Beurling zeta functions, generalised primes, and fractal membranes

We study generalised prime systems P (1 < p(1) <= p(2) <= ..., with p(j) is an element of R tending to infinity) and the associated Beurling zeta function zeta p(s) = Pi(infinity)(j=1)(1 - p(j)(-s))(-1). Under appropriate assumptions, we establish various analytic properties of zeta p(s), including its analytic continuation, and we characterise the existence of a suitable generalised functional equation. In particular, we examine the relationship between a counterpart of the Prime Number Theorem (with error term) and the properties of the analytic continuation of zeta p(s). Further we study 'well-behaved' g-prime systems, namely, systems for which both the prime and integer counting function are asymptotically well-behaved. Finally, we show that there exists a natural correspondence between generalised prime systems and suitable orders on N-2. Some of the above results are relevant to the second author's theory of 'fractal membranes', whose spectral partition functions are given by Beurling-type zeta functions, as well as to joint work of that author and R. Nest on zeta functions attached to quasicrystals.

Complexity theory and planning: examining 'fractals' for organising policy domains in planning practice

This article examines selected methodological insights that complexity theory might provide for planning. In particular, it focuses on the concept of fractals and, through this concept, how ways of organising policy domains across scales might have particular causal impacts. The aim of this article is therefore twofold: (a) to position complexity theory within social science through a ‘generalised discourse’, thereby orienting it to particular ontological and epistemological biases and (b) to reintroduce a comparatively new concept – fractals – from complexity theory in a way that is consistent with the ontological and epistemological biases argued for, and expand on the contribution that this might make to planning. Complexity theory is theoretically positioned as a neo-systems theory with reasons elaborated. Fractal systems from complexity theory are systems that exhibit self-similarity across scales. This concept (as previously introduced by the author in ‘Fractal spaces in planning and governance’) is further developed in this article to (a) illustrate the ontological and epistemological claims for complexity theory, and to (b) draw attention to ways of organising policy systems across scales to emphasise certain characteristics of the systems – certain distinctions. These distinctions when repeated across scales reinforce associated processes/values/end goals resulting in particular policy outcomes. Finally, empirical insights from two case studies in two different policy domains are presented and compared to illustrate the workings of fractals in planning practice.

Fractal spaces in planning and governance

This paper discusses concepts of space within the planning literature, the issues they give rise to and the gaps they reveal. It then introduces the notion of 'fractals' borrowed from complexity theory and illustrates how it unconsciously appears in planning practice. It then moves on to abstract the core dynamics through which fractals can be consciously applied and illustrates their working through a reinterpretation of the People's Planning Campaign of Kerala, India. Finally it highlights the key contribution of the fractal concept and the advantages that this conceptualisation brings to planning.

Fractal geometry and architecture design: case study review

The idea of buildings in harmony with nature can be traced back to ancient times. The increasing concerns on sustainability oriented buildings have added new challenges in building architectural design and called for new design responses. Sustainable design integrates and balances the human geometries and the natural ones. As the language of nature, it is, therefore, natural to assume that fractal geometry could play a role in developing new forms of aesthetics and sustainable architectural design. This paper gives a brief description of fractal geometry theory and presents its current status and recent developments through illustrative review of some fractal case studies in architecture design, which provides a bridge between fractal geometry and architecture design.