Optimizing for the Rupert property

A polyhedron is Rupert if it is possible to cut a hole in it and thread an identical polyhedron through the hole. It is known that all 5 Platonic solids, 10 of the 13 Archimedean solids, 9 of the 13 Catalan solids, and 82 of the 92 Johnson solids are Rupert. Here, a nonlinear optimization method is devised that is able to validate the previously known results in seconds. It is also used to show that 2 additional Catalan solids -- the triakis tetrahedron and the pentagonal icositetrahedron -- and 5 additional Johnson solids are Rupert

Extended abstract for: Solving Rupert’s problem algorithmically

International audienc

How do planets find their way? Laws of nature and the transformations of knowledge in the Scientific Revolution

Laws of nature are perceived as playing a central role in modern science. This thesis investigates the introduction of laws of nature into natural philosophy in the seventeenth century, from which modern science arguably evolved. Previous work has indicated that René Descartes was responsible for single-handedly introducing a mathematical concept of laws into physics under the form of ‘laws of nature’. However, there is less agreement on the originality, causes and aftermath of this manoeuvre. This thesis is sensitive to the circumstance that the introduction of ‘laws of nature’ in the seventeenth century is a problem for us given our hindsight perspective of the origins of modern science, not an explicit concern of the actors; ‘laws of nature’ emerged as part of a network of problems and possibilities converging in Descartes’ reform of natural philosophy. Then, the appropriation of his laws was not an assessment of isolated statements on nature, but a process bounded by critical stances towards the Cartesian enterprise involving theological and social underpinnings. Accordingly, this thesis approaches ‘laws of nature’ as by-products of the changing boundaries between mechanics, mathematics and natural philosophy in the seventeenth century and interprets them as embedded within the circumstances and interactions among the practitioners of these disciplines in which these laws were introduced, criticised and appropriated. Based on this approach, this thesis tracks the background of Descartes’s project of reform of physics from the sixteenth-century fascination for machines that led to codifications of mechanics as a mixed-mathematical science, generating quantitative ways to design and fabricate physical (artificial) objects (Chapter 1). This approach was picked up by Galileo, who transformed it to include natural motion. In so doing, Galileo developed a mathematical approach to natural philosophy—a mathematical science of motion—which ultimately relied on the physical assumption of the motion of the Earth (Chapter 2). An alternative reorganization of mathematics and natural philosophy was put forward by the Lutheran theologian Kepler, wh o considered that the natural knowledge of the world may be founded a priori by deciphering the archetypes that God followed when creating the world. His archetypal cosmology provided a link between geometry and natural philosophy, involving mechanics (Chapter 3). However, Descartes moved in a different direction. Instead of connecting mathematics to natural philosophy, he tried to anchor both mathematics and natural philosophy on certainty, claiming that matter is but extension and that a few principles codified all possible interactions among parts of this geometrical matter. These principles were three ‘laws of nature’ erected as foundations of an a priori physics (Chapter 4). These ‘laws of nature’ received considerable attention in England. Informed by local traditions, English writers rejected the causal role attributed to laws but reworked their contents in laws of motion that were moved to mechanics and extended to astronomy, in line with the local practices of the ‘elliptical astronomy’ (Chapter 5). The relocation of ‘laws of nature’ from physics to mechanics was connected with English debates concerning the role of motion in geometry. These discussions drew different consequences for the connections between mathematics and nature (Chapter 6). In line with the English appropriation of Descartes, the young Newton assumed laws of motion as mathematical explanations in mechanics. When asked by Halley about orbital motion, his answer displayed characteristics of the English disciplinary setting. However, in connection with his historical studies, Newton realised that his laws of motion were capable of accounting for the true system of the world and then they were transformed into mathematical principles of natural philosophy, redrawing the contours of mathematics, natural philosophy and mechanics. The most important outcome of this reorganization—the law of gravitation—raised suspicions for going beyond the boundaries of established practices in the Continent (Chapter 7). The thesis concludes that ‘laws of nature’ did not emerge as a generic label to denominate findings in science. On the contrary, they appeared as concrete achievements with an operative function within Descartes’ reform of natural philosophy and consequently embedded within a network of assumptions, traditions and practices that were central to the appropriation of ‘laws of nature’. English natural philosophers and mathematicians reworked these ‘laws of nature’ within different disciplinary settings and put forward alternative ‘laws of motion’ in ways not previously noticed. The picture that emerges is not that of an amalgamation of previous meanings into a more complex one that was subsequently disseminated. Instead of a unified concept of ‘laws of nature’, Descartes’ project triggered reactions framed within local traditions and therefore it is hard to claim that at the end of the seventeenth century there was any agreement on the meaning of ‘laws of nature’ or even laws of motion beyond the narrow circles that shared disciplinary commitments and values. It was during the appropriation of Newton in the eighteenth century that his achievements and those honoured as his peers were labelled with a non- Newtonian concept of ‘laws of nature’, creating a foundational myth of the origins of modern science that reached up to the twentieth century

Concepts of the 'Scientific Revolution': An analysis of the historiographical appraisal of the traditional claims of the science

´Scientific revolution´, as a concept, is both ´philosophically general´ and ´historically unique´. Both dual-sense of the term alludes to the occurrence of great changes in science. The former defines the changes in science as a continual process while the latter designate them, particularly, as the ´upheaval´ which took place during the early modern period. This research aims to demonstrate how the historicists´ critique of the justification of the traditional claims of science on the basis of the scientific processes and norms of the 16th and 17th centuries, illustrates the historical/local determinacy of the science claims. It argues that their identification of the contextual and historical character of scientific processes warrants a reconsideration of our notion of the universality of science. It affirms that the universality of science has to be sought in the role of such sources like scientific instruments, practical training and the acquisition of methodological routines"Revolución científica", como concepto, se refiere a la vez a algo «filosóficamente general» e « históricamente único". Ambos sentidos del término aluden a la ocurrencia de grandes cambios en la ciencia. El primero define los cambios en la ciencia como un proceso continuo, mientras que el último los designa, en particular, como la "transformación", que tuvo lugar durante la Edad Moderna. Esta investigación tiene como objetivo demostrar cómo la crítica de los historicistas a la justificación de las características tradicionales de la ciencia sobre la base de los procesos y normas científicos de los siglos XVI y XVII, ilustra la determinación histórica y local de los atributos de la ciencia. Se argumenta que la identificación del carácter contextual e histórico de los procesos científicos justifica una reconsideración de nuestra noción de la universalidad de la ciencia. Se afirma que la universalidad de la ciencia se ha de buscar en el papel de tales fuentes como instrumentos científicos, la formación práctica y la adquisición de rutinas metodológica

Representations of Space in Seventeenth Century Physics

The changing understanding of the universe that characterized the birth of modern science included a fundamental shift in the prevailing representation of space - the presupposed conceptual structure that allows one to intelligibly describe the spatial properties of physical phenomena. At the beginning of the seventeenth century, the prevailing representation of space was spherical. Natural philosophers first assumed a spatial center, then specified meanings with reference to that center. Directions, for example, were described in relation to the center, and locations were specified by distance from the center. Through a series of attempts to solve problems first raised by the work of Copernicus, this Aristotelian, spherical framework was replaced by a rectilinear representation of space. By the end of the seventeenth century, descriptions were understood by reference to linear orientations, as parallel or oblique to a presupposed line, and locations were identified without reference to a privileged central point. This move to rectilinear representations of space enabled Gilbert, Kepler, Galileo, Descartes, and Newton to describe and explain the behavior of the physical world in the novel ways for which these men are justly famous, including their theories of gravitational attraction and inertia. In other words, the shift towards a rectilinear representation of space was essential to the fundamental reconception of the universe that gave rise to both modern physical theory and, at the same time, the linear way of experiencing the world that characterizes modern science

Inductive Pattern Formation

With the extended computational limits of algorithmic recursion, scientific investigation is transitioning away from computationally decidable problems and beginning to address computationally undecidable complexity. The analysis of deductive inference in structure-property models are yielding to the synthesis of inductive inference in process-structure simulations. Process-structure modeling has examined external order parameters of inductive pattern formation, but investigation of the internal order parameters of self-organization have been hampered by the lack of a mathematical formalism with the ability to quantitatively define a specific configuration of points. This investigation addressed this issue of quantitative synthesis. Local space was developed by the Poincare inflation of a set of points to construct neighborhood intersections, defining topological distance and introducing situated Boolean topology as a local replacement for point-set topology. Parallel development of the local semi-metric topological space, the local semi-metric probability space, and the local metric space of a set of points provides a triangulation of connectivity measures to define the quantitative architectural identity of a configuration and structure independent axes of a structural configuration space. The recursive sequence of intersections constructs a probabilistic discrete spacetime model of interacting fields to define the internal order parameters of self-organization, with order parameters external to the configuration modeled by adjusting the morphological parameters of individual neighborhoods and the interplay of excitatory and inhibitory point sets. The evolutionary trajectory of a configuration maps the development of specific hierarchical structure that is emergent from a specific set of initial conditions, with nested boundaries signaling the nonlinear properties of local causative configurations. This exploration of architectural configuration space concluded with initial process-structure-property models of deductive and inductive inference spaces. In the computationally undecidable problem of human niche construction, an adaptive-inductive pattern formation model with predictive control organized the bipartite recursion between an information structure and its physical expression as hierarchical ensembles of artificial neural network-like structures. The union of architectural identity and bipartite recursion generates a predictive structural model of an evolutionary design process, offering an alternative to the limitations of cognitive descriptive modeling. The low computational complexity of these models enable them to be embedded in physical constructions to create the artificial life forms of a real-time autonomously adaptive human habitat

The Mathematics of Collision and the Collision of Mathematics in the 17th Century

Thesis (Ph.D.) - Indiana University, History and Philosophy of Science, 2015This dissertation charts the development of the quantitative rules of collision in the 17th century. These were central to the mathematization of nature, offering natural philosophy a framework to explain all the changes of nature in terms of the size and speed of bodies in motion. The mathematization of nature is a classic thesis in the history of early modern science. However, the significance of the dynamism within mathematics should not be neglected. One important change was the emergence of a new language of nature, an algebraic physico-mathematics, whose development was intertwined with the rules of collision. The symbolic equations provided a unified system to express previously diverse kinds of collision with a new representation of speed with direction, while at the same time collision provided a practical justification of the otherwise "impossible" negative numbers. In private manuscripts, Huygens criticized Descartes's rules of collision with heuristic use of Cartesian symbolic algebra. After he successfully predicted the outcomes of experiments using algebraic calculations at an early meeting of the Royal Society, Wallis and Wren extended the algebraic investigations in their published works. In addition to the impact of the changes in mathematics itself, the rules of collision were shaped by the inventive use of principles formulated by 'thinking with objects,' such as the balance and the pendulum. The former provided an initial framework to relate the speeds and sizes of bodies, and the latter was key both in the development of novel conservation principles and made possible experimental investigations of collision. This dissertation documents the formation of concepts central to modern physical science, and re-evaluates the mathematics of collision, with implications for our understanding of major figures in early modern science, such as Descartes and Huygens, and repercussions for the mathematization of nature

\u3cem\u3eWork\u3c/em\u3e 2006/2007

WORK is an annual publication of the Department of Architecture that documents student work in design studios and courses in the Master of Architecture and Post-Professional programs, as well as events, faculty news and student awards. It also includes abstracts of PhD dissertations defended that year. It provides an opportunity to explore the creative work of our students and is a permanent record of work in the Department

Guidobaldo dal Monte's mechanics in context

The doctoral thesis "Guidobaldo dal Monte's mechanics in context" presents reseaches on Guidobaldo's mechanics, against the background of his biography and technical-scientific environment. After three introductory chapters concerning Guidobaldo's life, his cultural milieu and the background of sixteenth-century mechanics, the thesis deals with the analysis of Guidobaldo's major works on mechanics. The second part of the thesis is dedicated to two crucial topics and problems of Guidobaldo's mechanics. The appendixes present ample documentary material

The Unity of Science in Early-Modern Philosophy: Subalternation, Metaphysics and the Geometrical Manner in Scholasticism, Galileo and Descartes

The project of constructing a complete system of knowledge---a system capable of integrating all that is and could possibly be known---was common to many early-modern philosophers and was championed with particular alacrity by René Descartes. The inspiration for this project often came from mathematics in general and from geometry in particular: Just as propositions were ordered in a geometrical demonstration, the argument went, so should propositions be ordered in an overall system of knowledge. Science, it was thought, had to proceed `more geometrico'. I offer a new interpretation of `science emph{more geometrico}' based on an analysis of the explanatory forms used in certain branches of geometry. These branches were optics, astronomy, and mechanics; the so-called subalternate, subordinate, or mixed-mathematical sciences. In Part I, I investigate the nature of the mixed-mathematical sciences according to Aristotle and some `liberal Jesuit' scholastic-Aristotelians. In Part II, the heart of the work, I analyze the metaphysics and physics of Descartes' "Principles of Philosophy" (1644, 1647) in light of the findings of Part I and an example from Galileo. I conclude by arguing that we must broaden our understanding of the early-modern conception of `science more geometrico' to include concepts taken from the mixed-mathematical sciences. These render the geometrical manner more flexible than previously thought