Wholeness as a Hierarchical Graph to Capture the Nature of Space

According to Christopher Alexander's theory of centers, a whole comprises numerous, recursively defined centers for things or spaces surrounding us. Wholeness is a type of global structure or life-giving order emerging from the whole as a field of the centers. The wholeness is an essential part of any complex system and exists, to some degree or other, in spaces. This paper defines wholeness as a hierarchical graph, in which individual centers are represented as the nodes and their relationships as the directed links. The hierarchical graph gets its name from the inherent scaling hierarchy revealed by the head/tail breaks, which is a classification scheme and visualization tool for data with a heavy-tailed distribution. We suggest that (1) the degrees of wholeness for individual centers should be measured by PageRank (PR) scores based on the notion that high-degree-of-life centers are those to which many high-degree-of-life centers point, and (2) that the hierarchical levels, or the ht-index of the PR scores induced by the head/tail breaks can characterize the degree of wholeness for the whole: the higher the ht-index, the more life or wholeness in the whole. Three case studies applied to the Alhambra building complex and the street networks of Manhattan and Sweden illustrate that the defined wholeness captures fairly well human intuitions on the degree of life for the geographic spaces. We further suggest that the mathematical model of wholeness be an important model of geographic representation, because it is topological oriented that enables us to see the underlying scaling structure. The model can guide geodesign, which should be considered as the wholeness-extending transformations that are essentially like the unfolding processes of seeds or embryos, for creating beautiful built and natural environments or with a high degree of wholeness.Comment: 14 pages, 7 figures, 2 table

THE USE OF FRACTAL GEOMETRY ELEMENTS AS A MEANS OF AESTHETIC EDUCATION OF PRIMARY SCHOOL STUDENTS

Currently, the issue of aesthetic education is becoming one of the key issues in the field of education. The relevance of the article is due to the fact that mathematics has great aesthetic potential, which is not always revealed in the process of teaching mathematics in elementary school. An important role in revealing the beauty of mathematical content in elementary school is played by the familiarity of schoolchildren with a young rapidly developing mathematical field - fractal geometry. In order to prove the effectiveness of using fractal geometry elements for the formation of aesthetic representations of younger schoolchildren, a formative experiment was carried out, with the help of testing we diagnosed changes in the level of formed aesthetic representations, feelings and judgments of younger schoolchildren during the experimental work. The tools and methods of using the elements of fractal geometry in the study of mathematics in elementary school are described. The results of the study allow us to conclude that the familiarity of primary schoolchildren with fractals increases the interest of students in mathematics. The amazing simplicity of fractals and the diversity of their forms contributes to the formation of aesthetic ideas and feelings among younger students. The construction of fractals in the study of geometric objects in elementary school makes it possible to show students that aesthetic pleasure is provided not only by art, but also by the joy of creativity in other fields of activity, including teaching.

Fractal Image Editing with PhotoFrac

In this paper, we describe the development and use of PhotoFrac, an application that allows artists and designers to turn digital images into fractal patterns interactively. Fractal equations are a rich source of procedural texture and detail, but controlling the patterns and incorporating traditional media has been difficult. Additionally, the iterative nature of fractal calculations makes implementation of interactive techniques on mobile devices and web apps challenging. We overcome these problems by using an image coordinate based orbit trapping technique that permits a user-selected image to be embedded into the fractal. Performance challenges are addressed by exploiting the processing power of graphic processing unit (GPU) and precomputing some intermediate results for use on mobile devices. This paper presents results and qualitative analyses of the tool by four artists (the authors) who used the PhotoFrac application to create new artworks from original digital images. The final results demonstrate a fusion of traditional media with algorithmic art

Methods in Visual Mathematics: Reductionism in Researching Mathematical Principles in Art

The visual arts offer a reflective source for understanding the processing of aesthetics and beauty that is significant to an awareness of brain function and the human psyche. Evaluating and determining what factors are integral to the experience of aesthetics holds value for understanding deeper psychological implications of perception. I conducted a survey of Western portraiture determined to be famed through repeated Internet searching of famous art and best-selling prints for the purpose of examining the works for mathematical attributes proposed to cause the experience of visual pleasure. While mathematical principles and patterns can be found within each example of portraiture, the overarching issue encountered is the validity of the methods that are present in the research that declares the merit of the principles and patterns used. As the data suggesting the value of the attributes sought in the works is flawed, so too is any conclusion based upon it. The ability to quantify the qualitative in an objective manner does not yet exist. Therefore, it is invalid and reductionist to assert the experience of visual pleasure as relates to fame is based on a singular attribute that cannot be empirically established. Attempts to discover adequate methods are not wasted, as the discussion generated by inquiry into the experience of aesthetics offers positive philosophical and critical thinking applications. Furthermore, the promising new frontier for aesthetic research involves utilization of social networking and the Internet as tools

A NATURALIZAÇÃO DO ARTIFICIAL: A UTOPIA COMO LOCUS DA INFINITA ESCOLHA

Numa época em que o artificial ameaça o lugar do biológico, é necessário desclassificar o computador, reduzi-lo a uma peça da engrenagem, excluir todos os critérios funcionais que regem a imagem e colocar o artista no centro. O perigo não reside em as máquinas imitarem os humanos, mas sim o contrário, humanos reduzidos a vidas-fluxograma compostas por escolhas binárias irrecusáveis. Pretendemos esconder o computador debaixo da arte e tornar infinita a escolha. Construo imagens nas quais a intervenção do computador não é evidenciada. Aqui, dada a natureza investigativa do programa doutoral, apresento os meandros das engrenagens algorítmicas do meu trabalho

Fluid Dynamics of Watercolor Painting : Experiments and Modelling

In his classic study in 1908, A.M. Worthington gave a thorough account of splashes and their formation through visualization experiments. In more recent times, there has been renewed interest in this subject, and much of the underlying physics behind Worthington\u27s experiments has now been clarified. One specific set of such recent studies, which motivates this thesis, concerns the fluid dynamics behind Jackson Pollock\u27s drip paintings. The physical processes and the mathematical structures hidden in his works have received serious attention and have made the scientific pursuit of art a compelling area of exploration. Our current work explores the interaction of watercolors with watercolor paper. Specifically, we conduct experiments to analyze the settling patterns of droplets of watercolor paint on wet and frozen paper. Variations in paint viscosity, paper roughness, paper temperature, and the height of a released droplet are examined from time of impact, through its transient stages, until its final, dry state. Observable phenomena such as paint splashing, spreading, fingering, branching, rheological deposition, and fractal patterns are studied in detail and classified in terms of the control parameters. Using the one-dimensional (1-D) Saint-Venant differential equations, which are a simplification of the three-dimensional (3-D) Navier-Stokes equations from fluid dynamics, we created a computer-simulated, mathematical model of a droplet splash of watercolor paint onto a flat surface. The mathematical model is analyzed using a MATLAB code which considered changes in droplet height, radius, and velocity of dispersal over time. We also implemented a stochastic version of the Saint-Venant equations which captured the random fingering patterns of a droplet splash. Initial conditions for height, radius, and velocity of a radially spreading droplet were given at the onset of the simulation. Dynamic viscosity and fluid density were parameters incorporated into this system of differential equations, which could be easily adjusted in the MATLAB code for the paint type to be simulated. The stochastic nature of our model was designed to recreate the complex behavior of water splashes, the non-homogeneity of the watercolor paper, and the resulting patterns. We then computed the fractal dimension of each computer-generated droplet image to compare theoretical and experimental values. Analysis of the set of data consisting of over 10,000 trials was conducted to determine any significant statistical correlations among the spreading pattern, the number of fingers, viscosity, density and fractal dimension. Finally, we extended the system of differential equations based on the Saint-Venant equations to include the effects of temperature upon the paint-spreading pattern. In a similar manner, we compared the theoretical values of fractal dimensions generated by our MATLAB model to the experimental results for paint droplets on a frozen substrate

Image Automation: Post-Conceptual Post-Photography and the Deconstruction of the Photographic Image

This PhD thesis delivers an artistic research practice based on a deconstruction of the photographic image. Photography in post-photographic, digital culture has, due to changes in technology, become a matter of style while neglecting its own traditional process base. This thesis claims that similar automated processes can be found within information technology, which in the artistic realm has a strong relation to Conceptual art. A post-conceptual critique of the notion of ‘information’ in Conceptual practice allows for a repositioning of the image. Focusing on visual, transformative reflection, the thesis resists the temptation to present generalising philosophical speculation in favour of an artistic research practice that focuses on the inner, transformative workings of artworks, or the work’s ‘figuration’ as I call it, following Jean-François Lyotard and Georges Didi-Huberman. This research project offers an artistic interrogation into the potential of post-photographic practices under post-conceptual conditions. Apart from photography, the practice employs drawing, installation art, painting and printmaking to produce work that is often conceptually developed on the computer. Much of the work consists of abstract, blob- like ‘figures’ appropriated from digital-image material, while other work is measurement- based. Figuration is advanced in each of these through constructive processes that remain visible. A developed understanding of process-oriented practices within the digital realm, which this PhD offers, allows present-day photography to connect to its traditional diversity. The necessary re-thinking of the image, which is a key result of the research, may affect artistic practices beyond photography, giving an extended contemporary photographic practice increased artistic relevance. The research is supported by art-historical discussions concerning the history of photography, the history of Conceptual art, and what Svetlana Alpers calls ‘the northern mode’ of painting. Technical discussions of post-photography and the notion of ‘information’ help clarify underlying processes, while philosophical considerations are used to give meaning to a changed concept of the image. Finally, a methodological discussion contextualises the research within current notions of practice-led research

Continuous and discrete properties of stochastic processes

This thesis considers the interplay between the continuous and discrete properties of random stochastic processes. It is shown that the special cases of the one-sided Lévy-stable distributions can be connected to the class of discrete-stable distributions through a doubly-stochastic Poisson transform. This facilitates the creation of a one-sided stable process for which the N-fold statistics can be factorised explicitly. The evolution of the probability density functions is found through a Fokker-Planck style equation which is of the integro-differential type and contains non-local effects which are different for those postulated for a symmetric-stable process, or indeed the Gaussian process. Using the same Poisson transform interrelationship, an exact method for generating discrete-stable variates is found. It has already been shown that discrete-stable distributions occur in the crossing statistics of continuous processes whose autocorrelation exhibits fractal properties. The statistical properties of a nonlinear filter analogue of a phase-screen model are calculated, and the level crossings of the intensity analysed. It is found that rather than being Poisson, the distribution of the number of crossings over a long integration time is either binomial or negative binomial, depending solely on the Fano factor. The asymptotic properties of the inter-event density of the process are found to be accurately approximated by a function of the Fano factor and the mean of the crossings alone