1 Introduction
1.1 Research background
Le Corbusier once stated in
Vers une Architecture that “the plan is the generator” (
Le Corbusier, 1924). The architectural plan determines the internal functional relationships and spatial experiences, while also shaping the building’s external outline and its relationship with the surrounding environment. Establishing the plan form of a building is often a crucial aspect of architectural design. Historically, the typological approach has occupied a pivotal position in the exploration of architectural plans. Different plan types embody distinct characteristics of architectural styles, providing crucial references for architects in their selection of forms. Since the 19th-century École des Beaux-Arts in Paris, the analysis and application of plan types have constituted a core component of classical architectural education.
Identifying the basic plan types of architecture has significant utility in architectural design and research. Basic plan types can serve as the starting point for design thinking, be used for comparative studies of multiple schemes, and evolve into more complex forms through mutual combinations. They also provide a systematic framework for the morphological analysis of buildings and cities. Particularly in contemporary computational design, they enhance the efficiency of form-finding and generative design processes through a structured database of basic shapes.
However, prior research on architectural plan types is marked by considerable limitations. Such studies tend to be confined to illustrating common types, with a lack of explicit substantiation concerning the boundaries of basic types. Moreover, the types enumerated are primarily derived from personal experiential insights and the inductive generalization of case studies, thereby exhibiting deficiencies in terms of logical coherence and systemic rigor. Consequently, this study re-examines the issue of basic plan types, remedying the gaps in previous theoretical inquiries while offering methodological support for design practice.
1.2 Research objectives and methods
This study focuses on singular, continuously connected architectural plans in which all interior spaces remain interconnected without traversing exterior areas; consequently, disconnected plan configurations (including point-connected cases) fall outside the scope of this research. While this article emphasizes planar morphology―predominantly determined by the outer contour―it also acknowledges the intrinsic relationship between plan geometry and interior space, a relationship that directly influences spatial connectivity and organization, as exemplified in the compositional and parti training of the École des Beaux-Arts.
The primary objective of this study is to establish a valid and systematic library of basic architectural plan types that provides a fundamental logical framework for the formal analysis of buildings and a rational, systematic methodology for formal exploration. This typological library must satisfy three key requirements:
(1) Mathematical Rigor: Typological divisions are grounded in explicit mathematical principles.
(2) Universal Coverage: Plan types are comprehensive, encompassing the majority of common architectural forms.
(3) Generative Capacity: Basic types serve as foundational elements from which complex configurations can be derived through combinatorial operations.
A secondary objective is the preliminary development of quantitative evaluation indicators that reflect the morphological characteristics of distinct types, thereby enabling data-driven form selection in computational design workflows.
In pursuit of these research objectives, this study employs an approach that integrates mathematical logic with traditional wisdom, proceeding with the logical deduction of basic types and subsequently validating their efficacy through historical cases. First, it analyzes the intrinsic properties of the typology concept and preliminarily delineates the quantitative scope of basic types by referencing linguistic symbolism. Subsequently, through an examination of early civilizations’ perceptions of basic forms, the basic plan types are taxonomically organized into two systems: regular polygon systems and grid systems. For the former, types are determined by the number of vertices of the regular polygon; for the latter, by the quantity and position of continuously filled grid cells. Algorithmic analysis of m × n grids within the grid system demonstrates that a 3 × 3 grid satisfies criteria for representativeness and finiteness of basic types. The two systems are combined to form a basic plan type library, followed by the calculation of quantitative evaluation indexes for each type and verification of their potential to derive complex forms. Finally, the universality and validity of these basic types are corroborated through canonical architectural case spanning different regions and cultures.
2 Previous studies on architectural plan types
2.1 Architectural typology
The term
Type originates from the Latin word
typus and the ancient Greek word
typos, carrying primal meanings such as image, form, category, imprint, and prototype. The early definition of
Type in architecture was described by the famous French architectural theorist Quatremère de Quincy in
Encyclopédie méthodique. He distinguished between
model and
type, establishing the concept of type as a generative principle of form (
De Quincy, 1788). For the purpose of architectural design operation and education, Durand classified architectural plans and summarized the form characteristics of the types, pioneering the use of abstract diagrams to explain architectural composition. Durand, in his book
Précis of the Lectures on Architecture, proposed a design analysis method that combines topological elements and axial grid, which is an architectural innovation that emerged around 1770 (
Durand, 1819;
Mallgrave, 2009) (Fig. 1). In terms of architectural plan, his classification is specifically manifested as a grid system. However, Durand’s theory lacked a discussion on the principles of formal organization itself, that is, what the classification logic is adopted and whether this experience-based classification has universal significance for architectural design (
Qu, 2005).
Composition is the core method of architectural design at the Ecole des Beaux-Arts in Paris. Its composition refers to the architect’s process of translating design conditions into concrete architectural forms. Based on Durand’s theory, Julien Guadet authored
Éléments et théorie de l’architecture, initiating the theorization of composition as a design method (
Guadet, 1901). Later, Nathaniel Cortlandt Curtis’s
Architectural Composition made a clearer discussion of the composition part of Guadet’s theory, exploring architectural elements and their combinatory principles, thereby further refining the theory and methodology of composition (
Curtis, 1923). Curti’s discussion on
parti reflects the thoughts of morphological typology (Fig. 2).
Later In the 1960s, based on the theories of de Quincey, Giulio Carlo Argan argued that type is not a definite form but a diagram or the outline of form. He describes type as the interior structure of a form or as a principle which contains the possibility of infinite formal variation and further structural modification of the type itself (
Argan, 1962). In addition, he mentioned the form variations of type. He believed that type is a posteriori and that there is a clear formal and functional comparability between type and architecture.
Since then, the research object of architectural typology has expanded to the urban level. Rob Krier focuses on the morphological study of urban and architectural space and analyzes urban space in a typological way. In
Urban Space, he proposed four archetypes of the intersection of urban streets and squares and 40 variant forms derived from them. He also proposed to use the three spatial forms of square, circle and triangle as the basis, and generate a large number of compound forms through deformation and control of scale, regularity and openness (
Krier and Rowe, 1979) (Fig. 3). Aldo Rossi defined the concept of type in
The Architecture of the City as something that is permanent and complex, a logical principle that is prior to form and that constitutes it, which, in spite of changes, has always imposed itself on the “feelings and reason” as the principle of architecture and of the city (
Rossi, 1984). In 1994, Anne Vernez Moudon proposed
Typomorphology, the crucial task of which is to recognize the variation of urban forms (
Moudon, 1994).
2.2 Research on plan types in contemporary computational design
In contemporary computational design, the issue of form type is still important but lacks systematic research. Under the guidance of early computational design ideas,
shape grammar, which developed in the 1980s, is a set of rules of transformation applied recursively to an initial shape, generating new shapes, where the
corpus refers to a limited yet diverse set of shapes for a singular design process, and
spatial relations describe recognizable configurations formed through operations like composition, arrangement, and rotation of basic shapes (
Stiny, 1980). G Stiny proposed a parametric shape grammar that generates the ground plans of Palladio’s villas which is well-known as the Palladian Grammar (
Stiny and Mitchell, 1978). Myrsini Mamoli proposed a shape grammar generation system consisting of 91 design rules for ancient libraries. This system can generate a large number of library floor plans that conform to historical characteristics, systematically summarize the design principles of ancient libraries from a visual level, and provide important technical support for the reconstruction of fragmentarily preserved ancient libraries (
Mamoli, 2020). In 1969, Superstudio proposed the
Histogram of Architecture, which is a catalogue consisting of 33 three-dimensional abstract forms to generate designs across scales. They used these diagrams to translate everything, including furniture, architecture, and landscapes. Immanuel Koh reinterpreted of the original
Histogram of Architecture, where the histograms are reperceived and rerepresented as shapeless and grammarless training set of forms for machine learning (
Koh, 2019). Philip Steadman proposed a binary coding method for plotting plans of rectangular built forms in a two-dimensional morphospace, offering a geometric tool to classify architectural forms and trace characteristic ‘morphological trajectories’ throughout architectural history. The research defines an (
x,
y) coordinate system within a grid, representing an architectural plan form by combining two sets of binary codes corresponding to the
x strings and
y strings, where the asterisks can be encoded using combinations of 0s and 1s to indicate internal spatial divisions. However, it does not systematically summarize and categorize fundamental architectural plan types from a morphological and typological perspective (
Steadman and Mitchell, 2010).
Since the 19th century, scholarly inquiry into architectural typology has been predominantly anchored in philosophical speculation and methodological frameworks, with exhaustive discussions on the significance of types. Contemporary computational design research primarily focuses on the application of digital tools and algorithms. However, for the types of architectural plan, there is a lack of systematic discussion from the perspective of mathematical logic, and no clear criteria have been established to define the basic types. A synthesis of historical and contemporary research reveals that inductive methods based on existing cases and architectural knowledge face limitations in obtaining a universal type library. This predicament necessitates a methodological pivot, calling for an exploration of logical deduction that originates from the graphic ontology of form itself.
3 Attributes of basic type
From a cognitive perspective, the meaning of basic type constitutes a fundamental mechanism of human pattern recognition, serving to abstract and simplify the complexity of the world into comprehensible categories. The definition of basic types enhances the efficiency of information processing and transmission. Basic types generally exhibit the following essential attributes: (1) Cultural Relativity: Classification criteria are context-dependent, varying across cultural frameworks and cognitive stages. Thus, no absolute or permanent classification standard exists. (2) Representativeness: Within a given category, the classification system must comprehensively encompass and effectively differentiate between distinct objects. The validity of such a system hinges on its applicability and logical consistency. (3) Finiteness: The number of types should remain constrained, as excessive distinctions undermine the cognitive utility of typological classification. This quantitative limitation arises from the bounded capacity of human memory and the need for efficient information processing. The basic types of architectural plan should likewise possess these attributes. Furthermore, the basic types of architectural plan that point to creation also need to have the ability to create complex types based on the basic plan types.
Among these attributes, the quantitative scope determined by finiteness requires particular clarification. From this perspective, the hierarchical structure and continuity of typological forms are evident. For example, at the most fundamental level, plan types can be as simple as the circle and square―analogous to the binary digits (0 and 1) in computational systems. Subsequent levels will then incorporate additional types through a process of continuous progression. For basic architectural plan types, an over-simplified classification system proves inadequate in distinguishing typical architectural forms, whereas an excessively complex system increases operational complexity in both selection and practical application. Research on finiteness can draw upon quantitative analyses of human linguistic symbols as a key reference. Analogies between architectural forms/spaces and languages have long been a distinctive feature of architectural theory. As an interesting and meaningful exercise, the illustrations of Architectural Alphabet by Johann David Steingruber in the eighteenth century adapted Baroque palaces into the different letters of the alphabet (
Steingruber, 1773). Alan Colquhoun’s typological theory was influenced by Ferdinand de Saussure’s structuralist linguistics (
Colquhoun, 1988). Space syntax analyzes spatial configurations by referencing the intrinsic relationships within language, breaking down space into combinations of relational structures between different spatial elements (
Hillier and Hanson, 1989).
As communicative symbol systems, writing systems universally employ a finite set of basic symbols combined under specific rules to produce complex expressions. Ferdinand de Saussure proposed a classification system for written symbols based on their modes of expression. Excluding primitive pictographic writing, mature writing systems worldwide can be categorized into two fundamental types: ideographic writing and phonetic writing (
De Saussure, 1980). The vast majority of writing systems worldwide are phonetic writing, with the most widely adopted basic symbols (letters) including the Latin, Cyrillic, and Arabic alphabets while Chinese characters stand as a typical example of ideographic writing (Table 1). Among them, five typical phonetic writings systems (without distinguishing between uppercase and lowercase) have 24, 26, 33, 28, and 24 letters respectively. The Korean alphabet, one of these systems, are created by scholars in 1446 based on pronunciation positions and mouth shapes. With its neat and logical structure, the Korean alphabet allows for precise spelling of all syllables through systematic combinations of consonants, vowels, and final tones (
Koehler, 2015). For Chinese characters, the stroke system also exhibits a similar scale. According to
Chinese Character Turning Stroke Standard of GB 13000.1 Character Set, Chinese characters have 32 types of strokes.
It is evident that the number of basic symbols, whether in ideographic or phonetic writing systems, tends to cluster around 20 to 40. Drawing on this observation, the optimal number of basic architectural plan types should be maintained within a similar range of roughly 20 to 40. This quantitative scope not only aligns with human cognitive capacities for efficient information processing but also offers sufficient combinatorial potential for generating diverse architectural forms.
4 Plan type classification system
The contemplation of basic forms possesses deep historical roots that continue to provide important references for contemporary exploration. Ancient Greek scholars made seminal contributions to this field, with Euclid’s Elements establishing a rigorous deductive system encompassing plan geometry, number theory, and solid geometry. Archimedes further advanced geometric understanding, representing the apex of classical Greek achievement. Significantly, Plato proposed five types of regular polyhedra based on regular polygons, which include the equilateral triangle, square, and regular pentagon (Fig. 4).
In the Eastern tradition, although the study of geometry was not highly developed, basic patterns still emerged based on the primitive cosmology, profoundly influencing the construction of physical space and metaphysical concepts.
Ancient China’s achievements in geometry include early geometric tools such as the mathematical totems of the Hetu and Luoshu, and the systematic geometric theories in The Nine Chapters on the Mathematical Art. The Zhou Bi Suan Jing describes the concept of a hemispherical sky and a square earth, forming a cosmic diagram composed of circle and square. Meanwhile, the culturally significant nine-palaces diagram―an ancient magic square diagram originating from the Hetu and Luoshu―served as the basis for both the geographical concept of the nine provinces in ancient statecraft and the spatial layout of early monumental architecture like the Mingtang, embodying a profound consciousness of cosmic order (Fig. 5).
The ancient Indian text
Vedas (circa 1500 BCE) documented the construction and measurement of fundamental geometric forms, including circles, squares, rectangles, and triangles. The
Śulba Sūtras further systematized practical geometry, trigonometry, algebraic geometry, and the cosmic symbolism of Maṇḍala. Among these, the Vāstupuruṣa Maṇḍala was extensively employed in Hindu temple layouts. The most typical division methods are the 64-cell and 81-cell grids, which are fundamental forms in the design of Hindu temple layouts. The even-numbered division, using a 2 × 2 grid as its prototype, features Shiva at its central core with surrounding sections representing his various manifestations. The odd-numbered 3 × 3 grid conversely served as a cosmological model (
Trivedi, 1989) (Fig. 6).
Islamic culture is deeply influenced by the
Quran and expresses the divine order of the cosmos through geometric language. In Islamic culture, the circle represents the indivisible whole, symbolizing the
oneness of Allah, while among the geometric Islamic patterns, the square, regular hexagon, regular octagon, and regular octagon are most prevalent, with star shapes as their variations reflecting the cultural reverence for the moon and stars. Traditional Islamic designs masterfully employ tessellation principles. For instance, by connecting intersection points of multiple circles, diverse polygons and stars can be generated, subsequently forming intricate patterns through tiling (
Critchlow, 1976).
By synthesizing the concepts of basic diagram from above early civilizations, two distinct systems of plan form can be extracted: regular polygon system and grid system. The former focuses on individual characteristics, emphasizing rigorous mathematical proportions and perfect geometric forms, while the latter focuses on spatial orientation and combinatorial relationships.
4.1 Regular polygon system
The regular polygon system derives from idealized geometric forms, including the circle, square, and equilateral triangle, conveying architectural purity and monumentality. These polygons can be systematically classified by their vertex count, progressing from three (equilateral triangle) to infinity (circle), with intermediate forms including the square, regular pentagon, and regular hexagon. In architectural plans, regular heptagons, nonagons, and polygons with more vertices are rarely used, though decagons and dodecagons can occasionally be seen. Therefore, the basic regular polygons (convex shapes) typically include equilateral triangle, square, regular pentagon, regular hexagon, regular octagon, and circle. This system can be extended to incorporate equilateral star polygons, which hold significant cultural meaning in classical symbolism. For example, the pentagram is the symbol of the Pythagorean school, the hexagram is the star of David, representing Judaism and Jewish culture, and the octagram is a common symbol in early China and is still found in ethnic minority regions today. These basic star shapes remain common in various logos and decorative designs to this day. Although the shapes of the star were historically used in defensive castles, they are very rare in modern architecture (Fig. 7).
The regular polygon system reflects the persistent pursuit of ideal forms in the Western architectural tradition. A typical example is the Castel Sant’Angelo in Rome, which is used as a typical example in Rossi’s
The Architecture of the City. Originally commissioned in 139 CE by Emperor Hadrian as a mausoleum for himself and his successors, the building later served as a military fortress, papal refuge, and prison. And it is now the national museum in Italy. Throughout its history, the building’s functional changes have led to successive additions and modifications, resulting in its rich and varied planar forms. The mausoleum’s core consists of a cylindrical structure atop a square base, enclosed by a square perimeter wall with four near-octagonal bastions at each corner. The adjoining plaza features a pentagram enclosing a pentagon (a remnant of the original pentagonal defensive walls built for military purposes). Thus, the plan simultaneously incorporates five distinct geometric typologies: circle, rectangle, pentagon, octagon, and pentagram (
Franceschini and Veneziano, 2015). In Eastern architecture, the regular polygonal system represents a synthesis of symbolism and order. Ancient China’s Mingtang, for instance, adhered to the geometric principle of circular exterior with square interior, embodying the cosmological concept of round heaven and square earth. It reflects the ritual system ideology of harmony between humanity and nature (Fig. 8).
4.2 Grid system
The grid system organizes space through orthogonal grids, which can be logically deduced from the perspective of the number of squares. In the grid system based on square fundamental units, the contour formed by a set of grid cells is an abstract representation of an architectural plan. This set of grid cells must satisfy spatial connectivity: the selected cells must form a single connected whole, with no isolated regions. Adjacency between grid cells is restricted to edge connections (squares touching along their sides), and diagonal connections are not allowed. A matrix can be used to represent the selection of grid cells, 1 denotes a selected cell, while 0 denotes an unselected cell. The prohibition of diagonal connections means that no 2 × 2 subgrid may exhibit either of the following patterns: or . For example, in a 3 × 3 grid, if the grid cell selection is as follows: . This configuration is invalid because it contains the prohibited 2 × 2 diagonal pattern.
Using algorithmic tools, combinations can be generated based on the logic of quantity and permutation. First, generate all possible selected grid combinations of m × n grids, and exclude those that do not meet the requirements according to the above conditions. For the cases that meet the conditions, their symmetry is further considered, and the cases that can be obtained by rotation, mirroring, or translation are regarded as the same type to exclude repeated types (Table 2).
In the determination of whether types are isomorphic, forms with the same number of outline vertices and the same sequence of concave and convex angles are classified as the same types. Within the same type, only the relative lengths of the sides change, while the number of vertices and the sequence of concave and convex angles remain consistent. For example, the two Z-shape forms on the left in Fig. 9 are considered the same type, and the cross-shape and windmill-shape on the right are also considered the same type.
4.2.1 The grid graph 2 × 2 and 2 × 3
After excluding repeated types (the same after rotation, mirroring, and translation), there are a total of 4 types in a 2 × 2 grid graph. When isomorphism is further taken into account, only two types, namely the rectangle and the L-shape, are included in the 2 × 2 grid graph, which is too limited in number. When the grid is expanded to 2 × 3, there are 10 types in the 2 × 3 grid graph after excluding repeated types, including Z-shape, T-shape, and U-shape. However, common architectural forms like the H-shape, cross-shape, and O-shape are not included. The limited number of types makes it difficult to fully reflect the diversity of architectural planar forms (Fig. 10).
4.2.2 The grid graph 3 × 3
After excluding repeated types, there are a total of 34 types in a 3 × 3 grid graph (arranged in Fig. 11 according to the number of occupied squares, ranging from 1 to 9), which include common architectural plan forms such as the T-shape, H-shape, and O-shape. By applying isomorphism to the 34 types, a total of 13 plan types can be obtained.
4.2.3 The grid graph m × n
As the grid scale increases, the number of types grows exponentially. After excluding repeated types, there are 184 types that meet the requirements in a 3 × 4 grid, while the number in a 4 × 4 grid is as high as 1078 (Table 2) (Fig. 12 shows some of the types). It is evident that in grid scales of 3 × 4 and above, the number of types has far exceeded the requirements and cognitive abilities of humans for basic types. Moreover, the majority of these plan types have overly complex contour topological structures that do not conform to the commonly used planar forms in architecture.
Based on the analysis of plan forms in the above two systems, basic plan types suitable for architecture could be identified. Within the regular polygon system, starshaped forms are rarely employed in architectural plans; thus, the selection is limited to six convex polygons: the equilateral triangle, square, regular pentagon, regular hexagon, regular octagon, and circle. For the grid system, the most appropriate planar forms are selected from 3 × 3 grids, a diagram of enduring cultural and architectural significance, equally prominent in ancient spatial diagrams and modern architectural design, as evidenced by seminal works like the Texas Rangers’ studies. By integrating these two systems, this study establishes a library of basic plan type comprising 18 types (excluding a repetitive square type which appears in both systems) (Fig. 13). Although the number of types is marginally less than the range of basic types obtained through linguistic symbol analysis in the previous section (20–40), considering that these types are constrained to individual building (single continuous spaces) and there are also types of discrete and non-continuous spaces, it can be considered an appropriate scale for this library.
Most of these 18 types have specific names in various languages. For example, in English and modern Chinese, regular polygons are named by their number of sides, while morphological types are often denoted by letter symbols, like L-shape, T-shape, H-shape, and so on. However, a small number of types lack specific names/symbols in English; thus, in this paper, they are represented by the Japanese symbol ｽ(su) and the Greek letter τ.
4.3 Characteristic evaluation of basic plan types
Following the establishment of the library, the 18 types within it are evaluated based on quantitative characteristics. According to the appearance characteristics usually considered in architectural design, the primary metrics for quantification are concentration, symmetry, and complexity, corresponding to the formal characteristics of centralized-decentralized, solemn-flexible, and simple-intricate, respectively. (1) Concentration reflects the centripetal quality of architectural form and space, calculated as area divided by perimeter, with all shapes normalized to unit area. (2) Symmetry is defined by the number of mirror symmetry axes (including vertical and diagonal axes, excluding rotational symmetry). (3) Complexity evaluates the tortuosity of plan, which equals the total number of concave and convex angles (equivalent to the vertex count of the shape). These characteristics can assist designers in selecting appropriate basic plan types based on project requirements (Fig. 14).
As illustrated in Fig. 14, the circle exhibits the highest concentration and symmetry, along with the lowest complexity, representing the most geometrically pure form. Regular polygons, cross-shape, H-shape, and O-shape have two or more axes of symmetry and are suitable for dignified or monumental architectural styles. Conversely, plan types with low concentration and symmetry but high complexity are ideal for organic or contextually integrated designs that blend with the environment.
4.4 Compound type
The basic plan types within this library can be flexibly combined to generate an extensive array of compound forms. In the polygon system, the same or different types can be combined in diverse ways, such as centralized and non-centralized compositions. The centralized composition arranges subordinate polygonal types around a dominant central polygon, resulting in configurations marked by strong memorial qualities and rigorous orderliness. By contrast, the non-centralized composition typically adopts an organizational structure to integrate multiple identical polygons. As a typical example, the works of Louis I. Kahn, a modern architectural master with a classical spirit, incorporate these two polygonal composition methods (Fig. 15).
In the grid system, each basic plan type can combine with itself or with other basic plan types to generate a variety of complex plan forms. However, due to the excessive number of the latter, and for the same reasons mentioned earlier, this study focuses exclusively on pairings of two identical basic plan types. The assembly process adheres to the following constraints: First, the grid area is limited to a 3 × 6 range, allowing units to combine in four directions and permitting rotational assembly of units, but overlapping of grid cells is not allowed. Additionally, the connection between the two units must be a straight line (interlocking is not permitted). Applying these rules and eliminating duplicates or types identical to basic plan types yields 75 compound types, of which 38 exhibit symmetrical properties (including rotational symmetry). While historical precedents suggest most compound types lack architectural prevalence, those with higher symmetry, such as the S-shape (D4), E-shape (B4), 土-shape (A3), 王-shape (A13), and 8-shape (A14), demonstrate recurrent use in design practice (Fig. 16).
The representativeness of the basic plan types obtained in the above studies can be verified through classic architectural cases. The selection of these cases is not confined to specific regions, cultures, or historical periods (Fig. 17). A comparative study of regular polygonal architectural plans reveals the following distinctions: (1) Square, hexagon and circle characterized by a high degree of symmetry, have been the most frequently used plan types from ancient times to the present. Among them, square and regular hexagon facilitate multi-directional splicing of units, while circle usually represent the simplest and sacred form. (2) Triangular plans remain uncommon in both Eastern and Western traditional architecture, primarily due to the difficulty in space utilization and structural component connection caused by sharp angles. However, in modern architectural practice, advancements in structural engineering have facilitated their limited adoption in modern large-scale buildings. (3) Octagonal plans, demonstrating an inverse historical trajectory, were deeply rooted in classical architecture due to their high symmetry and monumental associations, as seen in examples such as Buddhist pagodas in Asia and baptisteries in Europe, and their decline in contemporary practice may reflect the diminished emphasis on monumentality in modern design. Despite temporal shifts in stylistic preferences, the geometric principles underlying these polygonal systems persist as foundational design logic, maintaining significance in current architectural practice.
Among the 12 types in the grid system, 8 types (L-shape, T-shape, U-shape, Z-shape, H-shape, Y-shape, cross-shape, and O-shape) align closely with historical experience and conventional thinking. These 8 types, which have specific names or symbols across multiple languages, exhibit pronounced symmetry and identifiability, rendering them the most prevalent architectural plan configurations from antiquity to the present. Certain types acquire culturally specific connotations; for instance, the H-shaped plan frequently signifies official architectural traditions in ancient China, while cross-shape carries profound religious symbolism in Western contexts. These are also the types that are often mentioned in the formal analysis of modern architecture and urban space. In particular, when architects seek multiple formal variations in their designs, they are used to choose these types for comparison or juxtaposition consciously (
Wang, 2024).
As for the remaining 4 types, W-shape and stepped-shape exhibit single-axis symmetry, while the ｽ-shape and τ-shape are completely asymmetrical. These four types appear to fall outside the scope of previous experience-based basic types, representing novel formal discoveries through this research. Their external forms demonstrate organic formal qualities and enhanced environmental interface potential, making them suitable for designs that integrate with natural surroundings. Through the investigation of existing architectural cases, it is found that these so-called new types all have classic building examples, which to some extent proves their potential application value in architectural practice.
5 Conclusion
Focusing on the plan of a single building with continuous internal space, this study explores the issues of basic architectural plans and establishes a library of basic plan type based on a logical and systematic approach. Basic types generally exhibit the essential attributes such as cultural relativity, representativeness, and finiteness. Referring to the number of basic symbols in writing systems, the optimal number of basic types is estimated within a range of approximately 20–40. Combined with the analysis of different cultural diagrams, the basic plan types are divided into two categories: the regular polygon system and the grid system. The former includes triangle, square, pentagon, hexagon, octagon, and circle, while the latter is based on m × n grids. Through algorithmic and mathematical analysis, the 3 × 3 grid system is demonstrated to achieve an optimal balance between finiteness and representativeness, yielding 12 additional basic types excluding the repetitive square type. Combining these results, a consistent and universal basic plan type library encompassing 18 types is established. Considering the existence of discrete and non-continuous plan types, this library can be regarded as having a reasonable scale.
According to the requirements of architectural design for exterior styles, a preliminary set of character evaluation indicators for basic plan types, consisting of concentration, symmetry, and complexity, has been established. The validity of this library is verified through case studies of classic works. 4 of the 18 basic types proposed in this study seem to be beyond general experience, but case studies confirm that they are often used as irregularities in natural and organic style architecture. Furthermore, combinations of basic types can yield diverse compound types. Within the polygon system, combinations of identical or different polygons are highly prevalent in architectural design. And based on the 12 types of grid system, combining one basic type with itself generates 75 compound types, among which 38 exhibit symmetry. These include common complex plan shapes such as the S-shape, E-shape, 王-shape, 土-shape, and 8-shape.
The basic plan type library proposed in this study has significant application value. For architectural design, it offers a framework for the form exploration and optimization processes. It can help designers explore different forms and possibilities within a limited library of basic shapes, or generate rich and diverse forms through combination. For architectural research, it provides a typological recognition system and promotes the scientific analysis of various forms. Beyond architectural applications, the library also offers methodological support for urban morphology and urban design. For example, the shape of land block or public space can be analyzed or generated based on these basic plan types. The library can also serve as a comprehensive database for form-seeking in computational design based on formal character evaluation. It can also benefit other related disciplines, including graph design and product design.
Nevertheless, the study of basic plan types remains an open subject and challenging field. Beyond the polygonal system and grid system mentioned in this paper, there are other systems and generative methods worthy of exploration. For instance, while the grid system in this paper adopts a combinatorial additive operation, subtractive operation system can also yield representative plan types. The basic idea of the latter is to gradually subtract some simple shapes from a single shape, which may produce novel results under certain logical conditions.
The methodology of this study also has certain limitations. As the currently established library of plan types consists of two formal systems, namely the polygon system and the grid system, it does not yet have fully unified mathematical standards. Furthermore, although the typological derivation exhibits logical consistency and an objective methodological approach, it has not been entirely divorced from subjective judgments rooted in historical experience and case studies. These challenges inherently require further studies in the future through more profound formal analyses and the enhancement of algorithmic modeling capabilities.
2095-2635/2025 The Authors. Publishing services by Elsevier B.V. on behalf of KeAi Communications Co. Ltd.
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