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A special issue of Geometry (ISSN 3042-402X).

Deadline for manuscript submissions: closed (31 March 2026) | Viewed by 19863

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Prof. Dr. Yang-Hui He

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Guest Editor

1. London Institute for Mathematical Sciences, Royal Institution, London W1S 4BS, UK
2. Merton College, University of Oxford, Oxford OX1 4JD, UK
Interests: AI-assisted mathematics; mathematical physics; string theory; algebraic geometry; number theory
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Special Issue Information

Dear Colleagues,

As the Editor-in-Chief of Geometry, I am pleased to announce the Special Issue "Feature Papers in Geometry", which will be a collection of high-quality papers (original research articles or comprehensive reviews) from top academics addressing the nature of geometry. I welcome the submission of manuscripts from Editorial Board Members and from outstanding scholars invited by the Editorial Board and the Editorial Office related to any of the topics covered in the scope of the journal (https://www.mdpi.com/journal/geometry).

You are invited to send short proposals for submissions to our Editorial Office (geometry@mdpi.com) for evaluation. Please note that selected full papers will still be subject to a thorough and rigorous peer-review.

Prof. Dr. Yang-Hui He
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 250 words) can be sent to the Editorial Office for assessment.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Geometry is an international peer-reviewed open access quarterly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1000 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

Euclidean geometry
differential geometry
algebraic geometry
complex geometry
discrete geometry
computational geometry
geometric group theory
convex geometry
geometric theory/algorithm
mathematical physics

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Published Papers (15 papers)

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Research

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29 pages, 2822 KB

Open AccessFeature PaperArticle

Wessel’s Algebra and Morley’s Theorem

by Sebastian Xambó-Descamps

Geometry 2026, 3(2), 9; https://doi.org/10.3390/geometry3020009 - 8 May 2026

Viewed by 159

Abstract

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

E

(called Wessel’s algebra). This algebra is studied [...] Read more.

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

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. Full article

(This article belongs to the Special Issue Feature Papers in Geometry)

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15 pages, 285 KB

Open AccessArticle

Pseudoconvexity and Steinness of Connected Complex Lie Groups: A Concise Lie-Theoretic Approach

by Abdel Rahman Al-Abdallah

Geometry 2026, 3(2), 7; https://doi.org/10.3390/geometry3020007 - 1 Apr 2026

Viewed by 281

Abstract

We give new concise Lie-theoretic proofs of basic analytic–geometric properties of connected complex Lie groups. Using Matsushima’s biholomorphic splitting

G ≃ C^{n} \times \tilde{K}

together with a refined analysis of the center via its Cousin factor, we show that every connected [...] Read more.

We give new concise Lie-theoretic proofs of basic analytic–geometric properties of connected complex Lie groups. Using Matsushima’s biholomorphic splitting

G ≃ C^{n} \times \tilde{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. Full article

(This article belongs to the Special Issue Feature Papers in Geometry)

14 pages, 438 KB

Open AccessArticle

Properties of Elliptic Cycloids

by Matthew A. Pons and Nicholas D. White

Geometry 2026, 3(1), 4; https://doi.org/10.3390/geometry3010004 - 9 Feb 2026

Viewed by 627

Abstract

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 [...] Read more.

(This article belongs to the Special Issue Feature Papers in Geometry)

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13 pages, 577 KB

Open AccessFeature PaperArticle

Family of Non-Minkowski Measurable Fractals in

R^{2}

by Uta Freiberg and Jonas Lippold

Geometry 2026, 3(1), 3; https://doi.org/10.3390/geometry3010003 - 2 Feb 2026

Viewed by 455

Abstract

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 [...] Read more.

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

R^{2}

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. Full article

(This article belongs to the Special Issue Feature Papers in Geometry)

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15 pages, 302 KB

Open AccessArticle

Witten Deformation and Divergence-Free Symmetric Killing 2-Tensors

by Kwangho Choi and Junho Lee

Geometry 2026, 3(1), 2; https://doi.org/10.3390/geometry3010002 - 13 Jan 2026

Viewed by 479

Abstract

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

. Full article

(This article belongs to the Special Issue Feature Papers in Geometry)

15 pages, 424 KB

Open AccessArticle

Some Nice Configurations of Golden Triangles

by Aldo Scimone

Geometry 2025, 2(4), 21; https://doi.org/10.3390/geometry2040021 - 10 Dec 2025

Viewed by 1308

Abstract

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. Full article

(This article belongs to the Special Issue Feature Papers in Geometry)

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22 pages, 1109 KB

Open AccessArticle

Drapeability and Λ-Frames

by Yevgenya Movshovich and John Wetzel

Geometry 2025, 2(4), 18; https://doi.org/10.3390/geometry2040018 - 4 Nov 2025

Viewed by 773

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Feature Papers in Geometry)

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19 pages, 345 KB

Open AccessFeature PaperArticle

On d and M Problems for Newtonian Potentials in Euclidean n Space

by John Lewis

Geometry 2025, 2(3), 14; https://doi.org/10.3390/geometry2030014 - 2 Sep 2025

Viewed by 995

Abstract

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. Full article

(This article belongs to the Special Issue Feature Papers in Geometry)

10 pages, 304 KB

Open AccessFeature PaperArticle

On the Relation Between a Locus and Poncelet’s Closure Theorem

by Jiří Blažek

Geometry 2025, 2(2), 8; https://doi.org/10.3390/geometry2020008 - 9 Jun 2025

Viewed by 1918

Abstract

This article contains a synthetic proof of the following proposition: consider a conic

c_{1}

and its variable chord

A B

, which subtends a right angle at a given point P. Then, the foot E of the perpendicular dropped from P [...] Read more.

This article contains a synthetic proof of the following proposition: consider a conic

c_{1}

and its variable chord

A B

, which subtends a right angle at a given point P. Then, the foot E of the perpendicular dropped from P onto the line

A B

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. Full article

(This article belongs to the Special Issue Feature Papers in Geometry)

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27 pages, 1329 KB

Open AccessFeature PaperArticle

Defining and Visualizing the Geometry of Relativistic Physics

by Yaakov Friedman and Tzvi Scarr

Geometry 2025, 2(2), 7; https://doi.org/10.3390/geometry2020007 - 14 May 2025

Viewed by 2721

Abstract

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. Full article

(This article belongs to the Special Issue Feature Papers in Geometry)

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12 pages, 7101 KB

Open AccessFeature PaperArticle

Hyperbolic Cords and Wheels

by Andrew J. Simoson

Geometry 2025, 2(2), 6; https://doi.org/10.3390/geometry2020006 - 6 May 2025

Viewed by 1519

Abstract

The cycloidal family of curves in

R^{2}

The cycloidal family of curves in

R^{2}

, 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. Full article

(This article belongs to the Special Issue Feature Papers in Geometry)

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25 pages, 368 KB

Open AccessArticle

LU Factorizations for ℕ × ℕ-Matrices and Solutions of the k[S]-Hierarchy and Its Strict Version

by G. F. Helminck and J. A. Weenink

Geometry 2025, 2(2), 4; https://doi.org/10.3390/geometry2020004 - 15 Apr 2025

Viewed by 1442

Abstract

Let S be the

N \times 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 [...] Read more.

Let S be the

N \times 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

{S^{i}, 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

G_{c p t} (B)

GL (B)

such that the procedure with LU factorizations works for each

g \in G_{c p t} (B)

. Full article

(This article belongs to the Special Issue Feature Papers in Geometry)

17 pages, 7219 KB

Open AccessArticle

A Laguerre-Type Action for the Solution of Geometric Constraint Problems

by Nefton Pali

Geometry 2025, 2(1), 2; https://doi.org/10.3390/geometry2010002 - 18 Feb 2025

Viewed by 1023

Abstract

A well-known idea is to identify spheres, points, and hyperplanes in Euclidean space

R^{n}

A well-known idea is to identify spheres, points, and hyperplanes in Euclidean space

R^{n}

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

R^{n}

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. Full article

(This article belongs to the Special Issue Feature Papers in Geometry)

16 pages, 304 KB

Open AccessArticle

Trigonometric Polynomial Points in the Plane of a Triangle

by Clark Kimberling and Peter J. C. Moses

Geometry 2024, 1(1), 27-42; https://doi.org/10.3390/geometry1010005 - 23 Dec 2024

Cited by 1 | Viewed by 1961

Abstract

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

A B C

. For example, the circumcenter is represented by [...] Read more.

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

A B C

. For example, the circumcenter is represented by the polynomial

a (b^{2} + c^{2} - a^{2})

. It is not so well known that triangle centers have barycentric coordinates, such as

\tan A : \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 (a^{2} + b^{2} - c^{2}) (a^{2} + c^{2} - b^{2})

. 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. Full article

(This article belongs to the Special Issue Feature Papers in Geometry)

Review

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17 pages, 1278 KB

Open AccessReview

The Multiple Utility of Kelvin’s Inversion

by Eleftherios Protopapas

Geometry 2025, 2(3), 11; https://doi.org/10.3390/geometry2030011 - 9 Jul 2025

Viewed by 1892

Abstract

Inversion with respect to a unit sphere is a powerful tool when dealing with many problems in Mathematics. This inversion preserves harmonicity in

R^{2},

but it does not in

R^{n},

for

n > 2 .

Lord Kelvin overcame this [...] Read more.

Inversion with respect to a unit sphere is a powerful tool when dealing with many problems in Mathematics. This inversion preserves harmonicity in

R^{2},

but it does not in

R^{n},

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. Full article

(This article belongs to the Special Issue Feature Papers in Geometry)

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