Humans have always constructed spaces, through Mythos and Logos, as part of an aspiration to capture the essence of the changing world. This has been a permanent endeavour since the invention of language. By doing this, in fact, Humankind started con
structing itself: we are beings in constant evolutionary process in real and imaginary spaces. Our concepts of Space and our anthropological ideas, specially the fundamental concepts of subject and subjectivity, are intertwined and intimately connected. We believe that the great narratives about Humanity, which ultimately define our view of ourselves, are entangled with those concepts that Cassirer identified as the cornerstones of culture: space, time, and number. To explore these ideas, the authors wrote an essay, in 2017, in a book format, in which the fundamental role of real and imaginary spaces (and especially of their dimensionalities) in the History of Culture was discussed. This book, titled O Livro, o Espac{c}o e a Natureza: Ensaio Sobre a Leitura do Mundo, as Mutac{c}~oes da Cultura e do Sujeito, has a preface written by Francisco Antonio Doria. As many of the issues treated there are among his multiple interests, it was decided to revisit here the problems of subjectivity and subjects relationship with the dimensionality of space including the question of the architecture of books and other writing supports.
In this article, the evolution of the ideas about the fourth spatial dimension is presented, starting from those which come out within classical Euclidean geometry and going through those arose in the framework of non-Euclidean geometries, like those
of Riemann and Minkowski. Particular attention is given to the moment when real time is effectively considered as a fourth dimension, as introduced by Einstein.
Hydrogen atom is supposed to be described by a generalization of Schrodinger equation, in which the Hamiltonian depends on an iterated Laplacian and a Coulomb-like potential $r^{-beta}$. Starting from previously obtained solutions for this equation u
sing the $1/N$ expansion method, it is shown that new light can be shed on the problem of understanding the dimensionality of the world as proposed by Paul Ehrenfest. A surprisingly new result is obtained. Indeed, for the first time, we can understand that not only the sign of energy but also the value of the ground state energy of hydrogen atom is related to the threefold nature of space.
Bidimensional muonic and electronic atoms, with nuclei composed of a proton, deuteron, and triton, and governed by Chern-Simons potential, are numerically solved. Their eigenvalues and eigenfunctions are determined with a slightly modified Numerov me
thod. Results are compared with those assuming that the same atoms are governed by the usual $1/r$ potential even in a two-dimensional space, as well as with its three-dimensional analogs.
Within the general discussion of space and its dimensionality, Aristotles position is of the greatest relevance, as one will have the opportunity to argue and discuss in this article.
In this paper, a quantum dot mathematical model based on a two-dimensional Schrodinger equation assuming the 1/r inter-electronic potential is revisited. Generally, it is argued that the solutions of this model obtained by solving a biconfluent Heun
equation have some limitations. The known polynomial solutions are confronted with new numerical calculations based on the Numerov method. A good qualitative agreement between them emerges. The numerical method being more general gives rise to new solutions. In particular, we are now able to calculate the quantum dot eigenfunctions for a much larger spectrum of external harmonic frequencies as compared to previous results. Also the existence of bound state for such planar system, in the case l=0, is predicted and its respective eigenvalue is determined.
The model of a two-electron quantum dot, confined to move in a two dimensional flat space, in the presence of an external harmonic oscillator potential, is revisited for a specific purpose. Indeed, eigenvalues and eigenstates of the bound state solut
ions are obtained for any oscillation frequency considering both the $1/r$ and $ln r$ Ansatze for inter-electronic Coulombic-like potentials in 2$D$. Then, it is pointed out that the significative difference between measurable quantities predicted from these two potentials can shed some light on the problem of space dimensionality as well as on the physical nature of the potential itself.
From arguments based on Heisenbergs uncertainty principle and Paulis exclusion principle, the molar specific heats of degenerate ideal gases at low temperatures are estimated, giving rise to values consistent with the Nerst-Planck Principle (third la
w of Thermodynamics). The Bose-Einstein condensation phenomenon based on the behavior of specific heat of massive and non-relativistic boson gases is also presented.
A general sketch on how the problem of space dimensionality depends on anthropic arguments is presented. Several examples of how life has been used to constraint space dimensionality (and vice-versa) are reviewed. In particular, the influences of thr
ee-dimensionality in the solar system stability and the origin of life on Earth are discussed. New constraints on space dimensionality and on its invariance in very large spatial and temporal scales are also stressed.
The model of a two-electron quantum dot, confined to move in a two dimensional flat space, is revisited. Generally, it is argued that the solutions of this model obtained by solving a biconfluent Heun equation have some limitations. In particular, so
me corrections are also made in previous theoretical calculations. The corrected polynomial solutions are confronted with numerical calculations based on the Numerov method, in a good agreement between both. Then, new solutions considering the $1/r$ and $ln r$ Coulombian-like potentials in (1+2)D, not yet obtained, are discussed numerically. In particular, we are able to calculate the quantum dot eigenfunctions for a much larger spectrum of external harmonic frequencies as compared to previous results. Also the existence of bound states for such planar system in the case $l=0$ is predicted and the respective eigenvalues are determined.
