Related papers: Challenges in Plane Symmetry: From Theory to Perce…
Symmetry is a powerful tool for finding analytical solutions to differential equations, both partial and ordinary, via the similarity variables or via the invariance of the equation under group transformations. It is the largest group of…
Classical mathematics are founded within set theory, but sets don't have \emph{symmetries}. We conjecture that if we allow sets with symmetries, then many problems such as \emph{Mirror symmetry} or \emph{Homological mirror symmetry} can be…
This paper introduces a new approach for the automated reconstruction - reassembly of fragmented objects having one surface near to plane, on the basis of the 3D representation of their constituent fragments. The whole process starts by 3D…
Statistical models that possess symmetry arise in diverse settings such as random fields associated to geophysical phenomena, exchangeable processes in Bayesian statistics, and cyclostationary processes in engineering. We formalize the…
Robots exhibit a rich variety of symmetries arising from their mechanical structure and the properties of their tasks. Although many robotics problems exhibit several symmetries simultaneously, existing approaches typically treat them in…
Motivated by a question in origami, we consider sets of points in the complex plane constructed in the following way. Let $L_\alpha(p)$ be the line in the complex plane through $p$ with angle $\alpha$ (with respect to the real axis). Given…
Many mechanical systems have configuration spaces that admit symmetries. Mathematically, such symmetries are modelled by the action of a group on a topological space. Several variations of topological complexity have emerged that take…
The concepts of symmetry and its breakdown are investigated in two different terms according to whether the resulting asymmetry is universal or only obtained for a special configuration: we shall illustrate this by considering in the first…
Correctly capturing the symmetry transformations of data can lead to efficient models with strong generalization capabilities, though methods incorporating symmetries often require prior knowledge. While recent advancements have been made…
We construct the quaternion algebra [10] "geometrically" by a three dimensional analogue of the classic two dimensional geometric description of the complex field. The algebraic description of the multiplication operation in three…
A projective rectangle is like a projective plane that has different lengths in two directions. We develop harmonic conjugation in projective rectangles. We construct projective rectangles in some harmonic matroids (matroids where harmonic…
An idea to present a classical Lie group of positive dimension by generators and relations sounds dubious, but happens to be fruitful. The isometry groups of classical geometries admit elegant and useful presentations by generators and…
In this paper we present the results from a program developed by the author that finds the unitals of the known 193 projective planes of order 25.. There are several planes for which we have not found any unital. One or more than one…
Transversal structures (also known as regular edge labelings) are combinatorial structures defined over 4-connected plane triangulations with quadrangular outer-face. They have been intensively studied and used for many applications…
We describe a categorical approach to finite noncommutative geometries. Objects in the category are spectral triples, rather than unitary equivalence classes as in other approaches. This enables to treat fluctuations of the metric and…
The second-order differential equation describes harmonic oscillators, as well as currents in LCR circuits. This allows us to study oscillator systems by constructing electronic circuits. Likewise, one set of closed commutation relations…
Man-made environments typically comprise planar structures that exhibit numerous geometric relationships, such as parallelism, coplanarity, and orthogonality. Making full use of these relationships can considerably improve the robustness of…
Perceptual geometry refers to the interdisciplinary research whose objectives focuses on study of geometry from the perspective of visual perception, and in turn, applies such geometric findings to the ecological study of vision. Perceptual…
We investigate a special kind of contraction of symmetric spaces (respectively, of Lie triple systems), called homotopy. In this first part of a series of two papers we construct such contractions for classical symmetric spaces in an…
This paper develops a formal theory of musical scales and their harmonic coverings and introduces orbit covers: coverings obtained by translating a fixed subset across a scale via a group action. Orbit covers generalize familiar…