Geometry

29 pages, 2822 KB

Open AccessFeature PaperArticle

by Sebastian Xambó-Descamps

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

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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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14 pages, 300 KB

Open AccessFeature PaperArticle

Translative Covering a Square with Isosceles Right Triangles

by Janusz Januszewski and Łukasz Zielonka

Geometry 2026, 3(2), 8; https://doi.org/10.3390/geometry3020008 - 13 Apr 2026

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Abstract

A translative covering the rectangle

a \times b

with homothetic copies of a right isosceles triangle T (of the legs parallel to the sides of

a \times b

) is considered. It is shown that any collection of equal triangles homothetic to T [...] Read more.

A translative covering the rectangle

a \times b

with homothetic copies of a right isosceles triangle T (of the legs parallel to the sides of

a \times 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 \times 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 \times 1

. Moreover, it is proven that if

a \geq \frac{5 + \sqrt{33}}{4} b

, then any collection of triangles homothetic to T with the total area at least

\frac{1}{2} {(a + b)}^{2}

permits a translative covering of

a \times b

; this bound is tight. Full article

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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 275

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)

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Geometry, Volume 3, Issue 2 (June 2026) – 3 articles

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