Recent zbMATH articles in MSC 51M https://www.zbmath.org/atom/cc/51M 2022-06-24T15:10:38.853281Z Werkzeug Lunes areas and quadratures -- a problem and many centuries in the history of mathematics https://www.zbmath.org/1485.01006 2022-06-24T15:10:38.853281Z "Galvão, M. Elisa E. L." https://www.zbmath.org/authors/?q=ai:galvao.maria-elisa-esteves-lopes "De Souza, Vera H. G." https://www.zbmath.org/authors/?q=ai:de-souza.vera-h-g Summary: In this paper we describe the main contributions in the historical trajectory of this problem, from Hippocrates to al-Haytham in the 10th century and Wallenius, in 18th century, to the final solution in the 20th century, based on bibliographical and documental research. In ancient times, techniques or practices developed for measuring areas of plane figures were associated to simple geometric shapes as triangles, rectangles, quadrilaterals, or polygons. After pythagorean times, Greek geometers adopted the geometric algebra and constructions using straightedge and compass. Measures and area calculations were then changed to the quadrature process. Since approximately 500 b.C. up to the XIX, some questions remained unanswered, such as: Can we build a square equivalent to a given disk, using only an unmarked rule and a compass? -- that is, how one is supposed to find out, by means of geometric constructions, what is the side of the square of which the surface area is equal to the circle surface area? In his attempts to square the circle, Hippocrates of Chios was able to describe quadrature processes for the first non polygonal figures. He exhibited the quadrature for three lunes, as are called the non convex regions limited by two intersecting circles. Tracking the Hippocrates' problem we will find the works of al-Haytham in the tenth century and Wallenius' XVIII century work. Wallenius' advances in discovering two new quadrable lunes depends on Viète's trigonometrical formulas, another important reference in mathematical history that appears in the basic studies in trigonometry. The problem of the squarable lunes was completely answered in the first part of the last century. When we bring to life the original ideas and questions and track their developments down or discuss its partial or complete answers, we have an unique opportunity to analyse how the evolution of mathematical thinking, concepts and approaches that has occurred and bring to light the importance of new techniques for its progress. Remarks on Joachimsthal integral and Poritsky property https://www.zbmath.org/1485.37022 2022-06-24T15:10:38.853281Z "Arnold, Maxim" https://www.zbmath.org/authors/?q=ai:arnold.maxim "Tabachnikov, Serge" https://www.zbmath.org/authors/?q=ai:tabachnikov.serge-l Summary: The billiard in an ellipse has a conserved quantity, the Joachimsthal integral. We show that the existence of such an integral characterizes conics. We extend this result to the spherical and hyperbolic geometries and to higher dimensions. We connect the existence of Joachimsthal integral with the Poritsky property, a property of billiard curves, called so after \textit{H. Poritsky} whose important paper [Ann. Math. (2) 51, 446--470 (1950; Zbl 0037.26802)] was one of the early studies of the billiard problem. New hedonistic stories of groups and geometries. Vol. 1 https://www.zbmath.org/1485.51001 2022-06-24T15:10:38.853281Z "Caldero, Philippe" https://www.zbmath.org/authors/?q=ai:caldero.philippe "Germoni, Jérôme" https://www.zbmath.org/authors/?q=ai:germoni.jerome See the review of the first edition in [Zbl 1275.51001]. For Vol. 2 see [Zbl 1485.51002]. New hedonistic stories of groups and geometries. Vol. 2 https://www.zbmath.org/1485.51002 2022-06-24T15:10:38.853281Z "Caldero, Philippe" https://www.zbmath.org/authors/?q=ai:caldero.philippe "Germoni, Jérôme" https://www.zbmath.org/authors/?q=ai:germoni.jerome See the review of the first edition in [Zbl 1321.51001]. For Vol. 1 see [Zbl 1485.51001] Useful and beautiful geometry. A somewhat different introduction to Euclidean geometry. https://www.zbmath.org/1485.51003 2022-06-24T15:10:38.853281Z "Zeuge, Wolfgang" https://www.zbmath.org/authors/?q=ai:zeuge.wolfgang This second edition of a textbook (for the first edition, see, [Nützliche und schöne Geometrie. Eine etwas andere Einführung in die Euklidische Geometrie. Wiesbaden: Springer Spektrum (2018; Zbl 1446.51001)] meant to be read by contemporary high-school students, against the dominant trend of steering high-school students away from geometry and from anything proof-based or requiring an approach outside of an algorithm. Beside some corrections, this second edition contains some added material, and a few (still, way too few) problems, some of which are solved in a final chapter. It contains the \textit{C. Alsina} and \textit{R. B. Nelsen} [Forum Geom. 7, 99--102 (2007; Zbl 1123.51024)] proof of the Erdős-Mordell inequality. Some errors can still be found. On Page 32, it is claimed that A. W. Dorodnow proved in 1947 that, except for the few moons known to be squarable since antiquity, there are no other ones. Only three were known in antiquity, the other two were discovered by Daniel Wijnquist in 1766. On Page 10, the unsuspecting reader is told that the statement There exist rectangles'' is equivalent to the Euclidean parallel postulate. The equivalence is valid only in the presence of the Archimedean axiom, an axiom which is not mentioned at all, for geometry, as presented here, unfolds against the background of real numbers. Reviewer: Victor V. Pambuccian (Glendale) Delone sets on spirals https://www.zbmath.org/1485.51009 2022-06-24T15:10:38.853281Z "Akiyama, Shigeki" https://www.zbmath.org/authors/?q=ai:akiyama.shigeki Summary: Motivated by phyllotaxis in botany, the angular development of plants widely found in nature, we give a simple mathematical characterization of Delone sets on spirals. For the entire collection see [Zbl 1459.37002]. Important result about right angles https://www.zbmath.org/1485.51010 2022-06-24T15:10:38.853281Z "Atkinson, Adam" https://www.zbmath.org/authors/?q=ai:atkinson.adam For the entire collection see [Zbl 1360.00138]. Extending a theorem of van Aubel to the simplex https://www.zbmath.org/1485.51011 2022-06-24T15:10:38.853281Z "Hung, Tran Quang" https://www.zbmath.org/authors/?q=ai:hung.tran-quang The author extends van Aubel's theorem for a triangle in the plane to a simplex in the $n$-dimensional Euclidean space. Reviewer: Cătălin Barbu (Bačau) Triangle geometry, Oroslan and Ravello https://www.zbmath.org/1485.51012 2022-06-24T15:10:38.853281Z "Hvala, Bojan" https://www.zbmath.org/authors/?q=ai:hvala.bojan Summary: In this article, we present Seebach's theorem, which is a topic in triangle geometry. The events, as during guided online film screenings, take place on two channels. On one, there is a mathematical presentation of the topic, and on the other, a chat about possibilities, dilemmas, and backgrounds. Multiplying the hypercube https://www.zbmath.org/1485.51013 2022-06-24T15:10:38.853281Z "Razpet, Marko" https://www.zbmath.org/authors/?q=ai:razpet.marko Summary: The ancient geometric problem of doubling the cube can be extended to a problem of an arbitrary multiplying. Moreover, the problem can be extended to multiplying the hypercube. To this purpose we will use a suitable generalized cissoid. Rotation of mirrors https://www.zbmath.org/1485.51014 2022-06-24T15:10:38.853281Z "Razpet, Nada" https://www.zbmath.org/authors/?q=ai:razpet.nada Summary: We first consider rotation about a vertical axis of two mutually perpendicular mirrors making an angle of $$45^\circ$$ with the horizontal plane, then we consider rotation of a concave cylindrical mirror. We show that, in both cases, rotating mirrors by an angle $$\alpha$$ produces rotation of the image by an angle $$2\alpha$$. Paper folding, duplication of cube and conchoid of de Sluze https://www.zbmath.org/1485.51015 2022-06-24T15:10:38.853281Z "Razpet, Marko" https://www.zbmath.org/authors/?q=ai:razpet.marko "Razpet, Nada" https://www.zbmath.org/authors/?q=ai:razpet.nada Summary: By paper folding we can determine a point that divides the side of a square in ratio $$1:\sqrt{2}$$. This point is the intersection of this side and a conchoid of de Sluze which is also the pedal curve of a parabola. Paper folding, angle trisection and trisectrix of Maclaurin https://www.zbmath.org/1485.51016 2022-06-24T15:10:38.853281Z "Razpet, Marko" https://www.zbmath.org/authors/?q=ai:razpet.marko "Razpet, Nada" https://www.zbmath.org/authors/?q=ai:razpet.nada Summary: By paper folding we can trisect an angle and at the same time we find the trisectrix of Maclaurin in a natural way. The third of an angle can be constructed also by the trisectrix of Maclaurin which is the pedal curve of a parabola, and also the inverse of hyperbola with semiaxes ratio $$1/\sqrt{3}$$ on a suitable circle. Escobar constants of planar domains https://www.zbmath.org/1485.51017 2022-06-24T15:10:38.853281Z "Hassannezhad, Asma" https://www.zbmath.org/authors/?q=ai:hassannezhad.asma "Siffert, Anna" https://www.zbmath.org/authors/?q=ai:siffert.anna This is a study of a conjecture for the $$k$$-th Escobar constant $$I_k(M)$$ of a Riemannian manifold $$(M, g)$$, for $$k\geq 3$$. The isomperimetric constant $$I_k$$ was introduced in [\textit{A. Hassannezhad} and \textit{L. Miclo}, Ann. Sci. Éc. Norm. Supér. (4) 53, No. 1, 43--88 (2020; Zbl 1456.58021)], as an extension of $$I_2(M)$$, which had been introuced in [\textit{J. F. Escobar}, J. Funct. Anal. 150, No. 2, 544--556 (1997; Zbl 0888.58066)]. It is known that the unit disk $${\mathbb D}$$ maximizes $$I_2$$ among all bounded domains $$M$$ in $${\mathbb R}^2$$ with rectifiable boundary, i.e., $$I_2(M) \leq I_2({\mathbb D})$$, and that this holds for higher dimensions as well. This leads the authors to conjecture that the same inequality holds for all $$k\geq 3$$, i.e., $$I_k(M) \leq I_k({\mathbb D})$$. The conjecture is proved when $$M$$ is a polygon in $${\mathbb R}^2$$ and $$k$$ is greater or equal than the number of vertices of $$M$$, and for a family of curvilinear polygons. Reviewer: Victor V. Pambuccian (Glendale) On the density of shapes in three-dimensional affine subdivision https://www.zbmath.org/1485.51018 2022-06-24T15:10:38.853281Z "Luo, Qianghua" https://www.zbmath.org/authors/?q=ai:luo.qianghua "Wang, Jieyan" https://www.zbmath.org/authors/?q=ai:wang.jieyan Summary: The affine subdivision of a simplex $$\Delta$$ is a certain collection of $$(n+1)!$$ smaller $$n$$-simplices whose union is $$\Delta$$. Barycentric subdivision is a well know example of affine subdivision(see). \textit{R. E. Schwartz} [Discrete Comput. Geom. 30, No. 3, 373--377 (2003; Zbl 1048.52005)] proved that the infinite process of iterated barycentric subdivision on a tetrahedron produces a dense set of shapes of smaller tetrahedra. We prove that the infinite iteration of several kinds of affine subdivision on a tetrahedron produce dense sets of shapes of smaller tetrahedra, respectively. Polyhedrons the faces of which are special quadric patches https://www.zbmath.org/1485.51019 2022-06-24T15:10:38.853281Z "Stavrić, Milena" https://www.zbmath.org/authors/?q=ai:stavric.milena "Wiltsche, Albert" https://www.zbmath.org/authors/?q=ai:wiltsche.albert "Weiss, Gunter" https://www.zbmath.org/authors/?q=ai:weiss.gunter-m-t|weiss.gunter Summary: We seize an idea of \textit{O. Giering} (see [Mit HP-Flächen variierte Platonische Polyeder'', IBDG Inf. Bl. Geom. 39, No. 2, 28--31 (2015); Mit HP-Flächen variierte Platonische Polyeder. II'', ibid. 40, No. 1, 37--38 (2021)]), who replaced pairs of faces of a polyhedron by patches of hyperbolic paraboloids and linked up edge-quadrilaterals of a polyhedron with the pencil of quadrics determined by that quadrilateral. Obviously only ruled quadrics can occur. There is a simple criterion for the existence of a ruled hyperboloid of revolution through an arbitrarily given quadrilateral. Especially, if a (not planar) quadrilateral allows one symmetry, there exist two such hyperboloids of revolution through it, and if the quadrilateral happens to be equilateral, the pencil of quadrics through it contains even three hyperboloids of revolution with pairwise orthogonal axes. To mention an example, for right double pyramids, as for example the octahedron, the axes of the hyperboloids of revolution are, on one hand, located in the plane of the regular guiding polygon, and on the other, they are parallel to the symmetry axis of the double pyramid. \par Not only for platonic solids, but for all polyhedrons, where one can define edge-quadrilaterals, their pairs of face-triangles can be replaced by quadric patches, and by this one could generate roofing of architectural relevance. Especially patches of hyperbolic paraboloids or, as we present here, patches of hyperboloids of revolution deliver versions of such roofing, which are also practically simple to realize. Generalized dissections and Monsky's theorem https://www.zbmath.org/1485.51020 2022-06-24T15:10:38.853281Z "Abrams, Aaron" https://www.zbmath.org/authors/?q=ai:abrams.aaron "Pommersheim, Jamie" https://www.zbmath.org/authors/?q=ai:pommersheim.jamie Summary: Monsky's celebrated equidissection theorem follows from his more general proof of the existence of a polynomial relation $$f$$ among the areas of the triangles in a dissection of the unit square. More recently, the authors studied a different polynomial $$p$$, also a relation among the areas of the triangles in such a dissection, that is invariant under certain deformations of the dissection. In this paper we study the relationship between these two polynomials. We first generalize the notion of dissection, allowing triangles whose orientation differs from that of the plane. We define a deformation space of these generalized dissections and we show that this space is an irreducible algebraic variety. We then extend the theorem of Monsky to the context of generalized dissections, showing that Monsky's polynomial $$f$$ can be chosen to be invariant under deformation. Although $$f$$ is not uniquely defined, the interplay between $$p$$ and $$f$$ then allows us to identify a canonical pair of choices for the polynomial $$f$$. In many cases, all of the coefficients of the canonical $$f$$ polynomials are positive. We also use the deformation-invariance of $$f$$ to prove that the polynomial $$p$$ is congruent modulo 2 to a power of the sum of its variables. Poncelet plectra: harmonious curves in cosine space https://www.zbmath.org/1485.51021 2022-06-24T15:10:38.853281Z "Jaud, Daniel" https://www.zbmath.org/authors/?q=ai:jaud.daniel "Reznik, Dan" https://www.zbmath.org/authors/?q=ai:reznik.dan-s "Garcia, Ronaldo" https://www.zbmath.org/authors/?q=ai:garcia.ronaldo-a Summary: It has been shown that the family of Poncelet $$N$$-gons in the confocal pair (elliptic billiard) conserves the sum of cosines of its internal angles. Curiously, this quantity is equal to the sum of cosines conserved by its affine image where the caustic is a circle. We show that furthermore, (i) when $$N = 3$$, the cosine triples of both families sweep the same planar curve: an equilateral cubic resembling a plectrum (guitar pick). We also show that (ii) the family of triangles excentral to the confocal family conserves the same product of cosines as the one conserved by its affine image inscribed in a circle; and that (iii) cosine triples of both families sweep the same spherical curve. When the triple of log-cosines is considered, this curve becomes a planar, plectrum-shaped curve, rounder than the one swept by its parent confocal family. The classification of convex polygons with triangular area or perimeter bisecting deltoids https://www.zbmath.org/1485.52002 2022-06-24T15:10:38.853281Z "Berele, Allan" https://www.zbmath.org/authors/?q=ai:berele.allan "Catoiu, Stefan" https://www.zbmath.org/authors/?q=ai:catoiu.stefan Summary: We classify all convex polygons whose area-bisecting deltoids or perimeter-bisecting deltoids are similar to those for a triangle, that is, they are tri-cusped and tri-concave-out closed curves. The additional condition that these two kinds of deltoids are segment-free makes no difference to the first classification and restricts the second to one that is much more similar to the first. We show that, up to similarity, the restricted second class is a complete system of representatives for the first class modulo affine equivalence. Generalized regularity and the symmetry of branches of botanological'' networks https://www.zbmath.org/1485.52003 2022-06-24T15:10:38.853281Z "Zachos, Anastasios N." https://www.zbmath.org/authors/?q=ai:zachos.anastasios-n Summary: We derive the generalized regularity of convex quadrilaterals in $$\mathbb{R}^2$$, which gives a new evolutionary class of convex quadrilaterals that we call generalized regular quadrilaterals in $$\mathbb{R}^2$$. The property of generalized regularity states that the Simpson line defined by the two Steiner points passes through the corresponding Fermat-Torricelli point of the same convex quadrilateral. We prove that a class of generalized regular convex quadrilaterals consists of convex quadrilaterals, such that their two opposite sides are parallel. We solve the problem of vertical evolution of a botanological'' thumb (a two way communication weighted network) w.r to a boundary rectangle in $$\mathbb{R}^2$$ having two roots, two branches and without having a main branch, by applying the property of generalized regularity of weighted rectangles. We show that the two branches have equal weights and the two roots have equal weights, if the thumb inherits a symmetry w.r to the midperpendicular line of the two opposite sides of the rectangle, which is perpendicular to the ground (equal branches and equal roots). The geometric, rotational and dynamic plasticity of weighted networks for boundary generalized regular tetrahedra and weighted regular tetrahedra lead to the creation of botanological'' thumbs and botanological'' networks (with a main branch) having symmetrical branches.