It has been pointed out that there is no primitive name in natural and formal languages for one corner of the famous square of oppositions. We have all, some and no, but no primitive name for not all. It is true also in the modal version of the square, we have necessary, possible and impossible, but no primitive name for not necessary. I shed here a new light on this mysterious non-lexicalisati...

The spectral action functional, considered as a model of gravity coupled to matter, provides, in its non-perturbative form, a slow-roll potential for inflation, whose form and corresponding slowroll parameters can be sensitive to the underlying cosmic topology. We explicitly compute the non-perturbative spectral action for some of the main candidates for cosmic topologies, namely the quaternion...

A packing of congruent balls of unit radius in three-dimensional Euclidean space is determined by and is identified with the set Λ of centers of the balls. A packing Λ determines a region called the Voronoi cell around each ball. The Voronoi cell Ω(Λ, v) around a ball at v ∈ Λ consists of points of space that are closer to v than to any other w ∈ Λ. The Voronoi cell is a convex polyhedron conta...

A new way of probing the large-scale structure of the universe is proposed. Space is partitioned into cells the shape of rhombic dodecahedron. The cells are labelled “filled” or “empty” according as they contain galaxies or not. The cell size is so chosen as to have nearly equal numbers of filled and empty cells for the given galaxy sample. Two observables on each cell are definable: the number...

Context. Two fundamental questions regarding our description of the Universe concern the geometry and topology of its 3-dimensional space. While geometry is a local characteristic that gives the intrinsic curvature, topology is a global feature that characterizes the shape and size of the 3-space. The geometry constrains, but does not dictate, the spatial topology. Aims. We show that besides de...

Three-dimensional (3D) displays having regular-polyhedron structures are proposed and their imaging characteristics are analyzed. Four types of conceptual regular-polyhedron 3D displays, i.e., hexahedron, octahedron, dodecahedron, and icosahedrons, are considered. In principle, regular-polyhedron 3D display can present omnidirectional full parallax 3D images. Design conditions of structural fac...

The antiferromagnetic Heisenberg model on icosahedral symmetry Ih fullerene clusters exhibits unconventional magnetic properties despite the lack of anisotropic interactions. At the classical level, and for number of sites n 720, the magnetization has two discontinuities in an external magnetic field, except from the dodecahedron where it has three, emphasizing the role of frustration introduce...

In this article we examine the conjecture of Neumann and Reid that the only hyperbolic knots in the 3-sphere which admit hidden symmetries are the figure-eight knot and the two dodecahedral knots. Knots whose complements cover hyperbolic reflection orbifolds admit hidden symmetries, and we verify the Neumann-Reid conjecture for knots which cover small hyperbolic reflection orbifolds. We also sh...

This investigation concerns the comparison of twoand three-dimensional data obtained on a cellular material. By quantitative micrography techniques and spatial measurements, the cellular structure corresponds most closely with the shape of pentagonal dodecahedra, twelve-faced cells having five edges per face. The cell volumes have a normal distribution. The areal distributions of planar section...

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 representation of the surface. Thus these regular polyhedra have particular flat “zipper pairs.” No such zipper pair exists for a dodecahedron, whose Hamilton...