The Structural Transformation of Concave Cupolae of Fourth Sort Using Different Variants of Constructive Procedure, 4th International Scientific Conference on Geometry and Graphics

Concave Cupolae of fourth sort (CC IV) are polyhedra which lateral surface is created by folding a quadruple strip of equilateral triangles, while the bases are regular polygons. The unit cell that forms the deltahedral lateral surface by radial sequenceing around the axis of the solid is composed of two spatial hexahedrals. These unit cells are linked together in its upper zone by spatial quadrihedals, which consist of four equilateral triangles grouped around a common vertex. Over the same polygonal base, there can be formed four diverse Concave Cupolae of IV sort, using different variants of constructive procedure. In this paper, the application of the program for modeling the lateral surface of concave cupolae of IV sort, using the software package MATLAB, allows visual monitoring of the structural transformation of the concave cupolae of IV sort, for different polygonal bases and different variants of constructive procedure

Concave Pyramids of Second Sort -The Occurrence, Types, Variations

Correspondingly to the method of generating the Concave Cupolae of second sort, the Concave Pyramids of second sort have the similar logic of origination, and their counterpart in regular faced convex pyramids (tetrahedron, Johnson's solids J1 and J2). The difference is that instead of onefold series of equilateral triangles in the lateral surface of the solid, there appear twofold series, forming deltahedral lateral surface with a common point, while bases are also regular polygons. This time, instead of the bases from n=3 to n=5, there are the basis from n=6 to n=9. The same lateral surface’s net can be folded and creased in two different ways, which produces the two types of Concave Pyramids of second sort: with a major and with a minor solid height. Combining and joining so obtained solids by the correspondent bases, the concave (ortho) bipyramids of second sort emerge, which then may be elongated, gyroelongated, and conca-elongated, creating a distinctive family of diverse concave polyhedral structures

Constructive - geometric generating of cupolae with concave polyhedral surfaces

Конкавне куполе су полиедри који следе начин генерисања куполa (Џонсонових тела), с тим што изостаје критеријум конвексности и што се у омотачу ових тела појављују два или више низа једнакостраничних троуглова.Concave Cupolae are polyhedra which follow the method of generating geometric cupolae (Johnson´s solids), except for the convexity criterion which is omitted, and the fact that in the lateral surface of the solids appear two or more series of equilateral triangle

Learning while playing - Througie platform for creating models of spatial structures

The paper shows a way to create a set of tools that allows the construction of different models of spatial deltahedral structures as a result of the game. It is a set of accessories with which participants (school children, students) through play, experimenting with different layouts and combinations of elements of the whole, create spatial forms from uniform elements: sticks of length "a" and ringshaped joint connections. The game uses a STEM learning approach and the "learning by doing" method to provide knowledge about the spatial relationships of elements and static properties of the structure, in the process of playing. The process of forming structures via mentioned connecting elements uses the same principle of connecting supports and nodes as in the real space truss. A similar method is used by some other gaming platforms, such as Geomag, Magna-tiles, Picasso, or Blockaroo. Unlike them, our solution does not use magnets, but prestressing.https://mongeometrija2023.ftn.uns.ac.rs/?page_id=1

Forming the Cupolae With Concave Polyhedral Surfaces by Corrugating a Fourfold Strip of Equilateral Triangles

The cupolae with concave polyhedral surfaces consist of: two regular polygons, n-gone and 2n-gone in parallel planes, interconnected by an envelope constituted of series of equilateral triangles. The paper describes cupolae which originate by corrugating of a fourfold strip of equilateral triangles, forming thereby the envelope of a cupola. In this manner, a non-convex polyhedron is emerged. Such a method of corrugating the envelope, allows the solutions for generating cupolae with base polygon which number of sides exceedes n=10, which was the maximal number of sides for cupole with the envelope consisted of twofold strip of equilateral triangles. By analyzing the elements of these polyhedra and by help of their paper models, we find geometric constructions and projection procedures by which it is possible to graphically display the cupolae.http://mongeometrija.com/media/mongeometrija/2010/moNGeometrija%202010%20-%20PAGINACIJA.pdf http://mongeometrija.com/component/content/article/52-slobodan-mii/115-misic-obradovic-forming-the-cupolae-with-concave-polyhedral-surfaces-by-corrugating-a-fourfold-s

Konkavna kupola nad hendekagonalnom osnovom

Predmet istraživanja je konstruktivno geometrijska geneza novih geometrijskih tela, kupola sa konkavnim poliedarskim površima, koje bi korišćenjem pravilnih n-tougaonika u svojoj mreži, obrazovale zatvorene prostorne celine. Ove poliedarske forme – kupole, za polazne ntougaonike imaju jedanaestougaonik i dvadesetdvougaonik u paralelnim horizontalnim ravnima. Način formiranja ovakve kupole zasniva se na nabiranju mreže koja obrazuje traku, a presavijanjem iste dobija se deltaedarski omotač koji čine nizovi pravilnih poligona – jednakostraničnih trouglova. Za geometrijsko određivanje osnovnih parametara tela korišćeni su preseci pramenova lopti sa centrima u karakterističnim tačkama prostornog sedmostranika ABCDEFG kao osnovne ćelije kupole nad hendekagonalnom osnovom. Objašnjene su geometrijske konstrukcije i projekcioni postupci pomoću kojih je moguće prikazati kupolu nad hendekagonalnom osnovom, kroz pronalaženje međusobnih relacija parametara, dimenzija i elemenata samog tela.The subject of the research is the constructive geometric genesis of new geometric solids, domes with concave polyhedral surfaces, which, by using regular n-gons in their net, would form closed spatial units. These polyhedral forms - cupolae, for the starting n-gons have a hendecagon (eleven sided polygon) and doicosagon (twenty two sided polygon) in parallel horizontal planes. The way of forming such a cupola is based on the folding the planar triangular net that forms the strip, so that a delta-shaped shell is obtained. It consists of rows of regular polygons - equilateral triangles. To geometrically determine the basic parameters of the solid, the cross sections of spheres with centers at the characteristic points of the spatial heptagon ABCDEFG were used as the basic cells of the cupola with the hendecagonal base. Geometric constructions and projection procedures are explained, by means of which it is possible to obtain the cupola over the hendecagonal base, through finding the mutual relations of parameters, dimensions and elements of the solid itself.http://mongeometrija.com/zbornici/2008/284-slobodan-mii-marija-obradovi-konkavna-kupola-nad-hendekagonalnom-osnovo

Prevođenje konkavnih kupola druge vrste u tridimenzionalne konstruktivne sisteme – prostorne rešetke

Cilj istraživanja ovog rada je bio da se iznađu geometrijska rešenja za formiranje prostornih rešetki nad pravilnim poligonalnim osnovama, tako da se postigne prostorna struktura koja ispunjava zahteve rigidnosti datog sklopa, po osnovi geometrijske stabilnosti trougla kao ravne figure. Takođe, težnja je bila i da se nađe što jednostavnije rešenje, sa upotrebom minimalnog broja različitih štapova u samoj konfiguraciji prostorne rešetke. U tom smislu, moguće je primeniti nekoliko varijanti rasporeda štapova. Najekonomičnije od njih se svodi upravo na geometriju konkavnih kupola druge vrste, buduci da se radi o jednakoivičnim poliedrima kao osnovama za dalju nadogradnju tridimenzionalnih stuktura. Omotač konkavnih kupola druge vrste nastaje nabiranjem ravne mreže, a prevođenjem ivica ovog omotača u sistem štapova, uz dodavanje minimalnog broja različitih tipskih štapova razupirača, dobijamo strukturu kompozitnog poliedra prostorne rešetke.The aim of this paper is to find geometric solutions for the formation of space frames over regular polygonal bases, so to achieve a spatial structure that meets the rigidity requirements of a given assembly, based on the geometric stability of a triangle as a flat figure. Also, the aim is to find the simplest possible solution, with the use of a minimum number of different struts in the configuration of the space frame. In that sense, it is possible to apply several variants of strut arrangement. The most material-efficient of them comes down to the geometry of concave cupolae of the second sort, since they are polyhedra of all the edges of the same length, as a basis for further upgrading of three-dimensional structures. The lateral surface of a concave cupola of the second sort is created by folding its triangular planar net, and by transposing the edges of this surface into a system of struts, having the minimum number of different type struts, we obtain the structure of a composite polyhedron in the role of a space frame.http://mongeometrija.com/zbornici/2008/262-marija-obradovi-slobodan-mii-prevoenje-konkavnih-kupola-druge-vrste-u-tridimenzionalne-konstruktivne-sisteme--prostorne-reetk

Pentagon-Based Radial Tiling with Triangles and Rectangles and Its Spatial Interpretation

The paper considers a type of radial pentagon-based tiling consisting of two shapes: triangle and rectangle. The ob tained solution has a spatial interpretation in a 3D arrangement of equilateral triangles and squares dictated by the particular array of concave cupolae of the second sort, minor type (CC-II 5.m). These cupolae are arranged so that their decagonal bases partly overlap, making a pentagonal pattern (similar to the one of the Penrose tiling). Covering the folds between the faces of such a polyhedral structure with polygons, we use exactly equi lateral triangles and squares, thanks to the trigonometric prop erties of CC-II-5.m. Observed in the orthogonal projection onto the plane of the polygonal bases, this 3D “covering” is viewed as a pentagonal-based radial tiling in the Euclidean plane. Equilateral triangles will be projected into congruent isosceles triangles corresponding to those obtained by the radial sec tion of a regular pentagon in 5 parts. The squares are project ed into rectangles whose ratio is: a:b = 1:φ/√(1+φ2), where φ is the golden ratio. These triangles and rectangles form a ra dial tiling consisting of 5 sectors of the plane, where the pat terns of the established tiles are repeated locally periodically. However, with 5-fold rotation of the pattern, the tiling itself is non-periodic. The various tiling solutions that can be obtained in this way may serve as inspiration for the geometric design, especially interesting in architecture and applied arts, e.g. for rosettes, brise soleils, mosaics, stained glass, fences, partition screens and the likehttps://smartart-conference.rs/sr/%d1%82%d0%b5%d0%bc%d0%b0%d1%82%d1%81%d0%ba%d0%b8-%d0%b7%d0%b1%d0%be%d1%80%d0%bd%d0%b8%d0%ba-2021/ http://doi.fil.bg.ac.rs/pdf/eb_ser/smartart/2022-2/smartart-2022-2-ch19.pd

Konstruktivna obrada hiperboličke spirale kao centralne projekcije cilindrične zavojnice

Centralna projekcija cilindrične zavojnice jeste ravna transcedentna kriva, u najopštijem slučaju. U određenim slučajevima, ta kriva će biti upravo spiralna kriva. U specijalnom slučaju, kada se osovina cilindra poklapa sa glavnim očnim zrakom, ortogonalnim na likoravan, ova će spirala biti hiperbolička, budući da će imati beskonačno daleku tačku, pridruženu tački prodora zavojnice kroz očnu frontalnicu (Fs). Pravac određen ovom prodornom tačkom kroz Fs određuje asimptotu hiperboličke spirale. U radu su dati konstruktivni postupci za grafičko prikazivanje ove spirale.http://mongeometrija.com/zbornici/2006/337-obradovi-mii-konstruktivna-obrada-hiperbolike-spirale-kao-centralne-projekcije-cilindrine-zavojnic

Are Vauban's Geometrical Principles Applied in the Petrovaradin Fortress?

There is a widespread opinion in different sources, ranging from popular to scientific, that the project of the Petrovaradin Fortress was conceived under the influence of the most important European military engineer and innovator of the time, Sebastien de Vauban. By examining the historical context as well as by comparing Vauban's geometrical methods for determination of the fortification master line (la ligne magistrale) with Austrian plans and the actual state of the Petrovaradin fortress, especially its Wasserstadt part, we have examined how well-founded this claim is