The Distance Geometry of Music

We demonstrate relationships between the classic Euclidean algorithm and many other fields of study, particularly in the context of music and distance geometry. Specifically, we show how the structure of the Euclidean algorithm defines a family of rhythms which encompass over forty timelines (\emph{ostinatos}) from traditional world music. We prove that these \emph{Euclidean rhythms} have the mathematical property that their onset patterns are distributed as evenly as possible: they maximize the sum of the Euclidean distances between all pairs of onsets, viewing onsets as points on a circle. Indeed, Euclidean rhythms are the unique rhythms that maximize this notion of \emph{evenness}. We also show that essentially all Euclidean rhythms are \emph{deep}: each distinct distance between onsets occurs with a unique multiplicity, and these multiplicies form an interval $1,2,...,k-1$. Finally, we characterize all deep rhythms, showing that they form a subclass of generated rhythms, which in turn proves a useful property called shelling. All of our results for musical rhythms apply equally well to musical scales. In addition, many of the problems we explore are interesting in their own right as distance geometry problems on the circle; some of the same problems were explored by Erd\H{o}s in the plane.Comment: This is the full version of the paper: "The distance geometry of deep rhythms and scales." 17th Canadian Conference on Computational Geometry (CCCG '05), University of Windsor, Canada, 200

Diagrammatic Reasoning and Modelling in the Imagination: The Secret Weapons of the Scientific Revolution

Just before the Scientific Revolution, there was a "Mathematical Revolution", heavily based on geometrical and machine diagrams. The "faculty of imagination" (now called scientific visualization) was developed to allow 3D understanding of planetary motion, human anatomy and the workings of machines. 1543 saw the publication of the heavily geometrical work of Copernicus and Vesalius, as well as the first Italian translation of Euclid

A leap of faith: Abbott, Bellamy, Morris, Wells and the fin-de-siècle route to utopia

In the great surge of utopian writing that was produced during the fin de siècle, Edward Bellamy, William Morris and H. G. Wells among others imagined utopias that were global in scale and located in the future. They made a radical shift in utopian thinking by drawing a historical trajectory between their own time and that of utopia. A contemporaneous text that might seem to have little in common with these “historical utopias” is E. A. Abbott’s Flatland: A Romance of Many Dimensions (1884). This article shows how closely its ideas can bring into focus those of the specifically utopian texts being written alongside it. Flatland breaks the conventions of utopian narrative by removing the reader from the narrative plane and situating us instead in the “impossible” third dimension. The “leap of faith” necessary for scientific or religious revelation is simultaneously invoked as the route to utopia

Raphael\u27s School of Athens: A Theorem in a Painting?

Raphael\u27s famous painting The School of Athens includes a geometer, presumably Euclid himself, demonstrating a construction to his fascinated students. But what theorem are they all studying? This article first introduces the painting, and describes Raphael\u27s lifelong friendship with the eminent mathematician Paulus of Middelburg. It then presents several conjectured explanations, notably a theorem about a hexagram (Fichtner), or alternatively that the construction may be architecturally symbolic (Valtieri). The author finally offers his own null hypothesis : that the scene does not show any actual mathematics, but simply the fascination, excitement, and joy of mathematicians at their work

The Twilight of Land-Use Controls: A Paradigm Shift?

The subject chosen for this discussion is both timely and thought-provoking: the status and future of land-use regulations in the United States. In the hope of making the issues subsumed under this title as exciting to the general public as they are to the practitioners, Professor Michael Allan Wolf has taken the monumental Euclid decision of the United States Supreme Court in 1926 as the pivot of our deliberations. He has posed the question most dramatically with overtones of a swelling Wagnerian overture: Is It The Twilight of Environmental and Land-Use Regulation

The notion of dimension in geometry and algebra

This talk reviews some mathematical and physical ideas related to the notion of dimension. After a brief historical introduction, various modern constructions from fractal geometry, noncommutative geometry, and theoretical physics are invoked and compared.Comment: 29 pages, a revie

A Leap of Faith: Aesthetic Education in the Mathematics Education Classroom

We speak only for having been called, called by what there is to say, and yet we learn and hear what there is to say only in speech itself. Jean-Louis Chrétien (2004, p. 1) For the past four years I have hosted Lincoln Center Teaching Artists in a graduate mathematics education course I teach for elementary school teachers and elementary school pre-service teachers. I find the hosting experience enjoyable and informative and my impression, gained from comments and short written reflections, is that many of my students find it likewise. Simultaneously, I believe in the pragmatic worth of this experience. For instance, I hold that education in the aesthetic has the potential to deepen teachers’ awareness of how they intend—or grasp—the world they inhabit (Greene, 1973, p. 10)

Associating Mathematics to its History: Connecting the Mathematics we Teach to its Past

Across the USA and around the world now, globalization has taken a strong hold. The purpose of this paper is to explore the historical considerations that can be incorporated in the teaching of mathematics. The paper will also provide suggestions for teaching math by interweaving historical elements into the mathematics instruction. Teachers should strive to bridge the cultural and historical gap among all students by incorporating innovative ideas as well as historical and cultural connections into their teaching so to foster understanding, appreciation, and tolerance for the richness inherent in diversity and a sound understanding of mathematics and appreciation for other cultures and their contributions to the field of mathematics. It is often believed that no other subject besides math dissociates itself from its history. This paper makes an attempt to give teachers ideas to incorporate more history in the teaching of mathematics by showing what was going on at the time that may have influenced the development of such mathematics that we now teach in our classrooms of today