On affine images of regular pentagons and pentagrams

In this paper we explore pentagons that are affine images of the regular pentagon and the regular pentagram. We obtain their characterizations in terms of two mild forms of regularity that deal with the notions of medians for a pentagon and the natural requirement that they are concurrent. Using these characterizations we show that there are various values involving the number 5 (thus related to the golden section) for which a careful selection of division points on appropriate segments determined by any pentagon will result in a pentagon that is the affine image of either a regular pentagon or a regular pentagram.     

Extra two-fold Steiner pentagon systems

A two-fold pentagon system is a decomposition of the complete 2-multigraph (every two distinct vertices joined by two edges) into pentagons. A two-fold Steiner pentagon system is a two-fold pentagon system such that every pair of distinct vertices is joined by a path of length two in exactly two pentagons of the system. We consider two-fold Steiner pentagon systems with an additional property : for any two vertices, the two paths of length two joining them are distinct. We determine completely the spectrum for such systems, and point out an application of such systems to certain 4-cycle systems.     

五边方程的集合理论解

本文给出五边方程的集合理论解.假设V作用在有限群的张量积G(?)G上,满足五边方程V12V13V23=V23V12,则在给定条件下,V由三元组(a,d,p)惟一确定,其中a,d,p是G到自身的群同态。由此给出了V的分类.     

Embedding 5-cycle systems into pentagon triple systems

We show that the spectrum for pentagon triple systems is the set of all n≡1,15,21 or . We then construct a 5-cycle system of order 10n+1 which can be embedded in a pentagon triple system of order 30n+1 and also construct a 5-cycle system of order 10n+5 which can be embedded in a pentagon triple system of order 30n+15, with the possible exception of embedding a 5-cycle system of order 21 in a pentagon triple system of order 61.     

Steiner pentagon systems

A Steiner pentagon system is a pair (K_n, P) where K_n isthe complete undirected graph on n vertices. P is a collection of edge-disjoint pentagons which partition K_n, and such that every part of distinct vertices of K_n is joined by a path of length two in exactly one pentagon of the collection P. The number n is called the order of the system. This paper gives a somplete solution of the existence problem of Steiner pentagon systems. In particular it is shown that the spectrum for Steiner pentagon systems (=the set of all orders for which a Steiner pentagon system exists) is precisely the set of all n ≡ 1 or 5 (mod 10), except 15, for which no such system exists.     

Sub(L)中的五边形格特征及其数量不等式

五边形结构在刻画格的特征方面具有十分重要作用，应用格论及组合数学的方法讨论了格L与其子格格Sub(L)中所含五边形格之间的数量关系，给出了有限格L的子格格中三个元生成五边形格的充要条件，同时给出了Sub(L)所含不同五边形格数量的一个下界。     

Hyperbolic Polygons and Real Schottky Groups

In a work due to Aigon and Silhol (also Buser and Silhol) a construction of 10 genus two closed Riemann surfaces is done from a given right angled hyperbolic pentagon. In this note we construct real Schottky groups uniformizations of the corresponding constructions. In particular, we are able to write down the algebraic curves obtained in the above work in terms of the parameters of the real Schottky group. We generalize such a construction for any right angled hyperbolic polygon and also consider an example for a nonright angled pentagon in the last section.     

Super-Simple Twofold Steiner Pentagon Systems

A twofold pentagon system of order v is a decomposition of the complete undirected 2-multigraph 2K _v into pentagons. A twofold Steiner pentagon system of order v [TSPS(v)] is a twofold pentagon system such that every pair of distinct vertices is joined by a path of length two in exactly two pentagons of the system. A TSPS(v) is said to be super-simple if its underlying (v, 5, 4)-BIBD is super-simple; that is, if any two blocks of the BIBD intersect in at most two points. In this paper, it is shown that the necessary conditions for the existence of a super-simple TSPS(v); namely, v ≥ 15 and v ≡ 0 or 1 (mod 5) are sufficient. For these specified orders, the main result of this paper also guarantees the existence of a very special and interesting class of twofold and fourfold Steiner pentagon systems of order v with the additional property that, for any two vertices, the two or four paths of length two joining them are distinct.     

A matrix solution of the pentagon equation with anticommuting variables

We construct a solution of the pentagon equation with anticommuting variables on two-dimensional faces of tetrahedra. In this solution, matrix coordinates are assigned to tetrahedron vertices. Because matrix multiplication is noncommutative, this provides a “more quantum” topological field theory than in our previous works.     

Every Large Point Set contains Many Collinear Points or an Empty Pentagon

We prove the following generalised empty pentagon theorem for every integer ℓ ≥ 2, every sufficiently large set of points in the plane contains ℓ collinear points or an empty pentagon. As an application, we settle the next open case of the “big line or big clique” conjecture of Kára, Pór, and Wood [Discrete Comput. Geom. 34(3):497–506, 2005].     

Areas of polygons inscribed in a circle

Heron of Alexandria showed that the areaK of a triangle with sidesa,b, andc is given by $$K = \sqrt {s(s - a)(s - b)(s - c)} ,$$ wheres is the semiperimeter (a+b+c)/2. Brahmagupta gave a generalization to quadrilaterals inscribed in a circle. In this paper we derive formulas giving the areas of a pentagon or hexagon inscribed in a circle in terms of their side lengths. While the pentagon and hexagon formulas are complicated, we show that each can be written in a surprisingly compact form related to the formula for the discriminant of a cubic polynomial in one variable.     

A Classical Solution of the Pentagon Equation Related to the Group SL(2)

A new solution of the pentagon equation related to the flat geometry is obtained; it is invariant under the action of the group SL(2).     

THE HAUSDORFF MEASURE OF SIERPINSKI CARPETS BASING ON REGULAR PENTAGON

In this paper,we address the problem of exact computation of the Hausdorff measure of a class of Sierpinski carpets E the self-similar sets generating in a unit regular pentagon on the plane.Under some conditions,we show the natural covering is the best one,and the Hausdorff measures of those sets are euqal to | E | s,where s=dim H E.     

Note on Reflexive Algebras with Non—distributive Invariant Lattices

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Orthogonal triangulation of polygons

In this study, the question whether any convex polygon can be divided using an orthogonal grid into right-angled triangles is answered in the negative. Moreover, it is demonstrated that there exists a convex pentagon which cannot be even approximated by divisible pentagons.     

Pentagons inscribed in a closed convex curve

Two theorems are proved. Let the points A₁, A₂, A₃, A₄, and A₅ be the vertices of a convex pentagon inscribed in an ellipse, let Κ⊂ℝ² be a convex figure, and let A₀ be a fixed distinguished point of its boundary ϖK. If the sum of any two of the neighboring angles of the pentagon A₁A₂A₃A₄A₅ is greater than π or the boundary ϖK is C⁴-smooth and has positive curvature, then some affine image of the pentagon A₁A₂A₃A₄A₅ is inscribed in K and has A₀ as the image of the vertex A₁. (This is not true for arbitrary pentagons incribed in an ellipse and for arbitrary convex figures.) Bibliography: 4 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 246, 1997, pp. 184–190. Translated by N. Yu. Netsvetaev.     

On a Characterization Theorem of Semimodular Lattices

In general lattice theory,there are two basic criterion theorems:　 (1 ) A lattice L is modular if and only if L does not contain a pentagon N5(seeFig. 1 a) .　 (2 ) A lattice L is distributive ifand only if L contains neither a pentagon N5nor a dia-mond M3(see Fig. 1 b) .　 Now,the problem is that if L is semimodular,whether L possesses the similar char-acteristic?In this regard,the following conclusion is obtained in this paper:　 A lattice L is semimodular if and only if L does not co…     

Volkov pentagon for the modular quantum dilogarithm

The new form of pentagon equations suggested by Volkov (Int. Math. Res. Notices (2011); ) for the q-exponential on the basis of formal series is derived within the Hilbert space framework for the modular version of the quantum dilogarithm.     

A New by the Characterization of Mobius Transformations Use of Apollonius Points of （2n-1）-gons

In this paper, we give a new characterization of Mobius transformations. To do this, we extend the notion of Apollonius points of a triangle and of a pentagon, to the notion of Apollonius points of an arbitrary （2n-1）-gon.     

Use of a computer to find the number of regular pentagons that can simultaneously touch a given one

Around an initial regular pentagon one describes a contour L on which one introduces a measure m. One investigates the difference S(M)=1/7m(L)?m(L∩M) where M is a pentagon touching the initial one and congruent to it. The geometric part of the investigation reduces the proof of the inequality S(M)<0 for all M to the proof of the negativity of two effectively computable functions F(u,v) and G(v) in the compact domain of the variation of the arguments. By the method of demonstrative computations, one calculates on a computer the values of these functions at the nodes of a rectangular net of the domain of the variation of the arguments by taking into account the monotonicity and one estimates the computational error. The results of the computation show that we have the inequality S(M)<0, from where it follows that the desired number is equal to six.