Related papers: Two circles and only a straightedge

Jaeger's directed cycle double cover conjecture can be formulated as a problem of existence of special perfect matchings in a class of graphs that we call hexagon graphs. In this work, we explore the structure of hexagon graphs. We show…

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, specially, we prove that for the 2N-body problems, the twisted angles…

We describe convex hulls of the simplest compact space curves, reducible quartics consisting of two circles. When the circles do not meet in complex projective space, their algebraic boundary contains an irrational ruled surface of degree…

In this follow-up article to Symplectification of Circular Arcs and Arc Splines, biarc geometry is examined from a purely geometric point of view. Two given points together with their associated tangent vectors in the plane are sufficient…

We provide a complete description of the edge-to-edge tilings with a regular triangle and a shield-shaped hexagon with no right angle. The case of a hexagon with a right angle is also briefly discussed.

This is a simple way rigorously to construct Grassmann, Clifford and Geometric Algebras, allowing degenerate bilinear forms, infinite dimension, using fields or certain modules (characteristic 2 with limitation) - and characterize the…

In studying properties of simple drawings of the complete graph in the sphere, two natural questions arose for us: can an edge have multiple segments on the boundary of the same face? and is each face the intersection of sides of 3-cycles?…

Polyhedra are generically rigid, but can be made to flex under certain symmetry conditions. We generalise Raoul Bricard's 1897 method for making flexible octahedra to construct an infinite family of flexible polyhedra with…

We prove that two dimensional convex subsets of spherical buildings are either buildings or have a center.

In this note we consider the problem of manufacturing a convex polyhedral object via casting. We consider a generalization of the sand casting process where the object is manufactured by gluing together two identical faces of parts cast…

We are interested in the naive problem whether we can move a solid object in a solid box or not. We restrict move to rotation. In the case we can, the centre and the ``direction'' of rotation may be restricted. Simplifying, we consider…

This paper gives new solutions to the problem: 'Can we construct monohedral tilings of the disk such that a neighbourhood of the origin has trivial intersection with at least one tile?'

In their solution to the orchard-planting problem, Green and Tao established a structure theorem which proves that in a line arrangement in the real projective plane with few double points, most lines are tangent to the dual curve of a…

A cycle of elliptic curves is a list of elliptic curves over finite fields such that the number of points on one curve is equal to the size of the field of definition of the next, in a cyclic way. We study cycles of elliptic curves in which…

We construct special conics configurations from some points configurations which are the singularities of the dual of a quartic curve.

This article is a gentle introduction to the mathematical area known as circle packing, the study of the kinds of patterns that can be formed by configurations of non-overlapping circles. The first half of the article is an exposition of…

John Conway's Circle Theorem is a gem of plane geometry. The six points formed by continuing the sides of a triangle beyond every vertex by the length of its opposite side, are concyclic. The theorem has attracted several proofs. We present…

We determine all non-edge-to-edge tilings of the sphere by regular spherical polygons of three or more sides.

A cylinder will roll down an inclined plane in a straight line. A cone will roll around a circle on that plane and then will stop rolling. We ask the inverse question: For which curves drawn on the inclined plane $\mathbb{R}^2$ can one…

We give an example of a finite-dimensional algebra with a 2-cluster tilting module and a simple module which has infinite complexity. This answers a question of Erdmann and Holm.