Recent zbMATH articles in MSC 52Chttps://zbmath.org/atom/cc/52C2022-11-17T18:59:28.764376ZWerkzeugIslamic geometric patterns in higher dimensionshttps://zbmath.org/1496.000532022-11-17T18:59:28.764376Z"Moradzadeh, Sam"https://zbmath.org/authors/?q=ai:moradzadeh.sam"Nejad Ebrahimi, Ahad"https://zbmath.org/authors/?q=ai:ebrahimi.ahad-nejadSummary: The purpose of this paper is to develop the Islamic geometric patterns from planar coordinates to three or higher dimensions through their repeat units. We use historical plane methods, polygons in contact (PIC) and point-joined, in our deductive approaches. The mentioned approach makes use of a novel method of tessellation that generates 3D Islamic patterns called ``interior polyhedral stellations''. The outputs showed that both the PIC and point-joined methods have strengths and weaknesses. Point-joined stellations are more efficient for regular repeat units and PIC is suitable for complex designs. These two methods can produce a large range of patterns and can be employed simultaneously. This study effectively answers the question regarding the gap between planar design from Muslim achievements and contemporary demands in modern art and architecture. We also propose techniques for constructing aperiodic three-dimension Islamic geometric patterns tessellation and two-point family.Blossoming bijection for bipartite pointed maps and parametric rationality of general maps of any surfacehttps://zbmath.org/1496.050782022-11-17T18:59:28.764376Z"Dołęga, Maciej"https://zbmath.org/authors/?q=ai:dolega.maciej"Lepoutre, Mathias"https://zbmath.org/authors/?q=ai:lepoutre.mathiasSummary: We construct an explicit bijection between bipartite pointed maps of an arbitrary surface \(\mathbb{S} \), and specific unicellular blossoming maps of the same surface. Our bijection gives access to the degrees of all the faces, and distances from the pointed vertex in the initial map. The main construction generalizes recent work of the second author which covered the case of an orientable surface. Our bijection gives rise to a first combinatorial proof of a parametric rationality result concerning the bivariate generating series of maps of a given surface with respect to their numbers of faces and vertices. In particular, it provides a combinatorial explanation of the structural difference between the aforementioned bivariate parametric generating series in the case of orientable and non-orientable maps.On the Jacobian ideal of an almost generic hyperplane arrangementhttps://zbmath.org/1496.130052022-11-17T18:59:28.764376Z"Burity, Ricardo"https://zbmath.org/authors/?q=ai:burity.ricardo"Simis, Aron"https://zbmath.org/authors/?q=ai:simis.aron"Tohǎneanu, Ştefan O."https://zbmath.org/authors/?q=ai:tohaneanu.stefan-oLet \(\mathcal{A} = \{H_{1},\dots, H_{m}\}\) be a central hyperplane arrangement of rank \(n\) in \(\mathbb{K}^{n}\), where \(\mathbb{K}\) is a field of characteristic zero. For each \(i \in \{1,\dots, m\}\) let us denote by \(\ell_{i}\) a linear form such that \(\ker(\ell_{i})=H_{i}\), and consider \(f = f_{1} \cdots f_{m}\) the defining equation of \(\mathcal{A}\). Denote by \(J_{f}\) the Jacobian ideal associated with \(f\) generated by the partial derivatives of \(f\). The authors conjecture that \(J_{f}\) is a minimal reduction of the ideal \(\mathcal{I}\) which is generated by the \((m-1)\)-fold products of distinct forms among \(\ell_{1},\dots, \ell_{m}\). First of all, they verify this conjecture in the case when \(\mathcal{A}\) is almost generic, i.e., any \(n-1\) among the defining linear forms are linearly independent, and they confirm this conjecture for \(n=3\) unconditionally. Moreover, they show that in the case of \(n=3\) the ideal \(J_{f}\) is of linear type, i.e., the natural surjection from the symmetric algebra of \(J_{f}\) to its Rees algebra is an isomorphism. As a corollary, the authors show that if \(f \in \mathbb{K}[x,y,z]\) is the defining polynomial of a central hyperplane arrangement, then the Rees algebra of the Jacobian ideal \(J_{f}\) is Cohen-Macaulay.
Reviewer: Piotr Pokora (Kraków)A solution to two old problems by Menger concerning angle spaceshttps://zbmath.org/1496.510052022-11-17T18:59:28.764376Z"Prieto-Martínez, Luis Felipe"https://zbmath.org/authors/?q=ai:prieto-martinez.luis-felipeSummary: Around 1930, Menger expressed his interest in the concept of abstract angle function. He introduced a general definition of this notion for metric and semi-metric spaces. He also proposed two problems concerning conformal embeddability of spaces endowed with an angle function into Euclidean spaces. These problems received attention in later years but only for some particular cases of metric spaces. In this article, we first update the definition of angle function to apply to the larger class of spaces with a notion of betweenness, which seem to us a more natural framework. In this new general setting, we solve the two problems proposed by Menger.Strongly-Delaunay starshaped polygonshttps://zbmath.org/1496.520072022-11-17T18:59:28.764376Z"Bloch, Ethan D."https://zbmath.org/authors/?q=ai:bloch.ethan-dSummary: In the course of his study of simplexwise linear maps of disks, Ho gave conditions under which a single vertex of a starshaped polygon in the plane can be moved in such a way that the polygon stays starshaped. We show that the analog of this result does not hold with the added condition that for each starshaped polygon throughout the move there is a cone triangulation of the starshaped polygon that is a strongly-Delaunay triangulation. Nonetheless, we show that the space of all orientation-preserving starshaped polygons that have strongly-Delaunay cone triangulations has the homotopy type of \(S^1\).On lattice width of lattice-free polyhedra and height of Hilbert baseshttps://zbmath.org/1496.520152022-11-17T18:59:28.764376Z"Henk, Martin"https://zbmath.org/authors/?q=ai:henk.martin"Kuhlmann, Stefan"https://zbmath.org/authors/?q=ai:kuhlmann.stefan"Weismantel, Robert"https://zbmath.org/authors/?q=ai:weismantel.robertSeparating circles on the sphere by polygonal tilingshttps://zbmath.org/1496.520182022-11-17T18:59:28.764376Z"Bezdek, András"https://zbmath.org/authors/?q=ai:bezdek.andrasSummary: We say that a tiling separates discs of a packing in the Euclidean plane, if each tile contains exactly one member of the packing. It is a known elementary geometric problem to show that for each locally finite packing of circular discs, there exists a separating tiling with convex polygons. In this paper we show that this separating property remains true for circle packings on the sphere and in the hyperbolic plane. Moreover, we show that in the Euclidean plane circles are the only convex discs, whose packings with similar copies can be always separated by polygonal tilings. The analogous statement is not true on the sphere and it is not known in the hyperbolic plane.Coverings of planar and three-dimensional sets with subsets of smaller diameterhttps://zbmath.org/1496.520192022-11-17T18:59:28.764376Z"Tolmachev, A. D."https://zbmath.org/authors/?q=ai:tolmachev.a-d"Protasov, D. S."https://zbmath.org/authors/?q=ai:protasov.d-s"Voronov, V. A."https://zbmath.org/authors/?q=ai:voronov.vsevolod-aleksandrovichSummary: Quantitative estimates related to the classical Borsuk problem of splitting set in Euclidean space into subsets of smaller diameter are considered. For a given \(k\) there is a minimal diameter of subsets at which there exists a covering with \(k\) subsets of any planar set of unit diameter. In order to find an upper estimate of the minimal diameter we propose an algorithm for finding sub-optimal partitions. In the cases \(10 \leqslant k \leqslant 17\) some upper and lower estimates of the minimal diameter are improved. Another result is that any set \(M \subset \mathbb{R}^3\) of a unit diameter can be partitioned into four subsets of a diameter not greater than 0.966.Packings with geodesic and translation balls and their visualizations in \(\widetilde{\mathbf{SL}_2\mathbf{R}}\) spacehttps://zbmath.org/1496.520202022-11-17T18:59:28.764376Z"Molnár, Emil"https://zbmath.org/authors/?q=ai:molnar.emil"Szirmai, Jenő"https://zbmath.org/authors/?q=ai:szirmai.jenoSummary: Remembering on our friendly cooperation between the Geometry Departments of Technical Universities of Budapest and Vienna (also under different names) a nice topic comes into consideration: the ``Gum fibre model'' (see Fig. 1).
One point of view is the so-called kinematic geometry by Vienna colleagues, e.g., as in [\textit{H. Stachel}, Math. Appl., Springer 581, 209--225 (2006; Zbl 1100.52005)], but also in very general context. The other point is the so-called \(\mathbf{H}^2 \times \mathbf{R}\) geometry and \(\widetilde{\mathbf{SL}_2\mathbf{R}}\) geometry where -- roughly -- two hyperbolic planes as circle discs are connected with gum fibres, first: in a simple way, second: in a twisted way. This second homogeneous (Thurston) geometry will be our topic (initiated by Budapest colleagues, and discussed also in international cooperations).
We use for the computation and visualization of \(\widetilde{\mathbf{SL}_2\mathbf{R}}\) its projective model, as in our previous papers. We found seemingly extremal geodesic ball packing for \(\widetilde{\mathbf{SL}_2\mathbf{R}}\) group \(\mathbf{pq}_k \mathbf{o}_\ell\) (\(p = 9\), \(q = 3\), \(k = 1\), \(o = 2\), \(\ell = 1\)) with density
\(\approx 0.787758\) (Table 2). Much better translation ball packing is for group \(\mathbf{pq}_k \mathbf{o}_\ell\) (\(p = 11\), \(q = 3\), \(k = 1\), \(o = 2\), \(\ell = 1\)) with density \(\approx 0.845306\) (Table 3).On tilings of asymmetric limited-magnitude ballshttps://zbmath.org/1496.520212022-11-17T18:59:28.764376Z"Wei, Hengjia"https://zbmath.org/authors/?q=ai:wei.hengjia"Schwartz, Moshe"https://zbmath.org/authors/?q=ai:schwartz.mosheSummary: We study whether an asymmetric limited-magnitude ball may tile \(\mathbb{Z}^n\). This ball generalizes previously studied shapes: crosses, semi-crosses, and quasi-crosses. Such tilings act as perfect error-correcting codes in a channel which changes a transmitted integer vector in a bounded number of entries by limited-magnitude errors. A construction of lattice tilings based on perfect codes in the Hamming metric is given. Several non-existence results are proved, both for general tilings, and lattice tilings. A complete classification of lattice tilings for two certain cases is proved.Local structure of karyon tilingshttps://zbmath.org/1496.520222022-11-17T18:59:28.764376Z"Zhuravlev, V. G."https://zbmath.org/authors/?q=ai:zhuravlev.vladimir-gThe paper is devoted to karyon tilings of the torus \(\mathbb{T}^d\) of an arbitrary dimension \(d\). The prototypes of karyon tilings are the one-dimensional Fibonacci tilings and the two-dimensional fractal Rauzy tilings, described earlier by the author.
The set of \(d+1\) vectors in \(\mathbb{R}^d\) is a star if any \(d-1\) its vectors are linearly independent and if the hyperplane determined by these \(d-1\) vectors separates the two remaining vectors. The star is one of the basic concepts in the construction of karyon tilings given in previous works of the author.
In the paper under review, local properties of karyon tilings are studied. In particular, tilings of the torus are classified depending on polyhedral stars and ray stars, the connection between ray and polyhedral stars is established.
Karyon tilings have applcations to multidimensional continued fractions.
Reviewer: Elizaveta Zamorzaeva (Chişinău)Symmetry properties of karyon tilingshttps://zbmath.org/1496.520232022-11-17T18:59:28.764376Z"Zhuravlev, V. G."https://zbmath.org/authors/?q=ai:zhuravlev.vladimir-gKaryon tilings are multidimensional generalizations of the one-dimensional Fibonacci tilings and the two-dimensional Rauzy tilings. In the previous article [J. Math. Sci., New York 264, No. 2, 122--149 (2022; Zbl 1496.52022); translation from Zap. Nauchn. Semin. POMI 502, 32--73 (2021)], the author has investigated local properties of karyon tilings of the torus \(\mathbb{T}^d\) of an arbitrary dimension \(d\).
In the present paper, symmetry properties of karyon tilings of the torus \(\mathbb{T}^d\) are studied. The main results obtained are as follows:
\begin{itemize}
\item[1.] A karyon tiling is invariant with respect to canonical shift of the torus \(\mathbb{T}^d\). This is a fundamental property of karyon tilings. The action of shift consists in exchanging the karyon of the tiling, which is composed of \(d+1\) parallelepipeds.
\item[2.] A nondegenerate karyon tiling has \(2^d\) central symmetries.
\end{itemize}
In general the presence of symmetries is connected with optimality properties. The karyon tilings have applications to multidimensional continued fractions.
Reviewer: Elizaveta Zamorzaeva (Chişinău)Sharing pizza in \(n\) dimensionshttps://zbmath.org/1496.520242022-11-17T18:59:28.764376Z"Ehrenborg, Richard"https://zbmath.org/authors/?q=ai:ehrenborg.richard"Morel, Sophie"https://zbmath.org/authors/?q=ai:morel.sophie"Readdy, Margaret"https://zbmath.org/authors/?q=ai:readdy.margaret-aSummary: We introduce and prove the \(n\)-dimensional Pizza Theorem: Let \(\mathcal{H}\) be a hyperplane arrangement in \(\mathbb{R}^n \). If \(K\) is a measurable set of finite volume, the pizza quantity of \(K\) is the alternating sum of the volumes of the regions obtained by intersecting \(K\) with the arrangement \(\mathcal{H} \). We prove that if \(\mathcal{H}\) is a Coxeter arrangement different from \(A_1^n\) such that the group of isometries \(W\) generated by the reflections in the hyperplanes of \(\mathcal{H}\) contains the map \(-\text{id} \), and if \(K\) is a translate of a convex body that is stable under \(W\) and contains the origin, then the pizza quantity of \(K\) is equal to zero. Our main tool is an induction formula for the pizza quantity involving a subarrangement of the restricted arrangement on hyperplanes of \(\mathcal{H}\) that we call the even restricted arrangement. More generally, we prove that for a class of arrangements that we call even (this includes the Coxeter arrangements above) and for a sufficiently symmetric set \(K\), the pizza quantity of \(K+a\) is polynomial in \(a\) for \(a\) small enough, for example if \(K\) is convex and \(0\in K+a\). We get stronger results in the case of balls, more generally, convex bodies bounded by quadratic hypersurfaces. For example, we prove that the pizza quantity of the ball centered at \(a\) having radius \(R\geq \|a\|\) vanishes for a Coxeter arrangement \(\mathcal{H}\) with \(|\mathcal{H}|-n\) an even positive integer. We also prove the Pizza Theorem for the surface volume: When \(\mathcal{H}\) is a Coxeter arrangement and \(|\mathcal{H}| - n\) is a nonnegative even integer, for an \(n\)-dimensional ball the alternating sum of the \((n-1)\)-dimensional surface volumes of the regions is equal to zero.Topology of symplectomorphism groups and ball-swappingshttps://zbmath.org/1496.530012022-11-17T18:59:28.764376Z"Li, Jun"https://zbmath.org/authors/?q=ai:li.jun.2|li.jun.8|li.jun.3|li.jun.16|li.jun.1|li.jun.7|li.jun.6|li.jun|li.jun.11|li.jun.10"Wu, Weiwei"https://zbmath.org/authors/?q=ai:wu.weiwei|wu.weiwei.1|wu.weiwei.2Summary: In this survey article, we summarize some recent progress and problems on the symplectomorphism groups, with an emphasis on the connection to the space of ball-packings.
For the entire collection see [Zbl 1454.00057].Variational principles and combinatorial \(p\)-th Yamabe flows on surfaceshttps://zbmath.org/1496.531032022-11-17T18:59:28.764376Z"Li, Chunyan"https://zbmath.org/authors/?q=ai:li.chunyan"Lin, Aijin"https://zbmath.org/authors/?q=ai:lin.aijin"Yang, Chang"https://zbmath.org/authors/?q=ai:yang.changThe authors start giving details on PL-metrics (piecewise linear metrics) in Euclidean triangulated manifolds. For these types of metrics, the most natural curvature is the combinatorial Gauss curvature. Considering the combinatorial Gauss-Bonnet formula associated with this curvature one obtains the average of the total combinatorial Gauss curvature designated by \(K_{av}\). The constant combinatorial PL-metric has curvature \(K_{av}\) at all vertices. The Yamabe combinatorial problem deals with the existence of this metric.
The authors start by explaining previous studies on this problem done by other mathematicians. \textit{F. Luo} [Commun. Contemp. Math. 6, No. 5, 765--780 (2004; Zbl 1075.53063)] studied the discrete Yamabe problem through the combinatorial Yamabe flow. Luo showed that the combinatorial Yamabe flow is the negative gradient of a potential functional \(F\). He showed that \(F\) is locally convex and obtained the local rigidity for the curvature map and the local convergence of the combinatorial Yamabe flow. He also conjectured that the combinatorial Yamabe flow converges to a constant curvature PL-metric after a finite number of surgeries on the triangulation. \textit{H. Ge} and \textit{W. Jiang} [Calc. Var. Partial Differ. Equ. 55, No. 6, Paper No. 136, 14 p. (2016; Zbl 1359.53054)] introduced the extended combinatorial Yamabe algorithm to handle possible singularities along the combinatorial Yamabe flow and \textit{X. D. Gu} et al. [J. Differ. Geom. 109, No. 2, 223--256 (2018; Zbl 1396.30008); J. Differ. Geom. 109, No. 3, 431--466 (2018; Zbl 1401.30048)] started doing surgery by ``flipping the algorithm''.
In this paper the authors generalize results for both extended combinatorial Yamabe flow and combinatorial Yamabe flow with surgery. The generalization is done first for \(p>1\) introducing the extended combinatorial \(p\)-th Yamabe flow, which is the extended Yamabe flow when \(p=2\) introduced by Ge and Jiang and then generalize the main results they obtained. They show that the solution to the extended combinatorial \(p\)-th Yamabe flow exists for all time. The details are all explained in Section 2. In Section 3 the authors recall the discrete conformal theory and discrete unifomization theorem established by \textit{X. D. Gu} et al. [J. Differ. Geom. 109, No. 2, 223--256 (2018; Zbl 1396.30008); J. Differ. Geom. 109, No. 3, 431--466 (2018; Zbl 1401.30048)] and generalize their results for the combinatorial \(p\)-th Yamabe flow with surgery. It is shown that for the generalized \(p\)-th flows \(p>1\) and \(p\neq 2\) there exists only curvature convergence but no exponential convergence as in the case of \(p=2\).
Reviewer: Ana Pereira do Vale (Braga)