For Ford Circles on the real line, [0; 1], G.T. Williams and D.H. Browne discovered that this arrangement of infinite circles has an area-sum \pi+\pi\frac{\zeta(3)}{\zeta(4)}, where \zeta(s) is the Riemann-Zeta function from complex analysis and number theory. The purpose of this paper is to explore their findings in detail and provide alternative methods to prove the statements found in the paper. Then we will attempt to show similar results on the Apollonian Window packing using inversion through circles and the results of Williams and Browne.

The aim of this essay is to examine how Donna Tartt’s The Secret History expresses the sublime experience and what its’ primary function is in the novel. The essay begins with discussing the concept of the sublime through a historical perspective, connecting primarily to Edmund Burke’s ideas and further contrasting them with the feminist criticism of Kristina Fjelkestam. A critical aspect of this study is to examine the notion of terror in the definitions of the sublime and to analyze how Tartt incorporates it into the story by introducing the theme “beauty is terror”, in the beginning of the novel. The essay introduces different techniques used to express this terror, and simultaneously touches upon gothic conventions – such as sublime nature – as well as the postmodern convention of “the unpresentable”, put forward by Jean-Francois Lyotard. This leads to a minor presentation of gothic-postmodernism – a fairly new genre which highlights similarities in the terror experience of the French Revolution and the one we experience from terrorism and media in the modern world. Further, the essay wants to demonstrate how Tartt uses Friedrich Nietzsche’s idea of the Apollonian and Dionysian – how it is concretely embedded in the storyline but also how it functions as a theme throughout the novel, expressing not only a Nietzschean philosophy but also using this dichotomy to further establish a sublime expression.

The present study focuses on developing a Frenet-Serret frame and the Apollonian Window. In the first part of the study Apollonian disks are generated for first four generations by developing visual basic codes in excel. For the second part of the study, three orthonormal basis vectors, namely, tangent, normal, and binormal vectors have been calculated for the tangent points of Apollonian discs for the first three generations. Equations of the Normal, Osculating and Rectifying planes and Taylor Series approximation have been calculated for specific theta. Because Apollonian Window consists of planar curves with constant curvature, torsion is nowhere present. The planar Frenet-Serret Equations for the first three generations for the Apollonian Window is also shown.

Dans cette thèse, nous nous proposons d’analyser les composants apolliniens et dionysiaques qui dialoguent, s’affrontent ou complotent mutuellement dans la conception des techniques audiovisuelles. Pour ce faire nous suivons un parcours qui va du mythe de Narcisse jusqu’au cinématographe, en passant par la fixation de l’image issue de la chambre noire et la mise en mouvement des images avec le chronophotographe.Nous nous appuyons sur le distinguo de Friedrich Nietzsche développé dans La naissance de la tragédie entre ces deux impulsions (l'apollinien et le dionysiaque) dans la mesure où celles-ci ordonnent toute construction culturelle. Nous identifions ainsi dans l’archéologie des médias le combat entre les belles formes qui veulent être fixes, et le chaos ivre qui se perd dans le devenir actif.Dans ce contexte, nous aurons recours à des couples fondateurs indispensables pour aborder les techniques audiovisuelles depuis cet angle d’analyse : lumière/ombre, réalité/imagination, culture/nature, raison/corps et, bien entendu, art/technique.Bien que le combat entre l'apollinien et le dionysiaque donne lieu, selon Nietzsche, à un accouplement dans la tragédie grecque, nous verrons que tout au long de ce que nous appelons le « trajet de l’outil » – qui va du miroir au cinématographe –, les deux impulsions se rapprochent ou s’éloignent par intermittences.C’est à partir de cela que nous formulons l’hypothèse selon laquelle avec la naissance du cinématographe (en tant que technique) et du cinéma (en tant qu’art), Apollon et Dionysos vont à nouveau s’unir dans un accouplement qui engendre l’art et la technique de la civilisation des médias audiovisuels. / This dissertation aims to analyse the Apollonian and the Dionysian components that interact, compete or plot mutually in the design of audio-visual techniques. To achieve this, we follow a path from the myth of Narcissus to the cinematograph, to the process of fixing the image in the dark room such as the moving images of the chronophotograph.We rely on the distinction of Friedrich Nietzsche developed in The Birth of Tragedy between these two pulses (the Apollonian and the Dionysian) to the extent that they enjoin any cultural construction. We identify in the archeology of the media the battle between the beautiful forms that want to be fixed, and the drunken chaos that is lost in becoming active.In this context, we will use the essential founding couples to study audiovisual techniques from this angle of analysis: light / shadow, reality / fantasy, Culture / nature, reason / body and, of course, art / technique.Although the struggle between the Apollonian and the Dionysian elements results, according to Nietzsche, in a coupling in Greek tragedy, we see that throughout what we call the "Tool path" - which embraces all the instruments form the mirror to the cinematography - the two pulses towards or away intermittently.From this analysis, we assume that with the birth of cinema (as a technique) and film (as art), Apollo and Dionysus will unite again in a coupling that begets the art and technique of the audiovisual media civilization.

This dissertation examines the iconographical motif of the Apollonian triad in Attic art of the sixth and fifth centuries BC. Attic vase paintings constitute the chief evidence for this study, but other evidence, such as inscriptions, literary sources, sculptures and coins is considered, as well. My thesis focus on scenes without a clear mythological context, where the triad appears alone or accompanied by other, mostly, divine figures, and on what messages or information these images of the Apollonian triad convey. This study contributes to the ongoing discussion of the iconography and iconology of Attic vases, which enriches our understanding of Athenian socio-political and religious life and of Greek culture, more generally.

This essay analyzes the theme of the pursuit of beauty in The Secret History. It analyzes the main characters’ concept of beauty, their manner of seeking beauty, as well as the result of this search. For this analysis, I use Friedrich Nietzsche’s theories of the Apollonian and the Dionysian as outlined in The Birth of Tragedy and in scholarly texts that analyze TBT— which describe the Apollonian/Dionysian dichotomy as the opposed worlds of order and madness— to define the main characters’ concept of beauty. The narrator of the novel once says that “beauty is terror” (Tartt 45), a statement which paints beauty as harsh and shocking, and potentially destructive. Likewise, in this essay I argue that for these characters beauty is created through the interplay between the Apollonian and the Dionysian, and that its pursuit leads to destruction. I analyze this through the characters of Richard Papen, Henry Winter, and Bunny Corcoran. Richard and Henry pursue beauty in that the actions they take are aimed at embodying an aesthetic ideal. In Richard’s case, it is his longing for beauty which leads him to imitate and join the classicists— particularly by mimicking their socio-economic class— and which eventually places him in a disordered Dionysian world of madness and murder. Henry, on the other hand, is the embodiment of Apollonian order, and it is his search for beauty through a bacchanal which leads him to commit murder twice and, eventually, to take his own life. Lastly, Bunny is different in that he is neither beautiful nor interested in beauty as his peers define it. It is because of this that he is excluded from the others’ pursuit of beauty, that he is murdered, and that his murder is justifiable in the eyes of his murderers. This study finds that, in The Secret History, where beauty is defined as the dance between Apollonian order and Dionysian madness, the Dionysian ends up as the victorious half of the dichotomy, causing the loss of reason and the triumph of destruction and disaster. This portrayal of beauty as destruction and vice versa, rather than serving as the vehicle for a moral indictment, is instead the very purpose of the novel.

AN ABSTRACT OF THE DISSERTATION OFDinush Lanka Panditharathna Jayasooriya Arachchilage, forthe Doctor of Philosophy degree in MATHEMATICS, presented on June 22, 2020 at SouthernIllinois University Carbondale.TITLE: SPIN NETWORK EVALUATION AND THE ASYMPTOTIC BEHAVIORMAJOR PROFESSOR: Dr. Jerzy KocikGraphically, a spin network is a trivalent graph with weights on each edge. At anyof the vertices, the sum of all three weights is even and the sum of any two weights isgreater than or equal to the remaining weight. If the spin network has no free ends, thenwe can evaluate the spin network. Here, we propose a method to evaluate some basic spinnetworks using the idea of Stirling triangle.Tangent circles with integer curvatures are a natural source to make a spin network.In particular, there are spin networks corresponding the Apollonian circle packing and theFord circle packing. We obtain the recurrence relations using the Descartes circle theoremand we evaluate the Apollonian spin network and the Ford circle spin network. We alsodiscuss the asymptotic behavior of the Ford circle spin network.

If I relate beauty to nothingness, what happens? Is nothingness sort of an absence of beauty? Or it is portrayed by our culture and society, and, in such case, can I define this absence of beauty? Is it beauty that you cannot even catch? Is its appearance neutral, almost hidden? When we are born, we all have the same degree of beauty and purity, which is progressively lost as long as we start growing up. In a life span, we accumulate wrinkles, and defects and dirt which needs to be concealed in order to fit in certain social categories. But our bodies register all the marks, absorb all the signs and impurities, likewise filters. We don’t necessarly perceive our own dirt or impurities as disgusting, whereas, in the clash with the other, we automatically are ahsamed of it. Same way, we tend to regard the other’s dirt as disgusting, not our own, very private dirt. Dirt is matter out of place, so is ugliness. The stain must be cleansed, purified as it represents a threaten for beauty. It is subtracting clean space to beauty. We are part of a society that intimates us to clean up, shape up, hide your -very human- dirt under the carpet. But beauty, nor humanity, would not exist without that dirt. We do absorb impurities all life long. And that is what makes us what we ultimately are: humans. Dirt paradoxically works as a protection: the dirtier we are, the less afraid of getting dirty we will be. In the society we live in exist many difficulties when it comes to find an identity as humans and a position in it. We are often put in a situation of having to follow: a certain career, a living style, an ideology, somebody’s else opinion, what to consume, school systems. Etcetera. In such a society, and because of this “follower-like”, passive position, where we mostly have to repeat the same living patterns, it got harder and harder to retrieve the meaning of things and to understand where we come from. Who we really are. We tend to put on uniforms or masks to fit in different standardazied situations. Everything and everybody has to fit in its or his standard place. This way our intrinsic human beauty is concealed and somehow controlled. With my essai, I try to look under the carpet, undress, unmask and reach a new definition of beauty: a naked beauty, not concealed nor camouflaged. The beauty we all deeply share, unpretentious and honest. A beauty of nothingness: something I see or feel, but about which I keep wondering whether it is or it is not beauty. To develop such new definition of beauty, I recollect ideas and concepts of beauty from the past, with a main reference to western society: from beauty models in the ancient Greece, Apollonian VS Dionysian, to the Sublime, untill the present time. I try to define what purpose and non-purpose beauty is. What is ugliness and dirt and how they both are a prerequisites of beauty. I finally take a more personal look upon contemporary society and how its mechanisms define a beauty which is standard. It is starting from a reflexion about standard society and beauty, that I then define a more intrinsic human beauty. Such unevokable sensation of beauty is extremely subtle, hard to acknowledge: one needs to train ones eyes and go beyond the layers, to discover the beauty of nothingness.

This study offers a comparative investigation into selected figurative and nonfigurative, monochromatic (and mainly monochromatic) artworks by Ad Reinhardt, Ultimate Painting nr 39 (1963), Yves Klein, Monochrome bleu (IKB 3) (1956) and The thin red line (1970) by Douglas Portway. The aim of this research is to examine the possible subjective, meaningful function of the seemingly objective artworks. The selected artworks represent the formalist tendency of the high-Modernist conception of art-as-art, or the artwork as an autonomous objective object. At first sight the objective artworks seem to refute the subjective intentions of the artists who present them as both an externalisation of subjective experience and as possibly meaningful to the viewer. The investigation into the possible subjectively meaningful artworks is guided by an Existential approach to the aesthetic experience, as proposed by Nietzche’s Dionysian and Apollonian concepts as well as Sartre’s conceptualisation of néantisation and the imagination respectively. Both philosophers describe aesthetic experience as a meeting between both subjective and objective elements of their philosophy. The experience of the aesthetic (in the artworks) ultimately leads to a subjective space within which the seemingly objective artworks function as a subjective platform on which the Existential search for meaning can be considered (and possibly relieved). / MA (History of Art), North-West University, Potchefstroom Campus, 2014

This study offers a comparative investigation into selected figurative and nonfigurative, monochromatic (and mainly monochromatic) artworks by Ad Reinhardt, Ultimate Painting nr 39 (1963), Yves Klein, Monochrome bleu (IKB 3) (1956) and The thin red line (1970) by Douglas Portway. The aim of this research is to examine the possible subjective, meaningful function of the seemingly objective artworks. The selected artworks represent the formalist tendency of the high-Modernist conception of art-as-art, or the artwork as an autonomous objective object. At first sight the objective artworks seem to refute the subjective intentions of the artists who present them as both an externalisation of subjective experience and as possibly meaningful to the viewer. The investigation into the possible subjectively meaningful artworks is guided by an Existential approach to the aesthetic experience, as proposed by Nietzche’s Dionysian and Apollonian concepts as well as Sartre’s conceptualisation of néantisation and the imagination respectively. Both philosophers describe aesthetic experience as a meeting between both subjective and objective elements of their philosophy. The experience of the aesthetic (in the artworks) ultimately leads to a subjective space within which the seemingly objective artworks function as a subjective platform on which the Existential search for meaning can be considered (and possibly relieved). / MA (History of Art), North-West University, Potchefstroom Campus, 2014