Journal of Geometry

In this paper, we briefly introduce an algorithm, based on the standard basis of trades, which has proven successful in the complete classification of certain combinatorial objects.

In the plane of a triangle ABC, by using the circumcenter O and the orthocenter H, we can naturally construct certain triangles DEF and $$D'E'F'$$ that are similar to ABC. These triangles serve as a basis for the Kiss–Bíró configuration. Their circumcircles meet in the orthocenter, H, and also a remarkable second point, $$H'$$ , for which barycentric coordinates and properties are derived. A third triangle, $$D''E''F''$$ , is introduced and proved inversely similar to and orthologic to ABC, and other properties of this third triangle are given. The similarity transformation that maps $$D''E''F''$$ onto ABC, denoted by $${\mathcal {K}}$$ , is defined and shown, along with $${\mathcal {K}}^{-1}$$ , to play a natural and far-reaching role in relationships among triangles centers and loci in the plane of ABC.

This note clarifies the combinatorial nature of projective coordinate systems of modular upper continuous lattices. It generalizes the classical relationship between 3-dimensional Desarguesian configurations and coordinate systems of projective 3-spaces.

In [4] the author published the theory of curves in isotropic S2 × ℝ. New results of Pottmann [1] show, that isotropic geometry has a meaning in CAGD, especially in questions on scattered data and visualisation. These are not only considered in euclidean space, but also on manifolds. So it may be interesting to look at the theory of surfaces in isotropic manifolds. This will be done in this paper for the manifold S2 × ℝ by embedding it in I4. Special surfaces on isotropic S2 × ℝ will be geometrically interpreted.

$$E^{2}_{1}$$ be the 2-dimensional pseudo-Euclidean space of index 1, $$G=Sim_{L}(E^{2}_{1})$$ be the group of all linear pseudo-similarities of $$E^{2}_{1}$$ and $$G=Sim_{L}^{+}(E^{2}_{1})$$ be the group of all orientation-preserving linear pseudo-similarities of $$E^{2}_{1}$$ . In this paper, groups $$Sim_{GL}^{+}(E^{2}_{1})$$ and $$Sim_{GL}(E^{2}_{1})$$ are defined. For the groups $$G=Sim_{GL}^{+}(E^{2}_{1}),Sim_{GL}(E^{2}_{1})$$ , $$Sim_{L}(E^{2}_{1}),$$ $$Sim_{L}^{+}(E^{2}_{1})$$ , G-invariants of paths in $$E^{2}_{1}$$ are investigated. Using hyperbolic numbers, a method to detect G-similarities of paths and curves is presented. We give an evident form of a path and a curve in terms of their G-invariants. For two paths and two curves with the common G-invariants, evident forms of all linear pseudo-similarity transformations, carrying the paths and the curves, are obtained.

An isohedron is a 3-dimensional polyhedron all faces of which are equivalent under symmetries of the polyhedron. Many well known polyhedra are isohedra; among them are the Platonic solids, the polars of Archimedean polyhedra, and a variety of polyhedra important in crystallography. Less well known are isohedra with nonconvex faces. We establish that such polyhedra must be starshaped and hence of genus 0, that their faces must be star-shaped pentagons with one concave vertex, and that they are combinatorially equivalent to either the pentagonal dodecahedron, or to the polar of the snub cube or snub dodecahedron.

A point P in the plane of triangle ABC is said to be an isoperimetric point if PA + PB + AB = PB + PC + BC = PC + PA + CA, and is said to be a point of equal detour if PA + PB − AB = PB + PC − BC = PC + PA − CA. Incorrect conditions for the existence and uniqueness of such points were given by G. R. Veldkamp in Amer. Math. Monthly 92 (1985) 546-558. In this paper, we use a much simpler approach that yields correct versions of these conditions and that exhibits the relations of these points to the centers of the Soddy circles.