Complex Building Forms Roofed with Transformed Shell Units and Defined by Saddle Surfaces

A novel method and description of creating diversified complex original building forms roofed with a number of transformed folded shell units developed on the basis of a novel reference polyhedral network and arranged according to a reference surface with the negative Gaussian curvature is presented. For that purpose, specific reference polyhedral networks is are defined as a complex material deliberately composed of many regular tetrahedrons that are arranged regularly to obtain original attractive complex general building forms. The proposed method is a significant extension of the previous method for shaping roof structures with the positive Gaussian curvature and fills existing gaps in current scientific knowledge. The extended method enables the designer to significantly increase the variety of the created complex shell roof forms and plane-walled folded elevation forms of buildings and to define the shapes of their rod structural systems. It allows one to overcome the existing significant geometric and material limitations related to shape transformations of nominally flat rectangular folded steel sheets into different shell forms. The developed extension is based on formation of a set of properly connected tetrahedra as a material determining different (a) inclination of elevation walls to the vertical, and (b) distribution of many individual warped roof shells in accordance with the properties of a regular surface with negative Gaussian curvature. A number of the adopted specific sets of division coefficients (parameters) is used for determining the entire network and its complete tetrahedra. The presented description makes it possible to adopt appropriate assumptions and data and then employ the innovative method to obtain the expected characteristics of the unconventional building form shaped. The presented three different special forms created with the help of the novel method and the appropriately selected diversified values of the division coefficients of pairs of the vertices of a polyhedral reference network, a polygonal eaves network and points of a reference surface confirm the innovative scientific nature of the obtained results. The method has to be computationally aided due to the complexity of mathematical operations and the need to visualize the designed forms.


Introduction
Nominally flat folded rectangular steel sheets are characterized by relatively low transverse bending and longitudinal torsional stiffness [1]. This property enables the designer to exploit their diversified elastic transformations to shape diversified ruled shell forms [2] (Figure 1). However, there are significant geometric and material limitations in the formation of the shells due to a significant influence of these and other stiffnesses on the final shape of the transformed shells [3]. The limitations very often result in significant values of stresses and, for the cases of high transformation degrees, significant values of strains. The intentional spatial shape deformations are called initial transformations [2].
Further limitations are the straightness of all transformed shell folds and the rectangular shape of the sheets before transformation. On the one hand, these limitations simplify Further limitations are the straightness of all transformed shell f gular shape of the sheets before transformation. On the one hand, th plify the process of designing and assembling the transformed roof hand, they induce the edge lines of the developed building models form of various ruled surface sectors [4] each of which is limited by a rangle with vertex angles close to right [2]. Due to the limited length sheets used for roofing, single shell sheeting, being the result of joi sheets with their adjacent longitudinal edges, can cover buildings with 17 m. Therefore, for medium and long span roofs, edge roof structure single folded transformed shells must be accomplished. The simple complex building structures is to determine special sums of many formed into parabolic-hyperbolic folded sectors, the edge lines of whi quadrangles [4,5]. Moreover, all facade wall pieces of the structures gons ( Figure 2). In order to obtain a large variety of the designed bu çade walls are inclined to the vertical. Further limitations are the straightness of all transformed shell folds and th gular shape of the sheets before transformation. On the one hand, these limitati plify the process of designing and assembling the transformed roof shells. On t hand, they induce the edge lines of the developed building models to be shape form of various ruled surface sectors [4] each of which is limited by a closed spati rangle with vertex angles close to right [2]. Due to the limited length of complet sheets used for roofing, single shell sheeting, being the result of joining the in sheets with their adjacent longitudinal edges, can cover buildings with a span of u 17 m. Therefore, for medium and long span roofs, edge roof structures composed single folded transformed shells must be accomplished. The simplest way to sh complex building structures is to determine special sums of many single shel formed into parabolic-hyperbolic folded sectors, the edge lines of which are close quadrangles [4,5]. Moreover, all facade wall pieces of the structures have to be f gons ( Figure 2). In order to obtain a large variety of the designed building forms çade walls are inclined to the vertical. Bearing in mind the above-mentioned assumptions and limitations, a novel po network called a reference network was developed [6]. The network is used as a g material for modeling various unconventional forms of buildings characterized b walled folded facades and edge roof structures composed of many transforme folded shells. The structure of the layers used in the corrugated sheeting and their ature affect the mechanical and geometrical properties of these coating segments [7 Each configuration of the network is defined by an appropriate arrangement tetrads of vertices defining typical tetrahedral meshes connected by common tr Bearing in mind the above-mentioned assumptions and limitations, a novel polyhedral network called a reference network was developed [6]. The network is used as a geometric material for modeling various unconventional forms of buildings characterized by planewalled folded facades and edge roof structures composed of many transformed single folded shells. The structure of the layers used in the corrugated sheeting and their temperature affect the mechanical and geometrical properties of these coating segments [7,8].
Each configuration of the network is defined by an appropriate arrangement of many tetrads of vertices defining typical tetrahedral meshes connected by common triangular walls and adequately filling the three-dimensional space. These vertices, planes and side edges of the tetrahedra define the planes and edges of all facade walls of each modeled building. In the above planes, straight segments forming closed spatial quadrangles defining single shell roof are determined. The quadrangles constitute a polygonal network of the roof eaves structure and delimit all individual single roof shells. By varying the geometric properties of the polyhedral reference network and the polygonal network, in particular their subsequent meshes, some variety of building forms can be obtained, including multi-wall folded facades and multi-shell ribbed roofs [9].

Critical Analysis of the Present Knowledge
Many diversified unconventional building forms and structural system intended for these forms are presented by Abdel and Mungan [10]. The comprehensive classification of different types of a large number of unconventional structural systems is published by Saitoh [11]. Most of these systems can be employed for structural systems supporting complex building forms characterized by folded elevation walls and multi-segment shell structures [9].
Great theoretical possibilities in the field of shaping various ruled forms made up of the transformed folded sheets with an open profiles have resulted in extensive research development in the field of single shell sheeting transformations and creation of multisegment shell structures [2]. On the basis of the tests and analysis carried out, Bryan and Davis found that some significant material and technological limitations drastically reduce the diversification of the shell forms made up of transformed sheeting to a basic type of shallow hyperbolic paraboloids called hypars [12].
However, the developed novel Reichhart's method enables one to increase the diversity of shaping the unconventional roof shells and their structures composed of the nominally flat single-layer sheeting transformed into the position of the rigid skew directrices generating folded shell shapes [3].
The results of the tests performed by Abramczyk [13] indicate an important role played by a contraction of each transformed shell sheeting in shaping unconventional forms of buildings roofed with thin-walled folded sheeting transformed elastically into different ruled surfaces [4]. Abramczyk has added the condition related to the central location of the contraction in all effectively transformed folded shells to the Reichhart's algorithm.
The contraction of a roof shell can be modeled with a line of striction of a ruled surface. Thus, the Reichhart's method was extended with the boundary condition related to the line of striction. This action should result in a rational shaping of the shell roof forms so that they could be characterized by the relatively high visual attractiveness and the minimum effort [14]. If athe central location of the contraction, halfway along the length of each shell fold, is achieved by means of the arbitrary roof directrices, then the static-strength work of the transformed sheeting is stabilized and rationalized [4]. In this way, a freedom of designing of free-form roof and elevation structures should be achieved [2].
Biswas and Iffland [15] elaborated two trivial systems of many congruent rectilinear shell units distributed over a sphere with the help of bundles of planes, where the complete transformed shells are made of revolved hyperboloids or right hyperbolic paraboloids restricted by spatial straight quadrangles. The proposed method for shaping ribbed structures composed of many congruent shells can be extended to create complex systems of many complete ruled shells separated by sets of planes containing their edge lines.
Prokopska and Abramczyk have carried out some simple systems of reference tetrahedrons to model complete free forms with oblique and folded plane elevations, and roofed with many transformed complete shell sectors [16,17]. The presented building structures are often composed of quarters of hyperbolic paraboloids arranged symmetrically in dif-  The proposed diversified free form building structures roofed with  complex corrugated shells are characterized by medium spans. For the engineering developments, Abramczyk has proposed that each single shell segment is to be modeled with a sector of a warped surface [2] limited by a closed spatial line composed of four straight segments contained in four planes of a polyhedral reference network [6]. The planes of the reference network are a specific system dividing the designed roof structure into many complete shell segments arranged regularly in the three-dimensional space. In this way, all directrices of the shell sectors can be easily defined in these planes [9].
Pottman has proposed a few comprehensive methods for shaping the systems of planes separating subsequent plane and smooth shell sectors arranged on different regular surfaces [18]. If we want to shape transformed folded shell sheeting by means of the methods, their significant modifications taking into account the geometric and material constraints of the folded sheets must be performed. A few methods carried out by Attard [19], Vlasow [20] or Vasiri [21] can also be implemented to design thin-walled sheeting subjected to large displacements. Samyn has described a method for creating the transformed folded shells made up of aluminum or PVC [22].
The procedure for shaping complex building structures covered with transformed shell roof structures, the general form of which is characterized by a positive Gaussian curvature (Figure 3), has been described by Abramczyk [6] in a fairly accurate manner. This procedure has also been implemented in computer procedures [23]. However, there is no analogous procedure and implementation where the overall form of the shell roof structure complies with the negative Gaussian curvature. This issue is analyzed in detail in this article. designed roof structure into many complete shell segments arranged regularly in the three-dimensional space. In this way, all directrices of the shell sectors can be easily defined in these planes [9]. Pottman has proposed a few comprehensive methods for shaping the systems of planes separating subsequent plane and smooth shell sectors arranged on different regular surfaces [18]. If we want to shape transformed folded shell sheeting by means of the methods, their significant modifications taking into account the geometric and material constraints of the folded sheets must be performed. A few methods carried out by Attard [19], Vlasow [20] or Vasiri [21] can also be implemented to design thin-walled sheeting subjected to large displacements. Samyn has described a method for creating the transformed folded shells made up of aluminum or PVC [22].
The procedure for shaping complex building structures covered with transformed shell roof structures, the general form of which is characterized by a positive Gaussian curvature ( Figure 3), has been described by Abramczyk [6] in a fairly accurate manner. This procedure has also been implemented in computer procedures [23]. However, there is no analogous procedure and implementation where the overall form of the shell roof structure complies with the negative Gaussian curvature. This issue is analyzed in detail in this article. The analogous universal systems of planes called polyhedral reference networks are utilized in this article. On the basis of these systems, some derivative systems, including polygonal or shell sector networks can be defined to achieve different free-form buildings characterized by complex folded elevation walls and multi-sector shell roof structures ( Figure 4) [24]. The analogous universal systems of planes called polyhedral reference networks are utilized in this article. On the basis of these systems, some derivative systems, including polygonal or shell sector networks can be defined to achieve different free-form buildings characterized by complex folded elevation walls and multi-sector shell roof structures ( Figure 4) [24].  To increase the attractiveness of the building edge shell structures and the whole urban system, Prokopska has proposed green plant gardens located on the shell roof segments and specific communication routes between the segments [25]. The space around the designed building form, its physical form, urban greenways and cultural patterns of a whole spatial system have to be investigated. Sharma has described [26] the relation appearing between the formation of the urban space and the social experience of the human self. Hasgül [27] presents very important and interesting mathematics-based graph studies of patterns and shapes, thermal based photography and morphology related to the design syntax. Eekhout [28] provides the results of his research in the field of forms taking account of the relationships occurring between the function, structure, internal and external texture, static-strength work and comfort conditions. The systematic morphology leads to the rationalization of each design process.
A very interesting approach to exploit the mechanical properties of new materials making it possible to shape original forms of elevation and roof structures is used by Marin et al. [29], where the classical theory of elasticity developed by Green and Lindsay is extended in terms of the thermo-elasticity theory for dipolar bodies. The proposed novel method is based on a reciprocal theorem and not restrictive boundary conditions [30].
A method for designing complex architectonic forms and engineering rational systems has been developed by Ręebielak [31]. A number of attractive and effective structures adapted to human needs and the built environment has been provided by Tarczewski at al. [32].
The methods for shaping single and complex shell roofs presented in the above literature should be directed towards making each roof from many nominally flat rectangular thin-walled corrugated steel sheets. In order to use these sheets for shell roof covering, shape transformations are required. It is advisable to shape the edge line of each shell segment in the form of a spatial quadrangle with corner angles close to straight and take into account the shape changes and limitations resulting from the shape transformations.
The given references refer to single shells or simple ribbed structures composed of several single shells. The innovative method developed by Abramczyk allows one for a significant increase in the variety of general building forms. However, the method is limited only to structures, where the arrangement of all individual shell segments is consistent with the properties of surfaces characterized by the positive Gaussian curvature (saddle surfaces). In some articles published by the researcher, the possibility of using various structures composed of many single shells distributed on regular surfaces with the negative Gaussian curvature (saddle surfaces) is identified. To increase the attractiveness of the building edge shell structures and the whole urban system, Prokopska has proposed green plant gardens located on the shell roof segments and specific communication routes between the segments [25]. The space around the designed building form, its physical form, urban greenways and cultural patterns of a whole spatial system have to be investigated. Sharma has described [26] the relation appearing between the formation of the urban space and the social experience of the human self. Hasgül [27] presents very important and interesting mathematics-based graph studies of patterns and shapes, thermal based photography and morphology related to the design syntax. Eekhout [28] provides the results of his research in the field of forms taking account of the relationships occurring between the function, structure, internal and external texture, static-strength work and comfort conditions. The systematic morphology leads to the rationalization of each design process.
A very interesting approach to exploit the mechanical properties of new materials making it possible to shape original forms of elevation and roof structures is used by Marin et al. [29], where the classical theory of elasticity developed by Green and Lindsay is extended in terms of the thermo-elasticity theory for dipolar bodies. The proposed novel method is based on a reciprocal theorem and not restrictive boundary conditions [30].
A method for designing complex architectonic forms and engineering rational systems has been developed by Ręebielak [31]. A number of attractive and effective structures adapted to human needs and the built environment has been provided by Tarczewski at al. [32].
The methods for shaping single and complex shell roofs presented in the above literature should be directed towards making each roof from many nominally flat rectangular thin-walled corrugated steel sheets. In order to use these sheets for shell roof covering, shape transformations are required. It is advisable to shape the edge line of each shell segment in the form of a spatial quadrangle with corner angles close to straight and take into account the shape changes and limitations resulting from the shape transformations.
The given references refer to single shells or simple ribbed structures composed of several single shells. The innovative method developed by Abramczyk allows one for a significant increase in the variety of general building forms. However, the method is limited only to structures, where the arrangement of all individual shell segments is consistent with the properties of surfaces characterized by the positive Gaussian curvature (saddle surfaces). In some articles published by the researcher, the possibility of using various structures composed of many single shells distributed on regular surfaces with the negative Gaussian curvature (saddle surfaces) is identified.
However, a comprehensive novel method is presented in the present article and fills the gap in the current knowledge. This area is also important because torsional transformations of a single folded sheet lead to its various shell forms with the negative Gaussian curvature (saddle surfaces). After filling this gap, the authors intend to propose some innovative structural systems supporting the transformed folded shells and extend laboratory tests in the field of thin-walled folded sheeting transformed elastically into various forms of saddle surfaces.

The Aim
The aim is to present an innovative procedure leading to the creation of some building models characterized by unconventional folded forms of their façades and roofs, where the façade walls are inclined both inside and outside the designed building, and individual roof whose shell sectors are arranged in a three-dimensional space according to the properties of a surface with the negative Gaussian curvature. In order to achieve the abovementioned goal, a novel algorithm for creating a specific type of the polyhedral reference networks has to be used. The algorithm is implemented into a new procedure for shaping complex building forms in such a way that the created models are located between the vertices of the reference network. However, for the case of the previous building structures with the positive Gaussian curvature, the vertices are positioned on the same side of a reference surface.
The presented procedure allows for arranging the facade and roof elements in relation to the vertices of a reference network using appropriate coefficients expressing some proportions between the distances of the appropriate pairs of the adjacent vertices, and the distances of the characteristic points of the created models from the respective vertices of the reference network. The differentiation of the values of the coefficients adopted as independent variables leads to the differentiation of the obtained building forms. The degree of differentiation of the created models depends on the number of the assumed independent variables. In particular, all the coefficients used may depend on one parameter, which leads to adopting one independent variable and obtaining a single set of building forms with little differentiation.
It is decided that a detailed presentation of the proposed procedure is going to be made using some specific examples. A few different forms obtainable by the procedure are given to observe the possibilities of the procedure. A detailed description of the whole set of the examined diversified complex building forms goes beyond the scope of this article. There are only presented the main relationships governing the position of each mesh B vij (each shell unit Ω ij ) in the eaves network B v (the shell structure Ω) and three sets of the values of the partition coefficients assigned to the vertices of the same basic reference polyhedral network to analyze three specific types of the general building forms that are different to each other.

The Method's Concept
At first, a network Γ composed of a number appropriately arranged tetrahedra Γ ij must be defined. On the basis Γ, a model Σ of a complex folded building form can be created s as follows (Figure 3). The walls of Σ should be included in the planes of Γ. The elevation edges of Σ should be included in the side edges of Γ. The eaves lines of each single shell of the roof structure must be contained in the planes of Γ. The vertices of each eaves line must belong to the respective side edges of Γ. In the presented procedure, these vertices have to be located between the vertices of Γ.
The horizontal base plane P b of the whole structure Σ must intersect all side edges of Γ. The intersecting points should also belong to the side edges and lie between the vertices of Γ. The division coefficients of the side edges of Γ by the vertices of the base and Σ must ensure appropriate positions of these vertices. Construction of Σ begins with the determination of the first central tetrahedral mesh Γ 11 of Γ. Γ 11 is created on the basis of two arbitrary skew straight lines u 11 and v 11 perpendicular to each other, called the axes of Γ 11 (Figure 5a). An arbitrary distance d uv11 between u 11 and v 11 has to be adopted. It is measured along the straight line n 11 perpendicular to u 11 and v 11 between the points O u11 and O v11 of intersecting n 11 with u 11 and v 11 .
In order to determine the positions of four vertices W AB11 , W CD11 , W AD11 and W BC11 of Γ 11 , it is necessary to adopt the distance duv 11 between the above skew axes, the distance d u11 between W BC11 and W AD11 and the distance d v11 between W AB11 and W CD11 . Based on the above assumptions, it is possible to calculate the coordinates of the above vertices in the orthogonal coordinate system [x,y,z] and define four sides of Γ 11 (Figure 5a).
A few subsequent meshes are located in orthogonal and diagonal directions of Γ 11 ( Figure 6) as follows. For the case of the mesh Γ 12 , three vertices W AB12 , W CD12 and W AD12 are adopted to be identical to the vertices W AB11 , W CD11 and W BC11 of Γ 11 , respectively. The axis u 12 of Γ 12 is assumed to be identical to u 11 . Construction of Σ begins with the determination of the first central tetrahedral mesh Γ11 of Γ. Γ11 is created on the basis of two arbitrary skew straight lines u11 and v11 perpendicular to each other, called the axes of Γ11 (Figure 5a). An arbitrary distance duv11 between u11 and v11 has to be adopted. It is measured along the straight line n11 perpendicular to u11 and v11 between the points Ou11 and Ov11 of intersecting n11 with u11 and v11. In order to determine the positions of four vertices WAB11, WCD11, WAD11 and WBC11 of Γ11, it is necessary to adopt the distance duv11 between the above skew axes, the distance du11 between WBC11 and WAD11 and the distance dv11 between WAB11 and WCD11. Based on the above assumptions, it is possible to calculate the coordinates of the above vertices in the orthogonal coordinate system [x,y,z] and define four sides of Γ11 (Figure 5a).
A few subsequent meshes are located in orthogonal and diagonal directions of Γ11 ( Figure 6) as follows. For the case of the mesh Γ12, three vertices WAB12, WCD12 and WAD12 are adopted to be identical to the vertices WAB11, WCD11 and WBC11 of Γ11, respectively. The axis u12 of Γ12 is assumed to be identical to u11. The location of the vertex WBC12 is determined so that it is contained in the plane (y, z), distant from WBC11 by the adopted value of dv12 and distant from the axis u11 by the height dBC12 of the triangle WBC12WCD12WAB12 measured from WBC12, where WBC12WCD12WAB12 should be congruent to WBC11WCD11WAB11. The above activities lead to the creation of the form presented in Figure 7a.  The location of the vertex W BC12 is determined so that it is contained in the plane (y, z), distant from W BC11 by the adopted value of dv 12 and distant from the axis u 11 by the height d BC12 of the triangle W BC12 W CD12 W AB12 measured from W BC12 , where W BC12 W CD12 W AB12 should be congruent to W BC11 W CD11 W AB11 . The above activities lead to the creation of the form presented in Figure 7a. The location of the vertex WBC12 is determined so that it is contained in the plane (y, z), distant from WBC11 by the adopted value of dv12 and distant from the axis u11 by the height dBC12 of the triangle WBC12WCD12WAB12 measured from WBC12, where WBC12WCD12WAB12 should be congruent to WBC11WCD11WAB11. The above activities lead to the creation of the form presented in Figure 7a. It is more convenient to control the shape of Σ when the proportions ddBC12 = dv12/dv11 and ddBC12 = dBC12/dBC11 are used instead of the above-mentioned distances. DBC12 is the It is more convenient to control the shape of Σ when the proportions dd BC12 = d v12 /d v11 and dd BC12 = d BC12 /d BC11 are used instead of the above-mentioned distances. D BC12 is the distance of W BC12 from u 12 . Similarly, d BC11 is the distance of W BC11 from u 11 . However, the distances can be calculated with the help of these coefficients. Adoption of the proportions between all meshes makes it possible to parametrize and control the shape of Σ, and, consequently, create diversified forms Γ and Σ. For the mesh Γ 12 presented in Figure 7a, dd v12 = 1 and dd BC12 = 1. The parametrization should lead to a division of the forms Σ into different groups having similar geometric properties to predict the expected properties of the designed forms Σ. The subsequent tetrahedra Γ 1j (j = 1 to N, where N-the arbitrary natural number) belonging to the first orthogonal strip of Γ are constructed similar to Γ 12 .
All tetrahedra Γ i1 of the second orthogonal strip are formed in the same way as Γ 1j belonging to the first strip. The vertices W AD21 , W BC21 and W AB21 of the first Γ 21 tetrahedron, Figure 6, are assumed to be identical to the vertices of W AD11 , W BC11 and W CD11 of Γ 11 , respectively, when creating the network Γ.
The fourth W CD21 vertex of Γ 21 is constructed in the (x, z) plane in two arbitrary distances d u21 from W CD11 and d CD21 from v 11 . The method employed for parameterizing Γ consists in defining a certain number of coefficients expressing the proportions between the distances of the vertices of the subsequently created Γ ij 's axes. In the case of the Γ 21 tetrahedron, these proportions can be given as follows: dd u21 = d u21 /d u11 and dd CD21 = d CD21 /d CD11 . The above activities lead to the construction of the form presented in Figure 7b for which dd u21 = dd CD21 = 1.
In the presented algorithm, there is no freedom in determining the vertices of Γ 22 and other Γ ij tetrahedra arranged in diagonal strips (for i = 1 or j = 1). For the Γ 22 tetrahedron, it is assumed that W AD22 = W BC11 , W BC22 = W AD12 , W CD22 = W CD21 , W AB22 = W CD11 . The above activities lead to the creation of the form presented in Figure 8.
Generally, for diagonal tetrahedra, W ADij = W BCi−1j−1 , W BCij = W AD i−1j , W CDij = W CDij−1 , W ABij = W CDi−1j−1 . A slight modification of the described procedure allows one to set W ADij , W BCij , W CDij and W ABij at any points of the respective network side edges, not only at the vertices of the previously created tetrahedra. This reduction of the limitations in relation to the Γ network leads to fundamental changes in the proportions between the overall dimensions and the size of the elements of the building model shaped. The description of the modified procedure for shaping the Γ network goes beyond the scope of this work.
Each network Γ 1 (Figure 8) formed according to the proposed algorithm is composed of four tetrahedra Γ ij (i, j = 1, 2) and can be extended to a symmetrical forms Γ (Figure 9a,b) for which (x, z) and (y, z) are the planes of symmetry. Thus, Γ consists of four symmetrical parts Γ i (i = 1 to 4). Based on the symmetrical reference network Γ, a polygonal B v network is created to define a multi-shell roof structure. B v is created so that some respective relationships between the location of the vertices of the roof structure and the vertices of Γ are adopted. The relationships are defined by means of the division coefficients of the pairs {W ABij , W ADij }, {W ABij , W BCij }, {W ADij , W CDij }, {W ADij , W BCij } of each Γ ij tetrahedron by the vertices of the eaves polygonal network B v , reference surface and the base of the form Σ.
The first group {dd SAij , dd SBij , dd SCij and dd SDij } of the division coefficients is used to determine the points S Aij , S Bij , S Cij and S Dij on the side edges of Γ defining the reference surface ω r to search for the network B v . The second group {dd Aij , dd Bij , dd Cij and dd Dij } of the division coefficients is used to determine the points A ij , B ij , C ij and D ij constituting the vertices of the polygonal network B v . For the networks under consideration: (a) all acceptable values of the above-mentioned two types of the coefficients are in the range (0, 1), (b) the reference surfaces defined by the points S Aij , S Bij , S Cij and S Dij have negative Gaussian curvature. These limitations results from the geometric properties of the network Γ made. They proves the innovative nature of the performed analysis and the proposed procedure. In the case of the topics discussed so far in other articles, the values of these coefficients are within the range (1, +∞), and the reference surface is characterized by the positive Gaussian curvature. The case in which the values of the above-mentioned coefficients are within the range (−∞, 0) is similar to that with the range (1, +∞) but requires some modifications.
As it is convenient to define the positions of the vertices A ij , B ij , C ij and D ij with respect to the reference surface ω r , it is rational to use coefficients taking account of the difference in the levels of the vertices belonging to ω r and B v . For this purpose, the quotient of: (a) the respective values of the division coefficients of pairs {W ABij , W ADij }, {W ABij , W BCij }, {W ADij , W CDij } and {W ADij , W BCij } by: (a) points S Aij , S Bij , S Cij and S Dij , and (b) points A ij , B ij , C ij and D ij can be calculated. However, the exact description of this problem is presented by Abramczyk [24] and goes beyond the scope of the present article.
Materials 2022, 15, x FOR PEER REVIEW 9 distance of WBC12 from u12. Similarly, dBC11 is the distance of WBC11 from u11. However distances can be calculated with the help of these coefficients. Adoption of the proport between all meshes makes it possible to parametrize and control the shape of Σ, and, sequently, create diversified forms Γ and Σ. For the mesh Γ12 presented in Figure 7a, = 1 and ddBC12 = 1. The parametrization should lead to a division of the forms Σ into ferent groups having similar geometric properties to predict the expected properti the designed forms Σ. The subsequent tetrahedra Γ1j (j = 1 to N, where N-the arbit natural number) belonging to the first orthogonal strip of Γ are constructed similar to All tetrahedra Γi1 of the second orthogonal strip are formed in the same way a belonging to the first strip. The vertices WAD21, WBC21 and WAB21 of the first Γ21 tetrahed Figure 6, are assumed to be identical to the vertices of WAD11, WBC11 and WCD11 of Γ11, res tively, when creating the network Γ.
The fourth WCD21 vertex of Γ21 is constructed in the (x, z) plane in two arbitrary tances du21 from WCD11 and dCD21 from v11. The method employed for parameterizi consists in defining a certain number of coefficients expressing the proportions betw the distances of the vertices of the subsequently created Γij's axes. In the case of th tetrahedron, these proportions can be given as follows: ddu21 = du21/du11 and ddC dCD21/dCD11. The above activities lead to the construction of the form presented in Figu for which ddu21 = ddCD21 = 1.
In the presented algorithm, there is no freedom in determining the vertices of Γ22 other Γij tetrahedra arranged in diagonal strips (for i ≠ 1 or j ≠ 1). For the Γ22 tetrahedro is assumed that WAD22 = WBC11, WBC22 = WAD12, WCD22 = WCD21, WAB22 = WCD11. The above a ities lead to the creation of the form presented in Figure 8.   , and (b) points Aij, Bij, Cij and Dij can be calculated. However, the exact description of this problem is presented by Abramczyk [24] and goes beyond the scope of the present article.
In the considered examples, the values of the partition coefficients used for the meshes Bvij lead to small or big folding of the examined roof structure covered with complete transformed shells separated by ribs. On the other hand, the base of the modeled building is flat and horizontal, so the z-coordinates of all base points, belonging to the Pb plane, are the same. The arbitrary level of the Pb plane is adopted as constant zP from the interval <0, dduv11>. The coordinates of all points PAij, PBij, PCij and PDij of the base belonging In the considered examples, the values of the partition coefficients used for the meshes B vij lead to small or big folding of the examined roof structure covered with complete transformed shells separated by ribs. On the other hand, the base of the modeled building is flat and horizontal, so the z-coordinates of all base points, belonging to the P b plane, are the same. The arbitrary level of the P b plane is adopted as constant z P from the interval <0, dd uv11 >. The coordinates of all points P Aij , P Bij , P Cij and P Dij of the base belonging to the edges of the facade walls are obtained as a result of the intersection of the plane P b with all Γ's side edges.
The proportions taken into account, inter alia, the length, width and height of the entire Σ model and its fragments depend on the mutual position of the vertices of the reference polyhedral Γ network, the vertices of the polygonal B v network, the base plane P b and the reference surface used. These proportions are determined by the assumed values of the independent variables and the relationships between the values of the dependent and independent variables. In order to illustrate the impact of adopting different sets of values of the above-mentioned variables on the form of the Γ model, three examples of complex folded building forms covered with different shell roof structures characterized by the negative Gaussian curvature are presented in the next section.

Results
There are presented three examples showing the way of using the developed procedure in the process of shaping various unconventional complex forms of buildings, including the possibility of parameterizing these forms. In order to create the first tetrahedron Γ 11 of a polyhedral network Γ, the values of the following parameters have to be adopted: (1) the distances d v11 of two vertices W BC11 and W AD11 belonging to the first axis v 11 (Figure 6), (2) the ratio dd uv11 of the above distance d uv11 to the distance d v11 between the oblique axes u 11 and v 11, (3) the ratio dd u11 of the distance d u11 of two vertices W AB12 and W CD12 belonging to the second axis u 11 to d v11 . The values employed are given in Table 1. Identical values are adopted for the remaining congruent tetrahedra located in the orthogonal directions of the developed Γ 1 reference network. As a result of the respective composing of Γ 11 , Γ 21 and Γ 12 a network Γ 1 can be created. The vertices of Γ 22 are lain at four vertices of the above tetrahedrons obtained previously. The co-ordinates of these vertices are given in Table A1 in Appendix.
Then, to determine the positions of the vertices S A11 , S B11 , S C11 and S D11 of the first mesh of the reference surface ω r lying in the side edges of Γ 11 , the coefficients dd SA11 , dd SB11 , dd SC11 and dd SD11 constituting the division coefficients of the vertices of the Γ 11 tetrahedron by S A11 , S B11 , S C11 and S D11 should be adopted. The arbitrary values of these coefficients are presented in Table A2 in the Appendix. The calculated values of the coordinates of these vertices are presented in Table A3 in the Appendix.
Another operation provided for in the algorithm of the procedure is to determine the positions of the vertices A 11 , B 11 , C 11 and D 11 of the first mesh B v11 of the polygonal net B v1 . For the form shaped ( Figure 10) the appropriate values of the coefficients dd A11 , dd B11 , dd C11 and dd D11 of the vertices of the Γ 11 mesh by A 11 , B 11 , C 11 and D 11 are adopted. These values are given in Table A4 in Appendix. The p z level of the base plane of the considered form is equal to 15,920 mm in [x,y,z]. The individual vertices P A11 , P B11 , P C11 and P D11 of this base can be constructed as the points of intersection of the horizontal plane p z with the side edges of Γ 1 . The following meshes of the Γ1, Bv1 nets and the reference surface are created same way. Alternatively, some vertices must be adopted at the positions of the sponding vertices belonging to the previously constructed meshes.
A characteristic feature of the folded forms created by the procedure is the di inclination of the adjacent facade walls to the vertical. The form presented in Figur c is characterized by two opposite façade walls directed in the x-axis direction and in with their bases inwards, while the other two façade walls are inclined with their The following meshes of the Γ 1 , B v1 nets and the reference surface are created in the same way. Alternatively, some vertices must be adopted at the positions of the corresponding vertices belonging to the previously constructed meshes.
A characteristic feature of the folded forms created by the procedure is the different inclination of the adjacent facade walls to the vertical. The form presented in Figure 10a-c is characterized by two opposite façade walls directed in the x-axis direction and inclined with their bases inwards, while the other two façade walls are inclined with their bases outwards.
On the other hand, a specific feature distinguishing this form from two following forms is the fact that its elevation walls with their bases inwards are much longer than two other elevation walls with bases inclined outwards. As a result, the form has an elongated elliptical shape when projected onto a horizontal plane, and the x axis can be considered as the principal axis of the imaginary ellipse (Figure 10c).
Three exemplary forms Σ were created based on the same reference network Γ. For these forms, appropriate and different values of three sets of the partition coefficients related to the points of the created reference surfaces, eaves networks and the bases were adopted. The first set was adopted so that the form Σ has an elongated form in the direction of the axis x in the projection on the plane (x, y) and the base plane is closer to the lower vertices of the reference network Γ (Figure 10).
The second set of the partition coefficients was adopted so that the second form Σ has an elongated form in the direction of the y axis in the projection on the (x, y) plane, the plane of the base is halfway between the upper and lower vertices of the network Γ, and the points of the reference surface ω r lie close to the base ( Figure 11). The third set of the partition coefficients was adopted so that the third form Σ has a square form when projected onto the (x, y) plane, the base plane is halfway between the upper and lower vertices of the network Γ, and the points of the reference surface ω r lie further from the base than in the previous case ( Figure 12). The differences between the essential dimensions of the above forms are noticeable when comparing the respective views of each of the above-mentioned three forms.
Materials 2022, 15, x FOR PEER REVIEW the points of the reference surface ωr lie close to the base ( Figure 11). The third partition coefficients was adopted so that the third form Σ has a square form w jected onto the (x, y) plane, the base plane is halfway between the upper and lowe of the network Γ, and the points of the reference surface ωr lie further from the in the previous case ( Figure 12). The differences between the essential dimensio above forms are noticeable when comparing the respective views of each of th mentioned three forms.   The values of the relevant parameters and coordinates of these forms are g Tables A2-A10. It is worth noticing that some geometric properties of two new distinguishing them from the previously presented form are as follows. Two el walls of the form shown in Figure 11, whose direction is in line with the y-axis bases are tilted outwards, are much longer than the elevation walls titled with the inwards and running along the x-axis. The pz level of the base plane of the considere is equal to 31,010 mm in [x,y,z].
On the other hand, all façade walls of the form shown in Figure 12 are of length regardless of whether their bases are tilted outwards or inwards. As a res projection of the third form onto a horizontal plane has a shape similar to a cir square. The differentiation of three above-mentioned general forms is generated deliberate differentiation of the values of the single division coefficients used. A sion on this subject is presented in the next section. The pz level of the base plan considered form is equal to 23,194 mm in [x,y,z].

Discussion
The following relationships can be noticed from the observation of geometri erties of the orthogonal projections of three different structures presented in the p section.
The inclination of the facade walls with their bases towards the inside or outs considered building form depends on the orthogonal directions of two types of and uij of a polyhedral reference network Γ. For example, such a network is prese Figures 10-12. If the considered dimension of the building form is consistent with v11 of the Γ network, located above the base (x, y) of the form, as can be seen in The values of the relevant parameters and coordinates of these forms are given in Tables A2-A10. It is worth noticing that some geometric properties of two new forms, distinguishing them from the previously presented form are as follows. Two elevation walls of the form shown in Figure 11, whose direction is in line with the y-axis and the bases are tilted outwards, are much longer than the elevation walls titled with their bases inwards and running along the x-axis. The p z level of the base plane of the considered form is equal to 31,010 mm in [x,y,z].
On the other hand, all façade walls of the form shown in Figure 12 are of similar length regardless of whether their bases are tilted outwards or inwards. As a result, the projection of the third form onto a horizontal plane has a shape similar to a circle or a square. The differentiation of three above-mentioned general forms is generated by the deliberate differentiation of the values of the single division coefficients used. A discussion on this subject is presented in the next section. The p z level of the base plane of the considered form is equal to 23,194 mm in [x,y,z].

Discussion
The following relationships can be noticed from the observation of geometric properties of the orthogonal projections of three different structures presented in the previous section.
The inclination of the facade walls with their bases towards the inside or outside of a considered building form depends on the orthogonal directions of two types of axes v ij and u ij of a polyhedral reference network Γ. For example, such a network is presented in Figures 10-12. If the considered dimension of the building form is consistent with the axis v 11 of the Γ network, located above the base (x, y) of the form, as can be seen in Figures 10a-12a the projections show facade walls inclined towards the inside of the form.
However, if the dimension of the form is considered orthogonal to the axis v 11 located above the base of the building, as it is shown in Figures 10b-12b, then the elevation walls tilted with their bases outside the considered form are visible.
The overall dimensions of a building form in all three directions of the axes of its local system [x,y,z] are determined by the shape of the base edge line, the eaves line and their mutual position in the horizontal and vertical directions (Figures 10-12).
The To obtain an elongated horizontal projection of a building base along the axis x, the base level coefficient dd P should be taken significantly lower than 0.5 ( Figure 10). To generate a horizontal projection of the building base elongated along the axis y, the coefficient dd P should be assumed significantly greater than 0.5. However, in order to obtain a horizontal projection of the building base with a comparable length of two overall dimensions measured in the x and y directions, the dd P coefficient should be adopted close to 0.5.
If the examined polyhedral reference network Γ has properties analogous to the networks presented in Figures 11 and 12, i.e., u 11 is identical to x of [x,y,z], v 11 is perpendicular to u 11 and distant by d uv11 from z, then the following are true. To obtain an elongated plan view of the designed eaves line along the axis x, the coefficients dd Aij , dd Bij , dd Cij and dd Dij should be taken significantly less than 0.5. To obtain an elongated plan view of the eaves line along the axis y, the coefficients dd Aij , dd Bij , dd Cij and dd Dij should be significantly greater than 0.5. To generate a horizontal projection of roof eaves line characterized by comparable overall dimensions measured in the x and y directions, the arbitrary division coefficients close to 0.5 should be taken.
In order to generate an innovative form of a building having two façade walls inclined inwards and another two outwards of the form, and being close to a circle when projected onto a horizontal plane, the absolute values of the differences dd Aij -0.5, dd Bij -0.5, dd Cij -0.5 and dd Dij -0.5 should be adopted equal or close to an absolute value of the difference dd P -0.5, especially for all central meshes of the net Γ.
The examined general forms are to be influenced by corrugation of the polygonal eaves network Bv generated by the size of the differences between: (a) the division coefficients of the vertices of polyhedral reference network Γ by the vertices A ij , B ij , C ij and D ij of B v , and (b) the division coefficients of the same vertices of Γ by the corresponding points S Aij , S Bij , S Cij and S Dij of a reference surface. In addition, these forms are to be influenced by the ratio between the size of this corrugation and the height of the entire form, which depends on the division coefficients of the vertices of the polyhedral reference network by points belonging to three groups: (a) base vertex group, (b) reference surface point group, and (c) eaves vertex group. The discussion about the influence of some important proportions between the values of the above-mentioned coefficients and the geometrical properties of the building forms shaped goes beyond the scope of the article and requires definition of the so-called multiple division coefficients.

Conclusions
An innovative procedure significantly extending the method for modeling unconventional building forms with complex folded elevations and multi-segment transformed shell roof structure was developed. The specific features of the elaborated procedure is are: (a) an arrangement of many complete transformed shells of a roof structure in the three-dimensional space in accordance with the properties of a regular saddle surface with negative Gaussian curvature, (b) achievement of different inclinations of elevation walls to the vertical, including outside and inside of the same building. Thus, the final innovative method presented in the article fills the important gap existing in the current knowledge.
The algorithm of the procedure was presented on three specific examples of creating complex building forms using appropriate proportions between the distances of all characteristic vertices of their elevation walls and roofs. The procedure enables the designer to take some dependencies determining the respective interrelationships between the distances of the subsequent vertices of the designed reference network deciding on the complex shape of elevation walls. On the basis of the created reference network, the vertices of the polygonal eaves network determining the arrangement and forms of all multi-segment shells of the determined roof structures are defined by means of the division coefficients. The ranges of variability of the above coefficients are also defined so that it is possible to create a few specific types of the building forms under consideration.
A characteristic feature of the presented procedure is that the vertices of each polyhedral reference network are divided into two groups such that the vertices belonging to these groups define two families of the orthogonal axes lying on both sides of the reference surface employed. This leads to a complex shell roof structure the overall shape of which is characterized by the negative Gaussian curvature. This property of the designed roof form is generated by a model whose division coefficients of the vertices belonging to the reference polyhedral network by the roof eaves vertices are in the range (0, 1).
There is a need for searching for further unconventional forms of roofs and elevations as well as their innovative structural systems supporting the transformed folded shell roof sheeting. Some extend laboratory tests in the field of thin-walled folded sheeting transformed elastically into various forms of saddle surfaces are also going to be performed.