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Computer-aided architectural designs and associated covariants
Journal of Building Engineering 3 (2015) 127–134
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Journal of Building Engineering journal homepage: www.elsevier.com/locate/jobe
Computer-aided architectural designs and associated covariants Krishnendra Shekhawat n Department of Mathematics, University of Geneva, Geneva, Switzerland
art ic l e i nf o
a b s t r a c t
Article history: Received 18 April 2015 Received in revised form 7 July 2015 Accepted 15 July 2015 Available online 17 July 2015
To compare two architectural designs or to characterize them, some numbers are needed. These numbers are said to be covariants. In this paper, we present a software prototype that generates a floor plan design and its adjacency graph for a given set of data, and computes some mathematical covariants associated with the obtained design. In addition, we discuss and demonstrate the usefulness of covariants in comparing the architectural designs and in obtaining a best design among many possible solutions. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Covariants Floorplan Graph Prototype Rooms
1. Introduction The concept of an invariant appeared naturally in geometry in response to the need for the classification of figures. In concrete terms, what distinguishes the following three sub-figures in Fig. 1? Geometrically, the first sub-figure is a straight line, the second subfigure a rectangle and the third one a spiral. A general line in the same plane intersects in at most 1 and 2 points respectively for the first two figures, while it meets the spiral in infinitely many points. Although the numbers 1, 2 and ∞ do not fully describe these figures, yet they characterize and distinguish them from other geometrical shapes. Interestingly, these numbers remain unchanged if we apply certain geometric transformations, like isometries or scale changes. Thus the number of intersection points is an invariant with respect to a restricted set of transformations, which can be specified. If a number or a mathematical object associated with a geometric configuration remains unchanged with respect to a certain group of transformations then this number or mathematical object is said to be an invariant (with respect to the specified group of transformations). For the shapes in Fig. 1, the angles of intersection are certainly not invariant under rotation, but we can say how they behave. Therefore, the angles of intersection are relative invariants or covariants, with respect to rotation. If we know precisely how a number or a mathematical object n
Correspondence address: Present address: CIAUD, Faculdade de Arquitectura, Universidade de Lisboa, Portugal. E-mail address: [email protected] http://dx.doi.org/10.1016/j.jobe.2015.07.005 2352-7102/& 2015 Elsevier Ltd. All rights reserved.
associated with a geometric figure behaves with respect to a specified set of transformations, we call it a covariant. For a better understanding, consider the rectangular floor plan in Fig. 2A. After rotating it by 90°, as shown in Fig. 2B, its area remains unchanged while its width (dimension measured horizontally) and length (dimension measured vertically) are changed. Hence the area of the rectangular floor plan is an invariant while its width and length are covariants, with respect to the transformation rotation. In the literature, there exist some work where the concepts of invariants and covariants were introduced and studied [1–4]. In this paper, we will explore more about them. Also, we will demonstrate their usefulness in identifying and classifying the different architectural designs, which will present them as a very powerful tool for the architects. To proceed further, let us consider the plus-shape floor plan, illustrated in Fig. 3, which is generated using the algorithms given in [5]. A plus-shape, as shown in Fig. 4A, can be visualized as a shape made up of 5 rectangles, as illustrated in Fig. 4B. In this way, the plus-shape floor plan in Fig. 3 is generated by adjoining 5 rectangular blocks where each of them is best connected1 and is constructed using the spiral-based algorithm given in [6]. Therefore, we call the plus-shape floor plan described in this paper by spiral-based plus-shape floor plan and represent it by F SP (m) (floorplan plus-shape spiral-based) where m is the number of 1 A rectangular floor plan is best connected if it has 3n − 7 edges in its adjacency graph where n is the number of rooms i.e. there does not exist any other rectangular floor plan with n rooms whose adjacency graph has more than 3n − 7 edges (for details, refer [5]).
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Fig. 1. Understanding the concept of invariants and covariants.
Fig. 5. 8 different FRS congruent to each other.
Fig. 2. Invariants and covariants associated with a rectangular floor plan.
Fig. 3. A computer-generated F SP (16) .
different size rectangular pieces (without changing the width and length of rectangular pieces), undoubtedly there would be some empty spaces (or extra spaces) inside the produced rectangle. Therefore, we can see the presence of empty spaces as white rectangles inside the F SP (16) in Fig. 3, which we call by inner extra spaces. If 5 different size FRS are adjoined to generate a FPS, again there would be some empty spaces, which we call by outer extra spaces, as shown in Fig. 3. In [6], it has been illustrated that a FRS is congruent to 7 other FRS which are best connected and one can be derived from other by four types of mappings i.e. translations, reflections, rotations, and glide reflections (the last being a combination of a translation and a reflection). For better understanding, refer to Fig. 5 with eight congruent FRS (with no extra spaces), called by spiral1, spiral2, spiral3, spiral4, spiral5, spiral6, spiral7, and spiral8 FRS respectively. As an example, we can see that the F SP (16) in Fig. 3 is constructed by considering spiral8, spiral6, spiral4, spiral2 and spiral1 FRS for the central, left, upper, right and lower positions respectively (these positions are illustrated in Fig. 4B). From Fig. 3, we can notice that, for the construction of a FPS, five R FS are required which are placed at five different positions and each FRS can be generated in eight different ways, therefore, for a given set of data, 85 = 32, 768 FPS can be generated, which is a big number. For example, in Fig. 6, a FPS is generated from spiral1, spiral2, spiral3, spiral1 and spiral6 FRS for the central, left, upper,
Fig. 4. A plus-shape rectilinear polygon and its division into rectangles.
rooms given by the architects. Also, its graph is represented by G SP (m) (graph plus-shape spiral-based). A spiral-based rectangular block (or floor plan) is denoted by FRS (floorplan rectangular spiralbased). If it is required to produce a larger rectangle by arranging
Fig. 6. A computer-generated F SP (16) .
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right and lower positions respectively, which is geometrically different from the FPS in Fig. 3. But, for an architect, it would be very difficult to single out one solution out of so many possibilities. Also, it is not feasible to select a good or a best solution without some efficient criteria. Therefore, in this work, we present some mathematical covariants associated with the architectural designs. These covariants have the following functions: 1. They help to study and understand the problem. 2. They characterize and measure the difference between different solutions. 3. They are used to obtain the best solution among numerous solutions.This paper is divided into two main parts. The first part is concerned with the covariants associated with the architectural designs and the second part is about the prototype of a software which is developed to generate the required architectural designs and the associated covariants. For the explanation of the concepts presented in this paper, we only consider the plusshape floor plans.
2. Invariants and covariants associated with an architectural design In this section we study some graph invariants and covariants. Adjacency among the rooms that are present inside the FPS has already been defined in [5]. Using the definition of adjacency, G SP (16) corresponding to the F SP (16) in Fig. 3 is illustrated in Fig. 7. The readers who are not familiar with the definitions related to graphs (used in the upcoming subsections) are referred to Gross and Yellen [7]. 2.1. Adjacency matrix and degree of vertices The power of linear algebra is applied to graph theory through representation of graphs by matrices. An adjacency matrix is the first graph invariant to be elaborated. The adjacency matrix of a simple graph G, denoted by AG, is the symmetric matrix whose rows and columns are indexed by VG (in the same order) and is such that
AG [u, v] = 1
if u and v are adjacent, otherwise it is equal to 0.
For example, from the G SP (16) in Fig. 7, it's adjacency matrix can easily be derived using the definition given above. The degree (valency) of a vertex v in a graph G, denoted by deg(v ), is the number of vertices adjacent to v. The degree of each vertex is computed by adding the number of 1's in the corresponding row (or column) of the adjacency matrix. For the G SP (16) in Fig. 7, the degrees of all the vertices, which correspond to the rooms Ri ( i = 1, … , 16), are 5, 7, 4, 3, 3, 4, 4, 4, 3, 3, 2, 2, 3, 3, 7, 1 respectively. To study the distribution of degrees, we consider the methods
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similar to those commonly used in probability and statistics, namely mean, standard deviation, measures of dispersion, minimum and maximum. For real numbers {x1, x2, … , x n }, the arithmetic mean is denoted by x = (1/n) ∑ni=1 x i . The standard deviation σ = (1/n) ∑ni=1 (x i − x )2 gives the variation (or dispersion) from the corresponding mean. A low value of the standard deviation shows that the data points tend to be very close to the mean, whereas a high value of the standard deviation can sometimes indicate that the data points are spread out over a large range of values. The measure of statistical dispersion is a non-negative real number that is zero if all the data have the same value and increases if the data become more diverse. In percentage terms, it is given as ρ = (σ /x ) × 100. For all the degrees corresponding to the G SP (16) in Fig. 7, x = 3.62, s ¼1.58, ρ ¼43.48%, maximum ¼7 and minimum ¼ 1. 2.2. The distance matrix and shortest path The classic example of a real life graph is a network of roads connecting cities. One of the important algorithmic problems on such road networks is finding the shortest path between any two given vertices. For a graph G, the distance d (x, y ) between two vertices x and y is the length of the shortest path from x to y, considering all possible paths in G from x to y. The distance between any node and itself is 0 and if there is no path from x to y, then d (x, y ) is infinity. It is to be noted that for the distance and shortest path, we consider only non-weighted graphs. The distance matrix is a matrix (two-dimensional array) containing the distances, taken pairwise, of the set of vertices. This matrix has order n n, where n is the number of vertices. Once we know the adjacency matrix, we can obtain a shortest path (and its length) between any two vertices of the GPS and the distance matrix by using Floyd's Algorithm (see [8, Section A.2, Chap. 8]). Related to the G SP (16) in Fig. 7, some of the results are as follows: The distance between R1 and R4 is 3 and the distance between R5 and R12 is 4. The shortest path between R1 and R4 is R1 → R2 → R5 → R 4 and the shortest path between R5 and R12 is R5 → R2 → R15 → R13 → R12. 2.3. Eccentricity, radius, diameter and centre In this section, we discuss several graph invariants related to the all-pairs shortest-path matrix. The eccentricity of a graph vertex v in a graph G, denoted ecc(v) is the distance from v to a vertex farthest from v. That is,
ecc (v) = max {d (v, x)} x ∈ VG
The maximum eccentricity is the graph diameter, denoted by diam(G). That is,
diam (G) = max {ecc (x)} = max {d (x, y)} x ∈ VG
x, y ∈ VG
The minimum graph eccentricity is called the graph radius, denoted by rad(G). That is,
rad (G) = min {ecc (x)} x ∈ VG
Fig. 7. G SP (16) corresponding to the F SP (16) demonstrated in Fig. 3.
A central vertex of a graph G is a vertex with minimum eccentricity. Thus ecc (v ) = rad (G ).
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The centre of the graph is the set of all vertices with minimum eccentricity. For the G SP (16) in Fig. 7, the eccentricities of all the vertices corresponding to all the rooms are (4, 3, 3, 4, 4, 3, 3, 4, 4, 4, 4, 5, 4, 4, 3, 5) with radius¼3, diameter¼ 5 and centre {R2, R 3, R6, R7, R15 }.
independent from the other covariants. 2.6. Bipartite graphs A graph G = (V , E ) is called bipartite if V admits a partition into two classes such that every edge has one end in each class: vertices in the same partition class must not be adjacent.
2.4. Cut vertex A cut vertex of a connected graph G is a vertex whose removal renders G disconnected. Any graph with no cut vertices is said to be biconnected. Biconnectivity is an important property due to many reasons. For example, Menger's Theorem implies that any graph with at least 3 vertices is biconnected if and only if there are at least 2 vertex disjoint paths between any pair of vertices. If looked from the perspective of communication networks, the biconnected networks are more fault-tolerant since blowing away any single node does not cut off communication for any other node. To obtain the cut vertices, we eliminate each vertex, say vi, one by one and obtain the distance matrix for the remaining vertices. If any entry in the distance matrix is ∞ then vi is a cut vertex. For the G SP (16) in Fig. 7, the cut vertices are {R15, R9 }. 2.5. Characteristic polynomial and eigenvalues The next graph invariant is the characteristic polynomial of the adjacency matrix and from this polynomial, we derive eigenvalues. The characteristic polynomial of a matrix An × n is the polynomial p A (x ) = det (xI − A), where det is the determinant and I is the n n identity matrix. The eigenvalues of A are the roots of its characteristic polynomial. For the adjacency matrix corresponding to the G SP (16) in Fig. 7, the characteristic polynomial and the eigenvalues are: p (x ) = x16 − 29x14 − 26x13 + 278x12 + 422x11 − 955x10 − 2196x 9 .
+ 763x8 + 4044x7 + 1407x 6 − 2316x5 − 1538x 4 + 242x 3 + 315x 2 + 28x − 4 Eigenvalues: 2.52, 2.37, 2.12, 1.47, 1.12, 1, 0.85, 0.75, 0.22, 0.08, 0.51, 0.85, 1.46, 2.35, 2.79, 4.40. There are a lot of interesting results for the characteristic polynomial and the eigenvalues which can be associated with various geometrical interpretations. Some of the results are as follows. Theorem 1. Suppose p A (x ) is written as
p A (x) = x n + c1x n − 1 + ⋯ + cn. Then 1. c1 = 0. 2. −c2 is the number of edges in the graph G. 3. −c3 is twice the number of triangles in G. Proof. For a proof, see Biggs [9, Proposition 2.3].□ From above Proposition, we can say that the characteristic polynomial of the G SR (n) is of the form z n − E SR (n) z n − 2 − 2D SR (n) z n − 3 + ⋯, where E SR (n) and D SR (n) are the number of edges and number of triangles in the graph of an architectural design. Remark 1. We can see that the covariants defined in Sections 2.1– 2.5 are derived from the adjacency matrix directly or indirectly. In the upcoming sections we will introduce covariants which are
Theorem 2. A graph is bipartite if and only if its spectrum is symmetric about zero. Proof. For a proof, refer to Beineke and Wilson [10, Theorem 5.2].□ By using Theorem 2 for the obtained eigenvalues corresponding to the G SP (16) in Fig. 7, we conclude that the G SP (16) is not bipartite. 2.7. Chromatic number A vertex colouring is an assignment of labels or colours to each vertex of a graph such that no edge connects two identically coloured vertices. The chromatic number of a graph G is the smallest number of colours needed to colour the vertices of G so that no two adjacent vertices share the same colour. The G SP (16) illustrated in Fig. 7 has chromatic number 3. 2.8. Moment of Inertia Now we assume that the graph G is weighted, i.e., each vertex vi has a weight si which is a real number. In the case of an architectural design, the weight can for example be equal to the area of the corresponding room; therefore it is always strictly positive. For each vertex v, the moment of inertia of G relative to this vertex was introduced by Coray in [4,2]. In analogy with classical mechanics, it is given by
Iv =
1 s
n
∑ si d (v, vi )2, i=1
where n is the number of vertices, s = ∑ si the total weight, and d (v, vi ) the distance of vertex v to vertex vi. It is assumed that s > 0. In classical mechanics, the moment of inertia is a measure of a rigid body's resistance to rotational acceleration about an axis. But of course, it is out of question to rotate a graph around a vertex. However, in architectural terms, the moment of inertia is a good indication of the relative heaviness of a building. A large room in the centre contributes very little to the moment of inertia while the same room far from the centre leads to big numbers. I0 is defined as half the weighted average of the individual moments of inertia:
I0 =
1 2s
n
∑ si Ivi = i=1
1 2s 2
n
n
∑ ∑ si s j d (vi, v j )2. i=1 j=1
I0 can be regarded as a global covariant which expresses the quality of the construction, its compactness, relative ease of going from one room to another, etc. For a distribution of mass in space, the centre of gravity is the unique point with the property that the weighted position vectors relative to this point sum to zero. In the case of a rigid body, it is well known that its motion through space can be described completely in terms the motion of the total mass concentrated at the centre of gravity and rotation of the object about its centre of gravity. Interestingly, the centre of gravity is also the point u where the
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moment of inertia is minimal. In fact, one shows that Iv = Iu + d (u, v )2 for any other point (see Coray and Pellegrino [2]). Then it is easy to see that the moment of inertia at the centre of gravity is equal to I0. However, in the case of graphs, we can talk about vertices where the moment of inertia is minimal, but there is no other equivalent of a centre of gravity (see [4] for a detailed discussion). For the G SP (16) in Fig. 7, the moments of inertia with respect to all the vertices are given by (4.54, 2.70, 3.77, 7.83, 6.02, 4, 3.92, 5.86, 5.71, 11.49, 6.66, 8.16, 5.55, 4.05, 2.73, 10.70). Also, I0 ¼2.61. Clearly the rooms in the centre contributed less to the moments (e.g. R15, R2) in comparison to the rooms in the periphery (e.g. R10, R16).
3. Best solution associated with a covariant We have seen in the Introduction that 85 FPS can be generated for a given set of data. With the change of spiral used for each FRS, it is interesting to see that, the invariants defined in Section 2 behave as covariants for a FPS. Here we proceed with a standard example and obtain a best solution among 85 solutions on the basis of some of the covariants (e.g. area and moment of inertia). Fig. 8. A computer-generated minimum area F SP (16) .
3.1. The minimum and maximum area solutions In this section, from 85 possible FPS, we pick two, one with minimum area and one with maximum area. To obtain these two solutions, we first compute the area of all FPS. After obtaining the minimum and maximum values, the spiral used for each FRS is noted. After establishing the position and the spiral for each FRS, we construct the two required FPS. Consider the same set of data used to construct the F SP (16) in Fig. 3. We have given 16 rooms Ri ( i = 1, … , 16) having areas 48, 48, 48, 48, 24, 24, 6, 6, 144, 72, 48, 48, 48, 72, 72, 12 respectively and ratio between the width and length of each room is 1.618. The position for each group is as follows (these groups are formed on the basis of weighted adjacency matrix which is given by architects so that the adjacency relations among the rooms can be well established and satisfied; the weighted adjacency matrix for the F SP (16) in Fig. 3 is illustrated in Table 1, for details refer to [5]): 1. 2. 3. 4.
Using the developed prototype (see Section 4), we found that for these specific groups and their fixed positions, there are 1024 solutions with minimum area equivalent to 986.Since it is not feasible to display all the solutions, we select one and display it as an example (see Fig. 8). For this solution, the spirals for the central, left, upper, right and lower groups are spiral6, spiral8, spiral6, spiral6, spiral8 respectively. For the same groups and the same positions, there are 1024 solutions having maximum area equivalent to 1501.2. Again, only one solution is chosen and put forth as an example (see Fig. 9). From this solution, the spirals for the central, left, upper, right and lower groups are spiral8, spiral8, spiral6, spiral8, spiral6 respectively. 3.2. The minimum and maximum of minimum(moments of inertia)
Central group: R9, R16 , Left group: R1, R2, R7, R6, R15, Upper group: R 3, R 4 , R8, R5, Right group: R10, R11,
For the moments of inertia of the G SP (16) shown in Fig. 7 (relative to each of the 16 vertices), we determine their minimum. By doing this, we would have 85 values. Further, we compute minimum and maximum of these 85 values i.e. we obtain min(min
Table 1 A weighted adjacency matrix of order 16.
R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 R11 R12 R13 R14 R15 R16
5. Lower group: R12, R13, R14 .
R1
R2
R3
R4
R5
R6
R7
R8
R9
R10
R11
R12
R13
R14
R15
R16
0 8 6 6 8 6 9 6 4 5 3 2 2 2 8 6
8 0 6 6 8 6 9 6 4 5 3 2 2 2 8 6
6 6 0 8 6 8 6 9 4 4 3 6 6 4 4 6
6 6 8 0 6 8 6 9 4 4 3 4 4 4 4 6
8 8 6 6 0 6 9 10 2 2 2 2 2 2 4 2
6 6 8 8 6 0 10 9 6 2 2 2 2 2 4 2
9 9 6 6 9 10 0 6 2 2 2 2 2 2 4 2
6 6 9 9 10 9 6 0 6 6 4 4 4 4 4 6
4 4 4 4 2 6 2 6 0 8 6 6 6 6 6 9
5 5 4 4 2 2 2 6 8 0 10 4 4 4 6 4
3 3 3 3 2 2 2 4 6 10 0 2 2 2 4 4
2 2 6 4 2 2 2 4 6 4 2 0 8 10 2 9
2 2 6 4 2 2 2 4 6 4 2 8 0 10 2 9
2 2 4 4 2 2 2 4 6 4 2 10 10 0 2 4
8 8 4 4 4 4 4 4 6 6 4 2 2 2 0 6
6 6 6 6 2 2 2 6 9 4 4 9 9 4 6 0
Fig. 9. A computer-generated maximum area F SP (16) .
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computer-aided architectural designs (see [13,14]). In 2008, Terzidis [15] developed a computer program called autoPLAN that generates architectural plans. The program was written in the Processing computer language. In this section, we present a prototype called by CPAD(computer-generated plus-shape architectural design) which provides a FPS, corresponding GPS and all associated covariants (defined in Section 2) for a given set of data. 4.1. Processing language and Introduction
Fig. 10. A computer-generated F SP (16) with min(min(moments of inertia)).
(moments of inertia)) and max(min(moments of inertia)) for all 85 possibilities. After having the position and spiral for each group corresponding to min(min(moments of inertia)) and max(min (moments of inertia)) solutions respectively, we construct the required FPS. As an example, consider the same set of data used to construct the F SP (16) in Fig. 3. Using the developed prototype, we found that there are 64 solutions with min(min(moments of inertia)) which is equivalent to 1.79. The spirals for the central, left, upper, right and lower groups are respectively spiral7, spiral3, spiral3, spiral1, spiral1 and the area for the corresponding F SP (16) is 1252.32 (see Fig. 10). Similarly, we found that there are 8 solutions with max(min (moments of inertia)) which is equivalent to 3.89. The spirals for the central, left, upper, right and lower groups are respectively spiral1, spiral6, spiral5, spiral8, spiral8. And the area for the corresponding F SP (16) is 1165.68 (see Fig. 11).
4. A prototype for the architectural designs and associated covariants In 1974, Dietz [11] proposed the automatic generation of floor plans. The process involves a unit system, a site, a program, and an adjacency matrix and the computer system produces multiple solutions by trying various combinations of space allocation. In 2004, Kalay [12] mentioned various approaches related to the
Processing is an open source programming language and environment for people who want to create images, animations, and interactions (see [16]). It was developed by Casey Reas and Benjamin Fry, both formerly of the Aesthetics and Computation Group at the MIT Media Lab. Software written using Processing is in the form of the so-called sketches. These sketches are written in a specific text editor, which can have lots of tabs to manage different files. For syntax and other details about the Processing, refer to Terzidis [17]. After trying various other systems, we have written our code in the Processing. This code is subdivided into several components to make it more comprehensive and each component is written in a separate file (tab). CPAD has two external files and 16 tabs in total; one external file for input and another one for output. 4.2. Input for the CPAD The input for the code is extracted from an external file input. txt. While writing the code, input.txt is kept external so that it can be more user-friendly. The file has the following six different inputs given by the architect: 1. 2. 3. 4.
Weighted adjacency matrix Area of each room The ratio of width over length for each room Change of a room The formation of groups is done by using an algorithm given in [5] but it can sometimes happen that one is not pleased with the formed groups. Therefore, we kept an option which enables one to move a room from one group to another. 5. Position of the groups 6. Spiral for each group 4.3. Output of the CPAD CPAD is made up of 16 tabs which are displayed in Fig. 12. When the button ‘run’ is pressed, it opens a new window with a plus-shape floor plan and its adjacency graph as illustrated in Fig. 13. At the same time, a new file output.txt is generated, which contains the following results: 1. 2. 3. 4. 5. 6.
Fig. 11. A computer-generated F SP (16) with max(min(moments of inertia)).
The width and length of each room All the five groups and their members The number of inner and outer extra spaces The area of FPS The total area of all the extra spaces The adjacency matrix
7. The number of edges 8. The degrees of all rooms, their mean, standard deviation, dispersion, maximum and minimum 9. The eigenvalues of adjacency matrix and the corresponding polynomial 10. Whether the GPS is bipartite or not
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Fig. 12. Screen of CPAD with 16 tabs.
11. The distance matrix and the mean, standard deviation, dispersion, maximum and minimum of all distances 12. A shortest path between each pair of rooms 13. All the cut vertices and cut pairs 14. The eccentricities of all rooms, their mean, standard deviation, and dispersion. The radius, diameter and centre of GPS 15. The moments and their mean, standard deviation, dispersion, maximum and minimum 16. The chromatic numberTo run CPAD, Processing language is required which can be downloaded from [16]. The other details about CPAD are given in Table 2.
5. Discussion The work described in this paper can be seen as a part of a larger research which aims at developing design aids for architects
to be incorporated in intelligent CAD systems. Current work is concerned with the automated generation of layouts and associated covariants for the given set of data. The idea is to develop algorithms that are able to generate and compare candidate solutions according to predefined requirements, which can be further improved and adjusted by the architect, to provide a better solution to the user. In this paper, we have presented a software prototype that generates a plus-shape floor plan and its adjacency graph and computes the associated graph covariants. Here, we only considered plus-shape floor plans but this work is not restricted to a particular shape. By making small changes in the code of CPAD, many more different shape floor plans can be generated. For example, refer Fig. 14 with T-shape floor plan. In future, our aim is to cover many more aspects of the architectural designs, for example views, daylighting, doors and windows positioning, etc. The core emphasis of this work is on the covariants. They could
Fig. 13. A computer-generated file by CPAD.
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Table 2 Code metadata description. 1 2 3 4 5 6 7
Current code version Permanent link to code/repository used for this code version Legal Code License Code versioning system used Software code languages, tools, and services used Compilation requirements, operating environments and dependencies If available Link to developer documentation/manual
v1.0 krishnendrashekhawat.github.io/CPAD MIT git Processing Libraries (Jama and jgrapht) github.com/krishnendrashekhawat/ CPAD/blob/master/Documentation.pdf
architectural designs.
Acknowledgement This work was carried out during the research project ‘Formalisation et sens du projet architectural’ which was funded by the Swiss National Science Foundation (subsidy no. K-12K1120593). I am very grateful to Professors Daniel Coray and Pierre Pellegrino, University of Geneva, for their support and guidance. Fig. 14. A computer-generated T-shape floor plan.
be very helpful in the classification of two different architectural designs. For example, consider the FPS in Figs. 3 and 6. Clearly, they are geometrically different but it is hard to compute the amount by which they differ. Therefore, it is very interesting to compare them on the basis of different covariants, some of which are as follows: 1. The areas of FPS in Figs. 3 and 6 are 109480.19 and 102970.83 respectively. Clearly, the FPS in Fig. 6 has lesser area than the FPS in Fig. 3. 2. The number of edges of GPS corresponding to the FPS in Figs. 3 and 6 is 29 and 30 respectively. If connectivity is compared on the basis of number of edges (when the number of vertices are same), then the FPS in Fig. 6 is more connected than the FPS in Fig. 3. 3. The FPS in Fig. 6 has only one cut vertex i.e. R15. It means that by deleting R15, the FPS in Fig. 6 can be partitioned into two parts where the rooms R3, R4, R5 and R8 of first part cannot be reached by any other room of the second part through a path and vice-versa. FPS in Fig. 3 has two cut vertices i.e. R9 and R15. Generally, the topological constraints are given in terms of adjacency among the rooms. From this work, we can say that they can be further extended to some of the related covariants like the shortest path, the degree of each room, etc. to have a more realistic view of architecture. Also new measures (e.g. the moment of inertia) can be introduced to compare the quality of two
References [1] E. Nanchen, Distances entre classes d'isomorphisme de graphes (Thèse no. 3012), Section de Mathématiques, Université de Genève, 1998. [2] D. Coray, P. Pellegrino, E.P. Jeanneret, Arquitectura e Informática, Gustavo Gili, Barcelona, 1999. [3] P. Pellegrino et al., Formalisation et sens du projet architectural, 〈http://p3.snf. ch/Project-120593〉, Université de Genève, 2008-10. [4] D. Coray, Les invariants associés aux graphes en architecture, Preprint, Section de Mathématiques, Université de Genève, 2009, pp. 11–16. [5] K. Shekhawat, Automated space allocation using mathematical techniques, Ain Shams Eng. J. (2015) ASEJ402. [6] K. Shekhawat, Algorithm for constructing an optimally connected rectangular floor plan, Front. Archit. Res. 3 (2014) 324–330. [7] J.L. Gross, J. Yellen, Graph Theory and Its Applications, Second ed., Chapman & Hall/CRC, Boca Raton, 2006. [8] S. Pemmaraju, S. Skiena, Computational Discrete Mathematics (Combinatorics and Graph Theory with Mathematica), Cambridge University Press, New York, USA, 2003. [9] N. Biggs, Algebraic Graph Theory, Second ed., Cambridge University Press, New York, USA, 1993. [10] L.W. Beineke, R.J. Wilson, Topics in Algebraic Graph Theory, Cambridge University Press, Cambridge, United Kingdom, 2004. [11] A. Dietz, Dwelling House Construction, MIT Press, Cambridge, 1974. [12] Y.E. Kalay, Architecture's New Media (Principles, Theories, and Methods Of Computer-Aided Design), The MIT Press, USA, 2004. [13] S.A. Arvin, D.H. House, Modeling architectural design objectives in physically based space planning, Autom. Construct. 11 (2) (2002) 213–225. [14] L. March, P. Steadman, The Geometry of Environment (An Introduction to Spatial Organization in Design), Methuen & Co Ltd., London, 1971. [15] K. Terzidis, AutoPLAN: a stochastic generator of architectural plans from a building program, Form ·Z Joint Study Journal, 2007–2008, pp. 84–87. [16] 〈www.processing.org〉. [17] K. Terzidis, Algorithms For Visual Design Using the Processing Language, Wiley Publishing, Inc, Indianapolis, 2009.
Contents lists available at ScienceDirect
Journal of Building Engineering journal homepage: www.elsevier.com/locate/jobe
Computer-aided architectural designs and associated covariants Krishnendra Shekhawat n Department of Mathematics, University of Geneva, Geneva, Switzerland
art ic l e i nf o
a b s t r a c t
Article history: Received 18 April 2015 Received in revised form 7 July 2015 Accepted 15 July 2015 Available online 17 July 2015
To compare two architectural designs or to characterize them, some numbers are needed. These numbers are said to be covariants. In this paper, we present a software prototype that generates a floor plan design and its adjacency graph for a given set of data, and computes some mathematical covariants associated with the obtained design. In addition, we discuss and demonstrate the usefulness of covariants in comparing the architectural designs and in obtaining a best design among many possible solutions. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Covariants Floorplan Graph Prototype Rooms
1. Introduction The concept of an invariant appeared naturally in geometry in response to the need for the classification of figures. In concrete terms, what distinguishes the following three sub-figures in Fig. 1? Geometrically, the first sub-figure is a straight line, the second subfigure a rectangle and the third one a spiral. A general line in the same plane intersects in at most 1 and 2 points respectively for the first two figures, while it meets the spiral in infinitely many points. Although the numbers 1, 2 and ∞ do not fully describe these figures, yet they characterize and distinguish them from other geometrical shapes. Interestingly, these numbers remain unchanged if we apply certain geometric transformations, like isometries or scale changes. Thus the number of intersection points is an invariant with respect to a restricted set of transformations, which can be specified. If a number or a mathematical object associated with a geometric configuration remains unchanged with respect to a certain group of transformations then this number or mathematical object is said to be an invariant (with respect to the specified group of transformations). For the shapes in Fig. 1, the angles of intersection are certainly not invariant under rotation, but we can say how they behave. Therefore, the angles of intersection are relative invariants or covariants, with respect to rotation. If we know precisely how a number or a mathematical object n
Correspondence address: Present address: CIAUD, Faculdade de Arquitectura, Universidade de Lisboa, Portugal. E-mail address: [email protected] http://dx.doi.org/10.1016/j.jobe.2015.07.005 2352-7102/& 2015 Elsevier Ltd. All rights reserved.
associated with a geometric figure behaves with respect to a specified set of transformations, we call it a covariant. For a better understanding, consider the rectangular floor plan in Fig. 2A. After rotating it by 90°, as shown in Fig. 2B, its area remains unchanged while its width (dimension measured horizontally) and length (dimension measured vertically) are changed. Hence the area of the rectangular floor plan is an invariant while its width and length are covariants, with respect to the transformation rotation. In the literature, there exist some work where the concepts of invariants and covariants were introduced and studied [1–4]. In this paper, we will explore more about them. Also, we will demonstrate their usefulness in identifying and classifying the different architectural designs, which will present them as a very powerful tool for the architects. To proceed further, let us consider the plus-shape floor plan, illustrated in Fig. 3, which is generated using the algorithms given in [5]. A plus-shape, as shown in Fig. 4A, can be visualized as a shape made up of 5 rectangles, as illustrated in Fig. 4B. In this way, the plus-shape floor plan in Fig. 3 is generated by adjoining 5 rectangular blocks where each of them is best connected1 and is constructed using the spiral-based algorithm given in [6]. Therefore, we call the plus-shape floor plan described in this paper by spiral-based plus-shape floor plan and represent it by F SP (m) (floorplan plus-shape spiral-based) where m is the number of 1 A rectangular floor plan is best connected if it has 3n − 7 edges in its adjacency graph where n is the number of rooms i.e. there does not exist any other rectangular floor plan with n rooms whose adjacency graph has more than 3n − 7 edges (for details, refer [5]).
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Fig. 1. Understanding the concept of invariants and covariants.
Fig. 5. 8 different FRS congruent to each other.
Fig. 2. Invariants and covariants associated with a rectangular floor plan.
Fig. 3. A computer-generated F SP (16) .
different size rectangular pieces (without changing the width and length of rectangular pieces), undoubtedly there would be some empty spaces (or extra spaces) inside the produced rectangle. Therefore, we can see the presence of empty spaces as white rectangles inside the F SP (16) in Fig. 3, which we call by inner extra spaces. If 5 different size FRS are adjoined to generate a FPS, again there would be some empty spaces, which we call by outer extra spaces, as shown in Fig. 3. In [6], it has been illustrated that a FRS is congruent to 7 other FRS which are best connected and one can be derived from other by four types of mappings i.e. translations, reflections, rotations, and glide reflections (the last being a combination of a translation and a reflection). For better understanding, refer to Fig. 5 with eight congruent FRS (with no extra spaces), called by spiral1, spiral2, spiral3, spiral4, spiral5, spiral6, spiral7, and spiral8 FRS respectively. As an example, we can see that the F SP (16) in Fig. 3 is constructed by considering spiral8, spiral6, spiral4, spiral2 and spiral1 FRS for the central, left, upper, right and lower positions respectively (these positions are illustrated in Fig. 4B). From Fig. 3, we can notice that, for the construction of a FPS, five R FS are required which are placed at five different positions and each FRS can be generated in eight different ways, therefore, for a given set of data, 85 = 32, 768 FPS can be generated, which is a big number. For example, in Fig. 6, a FPS is generated from spiral1, spiral2, spiral3, spiral1 and spiral6 FRS for the central, left, upper,
Fig. 4. A plus-shape rectilinear polygon and its division into rectangles.
rooms given by the architects. Also, its graph is represented by G SP (m) (graph plus-shape spiral-based). A spiral-based rectangular block (or floor plan) is denoted by FRS (floorplan rectangular spiralbased). If it is required to produce a larger rectangle by arranging
Fig. 6. A computer-generated F SP (16) .
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right and lower positions respectively, which is geometrically different from the FPS in Fig. 3. But, for an architect, it would be very difficult to single out one solution out of so many possibilities. Also, it is not feasible to select a good or a best solution without some efficient criteria. Therefore, in this work, we present some mathematical covariants associated with the architectural designs. These covariants have the following functions: 1. They help to study and understand the problem. 2. They characterize and measure the difference between different solutions. 3. They are used to obtain the best solution among numerous solutions.This paper is divided into two main parts. The first part is concerned with the covariants associated with the architectural designs and the second part is about the prototype of a software which is developed to generate the required architectural designs and the associated covariants. For the explanation of the concepts presented in this paper, we only consider the plusshape floor plans.
2. Invariants and covariants associated with an architectural design In this section we study some graph invariants and covariants. Adjacency among the rooms that are present inside the FPS has already been defined in [5]. Using the definition of adjacency, G SP (16) corresponding to the F SP (16) in Fig. 3 is illustrated in Fig. 7. The readers who are not familiar with the definitions related to graphs (used in the upcoming subsections) are referred to Gross and Yellen [7]. 2.1. Adjacency matrix and degree of vertices The power of linear algebra is applied to graph theory through representation of graphs by matrices. An adjacency matrix is the first graph invariant to be elaborated. The adjacency matrix of a simple graph G, denoted by AG, is the symmetric matrix whose rows and columns are indexed by VG (in the same order) and is such that
AG [u, v] = 1
if u and v are adjacent, otherwise it is equal to 0.
For example, from the G SP (16) in Fig. 7, it's adjacency matrix can easily be derived using the definition given above. The degree (valency) of a vertex v in a graph G, denoted by deg(v ), is the number of vertices adjacent to v. The degree of each vertex is computed by adding the number of 1's in the corresponding row (or column) of the adjacency matrix. For the G SP (16) in Fig. 7, the degrees of all the vertices, which correspond to the rooms Ri ( i = 1, … , 16), are 5, 7, 4, 3, 3, 4, 4, 4, 3, 3, 2, 2, 3, 3, 7, 1 respectively. To study the distribution of degrees, we consider the methods
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similar to those commonly used in probability and statistics, namely mean, standard deviation, measures of dispersion, minimum and maximum. For real numbers {x1, x2, … , x n }, the arithmetic mean is denoted by x = (1/n) ∑ni=1 x i . The standard deviation σ = (1/n) ∑ni=1 (x i − x )2 gives the variation (or dispersion) from the corresponding mean. A low value of the standard deviation shows that the data points tend to be very close to the mean, whereas a high value of the standard deviation can sometimes indicate that the data points are spread out over a large range of values. The measure of statistical dispersion is a non-negative real number that is zero if all the data have the same value and increases if the data become more diverse. In percentage terms, it is given as ρ = (σ /x ) × 100. For all the degrees corresponding to the G SP (16) in Fig. 7, x = 3.62, s ¼1.58, ρ ¼43.48%, maximum ¼7 and minimum ¼ 1. 2.2. The distance matrix and shortest path The classic example of a real life graph is a network of roads connecting cities. One of the important algorithmic problems on such road networks is finding the shortest path between any two given vertices. For a graph G, the distance d (x, y ) between two vertices x and y is the length of the shortest path from x to y, considering all possible paths in G from x to y. The distance between any node and itself is 0 and if there is no path from x to y, then d (x, y ) is infinity. It is to be noted that for the distance and shortest path, we consider only non-weighted graphs. The distance matrix is a matrix (two-dimensional array) containing the distances, taken pairwise, of the set of vertices. This matrix has order n n, where n is the number of vertices. Once we know the adjacency matrix, we can obtain a shortest path (and its length) between any two vertices of the GPS and the distance matrix by using Floyd's Algorithm (see [8, Section A.2, Chap. 8]). Related to the G SP (16) in Fig. 7, some of the results are as follows: The distance between R1 and R4 is 3 and the distance between R5 and R12 is 4. The shortest path between R1 and R4 is R1 → R2 → R5 → R 4 and the shortest path between R5 and R12 is R5 → R2 → R15 → R13 → R12. 2.3. Eccentricity, radius, diameter and centre In this section, we discuss several graph invariants related to the all-pairs shortest-path matrix. The eccentricity of a graph vertex v in a graph G, denoted ecc(v) is the distance from v to a vertex farthest from v. That is,
ecc (v) = max {d (v, x)} x ∈ VG
The maximum eccentricity is the graph diameter, denoted by diam(G). That is,
diam (G) = max {ecc (x)} = max {d (x, y)} x ∈ VG
x, y ∈ VG
The minimum graph eccentricity is called the graph radius, denoted by rad(G). That is,
rad (G) = min {ecc (x)} x ∈ VG
Fig. 7. G SP (16) corresponding to the F SP (16) demonstrated in Fig. 3.
A central vertex of a graph G is a vertex with minimum eccentricity. Thus ecc (v ) = rad (G ).
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The centre of the graph is the set of all vertices with minimum eccentricity. For the G SP (16) in Fig. 7, the eccentricities of all the vertices corresponding to all the rooms are (4, 3, 3, 4, 4, 3, 3, 4, 4, 4, 4, 5, 4, 4, 3, 5) with radius¼3, diameter¼ 5 and centre {R2, R 3, R6, R7, R15 }.
independent from the other covariants. 2.6. Bipartite graphs A graph G = (V , E ) is called bipartite if V admits a partition into two classes such that every edge has one end in each class: vertices in the same partition class must not be adjacent.
2.4. Cut vertex A cut vertex of a connected graph G is a vertex whose removal renders G disconnected. Any graph with no cut vertices is said to be biconnected. Biconnectivity is an important property due to many reasons. For example, Menger's Theorem implies that any graph with at least 3 vertices is biconnected if and only if there are at least 2 vertex disjoint paths between any pair of vertices. If looked from the perspective of communication networks, the biconnected networks are more fault-tolerant since blowing away any single node does not cut off communication for any other node. To obtain the cut vertices, we eliminate each vertex, say vi, one by one and obtain the distance matrix for the remaining vertices. If any entry in the distance matrix is ∞ then vi is a cut vertex. For the G SP (16) in Fig. 7, the cut vertices are {R15, R9 }. 2.5. Characteristic polynomial and eigenvalues The next graph invariant is the characteristic polynomial of the adjacency matrix and from this polynomial, we derive eigenvalues. The characteristic polynomial of a matrix An × n is the polynomial p A (x ) = det (xI − A), where det is the determinant and I is the n n identity matrix. The eigenvalues of A are the roots of its characteristic polynomial. For the adjacency matrix corresponding to the G SP (16) in Fig. 7, the characteristic polynomial and the eigenvalues are: p (x ) = x16 − 29x14 − 26x13 + 278x12 + 422x11 − 955x10 − 2196x 9 .
+ 763x8 + 4044x7 + 1407x 6 − 2316x5 − 1538x 4 + 242x 3 + 315x 2 + 28x − 4 Eigenvalues: 2.52, 2.37, 2.12, 1.47, 1.12, 1, 0.85, 0.75, 0.22, 0.08, 0.51, 0.85, 1.46, 2.35, 2.79, 4.40. There are a lot of interesting results for the characteristic polynomial and the eigenvalues which can be associated with various geometrical interpretations. Some of the results are as follows. Theorem 1. Suppose p A (x ) is written as
p A (x) = x n + c1x n − 1 + ⋯ + cn. Then 1. c1 = 0. 2. −c2 is the number of edges in the graph G. 3. −c3 is twice the number of triangles in G. Proof. For a proof, see Biggs [9, Proposition 2.3].□ From above Proposition, we can say that the characteristic polynomial of the G SR (n) is of the form z n − E SR (n) z n − 2 − 2D SR (n) z n − 3 + ⋯, where E SR (n) and D SR (n) are the number of edges and number of triangles in the graph of an architectural design. Remark 1. We can see that the covariants defined in Sections 2.1– 2.5 are derived from the adjacency matrix directly or indirectly. In the upcoming sections we will introduce covariants which are
Theorem 2. A graph is bipartite if and only if its spectrum is symmetric about zero. Proof. For a proof, refer to Beineke and Wilson [10, Theorem 5.2].□ By using Theorem 2 for the obtained eigenvalues corresponding to the G SP (16) in Fig. 7, we conclude that the G SP (16) is not bipartite. 2.7. Chromatic number A vertex colouring is an assignment of labels or colours to each vertex of a graph such that no edge connects two identically coloured vertices. The chromatic number of a graph G is the smallest number of colours needed to colour the vertices of G so that no two adjacent vertices share the same colour. The G SP (16) illustrated in Fig. 7 has chromatic number 3. 2.8. Moment of Inertia Now we assume that the graph G is weighted, i.e., each vertex vi has a weight si which is a real number. In the case of an architectural design, the weight can for example be equal to the area of the corresponding room; therefore it is always strictly positive. For each vertex v, the moment of inertia of G relative to this vertex was introduced by Coray in [4,2]. In analogy with classical mechanics, it is given by
Iv =
1 s
n
∑ si d (v, vi )2, i=1
where n is the number of vertices, s = ∑ si the total weight, and d (v, vi ) the distance of vertex v to vertex vi. It is assumed that s > 0. In classical mechanics, the moment of inertia is a measure of a rigid body's resistance to rotational acceleration about an axis. But of course, it is out of question to rotate a graph around a vertex. However, in architectural terms, the moment of inertia is a good indication of the relative heaviness of a building. A large room in the centre contributes very little to the moment of inertia while the same room far from the centre leads to big numbers. I0 is defined as half the weighted average of the individual moments of inertia:
I0 =
1 2s
n
∑ si Ivi = i=1
1 2s 2
n
n
∑ ∑ si s j d (vi, v j )2. i=1 j=1
I0 can be regarded as a global covariant which expresses the quality of the construction, its compactness, relative ease of going from one room to another, etc. For a distribution of mass in space, the centre of gravity is the unique point with the property that the weighted position vectors relative to this point sum to zero. In the case of a rigid body, it is well known that its motion through space can be described completely in terms the motion of the total mass concentrated at the centre of gravity and rotation of the object about its centre of gravity. Interestingly, the centre of gravity is also the point u where the
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moment of inertia is minimal. In fact, one shows that Iv = Iu + d (u, v )2 for any other point (see Coray and Pellegrino [2]). Then it is easy to see that the moment of inertia at the centre of gravity is equal to I0. However, in the case of graphs, we can talk about vertices where the moment of inertia is minimal, but there is no other equivalent of a centre of gravity (see [4] for a detailed discussion). For the G SP (16) in Fig. 7, the moments of inertia with respect to all the vertices are given by (4.54, 2.70, 3.77, 7.83, 6.02, 4, 3.92, 5.86, 5.71, 11.49, 6.66, 8.16, 5.55, 4.05, 2.73, 10.70). Also, I0 ¼2.61. Clearly the rooms in the centre contributed less to the moments (e.g. R15, R2) in comparison to the rooms in the periphery (e.g. R10, R16).
3. Best solution associated with a covariant We have seen in the Introduction that 85 FPS can be generated for a given set of data. With the change of spiral used for each FRS, it is interesting to see that, the invariants defined in Section 2 behave as covariants for a FPS. Here we proceed with a standard example and obtain a best solution among 85 solutions on the basis of some of the covariants (e.g. area and moment of inertia). Fig. 8. A computer-generated minimum area F SP (16) .
3.1. The minimum and maximum area solutions In this section, from 85 possible FPS, we pick two, one with minimum area and one with maximum area. To obtain these two solutions, we first compute the area of all FPS. After obtaining the minimum and maximum values, the spiral used for each FRS is noted. After establishing the position and the spiral for each FRS, we construct the two required FPS. Consider the same set of data used to construct the F SP (16) in Fig. 3. We have given 16 rooms Ri ( i = 1, … , 16) having areas 48, 48, 48, 48, 24, 24, 6, 6, 144, 72, 48, 48, 48, 72, 72, 12 respectively and ratio between the width and length of each room is 1.618. The position for each group is as follows (these groups are formed on the basis of weighted adjacency matrix which is given by architects so that the adjacency relations among the rooms can be well established and satisfied; the weighted adjacency matrix for the F SP (16) in Fig. 3 is illustrated in Table 1, for details refer to [5]): 1. 2. 3. 4.
Using the developed prototype (see Section 4), we found that for these specific groups and their fixed positions, there are 1024 solutions with minimum area equivalent to 986.Since it is not feasible to display all the solutions, we select one and display it as an example (see Fig. 8). For this solution, the spirals for the central, left, upper, right and lower groups are spiral6, spiral8, spiral6, spiral6, spiral8 respectively. For the same groups and the same positions, there are 1024 solutions having maximum area equivalent to 1501.2. Again, only one solution is chosen and put forth as an example (see Fig. 9). From this solution, the spirals for the central, left, upper, right and lower groups are spiral8, spiral8, spiral6, spiral8, spiral6 respectively. 3.2. The minimum and maximum of minimum(moments of inertia)
Central group: R9, R16 , Left group: R1, R2, R7, R6, R15, Upper group: R 3, R 4 , R8, R5, Right group: R10, R11,
For the moments of inertia of the G SP (16) shown in Fig. 7 (relative to each of the 16 vertices), we determine their minimum. By doing this, we would have 85 values. Further, we compute minimum and maximum of these 85 values i.e. we obtain min(min
Table 1 A weighted adjacency matrix of order 16.
R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 R11 R12 R13 R14 R15 R16
5. Lower group: R12, R13, R14 .
R1
R2
R3
R4
R5
R6
R7
R8
R9
R10
R11
R12
R13
R14
R15
R16
0 8 6 6 8 6 9 6 4 5 3 2 2 2 8 6
8 0 6 6 8 6 9 6 4 5 3 2 2 2 8 6
6 6 0 8 6 8 6 9 4 4 3 6 6 4 4 6
6 6 8 0 6 8 6 9 4 4 3 4 4 4 4 6
8 8 6 6 0 6 9 10 2 2 2 2 2 2 4 2
6 6 8 8 6 0 10 9 6 2 2 2 2 2 4 2
9 9 6 6 9 10 0 6 2 2 2 2 2 2 4 2
6 6 9 9 10 9 6 0 6 6 4 4 4 4 4 6
4 4 4 4 2 6 2 6 0 8 6 6 6 6 6 9
5 5 4 4 2 2 2 6 8 0 10 4 4 4 6 4
3 3 3 3 2 2 2 4 6 10 0 2 2 2 4 4
2 2 6 4 2 2 2 4 6 4 2 0 8 10 2 9
2 2 6 4 2 2 2 4 6 4 2 8 0 10 2 9
2 2 4 4 2 2 2 4 6 4 2 10 10 0 2 4
8 8 4 4 4 4 4 4 6 6 4 2 2 2 0 6
6 6 6 6 2 2 2 6 9 4 4 9 9 4 6 0
Fig. 9. A computer-generated maximum area F SP (16) .
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computer-aided architectural designs (see [13,14]). In 2008, Terzidis [15] developed a computer program called autoPLAN that generates architectural plans. The program was written in the Processing computer language. In this section, we present a prototype called by CPAD(computer-generated plus-shape architectural design) which provides a FPS, corresponding GPS and all associated covariants (defined in Section 2) for a given set of data. 4.1. Processing language and Introduction
Fig. 10. A computer-generated F SP (16) with min(min(moments of inertia)).
(moments of inertia)) and max(min(moments of inertia)) for all 85 possibilities. After having the position and spiral for each group corresponding to min(min(moments of inertia)) and max(min (moments of inertia)) solutions respectively, we construct the required FPS. As an example, consider the same set of data used to construct the F SP (16) in Fig. 3. Using the developed prototype, we found that there are 64 solutions with min(min(moments of inertia)) which is equivalent to 1.79. The spirals for the central, left, upper, right and lower groups are respectively spiral7, spiral3, spiral3, spiral1, spiral1 and the area for the corresponding F SP (16) is 1252.32 (see Fig. 10). Similarly, we found that there are 8 solutions with max(min (moments of inertia)) which is equivalent to 3.89. The spirals for the central, left, upper, right and lower groups are respectively spiral1, spiral6, spiral5, spiral8, spiral8. And the area for the corresponding F SP (16) is 1165.68 (see Fig. 11).
4. A prototype for the architectural designs and associated covariants In 1974, Dietz [11] proposed the automatic generation of floor plans. The process involves a unit system, a site, a program, and an adjacency matrix and the computer system produces multiple solutions by trying various combinations of space allocation. In 2004, Kalay [12] mentioned various approaches related to the
Processing is an open source programming language and environment for people who want to create images, animations, and interactions (see [16]). It was developed by Casey Reas and Benjamin Fry, both formerly of the Aesthetics and Computation Group at the MIT Media Lab. Software written using Processing is in the form of the so-called sketches. These sketches are written in a specific text editor, which can have lots of tabs to manage different files. For syntax and other details about the Processing, refer to Terzidis [17]. After trying various other systems, we have written our code in the Processing. This code is subdivided into several components to make it more comprehensive and each component is written in a separate file (tab). CPAD has two external files and 16 tabs in total; one external file for input and another one for output. 4.2. Input for the CPAD The input for the code is extracted from an external file input. txt. While writing the code, input.txt is kept external so that it can be more user-friendly. The file has the following six different inputs given by the architect: 1. 2. 3. 4.
Weighted adjacency matrix Area of each room The ratio of width over length for each room Change of a room The formation of groups is done by using an algorithm given in [5] but it can sometimes happen that one is not pleased with the formed groups. Therefore, we kept an option which enables one to move a room from one group to another. 5. Position of the groups 6. Spiral for each group 4.3. Output of the CPAD CPAD is made up of 16 tabs which are displayed in Fig. 12. When the button ‘run’ is pressed, it opens a new window with a plus-shape floor plan and its adjacency graph as illustrated in Fig. 13. At the same time, a new file output.txt is generated, which contains the following results: 1. 2. 3. 4. 5. 6.
Fig. 11. A computer-generated F SP (16) with max(min(moments of inertia)).
The width and length of each room All the five groups and their members The number of inner and outer extra spaces The area of FPS The total area of all the extra spaces The adjacency matrix
7. The number of edges 8. The degrees of all rooms, their mean, standard deviation, dispersion, maximum and minimum 9. The eigenvalues of adjacency matrix and the corresponding polynomial 10. Whether the GPS is bipartite or not
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Fig. 12. Screen of CPAD with 16 tabs.
11. The distance matrix and the mean, standard deviation, dispersion, maximum and minimum of all distances 12. A shortest path between each pair of rooms 13. All the cut vertices and cut pairs 14. The eccentricities of all rooms, their mean, standard deviation, and dispersion. The radius, diameter and centre of GPS 15. The moments and their mean, standard deviation, dispersion, maximum and minimum 16. The chromatic numberTo run CPAD, Processing language is required which can be downloaded from [16]. The other details about CPAD are given in Table 2.
5. Discussion The work described in this paper can be seen as a part of a larger research which aims at developing design aids for architects
to be incorporated in intelligent CAD systems. Current work is concerned with the automated generation of layouts and associated covariants for the given set of data. The idea is to develop algorithms that are able to generate and compare candidate solutions according to predefined requirements, which can be further improved and adjusted by the architect, to provide a better solution to the user. In this paper, we have presented a software prototype that generates a plus-shape floor plan and its adjacency graph and computes the associated graph covariants. Here, we only considered plus-shape floor plans but this work is not restricted to a particular shape. By making small changes in the code of CPAD, many more different shape floor plans can be generated. For example, refer Fig. 14 with T-shape floor plan. In future, our aim is to cover many more aspects of the architectural designs, for example views, daylighting, doors and windows positioning, etc. The core emphasis of this work is on the covariants. They could
Fig. 13. A computer-generated file by CPAD.
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Table 2 Code metadata description. 1 2 3 4 5 6 7
Current code version Permanent link to code/repository used for this code version Legal Code License Code versioning system used Software code languages, tools, and services used Compilation requirements, operating environments and dependencies If available Link to developer documentation/manual
v1.0 krishnendrashekhawat.github.io/CPAD MIT git Processing Libraries (Jama and jgrapht) github.com/krishnendrashekhawat/ CPAD/blob/master/Documentation.pdf
architectural designs.
Acknowledgement This work was carried out during the research project ‘Formalisation et sens du projet architectural’ which was funded by the Swiss National Science Foundation (subsidy no. K-12K1120593). I am very grateful to Professors Daniel Coray and Pierre Pellegrino, University of Geneva, for their support and guidance. Fig. 14. A computer-generated T-shape floor plan.
be very helpful in the classification of two different architectural designs. For example, consider the FPS in Figs. 3 and 6. Clearly, they are geometrically different but it is hard to compute the amount by which they differ. Therefore, it is very interesting to compare them on the basis of different covariants, some of which are as follows: 1. The areas of FPS in Figs. 3 and 6 are 109480.19 and 102970.83 respectively. Clearly, the FPS in Fig. 6 has lesser area than the FPS in Fig. 3. 2. The number of edges of GPS corresponding to the FPS in Figs. 3 and 6 is 29 and 30 respectively. If connectivity is compared on the basis of number of edges (when the number of vertices are same), then the FPS in Fig. 6 is more connected than the FPS in Fig. 3. 3. The FPS in Fig. 6 has only one cut vertex i.e. R15. It means that by deleting R15, the FPS in Fig. 6 can be partitioned into two parts where the rooms R3, R4, R5 and R8 of first part cannot be reached by any other room of the second part through a path and vice-versa. FPS in Fig. 3 has two cut vertices i.e. R9 and R15. Generally, the topological constraints are given in terms of adjacency among the rooms. From this work, we can say that they can be further extended to some of the related covariants like the shortest path, the degree of each room, etc. to have a more realistic view of architecture. Also new measures (e.g. the moment of inertia) can be introduced to compare the quality of two
References [1] E. Nanchen, Distances entre classes d'isomorphisme de graphes (Thèse no. 3012), Section de Mathématiques, Université de Genève, 1998. [2] D. Coray, P. Pellegrino, E.P. Jeanneret, Arquitectura e Informática, Gustavo Gili, Barcelona, 1999. [3] P. Pellegrino et al., Formalisation et sens du projet architectural, 〈http://p3.snf. ch/Project-120593〉, Université de Genève, 2008-10. [4] D. Coray, Les invariants associés aux graphes en architecture, Preprint, Section de Mathématiques, Université de Genève, 2009, pp. 11–16. [5] K. Shekhawat, Automated space allocation using mathematical techniques, Ain Shams Eng. J. (2015) ASEJ402. [6] K. Shekhawat, Algorithm for constructing an optimally connected rectangular floor plan, Front. Archit. Res. 3 (2014) 324–330. [7] J.L. Gross, J. Yellen, Graph Theory and Its Applications, Second ed., Chapman & Hall/CRC, Boca Raton, 2006. [8] S. Pemmaraju, S. Skiena, Computational Discrete Mathematics (Combinatorics and Graph Theory with Mathematica), Cambridge University Press, New York, USA, 2003. [9] N. Biggs, Algebraic Graph Theory, Second ed., Cambridge University Press, New York, USA, 1993. [10] L.W. Beineke, R.J. Wilson, Topics in Algebraic Graph Theory, Cambridge University Press, Cambridge, United Kingdom, 2004. [11] A. Dietz, Dwelling House Construction, MIT Press, Cambridge, 1974. [12] Y.E. Kalay, Architecture's New Media (Principles, Theories, and Methods Of Computer-Aided Design), The MIT Press, USA, 2004. [13] S.A. Arvin, D.H. House, Modeling architectural design objectives in physically based space planning, Autom. Construct. 11 (2) (2002) 213–225. [14] L. March, P. Steadman, The Geometry of Environment (An Introduction to Spatial Organization in Design), Methuen & Co Ltd., London, 1971. [15] K. Terzidis, AutoPLAN: a stochastic generator of architectural plans from a building program, Form ·Z Joint Study Journal, 2007–2008, pp. 84–87. [16] 〈www.processing.org〉. [17] K. Terzidis, Algorithms For Visual Design Using the Processing Language, Wiley Publishing, Inc, Indianapolis, 2009.