Geometry

11 pages, 326 KB

Open AccessArticle

The Sum of the Solid Angles of an n-Simplex

by Harold R. Parks and Dean C. Wills

Geometry 2025, 2(3), 15; https://doi.org/10.3390/geometry2030015 - 19 Sep 2025

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 [...] Read more.

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, 2 π]

and those lower and upper bounds are the best possible. In 2020, H. Katsuura showed that

2 π

was unachievable. In this paper, we generalize these results to show that for a non-degenerate n-simplex in

R^{n}

with

n \geq 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

R^{n}

. 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

R^{n}

. Full article

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19 pages, 345 KB

Open AccessArticle

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 233

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)

8 pages, 284 KB

Open AccessArticle

Generalization of Napoleon–Barlotti Theorem

by Jiří Blažek and Pavel Pech

Geometry 2025, 2(3), 13; https://doi.org/10.3390/geometry2030013 - 19 Aug 2025

Viewed by 265

Abstract

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. [...] Read more.

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17 pages, 1159 KB

Open AccessFeature PaperArticle

The Largest Circle Enclosing n Interior Lattice Points

by Jianqiang Zhao

Geometry 2025, 2(3), 12; https://doi.org/10.3390/geometry2030012 - 11 Aug 2025

Viewed by 525

Abstract

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 [...] Read more.

x y

-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. Full article

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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 335

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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9 pages, 609 KB

Open AccessFeature PaperArticle

On Yiu’s Equilateral Triangles Associated with a Kiepert Hyperbola

by Cherng-tiao Perng

Geometry 2025, 2(3), 10; https://doi.org/10.3390/geometry2030010 - 1 Jul 2025

Viewed by 391

Abstract

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. Full article

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Geometry, Volume 2, Issue 3 (September 2025) – 6 articles

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