constructions concern the so-called extended Platonic polyhedra and their application to the molecular and theoretical characterization of chemical molecules [10] and minimizing the energy of point particles [11].

Examples of the direct use of Platonic solids or other polyhedra in architectural design have been encountered throughout the centuries. These include Egyptian pyramids (2700–2500 B.C.), Mayan pyramids (from 1000 B.C.), and contemporary structures: the Art Tower in Mito designed by Arata Isozaki [5,13], built in 1990, based on tetrahedrons; the Cubic Hauses in Rotterdam designed by Piet Blom [14] built in 1976; and the Nakagin Capsule Tower in Tokyo designed by architect Kisho Kurokawa [15], built on the basis of 140 cuboid capsules (2.3 m × 3.8 m × 2.1 m) over the years 1970–1972. The Orthodox Church of the Resurrection in Białystok, designed by Jerzy Uścinowicz and built in 1991–1994 (Figure 4), whose domes are the subject of analysis in this study, deviates from the direct applications of known shapes. It should be noted that the shapes of the domes go beyond the standard polyhedron types. We do not find such solutions or proposals in the excellent, extensive (724 pages) monograph [5]. It is also worth adding that in addition to the interesting architecture of the domes, the finial of the church tower is a spire polyhedron of the $c_8$ type, according to the classification of László Strommer [16].

It is also worth noting the use of polyhedra in describing roof geometry [17], classifying roof shapes [18], and determining roof shape in building design [19,20].

2. Problem Statement and Solution Methodology

Searching for a solution directly among the Platonic and Archimedean solids, two forms are indicated: a regular dodecahedron modified by replacing regular pentagons with quadrangles, and a modified truncated octahedron with the upper pyramid (Figure 5a,b).

In the previously mentioned papers [16,17,18,19,20], an attempt was made to describe selected classes of architectural objects in the language of specific types of polyhedra. In ref. [16], we find a classification of the spires of Western European medieval churches using specially defined pyramid shapes, the so-called spire polyhedra. In refs. [17,18], we have a description and classification of the shapes of roofs of contemporary buildings based on the so-called roof skeleton geometry, which has long been known from descriptive geometry [21]. Roof skeleton geometry was formulated later (in 1995, to be precise) and developed in computational geometry as the theory of so-called straight skeletons [22]. Algorithms creating straight skeletons and, accordingly, roof solids were implemented in CAD programs, including the AutoCAD v.2023 environment [23].

The roof skeleton (straight skeleton) is a flat object created in a plane. After adopting the angle of the roof’s slope, it defines the spatial form of the building’s roof. A similar principle of creating the spatial structure of a polyhedral dome will be adopted in this work. Here, the flat source structure will be a tessellation.

2.1. Assumptions Regarding the Construction of the Proposed Dome Shapes

To obtain a more subtle form of the flame of a burning candle, we continue the search for the shape by choosing a different geometric path. For these purposes, we assume that the generator of the desired geometric form of the shape of the flame of a burning candle is an initial ring composed of an even number of regular polygons adjacent to each other along the sides and perpendicular to a fixed horizontal plane. These polygons have an even number of sides and can be of one or two types, arranged alternately. We further assume that the orthogonal projections of the rings defined above are a regular polygon and a semiregular polygon, respectively, both with an even number of sides.

The justification for adopting such assumptions comes from the analysis of the structure of some planar regular and semiregular tessellations.

2.2. Regular and Semiregular Tessellations in the Design of Shapes

A tessellation on a plane covers the plane with Figures that are adjacent to each other but do not overlap [5,24]. Tessellations are an extremely broad class of geometric objects. We deal with some tessellations, for example, when laying floors and sidewalks. Among tilings, a small but important class consists of regular tilings, which are composed of one type of regular polygon (Figure 6), and semiregular tilings, in which the Figures covering the plane are regular but may be of different types (Figure 7 and Figure 8).

This is similar to the case of regular and semiregular polyhedra. In practice, tilings are often subjected to topological transformations, which means changing the proportions of the polygon sides to obtain a specific effect (Figure 8, Figure 9 and Figure 10). In such cases, semiregular tiling (Figure 11 and Figure 12) is only an inspiration in the design of, for example, floor tile patterns (Figure 9) or pavement tiles (Figure 10). So, when transitioning from a semiregular tiling (Figure 8) to the floor tile pattern in Figure 9, we reduce the length of the square side, while when transitioning from a regular tiling (Figure 10) to the pavement tile pattern, we increase the length of the square side and, by combining a square with an octagon, obtain a “hammer”-shaped tile.

Regular and semiregular polyhedra are therefore analogs of regular and semiregular tilings.

3. Geometric Analysis and Construction of Polyhedral Domes of the Orthodox Church

The architect of the Church of the Resurrection in Białystok, having decided on a polyhedral narrative of the building’s composition, despite the abundance of regular and semiregular polyhedra, did not select any of the Platonic or Archimedean solids [5,24,25]. Upon a very cursory inspection, the shape of a truncated octahedron appears to be that of a dome, but it is not (Figure 4). The designer of the church looked for another solution and achieved success. According to the author of this work, the resulting work is excellent. Working with the adopted concept of creating domes, he had great freedom to operate with proportions. For an architect, this is extremely important, but the geometry of Archimedean solids, and even more so Platonic solids, does not provide this possibility. The author of this work decided to describe the algorithm for creating this type of structures in reference to (flat) tessellations. It is an expansion and generalization of the problem formulated in ref. [26].

3.1. Method of Constructing a Dome Based on a Path Extracted from Tessellation

Now let us consider the tiling (Figure 7). From the existing tessellation, we can choose two sequences, 12-4-12-… and 6-4-6-…, to be expanded into the sequences 12-4-12-4-…, 6-4-6-4-…, respectively. Note that the 12-4-12 tessellation does not exist (${90}^{\mathrm{o}}+{150}^{\mathrm{o}}+{150}^{\mathrm{o}}$ > ${360}^{\mathrm{o}}$), and creating a spatial ring containing dodecagons would make it impossible to complete the dome with congruent polygons to crown the dome (the number of sides of the dodecagon is too large).

There is also no tessellation 4-6-6 (i.e., 6-4-6), because ${90}^{\mathrm{o}}+{120}^{\mathrm{o}}+{120}^{\mathrm{o}}$ < ${360}^{\mathrm{o}}$ (Figure 12), but creating a ring containing the hexagons would make it impossible to complete the dome with congruent polygons to crown the dome. Starting from the extended path the extended 4-6-4-6-4-6-4-6-4-6-4-6-4-6 path, we “wrap” (using, e.g., the command Polar Array in AutoCAD [5]) it so as to create an object (ring) with the projection of a semiregular polygon (a concept introduced by the author of this paper), according to the assumption made in Section 2.1. Then, we notice that it is possible to supplement the structure with a flat polygon because, due to symmetry, two sides belonging to the hexagons and one of the squares lying between them lie in one plane. A semiregular polygon is a 2n-sided polygon consisting of two types of side lengths alternately, with all angles equal to the angle of the 2n-sided regular polygon.

Such a polygon exists for any a and b. Indeed, let us consider two segments of arbitrary lengths a and b and a common endpoint, forming an angle $\phi ={\displaystyle \frac{\left(2n-2\right){180}^{\mathrm{o}}}{2n}}$. Of course, then $\phi <{180}^{\mathrm{o}}$ and these segments form a triangle, which uniquely describes a circle (Figure 13). Let us denote the corresponding angles by α, β, γ, δ, and the radius of the circle by r (Figure 10). By assumption, we have $\alpha +\beta ={\displaystyle \frac{\left(2n-2\right){180}^{\mathrm{o}}}{2n}}$ and from the properties of isosceles triangles: ${\gamma =180}^{\mathrm{o}}-2\alpha$, ${\delta =180}^o-2\beta$. We calculate the measure of the angle ${\gamma +{\delta =180}^{\mathrm{o}}-2\alpha +180}^{\mathrm{o}}-2\beta$ and obtain $\gamma +\delta ={\displaystyle \frac{{360}^{\mathrm{o}}}{n}}$. This means that the segment AB is a side of a n-sided regular polygon (Figure 13). Therefore, a 2n-sided semiregular polygon exists. Such a polygon has exactly n axes of symmetry, and thus n own rotations. The polygon has a center of symmetry when n is an even number (Figure 13). We obtain an object that has the shape of a ring and has n + 1 plane symmetries, including n vertical ones and one horizontal one, which induce a vertical axis of rotation. The base planes of plane symmetries are co-clustered and form the axis of a ring, which is the axis of the solid’s rotation. The plane symmetry of the constructed wreath (see Figure 14) allows us to state that each of the three upper consecutive sides of the hexagon, the square, and the hexagon are coplanar. We can therefore construct a flat pentagon as in Figure 15, and ultimately, a 3D model (Figure 16). As we will see in the next paragraph, in order to obtain a dome with the required proportions, it is necessary to choose the appropriate lengths of the polygon sides: squares can be replaced by congruent rectangles, regular hexagons by congruent axisymmetric hexagons. As a result, we achieve a dome structure (Figure 16).

3.2. Construction of the Dome Based on Path 4-6-4-6-4-6-4-6 (Type 1)

In order to create a model of the main dome of the church (Figure 4), we choose the path (4-6-4-6-4-6) and apply the method from Section 3.1. In the model we create, we assume squares and regular hexagons. In order to obtain the shapes of the domes specified in the church project, instead of a square, a rectangle with predetermined proportions is assumed, and instead of a regular hexagon, a hexagon with exactly one axis of symmetry. As can be seen in the photograph (Figure 4), the author of the project repeated the dome shapes in different proportions. The desired dimensions can be obtained using parametric design [5,27,28,29]. In refs. [27,28,29], we find examples of the use of modern tools, such as Grasshopper 3D and Karamba 3D operating in the Rhinoceros 3D v.5.0 environment, to support the shaping and optimization of structures. It should be added that the adoption of proportions in the rectangle and hexagon already induces the dimensions of the pentagons crowning the dome (Figure 17).

There is another interesting fact. If we take the sequence of hexagons 6-6-6-6-6-6 as the starting path, then by repeating the above algorithm we will obtain the structure of the church tower as a $c_6$ pointed polyhedron (see the tower in Figure 4).

3.3. Dome Construction Based on Tessellation (Type 2)

It turns out that the second type of dome, in contrast to the previous one, can be obtained on the basis of the 4-8-8 tessellation, which happens to exist here (Figure 8, Figure 18 and Figure 19). However, this does not prevent the creation of a spatial structure, the supplement does not have to be squares. This type of structure is, in a sense, topologically similar to a strip of flat tessellation (one layer of octagons and adjacent layers of squares). Here, we choose eight regular octagons as the path and create an octagonal ring that is based on a regular octagon. The adjacent sides of adjacent octagons of the ring of regular octagons generate a rhombus, one of whose diagonals is parallel to the horizontal plane. This rhombus is the equivalent of a square in the 4-8-8 tiling. Each octagon has two rhombuses adjacent to it. For any octagon, we then choose a polyline consisting of three segments: a rhombus side, an octagon side, and a rhombus side.

Due to the plane symmetry of the Figure composed of three segments, these segments lie on one plane. These are therefore the sides of a flat polygon. At the same time, it results from the construction that the sides of the rhombus are parallel to the sides of the regular octagon, which is the base of the constructed dome. Therefore, the projection angles of each rhombus have measures of ${45}^{\mathrm{o}}$ and ${135}^{\mathrm{o}}$. So, in the projection, we have a configuration of an isosceles trapezoid with angles having measures. From this, we conclude that the diagonal of the hexagon is twice as long as the side (Figure 18b). Therefore, it is a regular hexagon. This creates an interesting structure with hexagons (Figure 18c). However, in order to obtain a pointed dome, pentagons must be generated. To find the missing vertex of the pentagon, it is enough to construct a segment connecting the point of symmetry of the base with a point on the axis of the solid (Figure 18a). This segment is an extension of the segment connecting the midpoints of opposite sides of a regular hexagon. However, it was enough to use the diagonal of the hexagon. Notably, it is not important whether the hexagon is regular. We note the appearance of a regular hexagon only occasionally. As you can see, this fact has no effect on the algorithm for constructing the dome.

Although the analyses and constructions carried out were carried out on three examples, we indicate a universal method of creating similar geometric objects. In this way, we describe the important geometric forms occurring in the sacral building in question. They are analogous to domes known so far as surfaces of revolution. Here, the number of rotations is finite but can be any even number.

Finally, it is worth adding that a similar dome can be constructed based on a regular pentagon as a modified construction of a regular dodecahedron (Figure 20a,b). Taking a regular pentagon, we set up pentagons congruent to it, which are inclined at an angle $\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{s}(-{\displaystyle \frac{\sqrt{5}}{5}})\approx 116.6^{\mathrm{o}}$ [30,31] from the base (Figure 20b). The resulting ring of inclined pentagons is crowned with deltoid-shaped quadrangles (Figure 20c,d). Thus, we obtain a type 3 dome. In the context of the constructions presented earlier, a disadvantage of this polyhedron may be the odd number of sides of the base polygon.

Looking carefully at Figure 5a,b, we will see a dome (type 3) with a square base and a ring made of four regular hexagons, having the features of the domes from Figure 20, and, in a sense, the domes in Figure 17 and Figure 19 as well, in which the polygons generating the ring are perpendicular to the horizontal plane. Figure 5b and Figure 20d show domes obtained based on the Platonic and Archimedean polyhedra, respectively.

4. Results and Discussion

In both types of domes, 1 and 2, we have a main ring of polygons and other rings. The first type (type 1) is closed from the top by only one ring of crowning pentagons (Figure 17), while the second (type 2) is closed by two rings: quadrangles and crowning pentagons (Figure 19). Note that in both domes (type 1 and 2), we can introduce additional intermediate rings of congruent heptagons with one axis of symmetry, which will result in the closing polygons being congruent quadrangles. Heptagons and quadrangles can be obtained by cutting off all the corners of the part of the dome composed of the crowning pentagons; heptagons are formed by cutting the corners off pentagons. This will cause the dome to be flatter at the top.

From the presented construction, it follows that domes of types 1 and 2 can be built on the plan of a regular n-gon (type 2) and a semiregular 2n-gon (type 1), respectively. Thus, we have two infinite families of domes: type 1 and type 2.

The remaining two types of domes are type 3, a derivative of a truncated octahedron with a finial in the form of a ring of pentagons (Figure 5), and type 4, a derivative of a regular dodecahedron with a finial in the form of a ring of quadrangles (Figure 20). Slightly different than the previous two types, in a type 3 dome, we can introduce an additional ring of congruent heptagons with one axis of symmetry, while in a type 4 dome, we can introduce an additional ring of hexagons with one axis of symmetry; both are then topped with rings of congruent quadrangles. Hexagons and quadrangles can be obtained by cutting off all the corners of the part of the dome composed of the crowning pentagons; hexagons are formed by cutting off the corners of quadrangles. As with types 1 and 2, domes 3 and 4 will be flattened at the apex.

Note that there is a similarity in the number of rings between types 1 and 4 and between types 2 and 3.

The final dome shapes are obtained by changing the dimensions and proportions, e.g., using the parametric modeling method, examples of which can be found in refs. [27,28,29].

Let us also note that the spire on the church (Figure 4) can be treated as a dome in the sense of the constructions given in this article. This fact connects the spire polyhedra discussed in ref. [16] with the dome constructions proposed in this study.

A question remains: are there other types of domes built using this method? And what modifications to existing domes are possible?

5. Conclusions

• This study describes a general algorithm for creating axially symmetric polyhedral surfaces with an even number of rotations around the axis, previously unknown in the literature, which arose from polyhedra in the architecture of the Orthodox church dome.
• Transferring the structure of a flat tiling to a three-dimensional solid is an unexpected and interesting result in itself and can simultaneously be an inspiration for the creation of new architectural forms.
• The described algorithm allows for the creation of n-gonal base-truncated pyramids of the $c_n$ type (n-gonal base-truncated pyramids) [16].
• The concept of a semiregular polygon, previously unknown in the literature, was formulated in a natural way.
• The structures presented in this work show that inspiration for creating spatial structures can be sought in a seemingly distant area, e.g., among arrangements of flat Figures.