Related papers: An Etruscan Dodecahedron
Here we briefly discuss lattices in Euclidean spaces and spaces of lattices, which are basic objects that can be described in terms of matrices and are important settings in classical analysis.
Two books, the "De Architectura" by Vitruvius and the "Naturalis Historia" by Pliny the Elder, give us a portrait of the Materials Science, that is, the knowledge of materials, in Rome at the beginning of the Empire. Here, I am reporting…
We investigate the space of simplices in Euclidean Space
In the present work it is shown, by an examination of the Platonic dialogues Theaetetus, Sophistes, Politicus, and Philebus, that (a) a Platonic Idea is the philosophic analogue of a pair of lines incommensurable in length only, (b) the…
We consider the $N$-vortex problem on the sphere assuming that all vortices have equal strength. We develop a theoretical framework to analyse solutions of the equations of motion with prescribed symmetries. Our construction relies on the…
The article is an historical overview of some of the major contributions from different areas of Science with which, for centuries, it has been built up a scientific, sound and consistent vision of the atom. Some experiments that led us to…
Even though Plato's philosophy in ancient times was always closely associated with mathematics, modern Platonic scholarship, during the last five centuries, has moved steadily toward de-mathematization. The present work aims to outline a…
This paper introduces a new family of solids, which we call \textit{polycons}, which generalise the sphericon in a natural way. The static properties of the polycons are derived, and their rolling behaviour is described and compared to that…
We study translation covers of several triply periodic polyhedral surfaces that are intrinsically Platonic. We describe their affine symmetry groups and compute the quadratic asymptotics for counting saddle connections and cylinders,…
Experimental results on hadronic structures are discussed in view of our physics understanding. Achievements and challenges are noted.
There is only one type of tilings of the sphere by $12$ congruent pentagons. These tilings are isohedral.
We fantasize about an alternative history for theories of electromagnetism and gravitation.
The chapter advances a reformulation of the classical problem of the nature of mathematical objects (if any), here called "Plato's problem," in line with the program of a philosophy of mathematical practice. It then provides a sketch of a…
The regular icosahedron is connected to many exceptional objects in mathematics. Here we describe two constructions of the $\mathrm{E}_8$ lattice from the icosahedron. One uses a subring of the quaternions called the "icosians", while the…
A brief review of the history of the conic sections would not be complete without an exhaustively tolerable account of all the things related to the subject that can be found in the extensive work of the wise Archimedes. There is no strong…
Retaining the combinatorial Euclidean structure of a regular icosahedron, namely the 20 equiangular (planar) triangles, the 30 edges of length 1, and the 12 different vertices together with the incidence structure, we investigate variations…
Solids are rigid, which means that when left undisturbed, their structures are nearly static. It follows that these structures depend on history -- but it is surprising that they hold readable memories of past events. Here we review the…
We study quartic surfaces that admit a group of projective automorphisms isomorphic to icosahedron group.
These are some informal notes concerning topological vector spaces, with a brief overview of background material and basic notions, and emphasis on examples related to classical analysis.
We explore a novel link between two seemingly disparate mathematical concepts: Egyptian fractions and fractals. By examining the decomposition of rationals into sums of distinct unit fractions, a practice rooted in ancient Egyptian…