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This article provides a historical overview of Geometry of Numbers. 1. Figures, 2. The circuit problem and its relatives, 3. Minkowski lattice point set, 4. The young Hermann Minkowski, 5. The geometry of numbers develops, 6. Minkowski…
In this paper we deal analytically with a version of the so called clock paradox in which the moving clock performs a circular motion of constant radius. The rest clock is denoted as (1), the rotating clock is (2), the inertial frame in…
A geometrical pattern is a set of points with all pairwise distances (or, more generally, relative distances) specified. Finding matches to such patterns has applications to spatial data in seismic, astronomical, and transportation…
While modern optics is largely a physics of harmonic oscillators and two-by-two matrices, it is possible to learn about some hidden properties of the two-by-two matrix from optical systems. Since two-by-two matrices can be divided into…
A simple closed-form expression for the number of days between the winter solstice and the latest sunrise is derived. Formulas for the summer solstice and the sunset are derived as well. These approximate formulas make it easy to see the…
A better understanding of the formation of mass structures in the universe can be obtained by determining the amount and distribution of dark and luminous matter in spiral galaxies. To investigate such matters a sample of 12 galaxies, most…
Castor is a system of six stars in which the two brighter objects, Castor A and B, revolve around each other every $\sim$450 yr and are both short-period spectroscopic binaries. They are attended by the more distant Castor C, which is also…
The paper uses geometrical arguments to derive equations with relevance for cosmology; 5-dimensional spacetime is assumed because it has been shown in other works to provide a setting for significant unification of different areas of…
Solar system, exoplanet and stellar science rely on transits, eclipses and occultations for dynamical and physical insight. Often, the geometry of these configurations are modelled by assuming a particular viewpoint. Here, instead, I derive…
We have obtained empirical laws for the variation of the Earth parameters with geologic time that are in agreement with coral fossil data obtained by Wells and Runcorn. Our model predicts that the day is lengthening at a rate of 2…
The possibility of detecting the gravitomagnetic clock effect using artificial Earth satellites provides the incentive to develop a more intuitive approach to its derivation. We first consider two test electric charges moving on the same…
Circadian oscillation provides selection advantages through synchronization to the daylight cycle. However, a reliable clock must be designed through two conflicting properties: entrainability to synchronize internal time with periodic…
Consider an analytic map of a neighborhood of 0 in a vector space to a Euclidean space. Suppose that this map takes all germs of lines passing through 0 to germs of circles. Such a map is called rounding. We introduce a natural equivalence…
We construct a highly-symmetric periodic orbit of six bodies in three dimensions. In this orbit, binary collisions occur at the origin in a regular periodic fashion, rotating between pairs of bodies located on the coordinate axes.…
In this article we review some key time cycles in ancient Indian astronomy, especially those that have emerged from researches in the past couple of decades expressing knowledge of the changing frame of earth's axis. The article also…
The paper introduces cycles cross ratio, which extends the classic cross ratio of four points to various settings: conformal geometry, Lie spheres geometry, etc. Just like its classic counterpart cycles cross ratio is a measure of…
The Law of Universal Gravitation is part of middle and high school's general physics and astronomy curricula. This topic is included in the most popular physics textbooks available as a fact whose origin remains in the detailed work of Sir…
The possible existence of a complex structure on the 6-sphere has been a famous unsolved problem for over 60 years. In that time many "solutions" have been put forward, in both directions. Mistakes have always been found. In this paper I…
In this paper, the global qualitative analysis of planar quadratic dynamical systems is established and a new geometric approach to solving Hilbert's Sixteenth Problem in this special case of polynomial systems is suggested. Using geometric…
Timescales spanning 24 orders of magnitude smaller than one second can be studied experimentally, and each range is packed with different physical phenomena. This rich range of timescales offers a great context for an innovative…