Geometry

2307 Submissions

[3] viXra:2307.0053 [pdf] submitted on 2023-07-11 00:20:28

Every Convex Pentagon Has Some Vertex Such that the Sum of Distances to the Other Four Vertices is Greater Than Its Perimeter

Authors: Juan Moreno Borrallo
Comments: 4 Pages.

In this paper it is solved the case n = 5 of the problem 1.345 of the Crux Mathematicorum journal, proposed by Paul Erdös and Esther Szekeres in1988. The problem was solved for n ≥ 6 by János Pach and the solution published by the Crux Mathematicorum journal, leaving the case n = 5open to the reader. In september 2021, user23571113 posed this problem at the post https://math.stackexchange.com/questions/4243661/prove-thatfor-one-vertex-of-a-convex-pentagon-the-sum-of-distances-to-the-othe/4519514#4519514,and it has finally been solved.
Category: Geometry

[2] viXra:2307.0034 [pdf] submitted on 2023-07-06 18:36:56

One Piece

Authors: Volker W. Thürey
Comments: 4 Pages.

At first we tile the plane by 8-gons. Then we present a way to tile the plane by k-gons for a every fixed k for all natural numbers k larger than two. We use an infinite number of equal tiles to cover the plane.
Category: Geometry

[1] viXra:2307.0030 [pdf] submitted on 2023-07-07 03:10:23

Triedrul Lui Frenet de Ordinul al Doilea

Authors: Cavași Ioan Abel
Comments: 7 Pages.

Bazându-mă pe proprietatea remarcabilă a vectorului lui Darboux de a fi perpendicular pe normală, definesc un nou triedru asociat curbelor din spațiu și demonstrez că și acest triedru satisface formulele lui Frenet. Spre deosebire de lucrările anterioare, unde am folosit pentru simplitate forma trigonometrică a formulelor lui Frenet, în această lucrare construiesc o demonstrație bazată doar pe curbură și torsiune, respectiv, pe darbuzian și lancretian.
Category: Geometry