Related papers: An Etruscan Dodecahedron
We study quotients of projective and affine spaces by various actions of the icosahedral group. Basically we concentrate on the rationality questions.
This is a survey article on classical groups (over arbitrary division rings) and their geometries.
These are the handouts of an undergraduate minicourse at the Universit\`a di Bari, in the context of the 2017 INdAM Intensive Period "Contemporary Research in elliptic PDEs and related topics". Without any intention to serve as a throughout…
We show that four of the five Platonic solids' surfaces may be cut open with a Hamiltonian path along edges and unfolded to a polygonal net each of which can "zipper-refold" to a flat doubly covered parallelogram, forming a rather compact…
The notion of a spiral unfolding of a convex polyhedron, resulting by flattening a special type of Hamiltonian cut-path, is explored. The Platonic and Archimedian solids all have nonoverlapping spiral unfoldings, although among generic…
This paper constructs a Riemann surface associated to the icosahedron and discusses the geodesics associated to a flat metric on this surface. Because of the icosahedral symmetry, this is a distinguished special case of the example treated…
A brief review of the history of the discovery of the deuteron in provided. The current status of both experiment and theory for the elastic electron scattering is then presented.
Sol LeWitt famously enumerated all the incomplete open cubes, finding 122 of these connected, non-planar subsets of the edges of the cube. Since then, while several projects have revisited the cube enumeration, no such enumeration has been…
This contribution has the aim to review the basics of existing knowledge on asteroid interiors and to explore the possibility of advancing the field by introducing into it some ideas from solid state physics.
We characterize the Archimedean solids among the convex uniform polyhedra via face embeddings into a regular Tetrahedron. This result has been listed without proof in the literature.
A Platonic surface is a Riemann surface that underlies a regular map and so we can consider its vertices, edge-centres and face-centres. A symmetry (anticonformal involution) of the surface will fix a number of simple closed curves which we…
We present some arguments for the thesis that a set-theoretic inspired faith, in the ability of intuitive truth to faithfully reflect relationships between elements of a Platonic universe, may be as misplaced as an assumption that such…
There is a relationship between the Borromean rings, the icosahedron and something called the Poincar\'e homology sphere. This relationship is explored in a wandering path that introduces fundamental ideas from topology and a geometric…
In this paper, we study the so-called 'Mathematical part' of Plato's Theaetetus. Its subject concerns the incommensurability of certain magnitudes, in modern terms the question of the rationality or irrationality of the square roots of…
Bronze cuboctahedral weights dated to the VIII-X centuries were found in northwest Russia near Ladoga, one of the most important trading centers in Eastern Europe in the VIII-X centuries. The history of the mathematical study of…
This presentation starts with the regular polygons, of course, then with the Platonic and Archimedean solids. The latter ones are whose symmetry groups are transitive on the vertices, and in addition, whose faces are regular polygons (see…
The theory of the isoptic curves is widely studied in the Euclidean plane $\bE^2$ (see \cite{CMM91} and \cite{Wi} and the references given there). The analogous question was investigated by the authors in the hyperbolic $\bH^2$ and elliptic…
A problem about the present structure of dimensional analysis, and another one about the differences between solids and fluids are suggested. Both problems appear to have certain foundational aspects.
We present some methods for constructing connected spatial geometric configurations $(p_{q}, n_{k})$ of points and lines, preserved by the same rotations (and reflections) of Euclidean space $E^{3}$ as the chosen Platonic solid. In this…
This article is devoted to the study of classical and new results concerning equidistant sets, both from the topological and metric point of view. We start with a review of the most interesting known facts about these sets in the euclidean…