Are Points (Necessarily) Unextended?

Since Euclid defined a point as “that which has no part” it has been widely assumed that points are necessarily unextended. It has also been assumed that this is equivalent to saying that points or, more properly speaking, degenerate segments, have length zero. We challenge these assumptions by providing models of Euclidean geometry where the points are extended despite the fact that the degenerate segments have null lengths, and observe that whereas the extended natures of the points are not recognizable in the given models, they can be recognized and characterized by structures that are suitable expansions of the models.

Geometrization of the theory of electromagnetic and spinor fields on the background of the Schwarzschild spacetime

The geometrical Kosambi–Cartan–Chern approach has been applied to study the systems of differential equations which arise in quantum-mechanical problems of a particle on the background of non-Euclidean geometry. We calculate the geometrical invariants for the radial system of differential equations arising for electromagnetic and spinor fields on the background of the Schwarzschild spacetime. Because the second invariant is associated with the Jacobi field for geodesics deviation, we analyze its behavior in the vicinity of physically meaningful singular points r = M, ∞. We demonstrate that near the Schwarzschild horizon r = M the Jacobi instability exists and geodesics diverge for both considered problems.

RETROSPECTIVE AND CONTRADICTIONS OF CREATING ARCHITECTURAL PROJECTS IN THE CONTEXT OF DIGITALIZATION

The research article examines the retrospective of the creation of architectural projects, systematizes the experience of architects, reveals contradictions in the development of digital architecture and its relationship with traditional design. At the same time, the problems of the development of digital architecture in the context of the formation of digital civilization are noted. The tendencies of designing objects based on non-Euclidean geometry are revealed, the features of postmodernism and parametrism are systematized, the threats and consequences of the "from figure to form" approach are substantiated.

Lorentzian quantum cosmology goes simplicial

We employ the methods of discrete (Lorentzian) Regge calculus for analysing Lorentzian quantum cosmology models with a special focus on discrete analogues of the no-boundary proposal for the early universe. We use a simple 4-polytope, a subdivided 4-polytope and shells of discrete 3-spheres as triangulations to model a closed universe with cosmological constant, and examine the semiclassical path integral for these different choices. We find that the shells give good agreement with continuum results for small values of the scale factor and in particular for finer discretisations of the boundary 3-sphere, while the simple and subdivided 4-polytopes can only be compared with the continuum in certain regimes, and in particular are not able to capture a transition from Euclidean geometry with small scale factor to a large Lorentzian one. Finally, we consider a closed universe filled with dust particles and discretised by shells of 3-spheres. This model can approximate the continuum case quite well. Our results embed the no-boundary proposal in a discrete setting where it is possibly more naturally defined, and prepare for its discussion within the realm of spin foams.

Simultaneous determination of mass parameter and radial marker in Schwarzschild geometry

We show that mass parameter and radial marker values can be indirectly measured in thought experiments performed in Schwarzschild spacetime, without using the Newtonian limit of general relativity or approximations based on Euclidean geometry. Our approach involves different proper time quantifications as well as solutions to systems of algebraic equations, and aims to strengthen the conceptual independence of general relativity from Newtonian gravity.

Data with Non-Euclidean Geometry and its Characterization

Recently, dealing the Non-Euclidean data and its characterization is considered as one of the major issues by researchers. The first problem arises while distinction of among Euclidean and non-Euclidean geometry. The second problem arises with dealing the Non-Euclidean geometry in true, false and uncertain regions. The third problem arises while investigating some pattern in Non-Euclidean data sets. This paper focused on tackling these issues with some real life examples.

Non-Euclidean Geometry Lesson Promotes Mathematical Reasoning

A two-day lesson on taxicab geometry introduces high school students to a unit on proof.