Geometry doi: 10.3390/geometry3020009

Authors: Sebastian Xambó-Descamps

This paper is devoted to provide a proof of F. Morley’s theorem concerning triangles in the Euclidean plane E (see Theorem 1 in the Introduction section) phrased in terms of the geometric algebra G of E (called Wessel’s algebra). This algebra is studied in detail in Section 2, its uses in describing isometries of E in Section 3, its bearing on the geometry of Morley’s construction in Section 4, and the claimed proof in Section 5. Morley’s theorem can be extended by using all the trisectors (interior and exterior) of a triangle, and suitable intersections of them. These intersections form what we call Morley’s constellation and out of it 36 generalized Morley triangles can be formed. Among these triangles, 27 are equilateral and with sides parallel to the original Morley triangle (Appendix B). The 36 triangles are depicted in Appendix C. All graphics in this work have been created by the author.

Geometry doi: 10.3390/geometry3020008

Authors: Janusz Januszewski Łukasz Zielonka

A translative covering the rectangle a×b with homothetic copies of a right isosceles triangle T (of the legs parallel to the sides of a×b) is considered. It is shown that any collection of equal triangles homothetic to T with the total area at least 2 permits a translative covering of 1×1; this bound is tight. It is also demonstrated that any collection of positive homothetic copies of T with the total area at least 3 permits a translative covering of 1×1. Moreover, it is proven that if a≥5+334b, then any collection of triangles homothetic to T with the total area at least 12(a+b)2 permits a translative covering of a×b; this bound is tight.

Geometry doi: 10.3390/geometry3020007

Authors: Abdel Rahman Al-Abdallah

We give new concise Lie-theoretic proofs of basic analytic–geometric properties of connected complex Lie groups. Using Matsushima’s biholomorphic splitting G≃Cn×K˜ together with a refined analysis of the center via its Cousin factor, we show that every connected complex Lie group is pseudoconvex. Our approach is structural: it reduces to the reductive factor, separates the semisimple and central parts, and concludes using permanence of pseudoconvexity under products and finite quotients, together with standard triviality results for holomorphic principal bundles over Stein bases.

Geometry doi: 10.3390/geometry3010006

Authors: Satyanad Kichenassamy

We obtain improved regularity estimates on solutions of partial differential equations by combining the method of Fuchsian Reduction with geometric transformations. Examples include the meron problem and the regularity of the conformal radius. In each case, Reduction needs to be combined with a reinterpretation of the underlying geometry. We argue that the geometric meaning assigned to a problem has an influence, positive or negative, on the range of methods envisioned for its solution, and that the Euler–Poisson–Darboux (EPD) equation cannot be properly understood within a single geometric framework. This explains the central position of EPD-like equations.

Geometry doi: 10.3390/geometry3010005

Authors: Jiří Blažek

A focal circular cubic is a locus of the foci of conics tangent to a given quadrilateral. In this article, we derive some of the focal curve’s basic properties. We study the curve in the complex plane and prove that the complex coordinates of pairs of foci satisfy a quadratic equation. This equation can be expressed as a linear combination of two basic quadratic equations, which form a basis of a vector space. Furthermore, we give a nonstandard analytical condition, expressed in complex numbers, under which a circle can be inscribed in a quadrilateral. Finally, we leave the complex plane and show the construction of an arbitrary pair of foci of the curve by Euclidean means.

Geometry doi: 10.3390/geometry3010004

Authors: Matthew A. Pons Nicholas D. White

Given an ellipse rolling along a straight line without slipping and a point P on the ellipse, we will determine the shape of the elliptic cycloid traced by P as the ellipse rolls and compute the area under one arch of the elliptic cycloid. We also investigate the arc length, though we are only able to express it as an integral.

Geometry doi: 10.3390/geometry3010003

Authors: Uta Freiberg Jonas Lippold

A long-standing conjecture of Lapidus asserts that under certain conditions a self-similar fractal set is not Minkowski measurable if and only if it is of lattice-type. For self-similar sets in R, the Lapidus conjecture has been confirmed. However, in higher dimensions, it remains unclear whether all lattice-type self-similar sets are not Minkowski measurable. This work presents a family of lattice-type subsets in R2 that are not Minkowski measurable, hence providing further support for the conjecture. Furthermore, an argument is presented to illustrate why these sets are not covered by previous results.

Geometry doi: 10.3390/geometry3010002

Authors: Kwangho Choi Junho Lee

By using a Morse function and a Witten deformation argument, we obtain an upper bound for the dimensions of the space of divergence-free symmetric Killing p-tensors on a closed Riemannian manifold and explicitly calculate it for p=2.

Geometry doi: 10.3390/geometry3010001

Authors: Yu Chen Robert J. Fisher

Let ▵ABC be a triangle in the plane E, K be its symmedian point, and C be its circumcircle. Assume that P is a point on C such that it is not A, B, or C, it does not lie on the medians of ▵ABC, and the lines ℓAP, ℓBP, and ℓCP intersect ℓBC, ℓCA, and ℓAB at points Pa, Pb, and Pc, respectively. By the Ceva Concurrence Theorem, the harmonic conjugates Qa, Qb, and Qc of Pa, Pb, and Pc in BC¯, CA¯, and AB¯, respectively, are collinear. We prove that K lies on the line through Qa, Qb, and Qc, which provides a new characterization of the symmedian point. Moreover, this one-to-one correspondence extends to a bijection from the entire circumcircle C onto the set LK of all the lines in E that pass through K.

Geometry doi: 10.3390/geometry2040021

Authors: Aldo Scimone

It is well known among geometry scholars that the golden triangle, an isosceles triangle with sides and base in golden ratio, maintains a significant relationship with regular polygons, notably the regular pentagon, pentagram, and decagon. Extensive mathematical literature addresses this subject. Furthermore, its close association with the golden ratio—a mathematical concept describing a harmonious and proportionate relationship between segments—renders it a noteworthy element in the fields of geometry, art, and architecture. Nevertheless, the interrelationships among these mathematical constructs frequently reveal unexpected configurations, thereby accentuating intriguing patterns. The purpose of this investigation is to highlight these novel configurations, which indicate new connections between the golden triangle and regular polygons.

Geometry doi: 10.3390/geometry2040020

Authors: Ion Mihai

In the present paper, we give a simple proof of the Chen–Ricci inequality for submanifolds in Riemannian and Lorentzian space forms, respectively. Moreover, we extend the Chen–Ricci inequality to submanifolds in Lorentzian manifolds with a semi-symmetric non-metric connection.

Geometry doi: 10.3390/geometry2040019

Authors: John Donnelly

We prove that if we start with a non-continuous absolute plane, remove Side-Angle-Side as an axiom, and replace it with the three new axioms, Side-Angle-Angle, angle addition, and the existence of angle bisectors, then the result is also an absolute plane.

Geometry doi: 10.3390/geometry2040018

Authors: Yevgenya Movshovich John Wetzel

In recent years, two quite different tools have been employed to study global properties of arcs in the plane. The first is drapeability, which grew from ideas of J. R. Alexander in early 2000s defining an arc drapeable if it lies in the convex hull of a shorter convex arc. The second is Λ-configuration, where an arc travels from one line to another and back. We investigate interrelations between these notions and in the process find drapeability criteria for open arcs, necessary and sufficient drapeability conditions for three-segment z-shaped arcs, and new bounds for the width of non-drapeable arcs.

Geometry doi: 10.3390/geometry2040017

Authors: Gerard Thompson

We study the existence of Killing vector fields for right-invariant metrics on low-dimensional Lie groups. Specifically, Lie groups of dimension two, three and four are considered. Before attempting to implement the differential conditions that comprise Killing’s equations, the metric is reduced as much as possible by using the automorphism group of the Lie algebra. After revisiting the classification of the low-dimensional Lie algebras, we review some of the known results about Killing vector fields on Lie groups and add some new observations. Then we investigate indecomposable Lie algebras and attempt to solve Killing’s equations for each reduced metric. We introduce a matrix MM, that results from the integrability conditions of Killing’s equations. For n=4, the matrix MM is of size 20×6. In the case where MM has maximal rank, for the Lie group problem considered in this article, only the left-invariant vector fields are Killing. The solution of Killing’s equations is performed by using MAPLE, and knowledge of the rank of MM can help to confirm that the solutions found by MAPLE are the only linearly independent solutions. After finding a maximal set of linearly independent solutions, the Lie algebra that they generate is identified to one in a standard list.

Geometry doi: 10.3390/geometry2040016

Authors: Kenneth Stephenson

There exists an extensive and fairly comprehensive discrete analytic function theory which is based on circle packing. This paper introduces a faithful discrete analogue of the classical Schwarzian derivative to this theory and develops its basic properties. The motivation comes from the current lack of circle packing algorithms in spherical geometry, and the discrete Schwarzian derivative may provide for new approaches. A companion localized notion called an intrinsic schwarzian is also investigated. The main concrete results of the paper are limited to circle packing flowers. A parameterization by intrinsic schwarzians is established, providing an essential packing criterion for flowers. The paper closes with the study of special classes of flowers that occur in the circle packing literature. As usual in circle packing, there are pleasant surprises at nearly every turn, so those not interested in circle packing theory may still enjoy the new and elementary geometry seen in these flowers.

Geometry doi: 10.3390/geometry2030015

Authors: Harold R. Parks Dean C. Wills

It was known in antiquity that the sum of the three angles of a triangle equals π. Surprisingly, it was not until 1952 that the corresponding question for a tetrahedron was addressed. In that year, J.W. Gaddum proved that the sum of the four solid angles in a tetrahedron lies within the interval of [0,2π] and those lower and upper bounds are the best possible. In 2020, H. Katsuura showed that 2π was unachievable. In this paper, we generalize these results to show that for a non-degenerate n-simplex in Rn with n≥3, the solid angles at the vertices add up to a positive number that is less than one-half the (n−1)-dimensional area of the unit sphere in Rn. We also show that there are examples for which the sum can be made arbitrarily close to the extreme values of 0 and one-half the (n−1)-dimensional area of the unit sphere in Rn.

Geometry doi: 10.3390/geometry2030014

Authors: John Lewis

In this paper, we first make and discuss a conjecture concerning Newtonian potentials in Euclidean n space which have all their mass on the unit sphere about the origin and are normalized to be one at the origin. The conjecture essentially divides these potentials into subclasses whose criteria for membership is that a given member has its maximum on the closed unit ball at most M and its minimum at least d. It then lists the extremal potential in each subclass, which is conjectured to solve certain extremal problems. In Theorem 1, we show the existence of these extremal potentials. In Theorem 2, we prove an integral inequality on spheres about the origin, involving so-called extremal potentials, which lends credence to the conjecture.

Geometry doi: 10.3390/geometry2030013

Authors: Jiří Blažek Pavel Pech

The Napoleon–Barlotti theorem belongs to the family of theorems related to the Petr–Douglas–Neumann theorem. The Napoleon–Barlotti theorem states: On the sides of an affine-regular n-gon construct regular n-gons. Then the centers of these regular n-gons form a regular n-gon. In the paper we give a generalization of this theorem.

Geometry doi: 10.3390/geometry2030012

Authors: Jianqiang Zhao

In this paper, we propose a class of elementary plane geometry problems closely related to the title of this paper. Here, a circle is the one-dimensional curve bounding a disk. For any non-negative integer, a circle is called n-enclosing if it contains exactly n lattice points on the xy-plane in its interior. In this paper, we are mainly interested in when the largest n-enclosing circle exists and what the largest radius is. We study the small integer cases by hand and extend to all n<1100 with the aid of a computer. We find that frequently such a circle does not exist, e.g., when n=5,6. We then show a few general results on these circles including some regularities among their radii and an easy criterion to determine exactly when the largest n-enclosing circles exist. Further, from numerical evidence, we conjecture that the set of integers whose largest enclosing circles exist is infinite, and so is its complementary in the set of non-negative integers. Throughout this paper, we present more mysteries/problems/conjectures than answers/solutions/theorems. In particular, we list many conjectures and some unsolved problems including possible higher-dimensional generalizations at the end of the last two sections.

Geometry doi: 10.3390/geometry2030011

Authors: Eleftherios Protopapas

Inversion with respect to a unit sphere is a powerful tool when dealing with many problems in Mathematics. This inversion preserves harmonicity in R2, but it does not in Rn, for n>2. Lord Kelvin overcame this problem by defining a new (at the time) inversion, the so-called Kelvin’s inversion (or transformation). This inversion has many good properties, making it extremely useful in each case where the geometry of the original problem raises issues. But by using Kelvin’s inversion, these issues are transformed into easier ones, due to a simpler geometry. In this review paper, we study Kelvin’s inversion, deploying its basic properties. Moreover, we present some applications, where its use enables scientists to solve difficult problems in scattering, electrostaticity, thermoelasticity, potential theory and bioengineering.

Geometry doi: 10.3390/geometry2030010

Authors: Cherng-tiao Perng

In 2014, Paul Yiu constructed two equilateral triangles inscribed in a Kiepert hyperbola associated with a reference triangle. It was asserted that each of the equilateral triangles is triply perspective with the reference triangle, and in each case, the corresponding three perspectors are collinear. In this note, we provide proof of his assertions. Furthermore, as an analogue of Lemoine’s problem, we formulated and answered the question of how to recover the reference triangle given a Kiepert hyperbola, one of the two Fermat points and one vertex of the reference triangle.

Geometry doi: 10.3390/geometry2020009

Authors: Jae-Hyun Yang

In this paper, we consider six homogeneous manifolds GL(n,R)/O(n,R), SL(n,R)/SO(n,R), Sp(2n,R)/U(n),(GL(n,R)⋉R(m,n))/O(n,R), (SL(n,R)⋉R(m,n))/SO(n,R),(Sp(2n,R)⋉HR(n,m))/(U(n)×S(m,R)). They are homogeneous manifolds which are important geometrically and number theoretically. These first three spaces are well-known symmetric spaces and the other three are not symmetric spaces. It is well known that the algebra of invariant differential operators on a symmetric space is commutative. The algebras of invariant differential operators on these three non-symmetric spaces are not commutative and have complicated generators. We discuss invariant differential operators on these non-symmetric spaces and provide natural but difficult problems about invariant theory.

Geometry doi: 10.3390/geometry2020008

Authors: Jiří Blažek

This article contains a synthetic proof of the following proposition: consider a conic c1 and its variable chord AB, which subtends a right angle at a given point P. Then, the foot E of the perpendicular dropped from P onto the line AB lies on a certain circle (the line being the limiting case of the circle). To prove this proposition, we show how Poncelet’s closure theorem for quadrilaterals can be derived by elementary projective considerations only (without any computations, either in Cartesian or projective coordinates). Finally, the limiting case of the proposition, where the point P lies on the conic, is also mentioned. The problem can then be reduced to Frégier’s theorem and may represent a slightly different perspective on this theorem.

Geometry doi: 10.3390/geometry2020007

Authors: Yaakov Friedman Tzvi Scarr

We continue Riemann’s program of geometrizing physics, extending it to encompass gravitational and electromagnetic fields as well as media, all of which influence the geometry of spacetime. The motion of point-like objects—both massive and massless—follows geodesics in this modified geometry. To describe this geometry, we generalize the notion of a metric to local scaling functions which permit not only quadratic but also linear dependence on temporal and spatial separations. Our local scaling functions are defined on flat spacetime coordinates. We demonstrate how to construct various geometries directly from field sources, using symmetry and superposition, without relying on field equations. For each geometry, two key visualizations elucidate the connection between the geometry and the dynamics as follows: (1) the cross-sections of the ball of admissible velocities, and (2) the cross-sections of the local scaling function.

Geometry doi: 10.3390/geometry2020006

Authors: Andrew J. Simoson

The cycloidal family of curves in R2, also known as the trochoid family, are equivalently generated by two classic methods: bungee cords and rolling wheels. What about their counterpart families in the hyperbolic unit disk? We review the two methods in Euclidean space, outline pertinent hyperbolic geometry tools, using both the Klein and Poincaré models, and show that the two methods give distinct, yet similar, results in hyperbolic space.

Geometry doi: 10.3390/geometry2020005

Authors: Martin Held Stefan de Lorenzo

This paper presents a theoretical framework for constructing generalized weighted Voronoi diagrams (GWVDs) of weighted points and straight-line segments (“sites”) in the Euclidean plane, based on lower envelopes constructed in three-dimensional space. Central to our approach is an algebraic distance function that defines the minimum weighted distance from a point to a site. We also introduce a parameterization for the bisectors, ensuring a precise representation of Voronoi edges. The connection to lower envelopes allows us to derive (almost tight) bounds on the combinatorial complexity of a GWVD. We conclude with a short discussion of implementation strategies, ranging from leveraging computational geometry libraries to employing graphics hardware for approximate solutions.

Geometry doi: 10.3390/geometry2020004

Authors: G. F. Helminck J. A. Weenink

Let S be the N×N-matrix of the shift operator and let k denote the field of real or complex numbers. We consider two different deformations of the commutative algebra k[S], together with the evolution equations of the deformations of the powers {Si,i⩾1}. They are called the k[S]-hierarchy and the strict k[S]-hierarchy. For suitable Banach spaces B, we explain how LU factorizations in GL(B) can be used to produce dressing matrices of both hierarchies. These dressing matrices correspond to bounded operators on B, a class far more general than the one used at a prior construction. This wider class of solutions of both hierarchies makes it possible to treat reductions of both systems. The matrix coefficients of these matrices are shown to be quotients of analytic functions. Moreover, we present a subgroup Gcpt(B) of GL(B) such that the procedure with LU factorizations works for each g∈Gcpt(B).

Geometry doi: 10.3390/geometry2010003

Authors: Josef Mikeš Irena Hinterleitner Patrik Peška Lenka Vítková

In the paper, we consider holomorphically projective mappings of n-dimensional pseudo-Riemannian Kähler and hyperbolic Kähler spaces. We refined the fundamental linear equations of the above problems for metrics of differentiability class C2. We have found the conditions for n complete geodesics and their image that must be satisfied for the holomorphically projective mappings to be trivial, i.e., these spaces are rigid with precision to affine mappings.

Geometry doi: 10.3390/geometry2010002

Authors: Nefton Pali

A well-known idea is to identify spheres, points, and hyperplanes in Euclidean space Rn with points in real projective space. To address geometric constraints such as intersections, tangencies, and angle requirements, it is important to also encode the orientations of hyperplanes and spheres. The natural space for encoding such geometric objects is the real projective quadric with signature (n+1,2). In this article, we first provide a general formula for calculating the angles formed by the geometric objects encoded by the points of the quadric. The main result is the determination of a very simple parametrization of a Laguerre-type subgroup that acts transitively on the quadric while preserving the geometric nature of its points. That is, points of the quadric representing oriented spheres, points, and oriented hyperplanes in Rn are mapped by the action to points of the same geometric type. We also provide simple parametrizations of the isotropies of the action. The action described in this article provides the foundation for an effective solution to geometric constraint problems.

Geometry doi: 10.3390/geometry2010001

Authors: Lionel Garnier Jean-Paul Bécar Laurent Fuchs

This article points out the kinematics in tracing a Bézier curve defined by control mass points. A mass point is a point with a non-positive weight, a non-negative weight or a vector with a null weight. For any Bézier curve, the speeds at endpoints can be modified at the same time for both endpoints. The use of a homographic parameter change allows us to choose any arc of the curve without changing the degree but not offer to change the speeds at both endpoints independently. The homographic parameter change performs weighted points with any non-null real number as weight and also vectors. The curve is thus called a rational Bézier curve with control mass points. In order to build independent stationary points at endpoints, a quadratic parameter change is required. Adding null vectors in the Bézier representation is also an answer. Null vectors are obtained when converting any power function in a rational Bézier curve and their inverse. The authors propose a new approach on placing null vectors in the representation of the rational Bézier curve. It allows us to break free from projective geometry where there is no null vector. The paper ends with some examples of known curves and some perspectives.

Geometry doi: 10.3390/geometry1010005

Authors: Clark Kimberling Peter J. C. Moses

It is well known that the four ancient Greek triangle centers and others have homogeneous barycentric coordinates that are polynomials in the sidelengths a,b, and c of a triangle ABC. For example, the circumcenter is represented by the polynomial a(b2+c2−a2). It is not so well known that triangle centers have barycentric coordinates, such as tanA : tan B : tan C, that are also representable by polynomials, in this case, by p(a, b, c) : p(b, c, a) : p(c, a, b), where p(a, b, c)=a(a2+b2−c2)(a2+c2−b2). This paper presents and discusses the polynomial representations of triangle centers that have barycentric coordinates of the form f(a, b, c) : f(b, c, a) : f(c, a, b), where f depends on one or more of the functions in the set {cos, sin, tan, sec, csc, cot}. The topics discussed include infinite trigonometric orthopoints, the n-Euler line, and symbolic substitution.

Geometry doi: 10.3390/geometry1010004

Authors: Zoltán Szilasi

In this article, we provide elementary proofs of two projective generalizations of Hagge’s theorems. We describe Steiner’s correspondence as a projective generalization of inversion.

Geometry doi: 10.3390/geometry1010003

Authors: Andrej Hasilik

We describe a series of parallel lenses with constant proportions packed in a circle. To construct n lenses, a regular 2(n + 1)-gon is drawn with a central diagonal of 2r length, followed by an array of n parallel diagonals perpendicular to the former. These diagonals and the central angle of the pair of peripherals, the shortest diagonals, are used to construct n rhombi. The rhombi define the shape of lenses tangential to them. To construct the arcs of the lenses, beams perpendicular to the sides of each rhombus are drawn. Four beams radiating from the top and bottom vertices of each rhombus intersect in the centers of a pair of coaxal circles. Thus, the vertical axis of each rhombus coincides with the radical axis of the pair. The intersection of the pair represents the corresponding lens. All n lenses form a tangential sequence along the central diagonal. Their cusps circumscribe the polygon and the lenses themselves. The area covered by the lenses converges to (2/3) πr2.

Geometry doi: 10.3390/geometry1010002

Authors: Peter J. C. Moses Clark Kimberling

Suppose that X is a triangle center with homogeneous coordinates (barycentric or trilinear) x:y:z. Eight unary operations discussed in this paper include u1(X)=(y−z)/x:(z−x)/y:(x−y)/z. For each ui, there exist, formally, two points, P and U, such that ui(P)=ui(U)=X. To such pairs of inverses are applied nine binary operations, each resulting in a triangle center. If L is a line, then formally, ui(L) is a cubic curve that passes through the vertices A,B,C. If L passes through the point 1:1:1 (the centroid or incenter, assuming that the coordinates are barycentric or trilinear), then the cubic is degenerate as the union of a parabola and the line at infinity. The methods in this work are largely algebraic and computer-dependent.

Geometry doi: 10.3390/geometry1010001

Authors: Yang-Hui He

In the ancient realm of geometry, we have witnessed the ultimate display of mathematical abstract thought [...]