Symmetric tangled Platonic polyhedra

Conventional embeddings of the edge-graphs of Platonic polyhedra, {f, z}, where f, z denote the number of edges in each face and the edge-valence at each vertex, respectively, are untangled in that they can be placed on a sphere (S2) such that distinct edges do not intersect, analogous to unknotted loops, which allow crossing-free drawings of S1 on the sphere. The most symmetric (flag-transitive) realizations of those polyhedral graphs are those of the classical Platonic polyhedra, whose symmetries are *2fz, according to Conway’s two-dimensional (2D) orbifold notation (equivalent to Schönflies symbols Ih, Oh, and Td). Tangled Platonic {f, z} polyhedra—which cannot lie on the sphere without edge-crossings—are constructed as windings of helices with three, five, seven,… strands on multigenus surfaces formed by tubifying the edges of conventional Platonic polyhedra, have (chiral) symmetries 2fz (I, O, and T), whose vertices, edges, and faces are symmetrically identical, realized with two flags. The analysis extends to the “θz” polyhedra, {2,z}. The vertices of these symmetric tangled polyhedra overlap with those of the Platonic polyhedra; however, their helicity requires curvilinear (or kinked) edges in all but one case. We show that these 2fz polyhedral tangles are maximally symmetric; more symmetric embeddings are necessarily untangled. On one hand, their topologies are very constrained: They are either self-entangled graphs (analogous to knots) or mutually catenated entangled compound polyhedra (analogous to links). On the other hand, an endless variety of entanglements can be realized for each topology. Simpler examples resemble patterns observed in synthetic organometallic materials and clathrin coats in vivo.

Dream Screen: On Cinema and Painting, Blur and Absorption

This chapter explores some of the ways in which low definition or blur orchestrate the encounter between film and painting. Depending on the technique, film stock, the choice of analogue or digital filming, and the degree of experimentation, such an encounter may take an endless variety of forms. Focusing on the effects of blur on figurality and the representation of the human form, this chapter interrogates intermedial resonances as well as the regimes of identification or absorption that are produced when the depiction of the figure eschews visual clarity. With reference to films ranging from the silent to the digital era, the author considers the sense of absorption and empathetic response the softening of contours elicits from the viewer, as well as the contradictory effect of the blurring of lines, between intimacy and distance, closeness and concealment. In turn, M. Beugnet looks at the different kind of absorption offered by the chaotic universes of the cinema of sensation, before discussing the tension between theatricality and absorption in evidence in recent filmmaking.

Light: Toward a Transdisciplinary Science of Appearance and Atmosphere

To understand the processes behind seeing light, we need to integrate knowledge about the incoming optical structure, its perception, and how light interacts with material, shape, and space—objectively and subjectively. To that end, we need a novel approach to the science of light, namely, a transdisciplinary science of appearance, integrating optical, perceptual, and design knowledge and methods. In this article, I review existing literature as a basis for such a synthesis, which should discuss light in its full complexity, including its spatial properties and interactions with materials, shape, and space. I propose to investigate this by representing the endless variety of light, materials, shapes, and space as canonical modes and their combinations.

The Rules of Prose in Sixteenth-Century China: Tang Shunzhi (1507-1560) as an Anthologist

This paper examines the role of the anthologist in late imperial Chinese print culture. Specifically, it focuses on the sixteenth-century anthologist Tang Shunzhi. Tang’s first place finish in the 1529 metropolitan examinations came at a pivotal moment. As commercial anthology printers responded to an expanding reading public by applying readers’ aids such as punctuation and commentary to increasingly diverse textual corpora, Tang’s distinctive method of annotation was used to ‘reveal’ the rules of Ming examination prose operating universally across a seemingly endless variety of texts. At the same time, Tang’s own belief in universal rules of prose was the product of an educational movement to supplement the narrow and monotonous examination curriculum by providing students with anthologies of ancient literature. These two Tangs—one revealing uniformity within diversity, the other revealing diversity within uniformity—highlight contradictory trends toward both stereotypy and diversification within sixteenth-century print culture more broadly.

Conclusion

The potency of the animal as a symbol lies in humanity’s dualistic relationship with the animal. On the one hand, we ourselves are animals; on the other hand, we define ourselves in opposition to all other animals. There is thus both kinship and otherness, identity and difference, attraction and repulsion in humanity’s relationship to the animal. Through this dualistic interplay, animality becomes a fruitful resource for defining what it means to be human. As Buddhism—arguably, more so than any other major religion—is a profoundly human-centered tradition, we should not be surprised to see Buddhist texts from India making ample use of animality in expressing Buddhism’s vision of the project of being human. Animality manifests itself, moreover, through an endless variety of different species (with different qualities, characteristics, and habitats), and thus presents us with an extremely rich pool of symbolic possibilities. The availability and potency of this pool in the world of premodern India is perhaps difficult to imagine from our own vantage point in the modern, industrialized West, where animals are retreating more and more from our everyday view and experience. This book constitutes a very preliminary effort to gain a glimpse of some of the ways in which this pool of symbolic possibilities was employed....

Impact of Zero Tolerance Policies on American K-12 Education and Alternative School Models

Despite the original intent of zero tolerance policies in schools— to ensure guns and other dangerous weapons were kept out of schools—these policies have instead grown to encompass an endless variety of minor infractions that would, in previous generations, not necessarily result in the immediate removal of the student from the classroom. Zero tolerance policies do not proportionately discipline students and, instead, treats every child and situation the same. Further, studies confirm that as suspension, expulsion, and school-based arrests have increased since the mid-1990's, the majority of students being suspended, expelled, or arrested are predominately minority students. The goal of this chapter is to examine the application of zero tolerance policy in K-12 public schools and offer administrators and educators alternative school discipline models.