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The Structure of N Harmonic Points and Generalization of Desargues’ Theorems

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The Structure of N Harmonic Points and Generalization of Desargues’ Theorems mathematics Article The Structure of n Harmonic Points and Generalization of Desargues’ Theorems Xhevdet Thaqi * and Ekrem Aljimi Faculty of Applied Sciences, Public University “Kadri Zeka”, 60000 Gjilan, Kosovo; [email protected] * Correspondence: [email protected]; Tel.: +383-44132384 Abstract: In this paper, we consider the relation of more than four harmonic points in a line. For this purpose, starting from the dependence of the harmonic points, Desargues’ theorems, and perspectivity, we note that it is necessary to conduct a generalization of the Desargues’ theorems for projective complete n-points, which are used to implement the definition of the generalization of harmonic points. We present new findings regarding the uniquely determined and constructed sets of H-points and their structure. The well-known fourth harmonic points represent the special case 2 (n = 4) of the sets of H-points of rank 2, which is indicated by P4 . Keywords: projective transformations; perspectivity; harmonic points; generalization of Desargues’ theorems; complete plane n-point; set of H-points rank k 1. Introduction The idea of studying and investigating the possibility of construing more than four harmonic points was originally discussed in the article “The chords of the non-ruled quadratic in PG(3, 3)” (see [1]), where the forty-five chords of an “ellipsoid” in finite space Citation: Thaqi, X.; Aljimi, E. The PG(3, 3) are described, showing that they may be regarded as the edges of a notable graph Structure of n Harmonic Points and Generalization of Desargues’ that is in the group of automorphism of the symmetric group. The hexastigm is considered Theorems. Mathematics 2021, 9, 1018. to be formed by six points of general positions in any projective 4-space. The six vertices of https://doi.org/10.3390/math9091018 the hexastigm are joined in sets of two, three, or four, and each edge meets the opposite space in a diagonal point. Academic Editor: Yang-Hui He Three edges that together involve all six vertices are met by a unique line called a transversal. The correspondence between edges and transversals is seen in the fact that Received: 19 March 2021 each transversal meets three edges, and each edge belongs to three transversals, which Accepted: 27 April 2021 form a configuration 153 of a harmonic conjugate with fifteen diagonal points with respect Published: 30 April 2021 to the first and second vertices of the hexastigm. Aiming to generalize the harmony of n-points in the projective geometry, we con- Publisher’s Note: MDPI stays neutral ducted a study in 1996 titled “Generalization of the Desargues’ theorems” [2] and more with regard to jurisdictional claims in detailed presentation of set of harmonic points in 1998 titled “Harmonic points and Desar- published maps and institutional affil- gues’ Theorems” [3]. iations. In 2014, we expanded on our research with the study “Generalized Desargues’ the- orem” [4], where an arbitrary number of points on each one of the two distinct planes is considered, allowing the corresponding points on the two planes to coincide and three points on any of the planes to be collinear. In the work ([4], p. 3) we identified the Copyright: © 2021 by the authors. generalized Desargues’ theorem in the following form: Licensee MDPI, Basel, Switzerland. Let p, p’ be two distinct planes, (1), (2), ... ,(n), (n 2 N), distinct points on p, and (1’), This article is an open access article (2’), ... , (n’) distinct points on p’, both sets of points in general position. distributed under the terms and conditions of the Creative Commons (A) If all generalized lines (i)(i’) go through a common point, then all the intersections Attribution (CC BY) license (https:// of the pairs of lines (i)(j), (i’)(j’) for i 6= j are nonempty and lie on a common line (the creativecommons.org/licenses/by/ common line of p and p’). 4.0/). Mathematics 2021, 9, 1018. https://doi.org/10.3390/math9091018 https://www.mdpi.com/journal/mathematics Mathematics 2021, 9, x FOR PEER REVIEW 2 of 16 Mathematics 2021, 9, 1018 2 of 15 (B) If all the intersections of the pairs of lines ()(), (’)(’) for ≠ are nonempty, then all generalized lines ()(’) go through a common point ([4], pp. 3–4). The proof of the proposition is made by mathematical induction, and we assume that the (B)given If all points the intersections (1), (2), etc. of thelie pairson the of linesplane (i )(;j ),and (i’)( (1’),j’) for (2’),i 6= etc.,j are some nonempty, corresponding then pointsall generalizedlie on another lines plane (i)(i’) go’, throughallow for a commonthe possibility point. of some corresponding points co- inciding.([4], pp. This 3–4) restricts us from applying the generalization of the harmonic n-points, and we areThe also proof not of able the propositionto prove the is unicity made by ofmathematical the mentioned induction, generalized and Desargues’ we assume thattheo- therem. given points (1), (2), etc. lie on the plane p; and (1’), (2’), etc., some corresponding points lie onIn another this paper, plane wep’, explore allow for the the complete possibility n-poin of somets (the corresponding triangle is considered points coinciding. to have 3- Thispoints restricts as special us from case applying of complete the generalizationn-points and dual-figure of the harmonic complete n-points, n-lines) and and we areall of also the notcases able in to which prove the the intersection unicity of the line mentioned p is incident generalized with diagonal Desargues’ points. theorem. This allows us to defineIn thisgeneralized paper, we harmonic explore points, the complete mainly n-pointsbased on (the the works triangle [2,3]. is considered to have 3-pointsFirst as let special us remember case of complete some basic n-points concept and of dual-figure projective completegeometry n-lines)which we and will all use of theas casesa tool inin whichthis paper. the intersection line p is incident with diagonal points. This allows us to defineFollowing generalized Euclid, harmonic we usually points, denote mainly points based by on upper the works case [letters;2,3]. we denote lines by lowerFirst case let letters. us remember If P I g someis true, basic we conceptshall also of say projective that “'P geometry is incident which with we g”, will “'P useliesas on ag”, tool “g in passes this paper. through P”, and so on [5]. Instead of “P I g” we may also write “P ∈ g”. Now we introduceFollowing the Euclid, axioms we that usually are fundamental denote points for by the upper projective case letters; geometry we denote [5]: lines by lowerAxiom case letters. 1 (line If axiom): P I g is true,For any we shalltwo distinct also say points that “’P P isand incident Q there with is exactly g”, “’P one lies online g”,that “g is passes incident through with P P”, and and Q. so This on [line5]. Instead is denoted of “P by I g”PQ. we may also write “P 2 g”. Now we introduceAxiom 2 the (Veblen-Young): axioms that are Let fundamental A, B, C, and for D the be projective four points geometry such that [5]: AB intersects the lineAxiom CD. 1 Then (line AC axiom): also intersects For any two thedistinct line BD points(Figure P 1). and Q there is exactly one line that isWe incident find the with following P and Q. formulation This line is of denoted the Veblen-Young by PQ. axiom which is more concise: if a Axiomline (in 2our (Veblen-Young): case BD) intersects Let A, two B, C, “sides” and D be(namely four points PA and such PC) that of ABa triangle intersects (inthe our linecase CD. APC) Then then AC the also line intersects also intersects the line the BD third (Figure side1). (namely AC) (Figure 1). FigureFigure 1. 1.The The meaning meaning of of Veblen-Young Veblen-Young Axiom. Axiom. WeThe find Veblen-Young the following axiom formulation is a truly of theingenious Veblen-Young way of axiomsaying whichthat any is moretwo lines concise: of a ifplane a line meet (in our [5]. case A projective BD) intersects plane twois a “sides”nondegenerate (namely projective PA and PC) space of ain triangle which any (in our two caselines APC) have then at least the one line point also intersects in common. the third side (namely AC) (Figure1). TheWe Veblen-Youngwill continue working axiom is in a trulythis paper ingenious on the way real of projective saying that plane. any We two can lines observe of a planeany set meet X called [5]. A space, projective and planethe elements is a nondegenerate are the points projective of that space. space The in which subsets any of twoset X linesare called have atfigures, least one thus point figures in common. in the projective plane consists of any subsets of the points andWe lines will and continue the necessary working incidence in this paper relations on the among real projectivethem. The plane.transformation We can observe of set X anyis called set X the called bijective space, mapping and the elementsf of set X areto itself. the points We will of thattreat space. projective The subsetstransformations, of set X areto calledintroduce figures, specific thus concept figures inof theprojective projective geometry, plane consists taken from of any [6]. subsets A one-to-one of the points corre- andspondence lines and is thesaid necessary to exist between incidence the relations elements among of two them.simple The or complete transformation figures, of if set there X isis called associated the bijective with each mapping element f of of set one X tofigu itself.re a Weunique will treatelement projective of the transformations,other figure, and toreciprocally, introduce specific with each concept element of projectiveof the last a geometry, unique element taken fromof the [ 6firs].
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