A characterization of affinely regular polygons

In 1970, Coxeter gave a short and elegant geometric proof showing that if \(p_1, p_2, \ldots , p_n\) are vertices of an n-gon P in cyclic order, then P is affinely regular if, and only if there is some \(\lambda \ge 0\) such that \(p_{j+2}-p_{j-1} = \lambda (p_{j+1}-p_j)\) for \(j=1,2,\ldots , n\). The aim of this paper is to examine the properties of polygons whose vertices \(p_1,p_2,\ldots ,p_n \in \mathbb {C}\) satisfy the property that \(p_{j+m_1}-p_{j+m_2} = w (p_{j+k}-p_j)\) for some \(w \in \mathbb {C}\) and \(m_1,m_2,k \in \mathbb Z\). In particular, we show that in ‘most’ cases this implies that the polygon is affinely regular, but in some special cases there are polygons which satisfy this property but are not affinely regular. The proofs are based on the use of linear algebraic and number theoretic tools. In addition, we apply our method to characterize polytopes with certain symmetry groups.     

Steiner minimal trees for regular polygons

Fifty years ago Jarnik and Kössler showed that a Steiner minimal tree for the vertices of a regularn-gon contains Steiner points for 3 n5 and contains no Steiner point forn=6 andn13. We complete the story by showing that the case for 7n12 is the same asn13. We also show that the set ofn equally spaced points yields the longest Steiner minimal tree among all sets ofn cocircular points on a given circle.     

Model-theoretic properties of regular polygons

This work is devoted to results obtained in the model theory of regular polygons. We give a characterization of monoids with axiomatizable and model-complete class of regular polygons. We describe monoids with complete class of regular polygons that satisfy some additional conditions. We study monoids whose regular core is represented as a union of finitely many principal right ideals and all regular polygons over which have a stable and superstable theory. We prove the stability of the class of all regular polygons over a monoid provided this class is axiomatizable and model-complete. We also describe monoids for which the class of all regular polygons is superstable and ω-stable provided this class is axiomatizable and model-complete. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 10, No. 4, pp. 107–157, 2004.     

Numerical optimization method for packing regular convex polygons

An algorithm is presented for the approximate solution of the problem of packing regular convex polygons in a given closed bounded domain G so as to maximize the total area of the packed figures. On G a grid is constructed whose nodes generate a finite set W on G, and the centers of the figures to be packed can be placed only at some points of W. The problem of packing these figures with centers in W is reduced to a 0-1 linear programming problem. A two-stage algorithm for solving the resulting problems is proposed. The algorithm finds packings of the indicated figures in an arbitrary closed bounded domain on the plane. Numerical results are presented that demonstrate the effectiveness of the method.     

Equitransitive edge-to-edge tilings by regular convex polygons

In their article Tilings by regular polygons, B. Grünbaum and G. C. Shephard [1] conjecture that there are 19 equitransitive edge-to-edge tilings by regular convex polygons. We prove that there are 22 equitransitive edge-to-edge tilings by regular convex polygons, and it turns out that 3 of them are 1-equitransitive, 13 are 2-equitransitive, 5 are 3-equitransitive and 1 is 4-equitransitive.     

正多边形对称群的子群

利用Lagrange定理和正多边形对称群的性质,首先对正多边形对称群的子群的性质进行了研究,其次讨论了正多边形对称群的子群的结构,由此完全确定了正多边形对称群的子群,最后应用所得结论求出了正六边形对称群的所有子群.     

Half-regular and regular points in compact polygons

We investigate the derived structures of compact polygons at half-regular and at regular points. This enables us to give a geometric characterization of the real or complex split Cayley hexagon.     

The non-existence of certain regular generalized polygons

Senior Research Associate at the National Fund for Scientific Research (Belgium)     

A degree inequality for Lie algebras with a regular Poisson semi-center

For Lie algebras whose Poisson semi-center is a polynomial ring we give a bound for the sum of the degrees of the generating semi-invariants. This bound was previously known in many special cases.     

Periodic billiard trajectories in regular polygons and closed geodesics on regular polyhedra

We consider the relations between the lengths of periodic billiard trajectories in regular triangles, squares, and regular pentagons and those of closed geodesics on the surfaces of regular polyhedra. The cases of regular tetrahedra and octahedra are fully resolved in [3]. The cases of cubes and regular icosahedra are treated below. In the case of regular dodecahedra we can present only preliminary partial results.     

Twisted angles for central configurations formed by two twisted regular polygons

In this paper, we study the necessary conditions and sufficient conditions for the twisted angles of the central configurations formed by two twisted regular polygons, in particular, we prove that for the 2N-body problem, the twisted angles must be $\theta =0$ or $\theta =\pi /N$.     

Laplace eigenvalues on regular polygons: A series in 1/N

For regular polygons $P_N$ inscribed in a circle, the eigenvalues of the Laplacian converge as $N\to \infty$ to the known eigenvalues on a circle. We compute the leading terms of ${\lambda }_N/\lambda$ in a series in powers of $1/N$, by applying the calculus of moving surfaces to a piecewise smooth evolution from the circle to the polygon. The $O(1/N^2)$ term comes from Hadamard?s formula, and reflects the change in area. This term disappears if we “transcribe” the polygon, scaling it to have the same area as the circle.     

Compatible spreads of symmetry in near polygons

In De Bruyn [7] it was shown that spreads of symmetry of near polygons give rise to many other near polygons, the so-called glued near polygons. In the present paper we will study spreads of symmetry in product and glued near polygons. Spreads of symmetry in product near polygons do not lead to new glued near polygons. The study of spreads of symmetry in glued near polygons gives rise to the notion of ‘compatible spreads of symmetry'. We will classify all pairs of compatible spreads of symmetry for the known classes of dense near polygons. All these pairs of spreads can be used to construct new glued near polygons. Postdoctoral Fellow of the Research Foundation-Flanders.     

Characterizations of the Suzuki tower near polygons

In recent work, we constructed a new near octagon \(\mathcal {G}\) from certain involutions of the finite simple group \(G_2(4)\) and showed a correspondence between the Suzuki tower of finite simple groups, \(L_3(2)< U_3(3)< J_2< G_2(4) < Suz\), and the tower of near polygons, \(\mathrm {H}(2,1) \subset \mathrm {H}(2)^D \subset \mathsf {HJ} \subset \mathcal {G}\). Here we characterize each of these near polygons (except for the first one) as the unique near polygon of the given order and diameter containing an isometrically embedded copy of the previous near polygon of the tower. In particular, our characterization of the Hall–Janko near octagon \(\mathsf {HJ}\) is similar to an earlier characterization due to Cohen and Tits who proved that it is the unique regular near octagon with parameters (2, 4; 0, 3), but instead of regularity we assume existence of an isometrically embedded dual split Cayley hexagon, \(\mathrm {H}(2)^D\). We also give a complete classification of near hexagons of order (2, 2) and use it to prove the uniqueness result for \(\mathrm {H}(2)^D\).     

The structure of near polygons with quads

We develop a structure theory for near polygons with quads. Main results are the existence of sub 2j-gons for 2?j?d and the nonexistence of regular sporadic 2d-gons for d?4 with s>1 and t ₂>1 and t ₃≠t ₂(t ₂+1).