The Conic y = x2 in Moufang Planes
The connected components on the projective line over a ring.
The contributions of Hilbert and Dehn to non-archimedean geometries and their impact on the italian school
In this paper we investigate the contribution of Dehn to the development of non-Archimedean geometries. We will see that it is possible to construct some models of non-Archimedean geometries in order to prove the independence of the continuity axiom and we will study the interrelations between Archimedes’ axiom and Legendre’s theorems. Some of these interrelations were also studied by Bonola, who was one of the very few Italian scholars to appreciate Dehn’s work. We will see that, if Archimedes’...
The cross-ratio in Hjelmslev planes
The cross-ratio in Hjelmslev planes is defined. The cross-ratio in the Hjelmslev plane is independent of the choice of a coordinate system on a line.
The cyclic -clans with .
The cyclides of third order in pseudoisotropic space. II. (Die Zykliden 3. Ordnung im pseudoisotropen Raum. II.)
The degrees of certain strata of the dual variety
The Desargues configuration and inversive geometry
The development of the oloid.
The Double Tangency Symmetries in Laguerre Plane
The group generated by double tangency symmetries in a Laguerre plane is investigated. The geometric classification of involutions of a symmetric Laguerre plane is given. We introduce the notion of projective automorphisms using the double tangency and parallel perspectivities. We give the description of the groups of projective automorphisms and automorphisms generated by double tangency symmetries as subgroups of the group M(𝔽,ℝ) of automorphisms of a chain geometry Σ(𝔽,ℝ) following Benz.
The dynamics of two-circle and three-circle inversion
We study the dynamics of a map generated via geometric circle inversion. In particular, we define multiple circle inversion and investigate the dynamics of such maps and their corresponding Julia sets.
The edge-minimal polyhedral maps of Euler characteristic .
The Elation Group of a 4-DImensional Laguerre Plane.
The elementary geometry of a triangular world with hexagonal circles.
The Elements of Geometry [Book]
The envelope of the Wallace-Simpson lines of a triangle. A simple proof of the Steiner problem on the deltoid.
A simple proof is presented of a famous, and difficult, theorem by Jakob Steiner. By means of a straightforward transformation of the triangle, the proof of the theorem is reduced to the case of the equilateral triangle. Several relations of the Steiner deltoid with the Feuerbach circle and the Morley triangle appear then as obvious.
The Euclidean inscribed polygon.
The Euclidean plane kinematics
Restricting his considerations to the Euclidean plane, the author shows a method leading to the solution of the equivalence problem for all Lie groups of motions. Further, he presents all transitive one-parametric system of motions in the Euclidean plane.
The Euler Line Revisited.
The finite Moufang hexagons coordinatized.