Planar graphs : a historical perspective.

The field of graph theory has been indubitably influenced by the study of planar graphs. This thesis, consisting of five chapters, is a historical account of the origins and development of concepts pertaining to planar graphs and their applications. The first chapter serves as an introduction to the history of graph theory, including early studies of graph theory tools such as paths, circuits, and trees. The second chapter pertains to the relationship between polyhedra and planar graphs, specifically the result of Euler concerning the number of vertices, edges, and faces of a polyhedron. Counterexamples and generalizations of Euler\u27s formula are also discussed. Chapter III describes the background in recreational mathematics of the graphs of K5 and K3,3 and their importance to the first characterization of planar graphs by Kuratowski. Further characterizations of planar graphs by Whitney, Wagner, and MacLane are also addressed. The focus of Chapter IV is the history and eventual proof of the four-color theorem, although it also includes a discussion of generalizations involving coloring maps on surfaces of higher genus. The final chapter gives a number of measurements of a graph\u27s closeness to planarity, including the concepts of crossing number, thickness, splitting number, and coarseness. The chapter conclused with a discussion of two other coloring problems - Heawood\u27s empire problem and Ringel\u27s earth-moon problem

Relations between Modern Mathematics and Poetry: Czesław Miłosz; Zbigniew Herbert; Ion Barbu/Dan Barbilian

This doctoral thesis is an examination of the relationship between poetry and mathematics, centred on three twentieth-century case studies: the Polish poets Czesław Miłosz (1911-2004) and Zbigniew Herbert (1924-1998), and the Romanian mathematician and poet Dan Barbilian/Ion Barbu (1895-1961). Part One of the thesis is a review of current scholarly literature, divided into two chapters. The first chapter looks at the nature of mathematics, outlining its historical developments and describing some major mathematical concepts as they pertain to the later case studies. This entails a focus on non-Euclidean geometries, modern algebra, and the foundations of mathematics in Europe; the nature of mathematical truth and language; and the modern historical evolution of mathematical schools in Poland and Romania. The second chapter examines some existing attempts to bring together mathematics and poetry, drawing on literature and science as an academic field; the role of the imagination and invention in the languages of both poetics and mathematics; the interest in mathematics among certain Symbolist poets, notably Mallarmé; and the experimental work of the French groups of mathematicians and mathematician-poets, Bourbaki and Oulipo. The role of metaphor is examined in particular. Part Two of the thesis is the case studies. The first presents the ethical and moral stance of Czesław Miłosz, investigating his attitudes towards classical and later relativistic science, in the light of the Nazi occupation and the Marxist regimes in Poland, and how these are reflected in his poetry. The study of Zbigniew Herbert is structured around a wide selection of his poetic oeuvre, and identifying his treatment of evolving and increasingly more complex mathematical concepts. The third case study, on Dan Barbilian, who published his poetry under the name Ion Barbu, begins with an examination of the mathematical school at Göttingen in the 1920s, tracing the influence of Gauss, Riemann, Klein, Hilbert and Noether in Barbilian’s own mathematical work, particularly in the areas of metric spaces and axiomatic geometry. In the discussion, the critical analysis of the mathematician and linguist Solomon Marcus is examined. This study finishes with a close reading of seven of Barbu’s poems. The relationship of mathematics and poetry has rarely been studied as a coherent academic field, and the relevant scholarship is often disconnected. A feature of this thesis is that it brings together a wide range of scholarly literature and discussion. Although primarily in English, a considerable amount of the academic literature collated here is in French, Romanian, Polish and some German. The poems themselves are presented in the original Polish and Romanian with both published and working translations appended in the footnotes. In the case of the two Polish poets, one a Nobel laureate and the other a multiple prize-winning figure highly regarded in Poland, this thesis is unusual in its concentration on mathematics as a feature of the poetry which is otherwise much-admired for its politically-engaged and lyrical qualities. In the case of the Romanian, Dan Barbilian, he is widely known in Romania as a mathematician, and most particularly as the published poet Ion Barbu, yet his work is little studied outside that country, and indeed much of it is not yet translated into English. This thesis suggests at an array of both theoretical and specific starting points for examining the multi-stranded and intricate relationship between mathematics and poetry, pointing to a number of continuing avenues of further research

The Poetry of Logical Ideas: Towards a Mathematical Genealogy of Media Art

In this dissertation I chart a mathematical genealogy of media art, demonstrating that mathematical thought has had a significant influence on contemporary experimental moving image production. Rather than looking for direct cause and effect relationships between mathematics and the arts, I will instead examine how mathematical developments have acted as a cultural zeitgeist, an indirect, but significant, influence on the humanities and the arts. In particular, I will be narrowing the focus of this study to the influence mathematical thought has had on cinema (and by extension media art), given that mathematics lies comfortably between the humanities and sciences, and that cinema is the object par excellence of such a study, since cinema and media studies arrived at a time when the humanities and sciences were held by many to be mutually exclusive disciplines. It is also shown that many media scholars have been implicitly engaging with mathematical concepts without necessarily recognizing them as such. To demonstrate this, I examine many concepts from media studies that demonstrate or derive from mathematical concepts. For instance, Claude Shannon's mathematical model of communication is used to expand on Stuart Hall's cultural model, and the mathematical concept of the fractal is used to expand on Rosalind Krauss' argument that video is a medium that lends itself to narcissism. Given that the influence of mathematics on the humanities and the arts often occurs through a misuse or misinterpretation of mathematics, I mobilize the concept of a productive misinterpretation and argue that this type of misreading has the potential to lead to novel innovations within the humanities and the arts. In this dissertation, it is also established that there are many mathematical concepts that can be utilized by media scholars to better analyze experimental moving images. In particular, I explore the mathematical concepts of symmetry, infinity, fractals, permutations, the Axiom of Choice, and the algorithmic to moving images works by Hollis Frampton, Barbara Lattanzi, Dana Plays, T. Marie, and Isiah Medina, among others. It is my desire that this study appeal to scientists with an interest in cinema and media art, and to media theorists with an interest in experimental cinema and other contemporary moving image practices

Johanssonian Investigations

In the last decades, Ingvar Johansson has made a formidable contribution to the development of philosophy and particularly that of metaphysics. This volume consists of original papers written by 50 philosophers from all over the world to celebrate his 70th birthday. The papers cover traditional issues in metaphysics and the philosophy of mind, applied ethics, applied metaphysics, the nature of human rights, the philosophy of economics and sports

Johanssonian Investigations

In the last decades, Ingvar Johansson has made a formidable contribution to the development of philosophy and particularly that of metaphysics. This volume consists of original papers written by 50 philosophers from all over the world to celebrate his 70th birthday. The papers cover traditional issues in metaphysics and the philosophy of mind, applied ethics, applied metaphysics, the nature of human rights, the philosophy of economics and sports