| Convex lattice pentagon with three pairs of parallel sides and diagonals | 75$-$85 |
Abstract
This paper investigates convex lattice pentagons with at least three pairs $(a_i,d_i)$, where $a_i\parallel d_i$, i.e., diagonals parallel to sides. Based on the given conditions, we will form a system of Diophantine equations whose solutions we seek within the set of natural numbers or positive rational numbers. To characterize all obtained convex lattice pentagons of minimal area, we will use the concept of integer unimodular transformations. Specifically, these transformations of the plane preserve the parallelism of lattice segments, the number of lattice points inside a convex lattice polygon and on its boundary, as well as its area. We will then determine the minimum area of the pentagon in each resulting class and identify the pentagon with the smallest diameter. Finally, we will determine all convex lattice pentagons in which three sides are respectively parallel to three diagonals.
Keywords: Convex lattice pentagon; unimodular transformation; area minimiation.
MSC Subject Classification: 97G40
MathEduc Subject Classification: G44
| Angles and trigonometry | 86$-$99 |
Abstract
In this note, we introduce the sine and cosine functions on (abstract) angles that are defined as equivalence classes of vector pairs. We avoid power series, differential and integral calculus. The number $\pi$ emerges as the limit of repeated application of half-angle formula (the Vi\`{e}te formula). It is shown that the functions defined coincide with ordinary sine and cosine.
Keywords: Angle; trigonometric function; Euclidean dot product.
MSC Subject Classification: 97G60, 97I70
MathEduc Subject Classification: G65, I75
| The general change of variable formula for the Riemann integral | 100$-$106 |
Abstract
The change of variable theorem for functions that are Riemann integrable, i.e\. not obligatory continuous or monotonic, is established based on the definition of the integral and using nothing but the fundamentals of the Riemann theory. Specifically, the Lebesgue criterion for Riemann integrability or more advanced theories of integration are not required.
Keywords: Riemann integral; change of variable.
MSC Subject Classification: 97I50
MathEduc Subject Classification: I55
| From Feynman's triangle to Feynman's tetrahedra | 107$-$117 |
Abstract
We investigate a natural extension of so called planar $t$-Feynman configurations, referring to triangles, to three dimensional $t$-Feynman configurations, referring to tetrahedra. Our main result extends to three dimensions the well-known Routh's formula for planar $t$-Feynman configurations.
Keywords: Feynman's triangle; Feynman's tetrahedra; Euclidean geometry; integer sequences.
MSC Subject Classification: 97G40
MathEduc Subject Classification: G44
| Icosahedron and a paper dragon | 118$-$124 |
Abstract
We report on a mathematical video experiment involving the icosahedron. The animation created for this experiment was originally designed for the ``Živa matematika'' (Math Alive) project sponsored by the Mathematical Institute SASA (Belgrade) and the Belgrade Center for Promotion of Science. One of the motivations for the experiment was to create a simple, combinatorial geometric environment, involving a sequential blueprint that generates an icosahedron, in hope that this may eventually shed some new light on the mathematics behind the morphogenesis of icosahedral shapes in nature.
Keywords: Icosaherdon; Hamiltonian path.
MSC Subject Classification: 97G40
MathEduc Subject Classification: G44
| Icosahedron and a paper dragon revisited | 125$-$134 |
Abstract
This is a follow-up paper to the report [R.~T.~Živaljević and D.~R.~Živaljević, Icosahedron and a paper dragon, The Teaching of Mathematics 28, 2 (2025), 118--124] on an animated mathematical experiment (simulation) involving the icosahedron. The basic idea of the experiment was to create the simplest possible combinatorial geometric environment, for studying the mathematics behind the morphogenesis of icosahedral shapes in nature. Our objective is to present, in the form accessible to students, teachers, and non-specialists, some of the not so well-known facts about the geometry and combinatorics of the icosahedron, related to this mathematical simulation, emphasizing the unity of mathematics and the importance of the multidisciplinary approach in mathematical education.
Keywords: Icosahedron; Hamiltonian path; Kepler-Poinsot polyhedra; polyhedra nets.
MSC Subject Classification: 97G40, 51M20
MathEduc Subject Classification: G44, G45