# Angel PlazaUniversidad de Las Palmas de Gran Canaria | ULPGC · Department of Mathematics

Angel Plaza

PhD

Professor

## About

155

Publications

17,432

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1,567

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Citations since 2016

Introduction

Additional affiliations

December 1995 - January 1997

October 1989 - present

## Publications

Publications (155)

We introduce a new triangle transformation, the shortest-edge (SE) duplication, as a natural way of mesh derefinement suitable to those meshes obtained by iterative application of longest-edge bisection refinement. Metric properties of the SE duplication of a triangle in the region of normalised triangles endowed with the Poincare hyperbolic metric...

When the 8T-LE partition is recursively applied to any initial trirectangular tetrahedron T, only a finite number of dissimilar tetrahedra are generated. It implies the stability of the meshes. At each step of refinement the number of right-type or path tetrahedra grows, so the quality of the obtained meshes improves. The minimun angle condition an...

This paper is accepted and is scheduled to appear in the March/ 2023 issue of “ The Mathematical Gazette ” journal.

Many proofs of the arithmetic mean harmonic mean inequality have been proposed based on the rich connections between mathematics and physics. Sometimes the Arithmetic Mean Harmonic Mean inequality is proved by using electric networks. In this note, we use a simple set of two springs, instead of four springs which would be the equivalent set to thos...

We demonstrate visually that the sum of every other term in the (n + 1)st row of Pascal’s triangle is equal to the sum of all the terms in the previous row.

The barycentric partition of a 3D-cube into tetrahedra is carried out by adding a new node to the body at the centroid point and then, new nodes are progressively added to the centroids of faces and edges. This procedure generates three types of tetrahedra in every single step called, Sommerville tetrahedron number 3 (ST3), isosceles trirectangular...

104.14 More on zero-over-zero limits of special type - Volume 104 Issue 560 - Ángel Plaza

104.22 Proof without Words: Minimum perimeter of an inscribed quadrangle to a square - Volume 104 Issue 560 - Ángel Plaza

Recently, Hoseana proposed a technique to calculate complicated
zero-over-zero form limits of a particular type by expressing them as the
sum of simpler limits. Here, we adapt his idea to the case in which the
straightforward use of L'Hôpital's Rule implies the derivative of functions of the form f(x)^{g(x)}

Four of the six types of the regular triangulations of the 3D-cube, up to isomorphism, are obtuse. We study the eight-tetrahedra longest-edge partition (8T-LE) of the triangulations of the cube containing two regular right-type tetrahedra, two regular trirectangular tetrahedra and two quasi right-type tetrahedra.
We prove that the iterative 8T-LE p...

A recurrence formula for the k th Power of a Partial Sum is deduced and applied in some examples.

In this note we analyse the classical longest-edge n-section algorithm applied to the simplicial partition in Rd, and prove that an infinite sequence of simplices violating the Zlámal minimum angle condition, often required in finite element analysis and computer graphics, is unavoidably produced if n ≥ 4. This result implies the fact that the numb...

102.42 Proof without Words: An alternating geometrical series - Volume 102 Issue 555 - Ángel Plaza

We have introduced here the concept of Hamiltonian triangular refinement. For any Hamiltonian triangulation it is shown that there is a refinement which is also a Hamiltonian triangulation and the corresponding Hamiltonian path preserves the nesting condition of the corresponding space-filling curve. We have proved that the number of such Hamiltoni...

We give a visual proof of the geometric series of the constant negative one-half.

Visual proof of three arctangent identities involving arctan (√2 − 1) and arctan (√2 + 1).

We introduce an algorithm for obtaining a geometric bone model suitable for the analysis of bone mechanical properties. In the bone model construction, we use new patterns of the familiar Morton curve, a class of Space Filling Curves. By extending a previous idea for the Hilbert curve, we derive new curve patterns used two construct the space filli...

In a Pascal-like triangle, where each entry is the sum of the three numbers above them, we visually prove that the row sums are the Pell numbers, given by a two-step recursion.

It is proved without words that the golden ratio, φ, and the arctangent of 2 are related.

Using Pascal's identity, we visually demonstrate that the sum of entries in a row of Pascal's triangle is a power of two.

Based on the binomial property written as the sum of consecutive column entries of Pascal’s triangle is written as a difference of two binomial coefficients in the next column, which generalizes the so-called hockey stick identities.

By using the ellipse with foci at the extreme points of the base, we show wordlessly that the triangle with maximum area for a given base and perimeter is the isosceles triangle where the different edge is the base.

Based on the Pascal's identity, we visually demonstrate that the alternating sum of consecutive binomial coefficients in a row of Pascal's triangle is determined by two binomial coefficients from the previous row.

100.39 An olympiad mathematical problem, proof without words and generalisation - Volume 100 Issue 549 - Lucía Ma Li, Ángel Plaza

100.38 Proof without words: sum of a numerical series by telescoping - Volume 100 Issue 549 - Ángel Plaza

By using the ellipse with foci at the extreme points of the shortest diagonal and the minor axis being the longest diagonal, it is proved without words that the parallelogram with maximum perimeter for given diagonals is the rhombus.

Based on the Pascal’s identity, we visually demonstrate that the alternating sum of consecutive binomial coefficients in a row of Pascal’s triangle is determined by two binomial coefficients from the previous row.

The longest-edge (LE-) trisection of the given tetrahedron is obtained by joining two equally spaced points on its longest edge with the opposite vertices, and, thus, splitting the tetrahedron into three sub-tetrahedra. On the base such LE-trisections we introduce and numerically test the refinement algorithms for tetrahedral meshes. Computations c...

Visual proof that the limit of the recursive arithmetic mean sequence defined by is , where a1 and a2 are the initial values of the sequence.

Visual proof that the limit of the recursive root mean square sequence defined by an + 1 where a1 and a2 are the initial values of the sequence.

100.12 Visual proof of the limit of f-mean recurrence sequences - Volume 100 Issue 547 - Ángel Plaza

We provide a visual proof that the arithmetic mean of two positive numbers is greater or equal than the arithmetic mean of the geometric mean and the root mean square.

In this paper we survey all known (including own recent results) properties of the longest-edge nn-section algorithms. These algorithms (in classical and recently designed conforming form) are nowadays used in many applications, including finite element simulations, computer graphics, etc. as a reliable tool for controllable mesh generation. In add...

By the Law of Cosines and the arithmetic mean-root mean square inequality it is proved without words that The Parallelogram with Maximum Perimeter for given Diagonals is the Rhombus. As a corollary it also proved that for two positive numbers, their arithmetic mean is greater or equal than the arithmetic mean of their geometric mean and their root...

We review and discuss a method to normalize triangles by the longest-edge. A geometric diagram is described as a helpful tool for studying and interpreting the quality of triangle shapes during iterative mesh refinements. Modern CAE systems as those implementing the finite element method (FEM) require such tools for guiding the user about the quali...

We show that there is for every Hamiltonian triangulation a suitable local pattern for triangle subdivision such that the resulting refined triangulation is also Hamiltonian

In this note we introduce the conforming longest-edge n-section algorithm and show that for it produces a family of triangulations which does not satisfy the maximum angle condition.

The Longest-Edge (LE) bisection of a triangle is obtained by joining the midpoint of its longest edge with the opposite vertex. Here two properties of the longest-edge bisection scheme for triangles are proved. For any triangle, the number of distinct triangles (up to similarity) generated by longest-edge bisection is finite. In addition, if LE-bis...

Visual proof of the limit of a recursive arithmetic mean sequence.

The Longest-Edge (LE) trisection of a triangle is obtained by joining the two points which divide the longest edge in three with the opposite vertex. If LE-trisection is iteratively applied to an initial triangle, then the maximum diameter of the resulting triangles is between two sharpened decreasing functions. This paper mathematically answers th...

From an initial triangle, three triangles are obtained joining the two equally spaced points of the longest-edge with the opposite vertex. This construction is the base of the longest-edge trisection method. Let Δ be an arbitrary triangle with minimum angle α . Let Δ′ be any triangle generated in the iterated application of the longest-edge trisect...

In this work we study the diameters reduction rate for the iterative application of the longest edge (LE) n-section of triangles for n >= 4. The maximum diameter d(k)(n) of all triangles generated at the kth iteration of the LE n-section is closely connected with the properties of the triangular mesh generated by this refinement scheme. The upper a...

Proof Without Words: Fibonacci Triangles and Trapezoids

We prove that the longest-edge nn-section of triangles for n⩾4n⩾4 produces a sequence of triangle meshes with minimum interior angle converging to zero. The so called degeneracy property of LE for n⩾4n⩾4 is proved.

In this note, by using complex variable functions, we present a new simpler proof of the degeneracy property of the longest-edge n-section of triangles for n⩾4n⩾4. This means that the longest-edge n-section of triangles for n⩾4n⩾4 produces a sequence of triangles with minimum interior angle converging to zero.

In this paper it is shown the quality assessment implementation of longest-edge refinement algorithms for application in real-time terrain operations. The proposed refinement schemes are suitable tools for the subdivision of underlying triangle mesh. They pose quite acceptable properties as quality ratio, linear time operation and algorithm simplic...

We present empirical evidence of the convergence study in 3D and some other ongoing results will be given for the LE n-section of tetrahedra when n≥4. Our contribution helps to a better understanding of partitioning and refinement methods in 3 Dimensions.

In the refinement of meshes, one wishes to iteratively subdivide a domain following geometrical partition rules. The aim is to obtain a new discretized domain with adapted regions. We prove that the Longest Edge n-section of tri-angles for n 4 produces a finite sequence of triangle meshes with guaranteed convergence of diameters and review previous...

In this paper we present a local refinement algorithm based on the longest-edge trisection of triangles. Local trisection patterns are used to generate a conforming triangulation, depending on the number of non-conforming nodes per edge presented. We describe the algorithm and provide a study of the efficiency (cost analysis) of the triangulation r...

Let t be a triangle in R 2 . We find the Longest Edge (LE) of t, insert n − 1 equally-space points in the LE and connect them to the opposite vertex. This yields the generation of n new sub-triangles whose parent is t. Now, continue this process iteratively. Proficient algorithms for mesh refinement using this method are known when n = 2, but less...

A new edge-based partition for triangle meshes is presented, the Seven Triangle Quasi-Delaunay partition (7T-QD). The proposed partition joins together ideas of the Seven Triangle Longest-Edge partition (7T-LE), and the classical criteria for constructing Delaunay meshes. The new partition performs similarly compared to the Delaunay triangulation (...

The longest-edge (LE) trisection of a tetrahedron t is obtained by joining the two equally spaced points of the longest-edge of t with the opposite ver-tices. We introduce a new algorithm for the local refinement of conforming tetrahedral meshes. With the presented algorithm and using standard shape measures, it is shown empirical evidence on the n...

In this note, we present an affirmative answer to a question presented in the paper “Some inequalities in inner product spaces related to the generalized triangle inequality” by S.S. Dragomir et al. [Appl. Math. Comput. 217 (18) (2011) 7462–7468].

The longest-edge (LE) trisection of a triangle � is obtained by joining the two equally spaced points of its longest-edge with the opposite vertex. Let � > 0 be the smallest interior angle of � and �0 the smallest angle of any triangle obtained after iteration of the LE-trisection.

Let ¢ABC be a triangle with vertexes A, B, and C. The longest-edge trisection of ¢ABC is as follows: choose the longest side (say AB) of ¢ABC, let D and E be the points which divide in three AB, then replace ¢ABC by three triangles ¢ACD, ¢CDE and ¢BCE. If longest-edge trisection is iteratively applied to an initial triangle, then it is proved that...

94.18 Proof without words: Two inequalities proved by convexity - Volume 94 Issue 530 - Ángel Plaza, Sergio Falcón

The longest-edge (LE) trisection of a triangle t is obtained by joining the two equally spaced points of the longest-edge of t with the opposite vertex. In this paper we prove that for any given triangle t with smallest interior angle τ>0, if the minimum interior angle of the three triangles obtained by the LE-trisection of t into three new triangl...

In this paper, we apply the binomial, k-binomial, rising, and falling transforms to the k-Fibonacci sequence. Many formulas relating the so obtained new sequences are presented and proved. Finally, we define and find the inverse transforms of the sequences previously obtained.

We study the presence of the metallic ratios as limits of two complex valued transformations. These complex variable functions are introduced and related with the two geometric antecedents for each triangle in a particular triangle partition, the four-triangle longest-edge (4TLE) partition. In this way, the fractality of a geometric diagram for the...

We study here the period-length of the k-Fibonacci sequences taken modulo m. The period of such cyclic sequences is know as Pisano period, and the period-length is denoted by πk(m). It is proved that for every odd number k, πk(k2+4)=4(k2+4).

The triangle longest-edge bisection constitutes an efficient scheme for refining a mesh by reducing the obtuse triangles, since the largest interior angles are subdivided. In this paper we specifically introduce a new local refinement for triangulations based on the longest-edge trisection, the 7-triangle longest-edge (7T-LE) local refinement algor...

We present a refinement and coarsening algorithm for the adaptive representation of Right-Triangulated Irregular Network (RTIN) meshes. The refinement algorithm is very simple and proceeds uniformly or locally in triangle meshes. The coarsening algorithm decreases mesh complexity by reducing unnecessary data points in the mesh after a given error c...

In this paper, we study the sums of k-Fibonacci numbers with indexes in an arithmetic sequence, say for fixed integers a and r. This enables us to give in a straightforward way several formulas for the sums of such numbers.

The k-Fibonacci polynomials are the natural extension of the k-Fibonacci numbers and many of their properties admit a straightforward proof. Here in particular, we present the derivatives of these polynomials in the form of convolution of k-Fibonacci polynomials. This fact allows us to present in an easy form a family of integer sequences in a new...

We show that for any two continuous real valued functions f, g on [0,1], the problem
$$\int_{0}^1 f(x)dx \cdot \int_{0}^c w(x),g(x)dx = \int_{0}^1 g(x)dx \cdot int_{0}^c w(x) f(x)dx,$$
always has at least one solution $c\in (0,1)$, for a general class of weight-functions. Some applications are given.

Summary In this work we introduce a geometrical diagram to study the geometric quality of triangles generated by iterative application of the four Triangles Longest Edge (4TLE) partition. The diagram provides a convenient graphic tool to visualize the evolution and migration of element shapes leading to a better understanding of the improvement pro...

This paper gives two visual proofs of the following exponential inequalities:A B ⇒ eA+B/2eB−eA/B−A eA+eB/2.

The 3-dimensional k-Fibonacci spirals are studied from a geometric point of view. These curves appear naturally from studying the k-Fibonacci numbers {F-k,F-n}(n=0)(infinity) and the related hyperbolic k-Fibonacci functions. In this paper, after a summary of the main properties for the k-Fibonacci numbers, we focus on the geometry features (curvatu...

An extension of the classical hyperbolic functions is introduced and studied. These new k-Fibonacci hyperbolic functions generalize also the k-Fibonacci sequences, say {Fk,n}n=0∞, recently found by studying the recursive application of two geometrical transformations onto C¯=C∪{+∞} used in the well-known four-triangle longest-edge (4TLE) partition....

In this article, we consider some generalizations of Fibonacci numbers. We consider k-Fibonacci numbers (that follow the recurrence rule F k,n + 2 = kF k,n + 1 + F k,n ), the (k,ℓ)-Fibonacci numbers (that follow the recurrence rule F k,n + 2 = kF k,n + 1 + ℓF k,n ), and the Fibonacci p-step numbers ( , with , and p > 2). Then we provide combinatori...

A new triangle partition, the seven-triangle longest-edge partition, based on the trisection of the edges is presented and the associated mesh quality improvement property, discussed. The seven-triangle longest-edge (7T-LE) partition of a triangle t is obtained by putting two equally spaced points per edge. After cutting off three triangles at the...

This note shows a combinatorial approach to some identities for generalized Fibonacci numbers. While it is a straightforward task to prove these identities with induction, and also by arithmetical manipulations such as rearrangements, the approach used here is quite simple to follow and eventually reduces the proof to a counting problem.

The triangle longest-edge bisection constitutes an efficient scheme for refining a mesh by reducing the obtuse triangles, since the largest interior angles are subdivided. One of these schemes is the four-triangle longest-edge (4T-LE) partition. Moreover, the four triangle self-similar (4T-SS) partition of an acute triangle yields four sub-triangle...

The Kuhn triangulation of a cube is obtained by subdividing the cube into six right-type tetrahedra once a couple of opposite vertices have been chosen. In this paper, we explicitly define the eight-tetrahedra longest-edge (8T-LE) partition of right-type tetrahedra and prove that for any regular right-type tetrahedron t, the iterative 8T-LE partiti...

The general k-Fibonacci sequence were found by studying the recursive application of two geometrical transformations used in the well-known 4-triangle longest-edge (4TLE) partition. This sequence generalizes, between others, both the classical Fibonacci sequence and the Pell sequence. In this paper many properties of these numbers are deduced and r...

We introduce a general Fibonacci sequence that generalizes, between others, both the classic Fibonacci sequence and the Pell sequence. These general kth Fibonacci numbers were found by studying the recursive application of two geometrical transformations used in the well-known four-triangle longest-edge (4TLE) partition. Many properties of these nu...

We provide a visual solution for an alternating sum of odd squares.

We introduce a geometrical diagram to study the improvement in shape of triangles generated by iterative application of triangle subdivision. The four Triangles Longest Edge (4TLE) subdivision pattern and a new hybrid 4T Longest-Edge/Self-Similar (hybrid 4TLE-SS) scheme are investigated in this way. The map diagram provides a convenient way to visu...

In this paper we study and delimit a property of known four triangle subdivisions that is useful in tri to quad mesh conversion methods. We provide both theoretical results and empirical evidence showing that iterative application of the four triangles longest-edge subdivision and the four triangles similar subdivision produces block-balanced meshe...