Inscribing a regular m-gon in a regular n-gon

We find necessary and sufficient conditions on m and n for inscribing a regular m-gon in a regular n-gon.     

Regular polygons in taxicab geometry

A polygon of n sides will be called regular in taxicab geometry if it has n equal angles and n sides of equal taxicab length. This paper will show that there are no regular taxicab triangles and no regular taxicab pentagons. The sets of taxicab rectangles and taxicab squares will be shown to be the same, respectively, as the sets of Euclidean rectangles and Euclidean squares. A method of construction for a regular taxicab 2n-gon for any n will be demonstrated.     

Angle orders,regular n-gon orders and the crossing number

A finite poset P(X,<) on a set X={ x ₁,...,x _m} is an angle order (regular n-gon order) if the elements of P(X,<) can be mapped into a family of angular regions on the plane (a family of regular polygons with n sides and having parallel sides) such that x _ij if and only if the angular region (regular n-gon) for x _i is contained in the region (regular n-gon) for x _j. In this paper we prove that there are partial orders of dimension 6 with 64 elements which are not angle orders. The smallest partial order previously known not to be an angle order has 198 elements and has dimension 7. We also prove that partial orders of dimension 3 are representable using equilateral triangles with the same orientation. This results does not generalizes to higher dimensions. We will prove that there is a partial order of dimension 4 with 14 elements which is not a regular n-gon order regardless of the value of n. Finally, we prove that partial orders of dimension 3 are regular n-gon orders for n3.This research was supported by the Natural Sciences and Engineering Research Council of Canada, grant numbers A0977 and A2415.     

Geometrical Spines of Lens Manifolds

We introduce the concept of ‘geometrical spine’for 3-manifolds with natural metrics, in particular, for lensmanifolds. We show that any spine of L_p,q that is close enoughto its geometrical spine contains at least E(p,q) – 3vertices, which is exactly the conjectured value for the complexityc(L_p,q). As a byproduct, we find the minimal rotation distance(in the Sleator–Tarjan–Thurston sense) between atriangulation of a regular p-gon and its image under rotation.     

Strongly n-Trivial Knots

The authors prove in this paper that, given any knot k of genusg,k fails to be strongly n-trivial for all n with n 6g –3. 2000 Mathematics Subject Classification 57M99.     

Minimal regular polygons serving as universal covers in R 2

A universal cover is a set K with the property that each set of unit diameter is a subset of a congruent copy of K. It is shown that the smallest regular n-gon, for fixed n 4, which serves as an universal cover in R ² is the smallest regular n-gon covering a Reuleaux triangle of unit width.     

Convex nonagons with five intervertex distances

The vertices of a convex planar nonagon determine exactly five distances if and only if they are nine vertices of a regular 10-gon or a regular 11-gon. This result has important ties to related concerns, including the maximum number of points in the plane that determine exactly five distances and, for each n7, the samllest t for which there exists a convex n-gon whose vertices determine t distances and are not all on one circle.     

On a problem of Croft on optimally nested regular polygons

We present a solution for the largest regular m-gon contained in a regular n-gon. We find that the answer depends critically on the coprimality of m and n. We show that the optimal polygons are concentric if and only if gcd(m, n) > 1. Our principal result is a complete solution for the case where m and n share a common divisor. For the case of coprime m and n, we present partial results and a conjecture for the general solution. Our findings subsume some special cases which have previously been published on this problem.     

Minimal Convex k-Gons Containing a Given Convex Polygon

There is a k-gon of minimal area containing a given convex n-gon (k<n) such that k-1 sides of the n-gon lie on the sides of the k-gon. All midpoints of the sides of the k-gon belong to the n-gon. Bibliography: 3 titles.     

Axiomatizations of Euclidean geometry in terms of points,equilateral triangles or squares, and incidence

Dimension-free Euclidean geometry over Euclidean ordered fields can be axiomatized in a two-sortedfirst-order language, with points and regular n-gons (with n = 3 or 4) as variables, and with a binary predicate standing for the incidence of a point and a regular n-gon as the only primitive notions.     

Optimally nested regularN-gons

Various sequences of polygons are described and the radii are determined when each successive member of the sequence is the largest next member of the sequence containing its predecessor, or else the smallest next member of the sequence containing its predecessor. In particular, a correct solution of the problem of finding the area of the circle that is the limit of the largest regularn-gon fitting inside a regular (n?1)-gon, starting from an equilateral triangle of unit area, is found to be 0.0753105.     

Triangle group representations and constructions of regular maps

A regular map of type {m,n} is a 2-cell embedding of a graphin an orientable surface, with the property that for any twodirected edges e and e' there exists an orientation-preservingautomorphism of the embedding that takes e onto e', and in whichthe face length and the vertex valence are m and n, respectively.Such maps are known to be in a one-to-one correspondence withtorsion-free normal subgroups of the triangle groups T(2,m,n).We first show that some of the known existence results aboutregular maps follow from residual finiteness of triangle groups.With the help of representations of triangle groups in speciallinear groups over algebraic extensions of Z we then constructivelydescribe homomorphisms from T(2,m,n)=y,z|y^m=zⁿ=(yz)²=1 intofinite groups of order at most c^r where c=c(m,n), such thatno non-identity word of length at most r in x,y is mapped ontothe identity. As an application, for any hyperbolic pair {m,n}and any r we construct a finite regular map of type {m,n} ofsize at most C^r, such that every non-contractible closed curveon the supporting surface of the map intersects the embeddedgraph in more than r points. We also show that this result isthe best possible up to determining C=C(m,n). For r>m thegraphs of the above regular maps are arc-transitive, of valencen, and of girth m; moreover, if each prime divisor of m is largerthan 2n then these graphs are non-Cayley. 2000 Mathematics SubjectClassification: 05C10, 05C25, 20F99, 20H25.     

Affinely Regular Polygons as Extremals of Area Functionals

For any convex n-gon P we consider the polygons obtained by dropping a vertex or an edge of P. The area distance of P to such (n−1)-gons, divided by the area of P, is an affinely invariant functional on n-gons whose maximizers coincide with the affinely regular polygons. We provide a complete proof of this result. We extend these area functionals to planar convex bodies and we present connections with the affine isoperimetric inequality and parallel X-ray tomography.     

Rotatingn-gon/kn-gon vortex configurations

Summary We demonstrate the existence of stationary point-vortex configurations consisting ofk vortexn-gons and a vortexkn-gon. These configurations exist only for specific values of the vortex strengths; the relative vortex strengths of such a consiguration can be uniquely expressed as functions of the radii of the polygons. Thekn-gon must be oriented so as to be fixed by any reflection fixing one of then-gons; for sufficiently smallk, we show that then-gons must be oriented in such a way that the entire configuration shares the symmetries of any of then-gons. Necessary conditions for the formal stability of general stationary point-vortex configurations set conditions on the vortex strengths. We apply these conditions to then-gon/kn-gon configurations and carry out a complete linear and formal stability analysis in the casek=n=2, showing that linearly and nonlinearly orbitally stable configurations exist.     

On the minimality of polygon triangulation

The problem of triangulating a polygon using the minimum number of triangles is treated. We show that the minimum number of triangles required to partition a simplen-gon is equal ton+2w –d – 2, wherew is the number of holes andd is the maximum number of independent degenerate triangles of then-gon. We also propose an algorithm for constructing the minimum triangulation of a simple hole-freen-gon. The algorithm takesO(nlog² n+DK ²) time, whereD is the maximum number of vertices lying on the same line in then-gon andK is the number of minimally degenerate triangles of then-gon.     

Optimal Angle Bounds for Quadrilateral Meshes

We show that any simple planar n-gon can be meshed in linear time by O(n) quadrilaterals with all new angles bounded between 60 and 120 degrees.     

Conformal Mapping in Linear Time

We show that any simple planar n-gon can be meshed in linear time by O(n) quadrilaterals with all new angles bounded between 60 and 120 degrees.     

On the extension of real regular embedding

Let X be a real nonsingular affine algebraic variety of dimensionk. It is proved that any two regular (algebraic) embeddingsX ⁿ are regularly equivalent, provided that n 4k + 2.