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In this work, we measure the volumetric mean radius of the Earth, reproducing the historical experiment of Eratosthenes, carried out around 240 BC, in the ancient cities of Siena and Alexandria. Here, we perform measurements in the cities…
This is the fifth in a series of papers giving a proof of the Kepler conjecture, which asserts that the density of a packing of congruent spheres in three dimensions is never greater than $\pi/\sqrt{18}\approx 0.74048...$. This is the…
We show how the expectation values of geometrical quantities in 3d quantum gravity can be explicitly computed using grasping rules. We compute the volume of a labelled tetrahedron using the triple grasping. We show that the large spin…
The notorious `measurement problem' has been roving around quantum mechanics for nearly a century since its inception, and has given rise to a variety of `interpretations' of quantum mechanics, which are meant to evade it. We argue that no…
We present a survey article about the geometry of convex bodies on the $d$-dimensional sphere $S^d$. We concentrate on the results based on the notion of the width of a convex body $C \subset S^d$ determined by a supporting hemisphere of…
In this paper, we mainly consider the problem of spherical distribution of 5 points, that is, how to configure 5 points on a sphere such that the mutual distance sum attains the maximum. It is conjectured that the sum of distances is…
We provide some historical context to the study of solid angles carried out by Euler in his memoir \emph{De mensura angulorum solidorum} (On the measure of solid angles). We extend our study to the general notion of angle (not only solid).…
Spinors have played an essential but enigmatic role in modern physics since their discovery. Now that quantum-gravitational theories have started to become available, the inclusion of a description of spin in the development is natural and…
Three different representation of the proper Euclidean geometry are considered. They differ in the number of basic elements, from which the geometrical objects are constructed. In E-representation there are three basic elements (point,…
A fundamental problem in spherical distance geometry aims to recover an $n$-tuple of points on a 2-sphere in $\mathbb{R}^3$, viewed up to oriented isometry, from $O(n)$ input measurements. We solve this problem using algorithms that employ…
Even though it has been almost a century since quantum mechanics planted roots, the field has its share of unresolved problems. It could be the result of a wrong mathematical structure providing inadequate understanding of the quantum…
Several recent large length scale methods of detecting the topology of the Universe are briefly reviewed. Three-dimensional methods include those of Lehoucq, Luminet & Lachi\`eze-Rey (1996), Roukema (1996) and Roukema & Edge (1997). The…
Many classical geometric inequalities on functionals of convex bodies depend on the dimension of the ambient space. We show that this dimension dependence may often be replaced (totally or partially) by different symmetry measures of the…
The direct gravimetry problem is solved by dividing each deposit body into a set of vertical adjoining bars, whereas in the inverse problem, each deposit body is modelled by a homogeneous ellipsoid of revolution (spheroid). Well-known…
Here, I test the long-held assumption that the ancient Egyptian sky goddess Nut represented the Milky Way by examining Nut's visual depictions on ancient Egyptian coffins. I assemble a catalog of 555 coffin elements, including 118…
We produce a family of bodies in $\mathbb R^3$ parameterized by $\varepsilon > 0$, each bounded by a smooth topological sphere with principal curvatures in $[-1, 1]$, and having volume arbitrarily close to $ 16 - 4\sqrt 3 + \left(10 \sqrt 3…
In the middle of the third millennium BC the Sumerians must have noticed that the reciprocal of the number 7,in contrast to the numbers 1,2,3,4,5,and 6,could not be expressed by a finite sexagesimal fraction but it recurred every three…
Let $\Omega$ be a measurable Euclidean set in $\mathbb{R}^{n}$ that is symmetric, i.e. $\Omega=-\Omega$, such that $\Omega\times\mathbb{R}$ has the smallest Gaussian surface area among all measurable symmetric sets of fixed Gaussian volume.…
The three-particle quantization condition is partially diagonalized in the center-of-mass frame by using cubic symmetry on the lattice. To this end, instead of spherical harmonics, the kernel of the Bethe-Salpeter equation for…
It will be shown in this article that an ontological approach for some problems related to the interpretation of Quantum Mechanics could emerge from a re-evaluation of the main paradox of early Greek thought: the paradox of Being and…