Nikos A. Salingaros

A Universal Rule for the Distribution of Sizes

Human artifacts, ranging from small objects all the way up to large buildings and cities, display a variety and range of subdivisions. Repeating structural and design elements of the same size will define a particular scale. Most pleasing designs obey an inverse power-law distribution: the product of the relative multiplicity p of a substructure with an algebraic power of its size x is a constant, pxμ = constant, which is derived here from basic principles. This means that the logarithmic plot of p versus x has a slope of −μ, where typically 1 ≤ μ ≤ 2, and this is a widely observed relationship in both the natural and the social sciences. Departures from this rule apparently result in incoherent, alien structures. Three applications to urban systems are proposed. It is argued that in living cities (1) the distribution of path lengths, (2) the allocation of project funding, and (3) the distribution of built elements all follow the derived multiplicity rule. By violating all three, modernist cities create inhuman urban environments.

Theory of the urban web

This paper identifies fundamental processes behind urban design. Rules are derived from connective principles in complexity theory, pattern recognition and artificial intelligence. Any urban setting can be decomposed into human activity nodes and their interconnections. The connections are then treated as a mathematical problem (here in a qualitative manner). Urban design is most successful when it establishes a certain number of connections between activity nodes. Mathematics itself depends upon establishing relationships between ideas, this ability being a central component of the intelligence of human beings. The creation of the built environment is driven by forces analogous to those that lead us to do mathematics.

Complexity and Urban Coherence

Structural principles developed in biology, computer science and economics are applied here to urban design. The coherence of urban form can be understood from the theory of complex interacting systems. Complex large-scale wholes are assembled from tightly interacting subunits at many different levels of scale, in a hierarchy going down to the natural structure of materials. A variety of elements and functions at the small scale is necessary for large-scale coherence. Dead urban and suburban regions may be resurrected in part by reconnecting their geometry. If these suggestions are put into practice, new projects could even approach the coherence that characterizes the best-loved urban regions built in the past. The proposed design rules differ radically from ones in use today. In a major revision of contemporary urban practice, it is shown that grid alignment does not connect a city, giving only the misleading impression of doing so. Although these ideas are consistent with the New Urbanism, they come from science and are independent of traditional urbanist arguments.

Realization, extension, and classification of certain physically important groups and algebras

An associative algebra of differential forms with division has been constructed. The algebra of forms in each different space provides a practical realization of the universal Clifford algebra of that space. A classification of all such algebras is given in terms of two distinct types of algebras Nk and Sk. The former include the dihedral, quaternion, and Majorana algebras; the latter include the complex, spinor, and Dirac algebras. The associative product expresses Hodge duality as multiplication by a basis element. This makes possible the realization of higher order algebras in a calculationally useful algebraic setting. The fact that the associative algebras, as well as the enveloped Lie algebras, are precisely those arising in physics suggests that this formalism may be a convenient setting for the formulation of basic physical laws.

Urban space and its information field

Abstract This paper proposes an essentially new theory of urban space based on information theory and the laws of optics. The use of urban space is linked to the information field generated by surrounding surfaces, and on how easily the information can be received by pedestrians. Historical building exteriors usually present a piecewise concave, fractal aspect, which optimizes visual and acoustical signals that transmit information content. Successful urban spaces also offer tactile information from local structures meant for standing and sitting. The total information field in turn determines the optimal positioning of pedestrian paths and nodes. This complex interaction between human beings and the built environment, incredibly neglected in our times, explains why so many historical urban spaces provide an emotionally nourishing environment.

On the classification of Clifford algebras and their relation to spinors in n dimensions

A classification of all the Clifford algebras is given in terms of Kronecker products of the quaternion and dihedral groups. The relationship to spinors in n dimensions is explicitly determined. We show that the real Clifford algebra in Minkowski spacetime is distinct from both the algebra of Dirac matrices and the algebra of Majorana matrices, and cannot be realized by the spinor framework. The matrix representations of Clifford algebras are discussed, and are utilized to give a classification of the real forms of Lie algebras. We are thus able to relate Clifford, Lie, and spinor algebras in an intrinsic geometrical setting.

The relationship between finite groups and Clifford algebras

Clifford algebras are traditionally realized in terms of a specific set of representation matrices. This paper provides a more effective alternative by giving the finite group associated with each Clifford algebra. All the representation‐independent algebraic results, which are really direct consequences of the underlying group structure, can thus be derived in an easier and more general manner. There are five related but distinct classes of finite groups associated with the Clifford algebras. These groups are constructed from the complex, cyclic, quaternion, and dihedral groups in a way which is discussed here in detail. Of particular utility is a table which lists the order structure of each group: this permits the immediate identification of any Clifford algebra in any dimension.

Algebras with three anticommuting elements. I. Spinors and quaternions

A general construction of alternative algebras with three anticommuting elements and a unit is given. As an exhaustive result over the real and complex fields, we obtain the Clifford algebras H (quaternions), N1 (dihedral Clifford algebra which is related to real 2‐spinors), and S1 (algebra of Pauli matrices which is related to complex 2‐spinors). What is important is that the algebras N1 and S1 possess inverses everywhere except on a region akin to the light cone of the Minkowski space, while the quaternion algebra H has inverses everywhere except at the zero element. We discuss the reasons why the three algebras N1, H, and S1 are so difficult to distinguish in the representation space of 2×2 complex matrices.

A pattern measure

In this paper we propose numerical measures for evaluating the aesthetic interest of simple patterns. The patterns consist of elements (symbols, pixels, etc) in regular square arrays. The measures depend on two characteristics of the patterns: the number of different types of element, and the number of symmetries in their arrangement. We define two complementary composite measures L and C for the degree of pattern in a design, and compute them here for 2 × 2 and 6 × 6 arrays. The results distinguish simple from high-variation cases. We suspect that the measure L corresponds to the degree that human beings intuitively feel a design to be “interesting”, so this model would aid in quantifying the visual connection of two-dimensional designs with viewers. The other composite measure C based on these numerical properties characterizes the extent of randomness of an array. Combining symbol variety with symmetry calculations allows us to employ hierarchical scaling to count the relative impact of different levels of scale. By identifying substructures we can distinguish between organized patterns and disorganized complexity. The measures described here are related to verbal descriptors derived from work by psychologists on responses to visual environments.

The Laws of Architecture From a Physicist's Perspective

Three laws of architectural order are obtained by analogy from basic physical principles. They apply to both natural and man-made structures. These laws may be used to create buildings that match the emotional comfort and beauty of the worlds great historical buildings. The laws are consistent with Classical, Byzantine, Gothic, Islamic, Eastern, and Art Nouveau architectures; but not with the modernist architectural forms of the past 70 years. It seems that modernist 20th century architecture intentionally contradicts all other architectures in actually preventing structural order.