Nonagon Research Articles

Zig-Zag Paths and Neusis Constructions of a Heptagon and a Nonagon

Summary Although it is impossible to construct a regular heptagon and a regular nonagon using a compass and unmarked straightedge, it is possible to construct them with a compass and marked straightedge using the neusis technique. We give a geometric proof of Johnson’s neusis construction of the regular heptagon, which he had proven using trigonometry. We do so using so-called central triangles and zig-zag paths in the polygons. We then give efficient neusis constructions of the regular heptagon and the regular nonagon.

The Impact of the Cut-Out Shape on the Dynamic Behavior of Composite Thin Circular Plates

This study measures the impact of the cut-out shape on the fundamental frequency and the harmonic response of thin composite circular plates under fixed boundary conditions. For this purpose, the Finite Element Method has been employed using the SHELL181 element of ANSYS Workbench 18.2 software. Nine different circular plates having central equilateral triangle, square, regular pentagon, regular hexagon, regular heptagon, regular octagon, regular nonagon, regular decagon, and circular cut-outs have been designed to understand the changes in the vibrational behavior of a structure as the cut-out geometry approximates to circular. Additionally, a solid circular plate has been modelled for comparison purposes. To investigate the effect of the cut-out shape more deeply, all structures have been designed as cross-ply and angle-ply composites. The free vibration analysis has been performed to obtain the fundamental frequency of each structure. The harmonic response analysis problem has been solved by employing the Mode Superposition Method and considering a constant damping ratio. The results of the study have been interpreted by considering the displacement response, stress, and phase shifts at a certain frequency range. The results indicate that the shape of the cut-out considerably affects the fundamental frequency and harmonic response of the circular structure regardless of its fiber orientation.

Applying Peirce. From the Three Categories to the Semiotic Nonagon

Applying Peirce. From the Three Categories to the Semiotic Nonagon

Dashboard for Evaluating the Quality of Open Learning Courses

Universities are developing a large number of Open Learning projects that must be subject to quality evaluation. However, these projects have some special characteristics that make the usual quality models not respond to all their requirements. A fundamental part in a quality model is a visual representation of the results (a dashboard) that can facilitate decision making. In this paper, we propose a complete model for evaluating the quality of Open Learning courses and the design of a dashboard to represent its results. The quality model is hierarchical, with four levels of abstraction: components, elements, attributes and indicators. An interesting contribution is the definition of the standards in the form of fulfillment levels, that are easier to interpret and allow using a color code to build a heat map that serves as a dashboard. It is a regular nonagon, divided into sectors and concentric rings, in which each color intensity represents the fulfillment level reached by each abstraction level. The resulting diagram is a compact and visually powerful representation, which allows the identification of the strengths and weaknesses of the Open Learning course. A case study of an Ecuadorian university is also presented to complete the description and draw new conclusions.

Nonagons in the Hagia Sophia and the Selimiye Mosque

One of the classical topics in geometry is the ruler and compass construction of good approximations of the regular nonagon. We propose a method to choose a desired error of approximation, based on new linear third-order recurrence relations. It is related to patterns shown on the balustrade of minbar of Selimiye Mosque in Edirne (Turkey, 1569–1575), where apparently regular nonagons are placed over a net of regular hexagons. This kind of ornament also occurs in Hagia Sophia in the minbar, in freezes stucco carvings and in window grilles. In Medieval Islamic art and architecture the use of the nonagon is not frequent, but it is remarkable that this particular grid of interlocked nonagons and hexagons appears in the decorations of the works of Ottoman architect Mimar Sinan. The pattern is present in his masterpiece, the Selimiye Mosque, in the Hagia Sophia and in other works in the Istanbul city. Looking at practical ways for the construction of the pattern, we provide simple procedures to obtain angles close to 40° that could have been useful for a craftsman to realize the nonagonal geometric designs. In particular, almost regular nonagons are constructed using some elementary shapes that are related with semi-regular tessellations. We compare the patterns obtained through theoretical considerations to those displayed in the examples given above. Several hypotheses are proposed for the practical construction of the interlocked hexagonal patterns for nonagons.

Triangles in regular nonagons

Triangles in regular nonagons

Close pairs of relative equilibria for identical point vortices

Numerical solution of the classical problem of relative equilibria for identical point vortices on the unbounded plane reveals configurations that are very close to the analytically known, centered, symmetrically arranged, nested equilateral triangles. New numerical solutions of this kind are found for 3n + 1 vortices, where n = 2, 3, ..., 30. A sufficient, although apparently not necessary, condition for this phenomenon of close solutions is that the “core” of the configuration is marginally stable, as occurs for a central vortex surrounded by an equilateral triangle. The open, regular heptagon also has this property, and new relative equilibria close to the nested, symmetrically arranged, regular heptagons have been found. The centered regular nonagon is also marginally stable. Again, a new family of close relative equilibria has been found. The closest relative equilibrium pairs occur, however, for symmetrically nested equilateral triangles.

Convex polygons with few intervertex distances

Thirty years ago E. Altman proved that every convex n-gon ( n≥3) has at least ⌊ n 2 ⌋ different distances between vertices and that a convex n-gon for odd n has exactly (n−1) 2 intervertex distances if and only if it is regular. We prove that a convex n-gon for even n≥8 has exactly n 2 intervertex distances if and only if it is a regular n-gon or a regular ( n + 1)-gon with a vertex removed. Three hexagons have exactly three intervertex distances, and four quadrilaterals have exactly two intervertex distances. Moreover, 15 pentagons have exactly three intervertex distances, and five heptagons have exactly four intervertex distances. The latter five are the regular octagon minus a vertex and the four dissimilar versions of a regular nonagon with two vertices removed.

Discussions: An Approximate Construction of the Side of a Regular Inscribed Nonagon

Discussions: An Approximate Construction of the Side of a Regular Inscribed Nonagon