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A FORM ANALYSIS OF “SEAMLESS ARCHITECTURE”
1. Introduction
In recent years, an increasing number of architectural works characterized by distinct features have been constructed successively, particularly outside Japan (Fig. 1). These features include, for example, buildings whose roofs and exterior walls are finished with the same material, deliberately designed to minimize the visibility of joints as much as possible.
The purpose of this study is to provide a detailed analysis of such architectural works, which are herein referred to as "seamless architecture" [Note 1].
1-1. What Is “Seamless Architecture”?
For the purpose of this study, "seamless architecture" is provisionally defined as follows: "An architectural structure whose exterior surfaces are finished with a single identical material—specifically, a building where the roof and exterior walls (and in some cases, the floor) consist of the same material."
In short, it can be described as architecture enclosed within a single, continuous envelope. Just as our bodies are covered with seamless skin made of a single material. Although still in the hypothesis stage, we believe that this is 21st-century architecture modeled after "life-form," as opposed to 20th-century architecture modeled after "machines."
Since seamless architecture blurs the boundary between the roof and exterior walls, it is hypothesized that an ideal, fully realized seamless architecture would be one in which the distinction between roof, wall, and floor becomes faint, both internally and externally.
2. Methodology
2-1. Case Study Collection
Based on the provisional definition outlined above, this study conducts a comprehensive survey of both domestic and international (with an emphasis on international) case studies, followed by an analysis.
First, as many potential examples of seamless architecture as possible are selected from projects featured in daily updated international web magazines, specifically ArchDaily and Archello. Following this initial screening, more detailed documentation and materials are gathered for projects that have also been published in architectural journals. In addition, further case studies are compiled from the world-architects.com newsletter (eMagazine) and books containing a broad collection of contemporary architecture, such as Sky is The Limit.
2-2. Analytical Framework
The collected case studies will be examined through the following perspectives:
• A. Form (Overall Geometry and Surface Finish)
• B. Function (Program and Spatial Plan)
• C. Structure (Typology and Construction Method)
• D. Environment (Sustainability and Environmental Control)
• E. Relationship Between Parts and the Whole
To fully articulate the characteristics of seamless architecture, elucidating E (The relationship between parts and the whole) is considered particularly vital. However, as an initial phase, this paper focuses its analysis and reporting specifically on A (Form: Overall Geometry and Surface Finish).
2-3. Research Goals
The ultimate goal of this series of studies is to elucidate the distinct characteristics of "seamless architecture" and, based on the findings, to subsequently establish a refined redefinition of the concept.
2-4. Objective and Scope of This Paper
The specific objectives of this paper are twofold:
1. To clarify the characteristics and trends of seamless architecture from an external perspective by analyzing projects through the lenses of "form" and "surface finish."
2. To devise an analytical methodology (specifically, indicators for quantifying form) to achieve this purpose.
The scope of this study consists of a total of 100 selected architectural works (Table 1). These cases were compiled from projects published up to September 2015 [Note 2] in ArchDaily and Archello, works featured in the book The Sky is the Limit, projects documented in Japanese architectural journals such as Shinkenchiku, Shinkenchiku Jutakutokushu (Residential Architecture), and a+u (Architecture and Urbanism), as well as structures visited firsthand by the author.
2-5. Reserved Matters and Limitations
At present, how to classify buildings where only the interior surfaces (ceilings, walls, and floors) are finished with an identical material remains a reserved matter; thus, they are excluded from the analysis in this paper.
While there are other cases that are currently difficult to categorize, the immediate priority is to expand the collection of case studies. Moving forward, it may be necessary to introduce an additional criterion: rather than simply requiring the roof and exterior walls to share the same finishing material, the distinction between the roof and the walls must be visibly blurred or ambiguous.
3. Analysis of Form
At present, "seamless architecture" can be considered a concept independent of any specific form. This is because examples of seamless architecture can be identified across a wide variety of architectural geometries rather than being confined to a single typical shape. To verify this observation, a detailed analysis of the forms of seamless architecture is conducted below.
3-1. Classification of Form
3-1-1. Classification by Formal Characteristics
The 100 collected case studies of seamless architecture were categorized into the following six models based on their architectural geometry and formal characteristics:
[I] Box Model (Gabled/Rectilinear) This category comprises forms composed of cubes or rectangular parallelepipeds, including configurations where multiple boxes are combined. Generally, in a standard box model, the roof finish is difficult to visually perceive from the ground level. Consequently, its character as seamless architecture—in the sense that the exterior walls and roof are covered with the same material—tends to be less pronounced.
[II] House Model (Vernacular/Archetypal) This architectural form features distinct roof shapes such as gables or shed roofs (monopitch roofs). When a house model manifests as seamless architecture, it typically lacks eaves, resulting in a form where the roof and walls connect without a visible boundary. This category also includes cluster configurations where multiple house forms are grouped together.
[III] Polyhedral Model The overall geometry in this model is configured through a combination of regular or irregular polygons (flat planes).
[IV] Dome Model The dome model represents the historically oldest form of seamless architecture, inherently characterized by an ambiguous boundary between the roof and the exterior walls.
[V] 2D Curved Surface Model (Folding Model) Mathematically speaking, all surfaces (including planes) are two-dimensional. However, for the purpose of morphological classification, forms created by folding flat planes are designated here as the "2D curved surface model." Since buildings utilizing the design methodology known as "folding" fall into this category, it is also referred to as the folding model.
[VI] 3D Curved Surface Model This model encompasses forms composed of surfaces that cannot be generated simply by folding flat planes, such as egg-shaped (ovoid) geometries or organic, fluid forms.
3-1-2. Discussion: Formal Trends in Seamless Architecture
The Type [III] polyhedral model is the most prevalent, accounting for 37% of the total case studies. However, curved surface models also constitute a significant portion; when the 2D and 3D curved surface models are combined, they represent 36% of the cases, which is nearly equal to the proportion of the Type [III].
3-2. Quantification of Form
In the previous section, the forms of seamless architecture were categorized into six models based on their morphological characteristics. To enable a more quantitative analysis, this section explores methods for the quantification of form. First, a brief overview of the quantification of form in mathematics is provided below.
3-2-1. Quantification of Form in Elementary Geometry
In elementary geometry, metrics for quantifying form include the number of vertices, edges, and faces. While these can be applied to box, house, and polyhedral models that possess distinct edges and flat surfaces, they cannot be applied to architectural forms with curved surfaces.
3-2-2. Topology
Topology is a branch of geometry that developed rapidly in the 20th century [2, 3, 4]. To avoid unnecessary complexity, the following discussion excludes non-orientable surfaces, such as the projective plane and the Klein bottle, as they are considered irrelevant to standard architectural spaces addressed in this study.
a. Genus [4]
In topology, when considering two distinct forms, if they are imagined as being made of a rubber-like material and can be stretched or contracted into the same shape without tearing or gluing, the two figures are considered identical (homeomorphic). Conversely, a sphere and a doughnut shape (referred to as a torus, which has a single hole in the center) cannot be transformed into the same shape no matter how they are deformed; thus, they are not topologically identical.
In this context, the number of holes in a figure serves as a key clue for classification. A sphere has 0 holes, a torus has 1 hole, and so on. This number of holes corresponds to a topological invariant known as the "genus" [Note 3]. According to the classification of closed surfaces in topology, every surface is homeomorphic to either a sphere or a torus of genus n. A surface with a genus of 2 or more—meaning it has two or more holes—is sometimes referred to as a "biscuit surface" [3] to distinguish it from a standard doughnut shape.
b. Connectivity [1]
Let F be a continuously connected 3D solid object. If F can be divided into two separate parts by making a single cut at an arbitrary location, the connectivity of F is defined as 0 (also referred to as simply connected). If a solid can be rendered simply connected by making n+1 cuts, the connectivity of that solid is defined as n.
For example, the connectivity of an apple is 0, whereas the connectivity of a doughnut (torus) with one hole is 1, as it requires two cuts to be completely divided into two separate pieces. For the standard 3D solid objects dealt with in architectural design, the aforementioned "genus" and "connectivity" are equivalent.
c. Euler Characteristic [1]
When a 3D solid F is a polyhedron, let S, E, and V denote the number of faces, edges, and vertices, respectively. If the connectivity of F is n, the following equation holds:
χ=S-E+V=2(1-n) (3-1)
Here, χ is referred to as the Euler characteristic, or simply the Euler number.
The Euler characteristic can also be determined for figures that do not possess distinct vertices, edges, or faces (unlike polyhedrons) by appropriately applying simplicial decomposition [3]. The Euler characteristic of a sphere is 2, that of a torus with a single central hole is 0, and that of a biscuit surface with two holes is -2. While topologically identical figures share the same Euler characteristic, having the same Euler characteristic does not necessarily guarantee that the figures are topologically homeomorphic.
d. Betti Numbers [2]
When a closed surface K is decomposed into several connected closed surfaces, each separated component is called a connected component. The number of connected components of K, denoted as r, is defined as the 0-dimensional Betti number of the closed surface K and is written as p_0 (K).
Next, the number of non-torsional principal cutting lines (essential cycles) on a closed surface K is defined as the 1-dimensional Betti number of K and is written as p_1 (K). The p_1 of a sphere is 0, whereas the p_1 of a torus of genus n is 2n.
Furthermore, the 2-dimensional Betti number p_2 for both a sphere and a torus of genus n is 1. Regarding the relationship between Betti numbers and the Euler characteristic, the following Euler-Poincaré formula holds:
χ=p_0 - p_1+p_2 (3-2)
3-2-3. Proposal of the "Ridge Number"
Although the aforementioned Betti numbers are reportedly "applied to the analysis of complex curved surfaces such as biological tissues" [1], topology is fundamentally a branch of geometry that abstracts away fine morphological differences to "study figures based solely on the connectivity between points" [3]. Therefore, while it is useful for exploring the structure of the universe, it is inherently ill-suited for the analysis of architectural forms, where minute formal variations are of critical importance.
To address this limitation, this paper proposes the "Ridge Number" (R) as a metric applicable to both polyhedral and curved-surface architectural forms. The Ridge Number is defined as the total count of "ridges," where a "ridge" refers to a line formed by a collection of non-differentiable points on a three-dimensional solid object. In polyhedral architecture, this corresponds directly to the number of edges. Generally, in curved-surface architecture, a ridge represents sections that lack smoothness and are folded or sharp—akin to the ridgelines or crests of a mountain—as well as the boundaries (edges) of the form. The method for identifying and counting ridges and determining the Ridge Number on a curved surface is illustrated in the figure below.
The term "ridge" originates from the "Active Contour Theory (AC Theory)" [5, 6, 7] developed by Hara, Ashikawa, Fujii, et al. However, the concept of the "Ridge Number" (R) has been newly devised by the authors. Furthermore, this study extends the conventional definition of a ridge so that it can be applied not only to curved surfaces but also to polyhedral geometries. (For a detailed definition of a ridge (R^*) within the context of the Active Contour Theory, see Note 4).
Although the Ridge Number of a building resembling a sphere suspended in mid-air would be 0, such a morphology is practically non-existent in actual architecture. Therefore, the Ridge Number is defined as a natural number greater than or equal to 1:
1≦R ("where " R" is a natural number" ) (3-3)
The Ridge Numbers (R) corresponding to the six morphological classifications based on formal characteristics are illustrated in Fig. 5.
3-2-4. Treatment of Openings in the Calculation of the Ridge Number [Note 5]
How openings are designed within seamless architecture is a significant research theme in its own right. However, this section focuses exclusively on how openings are treated when calculating the Ridge Number.
(1) Openings such as entrances, windows, and skylights can be viewed as punctures within a solid volume. Consequently, the perimeter (edges) of these openings inherently constitutes ridges. However, if these perimetric ridges were included in the Ridge Number, buildings with the exact same "house model" geometry would yield different Ridge Numbers depending solely on the shape and quantity of their windows. Because the Ridge Number is introduced primarily as an indicator for morphological classification, this inconsistency is problematic. Therefore, in calculating the Ridge Number, ridges around openings that can be considered as holes punctured within the same continuous surface (including flat planes) are disregarded (Fig. 6).
(2) Exceptionally, protruding elements such as the skylights of the Kunsthaus Graz exert a significant impact on the overall architectural geometry; thus, the ridges of such protrusions are included in the Ridge Number count (Fig. 7).
(3) Furthermore, fixed glass screens are morphologically more appropriate to treat as walls rather than conventional openings. Therefore, the ridges along the perimeter of fixed glass screens are included in the Ridge Number count.
3-2-5. Proposal of the "Curved Architecture Degree"
While an architectural form with a larger Ridge Number can be described as more polyhedral, this paper proposes the "Curved Architecture Degree" (C) as an inverse indicator to determine how curved a form is. The Curved Architecture Degree (C) is defined as the reciprocal of the Ridge Number (R):
C=1/R ("where " 0<C≦1) (3-4)
Because the minimum value of the Ridge Number is 1, the maximum value of the Curved Architecture Degree is 1. Conversely, as the configuration becomes increasingly polyhedral, the Ridge Number approaches infinity (R→∞), which causes the Curved Architecture Degree to approach 0 (C→0). Therefore, the range is established as 0<C≦1.
A value of C=1 represents a form with the highest degree of curvature.
3-3. Distribution of Projects Based on the Ridge Number and the Curved Architecture Degree
This study has proposed the Ridge Number (R) and the Curved Architecture Degree (C) as indicators for classifying the forms of seamless architecture. This section examines the distribution of the analyzed architectural works from the perspective of each metric.
3-3-1. Distribution of the Ridge Number and the Number of Projects in Seamless Architecture
The distribution of the Ridge Number (R) and its corresponding number of architectural projects is illustrated in Fig. 8.
3-3-2. Relationship Between the Six Morphological Typologies and the Ridge Number
While the forms were previously categorized into six models based on their formal characteristics, this section investigates how the Ridge Number and the number of projects are distributed within each respective typology.
For the house model, dome model, and 3D curved surface model, a clear correspondence with specific Ridge Number ranges was identified. Conversely, the polyhedral model exhibited a wide distribution of Ridge Numbers spanning a broad range.
3-3-3. Distribution of the Curved Architecture Degree and the Number of Projects in Seamless Architecture
The distribution of the Curved Architecture Degree (C) and its corresponding number of projects is illustrated in Fig. 10. Note that the horizontal axis is plotted on a logarithmic scale. Additionally, the range representing a high degree of curvature is indicated by a gradient in the figure. (This range was back-calculated from the Ridge Numbers where projects are densely concentrated within the 3D curved surface model shown in Fig. 9. Furthermore, since a Ridge Number of 12 or greater increases the likelihood of a project falling into the box, house, or polyhedral models, a Curved Architecture Degree of approximately 0.1 or higher was deemed to be predominantly curved.)
4. Surface Finish
4-1. Constraints on Materials and Construction Methods
Because the surface finish requires the roof and exterior walls to be finished using the identical material and construction method, the selection of materials is strictly limited to those possessing waterproofing capabilities.
4-2. Classification of Overall Enclosure Methods
The methods used to enclose the architectural volume are categorized into the following three types:
[1] Monolithic Integration This category comprises finishes, such as seamless coatings or plasters, that deliberately eliminate joints as much as possible to present the entire structure as a single, unified mass.
[2] Tiling This approach configures the entire surface by tightly laying components of the identical shape and material without any gaps. It encompasses various geometric units, including circles, squares, rectangles, and polygons.
[3] Division In this method, the overall form is segmented into finely divided components that conform to the geometry of the building. Here, the joints themselves play a crucial role in the architectural design. Generally, these segments do not retain a uniform shape and are often curved, though they may occasionally be approximated using flat, planar components.
5. Conclusion
5-1. Compilation of the Summary Table
Table 1 [Right Figure] presents a comprehensive summary table compiling the 100 examined case studies, categorized according to the following metrics:
Morphological classification of the overall form based on formal characteristics,
Ridge Number (R),
Curved Architecture Degree (C), and
Classification of the overall enclosure method regarding the surface finish.
5-2. Future Research Directions (Primarily Regarding the Calculation of the Ridge Number)
Based on the research findings obtained thus far, the following points are identified as key issues for future investigation, focusing primarily on the refinement of the Ridge Number calculation:
(1) Treatment of Multi-Structure Ensembles: How to systematically address architectural projects that consist of a cluster or complex of multiple distinct buildings.
(2) Handling of Appendages and Connected Volumes: Determining the appropriate methodological approach for handling architectural appendages such as bridges (e.g., Selfridges Birmingham) or cases where separate volumes with entirely different geometries are interconnected (e.g., Kunsthaus Graz).
(3) Structural Variations within Curved Models: Addressing cases where certain "curved models" inherently yield a large Ridge Number depending on their specific geometric configuration (e.g., Heydar Aliyev Center).
(4) Hybrid Geometries: Investigating complex architectural forms that simultaneously exhibit both curved and polyhedral characteristics (e.g., FRAC Centre).
(5) Limitations of a Single Metric for Highly Complex Forms: Acknowledging that as architectural geometry increases in complexity, it becomes increasingly difficult to accurately discern and characterize specific formal features using the Ridge Number alone.
5-3. Summary
As the initial draft of the research on "seamless architecture," this study conducted an analysis from the perspectives of morphology and surface finish. The primary research findings are summarized as follows:
(1) Morphological Classification into Six Models: Based on formal characteristics, the configurations of seamless architecture were categorized into six distinct models. It was confirmed that seamless architecture encompasses a diverse range of forms, demonstrating that at this stage, seamless architecture remains a form-independent concept. Notably, however, the polyhedral model and the curved surface model each accounted for just over 35%, constituting relatively dominant proportions.
(2) Quantification of Forms via the "Ridge Number": To analyze the morphology in greater detail, geometric quantification was conducted, and the "Ridge Number" (R) was proposed as an indicator for this purpose. Although several challenges remain for future investigation, the metric was confirmed to be highly effective as a quantitative indicator.
(3) Evaluation of Curvature via the "Curved Architecture Degree": The "Curved Architecture Degree" (C) was proposed as an indicator to determine whether a form exhibits curved characteristics. It was established that an architectural work with a Curved Architecture Degree of approximately 0.1 or higher can be deemed predominantly curved.
(4) Three Types of Overall Enclosure Methods: Regarding the surface finish, it was revealed that the methods for enclosing the overall architectural volume can be classified into three distinct categories: [1] Monolithic Integration, [2] Tiling, and [3] Division.
Notes
[Note 1] In this paper, concepts, indicators, and terms newly discovered or devised by the authors are enclosed in double quotation marks in the English version (and indicated with angled brackets "〈 〉" in the Japanese text ). However, in the English captions of figures, tables, and photographs, italics are used without double quotation marks to avoid unnecessary complexity.
[Note 2] The reason for setting a specific time limit for the architectural works to be collected was that, because new examples of seamless architecture are emerging daily, it was necessary to temporarily halt the collection process to proceed with the analysis. In recent years, the number of seamless architectural works among major award-winning projects has been increasing. For instance, the Philharmonic Hall Szczecin by Barozzi/Veiga, which won the 2015 Mies van der Rohe Award (EU Prize for Contemporary Architecture), as well as one of the five finalist projects for the same award in 2016, are classified as seamless architecture (the bottom-center and rightmost works in Fig. 1).
[Note 3] For non-orientable surfaces whose true forms can only be visualized in a four-dimensional space, such as a projective plane or a Klein bottle, the meaning of the genus differs from the "number of holes."
"In general, let us consider n distinct circular disks on a sphere, discard the interiors of these disks (thereby creating holes), and attach one Möbius strip to the boundary circumference of each disk (resulting in a total of n strips). The non-orientable surface thus constructed is called a non-orientable surface of genus n. [...] A non-orientable surface of genus n has an Euler number of χ=2-n." (Reference [3], pp. 128–129)
[Note 4] In general, except for cases where a geometric figure can be explicitly represented by an equation, a vast amount of information contained within a shape can only be demonstrated in a simplified manner by replacing it with another figure. The ridge (R*) in Active Contour Theory can be understood as an extraction of the essence (central concept) of a potential surface—which should originally be represented in a 3D display—by transforming it into a 2D contour geometry and subsequently condensing it into 1D linear information.
Because this paper proposes the ridge and the Ridge Number as clues for classifying the diverse morphologies of seamless architecture, it shares a commonality with Active Contour Theory in that a ridge represents a set of non-differentiable singular points on a curved surface. However, its intended architectural meaning differs somewhat from the ridge (R*) defined above in Active Contour Theory.
[Note 5] In calculating the Ridge Number, the architectural morphology is evaluated macroscopically to determine whether an element constitutes a "ridge." Evaluating macroscopically means, for example, that if there is an overhanging roof or floor slab, the slab actually possesses thickness, making the physical shape of its leading edge rectangular. Evaluating this purely from a microscopic perspective would increase the Ridge Number by 4 at the edge alone. Therefore, when calculating the Ridge Number, the thickness of the slab is disregarded, and it is treated as a single thicknessless plate, counting the tip as only one ridge. This is because a microscopic calculation would cause the Ridge Number to increase infinitely, thereby failing to fulfill its original purpose as a morphological classification indicator.
1. Introduction
In recent years, an increasing number of architectural works characterized by distinct features have been constructed successively, particularly outside Japan (Fig. 1). These features include, for example, buildings whose roofs and exterior walls are finished with the same material, deliberately designed to minimize the visibility of joints as much as possible.
The purpose of this study is to provide a detailed analysis of such architectural works, which are herein referred to as "seamless architecture" [Note 1].
1-1. What Is “Seamless Architecture”?
For the purpose of this study, "seamless architecture" is provisionally defined as follows: "An architectural structure whose exterior surfaces are finished with a single identical material—specifically, a building where the roof and exterior walls (and in some cases, the floor) consist of the same material."
In short, it can be described as architecture enclosed within a single, continuous envelope. Just as our bodies are covered with seamless skin made of a single material. Although still in the hypothesis stage, we believe that this is 21st-century architecture modeled after "life-form," as opposed to 20th-century architecture modeled after "machines."
Since seamless architecture blurs the boundary between the roof and exterior walls, it is hypothesized that an ideal, fully realized seamless architecture would be one in which the distinction between roof, wall, and floor becomes faint, both internally and externally.
2. Methodology
2-1. Case Study Collection
Based on the provisional definition outlined above, this study conducts a comprehensive survey of both domestic and international (with an emphasis on international) case studies, followed by an analysis.
First, as many potential examples of seamless architecture as possible are selected from projects featured in daily updated international web magazines, specifically ArchDaily and Archello. Following this initial screening, more detailed documentation and materials are gathered for projects that have also been published in architectural journals. In addition, further case studies are compiled from the world-architects.com newsletter (eMagazine) and books containing a broad collection of contemporary architecture, such as Sky is The Limit.
2-2. Analytical Framework
The collected case studies will be examined through the following perspectives:
• A. Form (Overall Geometry and Surface Finish)
• B. Function (Program and Spatial Plan)
• C. Structure (Typology and Construction Method)
• D. Environment (Sustainability and Environmental Control)
• E. Relationship Between Parts and the Whole
To fully articulate the characteristics of seamless architecture, elucidating E (The relationship between parts and the whole) is considered particularly vital. However, as an initial phase, this paper focuses its analysis and reporting specifically on A (Form: Overall Geometry and Surface Finish).
2-3. Research Goals
The ultimate goal of this series of studies is to elucidate the distinct characteristics of "seamless architecture" and, based on the findings, to subsequently establish a refined redefinition of the concept.
2-4. Objective and Scope of This Paper
The specific objectives of this paper are twofold:
1. To clarify the characteristics and trends of seamless architecture from an external perspective by analyzing projects through the lenses of "form" and "surface finish."
2. To devise an analytical methodology (specifically, indicators for quantifying form) to achieve this purpose.
The scope of this study consists of a total of 100 selected architectural works (Table 1). These cases were compiled from projects published up to September 2015 [Note 2] in ArchDaily and Archello, works featured in the book The Sky is the Limit, projects documented in Japanese architectural journals such as Shinkenchiku, Shinkenchiku Jutakutokushu (Residential Architecture), and a+u (Architecture and Urbanism), as well as structures visited firsthand by the author.
2-5. Reserved Matters and Limitations
At present, how to classify buildings where only the interior surfaces (ceilings, walls, and floors) are finished with an identical material remains a reserved matter; thus, they are excluded from the analysis in this paper.
While there are other cases that are currently difficult to categorize, the immediate priority is to expand the collection of case studies. Moving forward, it may be necessary to introduce an additional criterion: rather than simply requiring the roof and exterior walls to share the same finishing material, the distinction between the roof and the walls must be visibly blurred or ambiguous.
3. Analysis of Form
At present, "seamless architecture" can be considered a concept independent of any specific form. This is because examples of seamless architecture can be identified across a wide variety of architectural geometries rather than being confined to a single typical shape. To verify this observation, a detailed analysis of the forms of seamless architecture is conducted below.
3-1. Classification of Form
3-1-1. Classification by Formal Characteristics
The 100 collected case studies of seamless architecture were categorized into the following six models based on their architectural geometry and formal characteristics:
[I] Box Model (Gabled/Rectilinear) This category comprises forms composed of cubes or rectangular parallelepipeds, including configurations where multiple boxes are combined. Generally, in a standard box model, the roof finish is difficult to visually perceive from the ground level. Consequently, its character as seamless architecture—in the sense that the exterior walls and roof are covered with the same material—tends to be less pronounced.
[II] House Model (Vernacular/Archetypal) This architectural form features distinct roof shapes such as gables or shed roofs (monopitch roofs). When a house model manifests as seamless architecture, it typically lacks eaves, resulting in a form where the roof and walls connect without a visible boundary. This category also includes cluster configurations where multiple house forms are grouped together.
[III] Polyhedral Model The overall geometry in this model is configured through a combination of regular or irregular polygons (flat planes).
[IV] Dome Model The dome model represents the historically oldest form of seamless architecture, inherently characterized by an ambiguous boundary between the roof and the exterior walls.
[V] 2D Curved Surface Model (Folding Model) Mathematically speaking, all surfaces (including planes) are two-dimensional. However, for the purpose of morphological classification, forms created by folding flat planes are designated here as the "2D curved surface model." Since buildings utilizing the design methodology known as "folding" fall into this category, it is also referred to as the folding model.
[VI] 3D Curved Surface Model This model encompasses forms composed of surfaces that cannot be generated simply by folding flat planes, such as egg-shaped (ovoid) geometries or organic, fluid forms.
3-1-2. Discussion: Formal Trends in Seamless Architecture
The Type [III] polyhedral model is the most prevalent, accounting for 37% of the total case studies. However, curved surface models also constitute a significant portion; when the 2D and 3D curved surface models are combined, they represent 36% of the cases, which is nearly equal to the proportion of the Type [III].
3-2. Quantification of Form
In the previous section, the forms of seamless architecture were categorized into six models based on their morphological characteristics. To enable a more quantitative analysis, this section explores methods for the quantification of form. First, a brief overview of the quantification of form in mathematics is provided below.
3-2-1. Quantification of Form in Elementary Geometry
In elementary geometry, metrics for quantifying form include the number of vertices, edges, and faces. While these can be applied to box, house, and polyhedral models that possess distinct edges and flat surfaces, they cannot be applied to architectural forms with curved surfaces.
3-2-2. Topology
Topology is a branch of geometry that developed rapidly in the 20th century [2, 3, 4]. To avoid unnecessary complexity, the following discussion excludes non-orientable surfaces, such as the projective plane and the Klein bottle, as they are considered irrelevant to standard architectural spaces addressed in this study.
a. Genus [4]
In topology, when considering two distinct forms, if they are imagined as being made of a rubber-like material and can be stretched or contracted into the same shape without tearing or gluing, the two figures are considered identical (homeomorphic). Conversely, a sphere and a doughnut shape (referred to as a torus, which has a single hole in the center) cannot be transformed into the same shape no matter how they are deformed; thus, they are not topologically identical.
In this context, the number of holes in a figure serves as a key clue for classification. A sphere has 0 holes, a torus has 1 hole, and so on. This number of holes corresponds to a topological invariant known as the "genus" [Note 3]. According to the classification of closed surfaces in topology, every surface is homeomorphic to either a sphere or a torus of genus n. A surface with a genus of 2 or more—meaning it has two or more holes—is sometimes referred to as a "biscuit surface" [3] to distinguish it from a standard doughnut shape.
b. Connectivity [1]
Let F be a continuously connected 3D solid object. If F can be divided into two separate parts by making a single cut at an arbitrary location, the connectivity of F is defined as 0 (also referred to as simply connected). If a solid can be rendered simply connected by making n+1 cuts, the connectivity of that solid is defined as n.
For example, the connectivity of an apple is 0, whereas the connectivity of a doughnut (torus) with one hole is 1, as it requires two cuts to be completely divided into two separate pieces. For the standard 3D solid objects dealt with in architectural design, the aforementioned "genus" and "connectivity" are equivalent.
c. Euler Characteristic [1]
When a 3D solid F is a polyhedron, let S, E, and V denote the number of faces, edges, and vertices, respectively. If the connectivity of F is n, the following equation holds:
χ=S-E+V=2(1-n) (3-1)
Here, χ is referred to as the Euler characteristic, or simply the Euler number.
The Euler characteristic can also be determined for figures that do not possess distinct vertices, edges, or faces (unlike polyhedrons) by appropriately applying simplicial decomposition [3]. The Euler characteristic of a sphere is 2, that of a torus with a single central hole is 0, and that of a biscuit surface with two holes is -2. While topologically identical figures share the same Euler characteristic, having the same Euler characteristic does not necessarily guarantee that the figures are topologically homeomorphic.
d. Betti Numbers [2]
When a closed surface K is decomposed into several connected closed surfaces, each separated component is called a connected component. The number of connected components of K, denoted as r, is defined as the 0-dimensional Betti number of the closed surface K and is written as p_0 (K).
Next, the number of non-torsional principal cutting lines (essential cycles) on a closed surface K is defined as the 1-dimensional Betti number of K and is written as p_1 (K). The p_1 of a sphere is 0, whereas the p_1 of a torus of genus n is 2n.
Furthermore, the 2-dimensional Betti number p_2 for both a sphere and a torus of genus n is 1. Regarding the relationship between Betti numbers and the Euler characteristic, the following Euler-Poincaré formula holds:
χ=p_0 - p_1+p_2 (3-2)
3-2-3. Proposal of the "Ridge Number"
Although the aforementioned Betti numbers are reportedly "applied to the analysis of complex curved surfaces such as biological tissues" [1], topology is fundamentally a branch of geometry that abstracts away fine morphological differences to "study figures based solely on the connectivity between points" [3]. Therefore, while it is useful for exploring the structure of the universe, it is inherently ill-suited for the analysis of architectural forms, where minute formal variations are of critical importance.
To address this limitation, this paper proposes the "Ridge Number" (R) as a metric applicable to both polyhedral and curved-surface architectural forms. The Ridge Number is defined as the total count of "ridges," where a "ridge" refers to a line formed by a collection of non-differentiable points on a three-dimensional solid object. In polyhedral architecture, this corresponds directly to the number of edges. Generally, in curved-surface architecture, a ridge represents sections that lack smoothness and are folded or sharp—akin to the ridgelines or crests of a mountain—as well as the boundaries (edges) of the form. The method for identifying and counting ridges and determining the Ridge Number on a curved surface is illustrated in the figure below.
The term "ridge" originates from the "Active Contour Theory (AC Theory)" [5, 6, 7] developed by Hara, Ashikawa, Fujii, et al. However, the concept of the "Ridge Number" (R) has been newly devised by the authors. Furthermore, this study extends the conventional definition of a ridge so that it can be applied not only to curved surfaces but also to polyhedral geometries. (For a detailed definition of a ridge (R^*) within the context of the Active Contour Theory, see Note 4).
Although the Ridge Number of a building resembling a sphere suspended in mid-air would be 0, such a morphology is practically non-existent in actual architecture. Therefore, the Ridge Number is defined as a natural number greater than or equal to 1:
1≦R ("where " R" is a natural number" ) (3-3)
The Ridge Numbers (R) corresponding to the six morphological classifications based on formal characteristics are illustrated in Fig. 5.
3-2-4. Treatment of Openings in the Calculation of the Ridge Number [Note 5]
How openings are designed within seamless architecture is a significant research theme in its own right. However, this section focuses exclusively on how openings are treated when calculating the Ridge Number.
(1) Openings such as entrances, windows, and skylights can be viewed as punctures within a solid volume. Consequently, the perimeter (edges) of these openings inherently constitutes ridges. However, if these perimetric ridges were included in the Ridge Number, buildings with the exact same "house model" geometry would yield different Ridge Numbers depending solely on the shape and quantity of their windows. Because the Ridge Number is introduced primarily as an indicator for morphological classification, this inconsistency is problematic. Therefore, in calculating the Ridge Number, ridges around openings that can be considered as holes punctured within the same continuous surface (including flat planes) are disregarded (Fig. 6).
(2) Exceptionally, protruding elements such as the skylights of the Kunsthaus Graz exert a significant impact on the overall architectural geometry; thus, the ridges of such protrusions are included in the Ridge Number count (Fig. 7).
(3) Furthermore, fixed glass screens are morphologically more appropriate to treat as walls rather than conventional openings. Therefore, the ridges along the perimeter of fixed glass screens are included in the Ridge Number count.
3-2-5. Proposal of the "Curved Architecture Degree"
While an architectural form with a larger Ridge Number can be described as more polyhedral, this paper proposes the "Curved Architecture Degree" (C) as an inverse indicator to determine how curved a form is. The Curved Architecture Degree (C) is defined as the reciprocal of the Ridge Number (R):
C=1/R ("where " 0<C≦1) (3-4)
Because the minimum value of the Ridge Number is 1, the maximum value of the Curved Architecture Degree is 1. Conversely, as the configuration becomes increasingly polyhedral, the Ridge Number approaches infinity (R→∞), which causes the Curved Architecture Degree to approach 0 (C→0). Therefore, the range is established as 0<C≦1.
A value of C=1 represents a form with the highest degree of curvature.
3-3. Distribution of Projects Based on the Ridge Number and the Curved Architecture Degree
This study has proposed the Ridge Number (R) and the Curved Architecture Degree (C) as indicators for classifying the forms of seamless architecture. This section examines the distribution of the analyzed architectural works from the perspective of each metric.
3-3-1. Distribution of the Ridge Number and the Number of Projects in Seamless Architecture
The distribution of the Ridge Number (R) and its corresponding number of architectural projects is illustrated in Fig. 8.
3-3-2. Relationship Between the Six Morphological Typologies and the Ridge Number
While the forms were previously categorized into six models based on their formal characteristics, this section investigates how the Ridge Number and the number of projects are distributed within each respective typology.
For the house model, dome model, and 3D curved surface model, a clear correspondence with specific Ridge Number ranges was identified. Conversely, the polyhedral model exhibited a wide distribution of Ridge Numbers spanning a broad range.
3-3-3. Distribution of the Curved Architecture Degree and the Number of Projects in Seamless Architecture
The distribution of the Curved Architecture Degree (C) and its corresponding number of projects is illustrated in Fig. 10. Note that the horizontal axis is plotted on a logarithmic scale. Additionally, the range representing a high degree of curvature is indicated by a gradient in the figure. (This range was back-calculated from the Ridge Numbers where projects are densely concentrated within the 3D curved surface model shown in Fig. 9. Furthermore, since a Ridge Number of 12 or greater increases the likelihood of a project falling into the box, house, or polyhedral models, a Curved Architecture Degree of approximately 0.1 or higher was deemed to be predominantly curved.)
4. Surface Finish
4-1. Constraints on Materials and Construction Methods
Because the surface finish requires the roof and exterior walls to be finished using the identical material and construction method, the selection of materials is strictly limited to those possessing waterproofing capabilities.
4-2. Classification of Overall Enclosure Methods
The methods used to enclose the architectural volume are categorized into the following three types:
[1] Monolithic Integration This category comprises finishes, such as seamless coatings or plasters, that deliberately eliminate joints as much as possible to present the entire structure as a single, unified mass.
[2] Tiling This approach configures the entire surface by tightly laying components of the identical shape and material without any gaps. It encompasses various geometric units, including circles, squares, rectangles, and polygons.
[3] Division In this method, the overall form is segmented into finely divided components that conform to the geometry of the building. Here, the joints themselves play a crucial role in the architectural design. Generally, these segments do not retain a uniform shape and are often curved, though they may occasionally be approximated using flat, planar components.
5. Conclusion
5-1. Compilation of the Summary Table
Table 1 [Right Figure] presents a comprehensive summary table compiling the 100 examined case studies, categorized according to the following metrics:
Morphological classification of the overall form based on formal characteristics,
Ridge Number (R),
Curved Architecture Degree (C), and
Classification of the overall enclosure method regarding the surface finish.
5-2. Future Research Directions (Primarily Regarding the Calculation of the Ridge Number)
Based on the research findings obtained thus far, the following points are identified as key issues for future investigation, focusing primarily on the refinement of the Ridge Number calculation:
(1) Treatment of Multi-Structure Ensembles: How to systematically address architectural projects that consist of a cluster or complex of multiple distinct buildings.
(2) Handling of Appendages and Connected Volumes: Determining the appropriate methodological approach for handling architectural appendages such as bridges (e.g., Selfridges Birmingham) or cases where separate volumes with entirely different geometries are interconnected (e.g., Kunsthaus Graz).
(3) Structural Variations within Curved Models: Addressing cases where certain "curved models" inherently yield a large Ridge Number depending on their specific geometric configuration (e.g., Heydar Aliyev Center).
(4) Hybrid Geometries: Investigating complex architectural forms that simultaneously exhibit both curved and polyhedral characteristics (e.g., FRAC Centre).
(5) Limitations of a Single Metric for Highly Complex Forms: Acknowledging that as architectural geometry increases in complexity, it becomes increasingly difficult to accurately discern and characterize specific formal features using the Ridge Number alone.
5-3. Summary
As the initial draft of the research on "seamless architecture," this study conducted an analysis from the perspectives of morphology and surface finish. The primary research findings are summarized as follows:
(1) Morphological Classification into Six Models: Based on formal characteristics, the configurations of seamless architecture were categorized into six distinct models. It was confirmed that seamless architecture encompasses a diverse range of forms, demonstrating that at this stage, seamless architecture remains a form-independent concept. Notably, however, the polyhedral model and the curved surface model each accounted for just over 35%, constituting relatively dominant proportions.
(2) Quantification of Forms via the "Ridge Number": To analyze the morphology in greater detail, geometric quantification was conducted, and the "Ridge Number" (R) was proposed as an indicator for this purpose. Although several challenges remain for future investigation, the metric was confirmed to be highly effective as a quantitative indicator.
(3) Evaluation of Curvature via the "Curved Architecture Degree": The "Curved Architecture Degree" (C) was proposed as an indicator to determine whether a form exhibits curved characteristics. It was established that an architectural work with a Curved Architecture Degree of approximately 0.1 or higher can be deemed predominantly curved.
(4) Three Types of Overall Enclosure Methods: Regarding the surface finish, it was revealed that the methods for enclosing the overall architectural volume can be classified into three distinct categories: [1] Monolithic Integration, [2] Tiling, and [3] Division.
Notes
[Note 1] In this paper, concepts, indicators, and terms newly discovered or devised by the authors are enclosed in double quotation marks in the English version (and indicated with angled brackets "〈 〉" in the Japanese text ). However, in the English captions of figures, tables, and photographs, italics are used without double quotation marks to avoid unnecessary complexity.
[Note 2] The reason for setting a specific time limit for the architectural works to be collected was that, because new examples of seamless architecture are emerging daily, it was necessary to temporarily halt the collection process to proceed with the analysis. In recent years, the number of seamless architectural works among major award-winning projects has been increasing. For instance, the Philharmonic Hall Szczecin by Barozzi/Veiga, which won the 2015 Mies van der Rohe Award (EU Prize for Contemporary Architecture), as well as one of the five finalist projects for the same award in 2016, are classified as seamless architecture (the bottom-center and rightmost works in Fig. 1).
[Note 3] For non-orientable surfaces whose true forms can only be visualized in a four-dimensional space, such as a projective plane or a Klein bottle, the meaning of the genus differs from the "number of holes."
"In general, let us consider n distinct circular disks on a sphere, discard the interiors of these disks (thereby creating holes), and attach one Möbius strip to the boundary circumference of each disk (resulting in a total of n strips). The non-orientable surface thus constructed is called a non-orientable surface of genus n. [...] A non-orientable surface of genus n has an Euler number of χ=2-n." (Reference [3], pp. 128–129)
[Note 4] In general, except for cases where a geometric figure can be explicitly represented by an equation, a vast amount of information contained within a shape can only be demonstrated in a simplified manner by replacing it with another figure. The ridge (R*) in Active Contour Theory can be understood as an extraction of the essence (central concept) of a potential surface—which should originally be represented in a 3D display—by transforming it into a 2D contour geometry and subsequently condensing it into 1D linear information.
Because this paper proposes the ridge and the Ridge Number as clues for classifying the diverse morphologies of seamless architecture, it shares a commonality with Active Contour Theory in that a ridge represents a set of non-differentiable singular points on a curved surface. However, its intended architectural meaning differs somewhat from the ridge (R*) defined above in Active Contour Theory.
[Note 5] In calculating the Ridge Number, the architectural morphology is evaluated macroscopically to determine whether an element constitutes a "ridge." Evaluating macroscopically means, for example, that if there is an overhanging roof or floor slab, the slab actually possesses thickness, making the physical shape of its leading edge rectangular. Evaluating this purely from a microscopic perspective would increase the Ridge Number by 4 at the edge alone. Therefore, when calculating the Ridge Number, the thickness of the slab is disregarded, and it is treated as a single thicknessless plate, counting the tip as only one ridge. This is because a microscopic calculation would cause the Ridge Number to increase infinitely, thereby failing to fulfill its original purpose as a morphological classification indicator.