Burkard Polster

On the Kleinewillinghöfer types of flat Laguerre planes

Kleinewillinghöfer classified in [7] Laguerre planes with respect to central automorphisms and obtained a multitude of types. For finite Laguerre planes many of these types are known to be empty. In this paper we investigate the Kleinewillinghöfer types of flat Laguerre planes with respect to the full automorphism groups of these planes and completely determine all possible types of flat Laguerre planes with respect to Laguerre translations.

Some Constructions of Small Generalized Polygons

We present several new constructions for small generalized polygons using small projective planes together with a conic or a unital, using other small polygons, and using certain graphs such as the Coxeter graph and the Pappus graph. We also give a new construction of the tilde geometry using the Petersen graph.

A family of 2-dimensional Laguerre planes of generalised shear type

We construct a family of 2-dimensional Laguerre planes that generalises ovoidal Laguerre planes and the Laguerre planes of shear type, as described by Lowen and Pfuller, by gluing together circle sets from up to eight different ovoidal Laguerre planes. Each plane in this family admits all maps ( x, y ) ↦ ( x , ry ) for r > 0 as central automorphisms at the circle y = 0.

Semi-biplanes on the cylinder

We construct semi-biplanes from 2-dimensional projective planes and 2-dimensional circle planes.

Semi-Biplanes and Antiregular Generalied Quadrangles

A natural method to construct semi-biplanes from antiregular generalized quadrangles is introduced. Properties of the semi-biplanes constructed are discussed. In the finite case and in the topological case the semi-biplanes that arise bear a strong resemblance to semi-biplanes that arise in the natural way from projective planes admitting an involutory homology.

THe shoelace book: A mathematical guide to the best (and worst) ways to lace your shoes

Setting the stage One-column lacings Counting lacings The shortest lacings Variations on the shortest lacing problem The longest lacings The strongest lacings The weakest lacings Related mathematics Loose ends References Index.

Centering small generalized polygons-projective pottery at work

The generalized triangle, quadrangle, and hexagons of order 2 are small point-line geometries that play a role in the theory of generalized polygons and buildings that is comparable to that of the Fano plane in the theory of projective planes. Virtually everybody working in discrete mathematics is familiar with the generalized triangle of order 2 aka the Fano plane, the smallest projective plane, PG(2,2), or the unique symmetric block design with parameters 2-(7, 3, 1), and its elementary representation as the geometrical configuration of an equilateral triangle together with its three medians and inscribed circle. The main purpose of this paper is to derive an elementary representation of the generalized hexagons of order 2 which extends naturally to a special representation of the projective space PG(5,2). Along the way, we also derive a similar representation of the generalized quadrangle of order 2 which turns out to be equivalent to the well-known representation of the quadrangle in terms of synthems and duads and which extends naturally to a representation of PG(3,2).

unisolvent sets and flat incidence structures

For the past forty years or so topological incidence geometers and mathematicians interested in interpolation have been studying very similar objects. Nevertheless no communication between these two groups of mathematicians seems to have taken place during that time. The main goal of this paper is to draw attention to this fact and to demonstrate that by combining results from both areas it is possible to gain many new insights about the fundamentals of both areas. In particular, we establish the existence of nested orthogonal arrays of strength n, for short nested n-OAs, that are natural generalizations of flat affine planes and flat Laguerre planes. These incidence structures have point sets that are “flat” topological spaces like the Möbius strip, the cylinder, and strips of the form I ×R, where I is an interval of R. Their circles (or lines) are subsets of the point sets homeomorphic to the circle in the first two cases and homeomorphic to I in the last case. Our orthogonal arrays of strength n arise from n-unisolvent sets of half-periodic functions, n-unisolvent sets of periodic functions, and n-unisolvent sets of functions I → R, respectively. Associated with every point p of a nested n-OA, n > 1, is a nested (n− 1)OA—the derived (n−1)-OA at the point p. We discover that, in our examples that arise from n-unisolvent sets of n − 1 times differentiable functions that solve the Hermite interpolation problem, deriving in our geometrical sense coincides with deriving in the analytical sense. 1. Flat Laguerre planes, flat affine planes and unisolvent sets of functions Consider the incidence structure of non-vertical plane sections of the cylinder {(cos t, sin t, z) ∈ R3|t ∈ [−π, π], z ∈ R} over the unit circle S in the xy-plane (see Figure 1). The point set of this incidence structure (P,C) is the cylinder P = S1×R and its “circle set” C is the set of all graphs of functions in F , where F is the set of functions S → R : t 7→ a + b cos t + c sin t, a, b, c ∈ R (here we identify S with the interval [−π, π) in the usual way). Furthermore, in this context, two points (x1, y1), (x2, y2) ∈ S ×R are called parallel if and only if x1 = x2. It is easy to see that (P,C) satisfies the following axioms: (L1) Three pairwise non-parallel points can be uniquely joined by a circle. (L2) For two non-parallel points p, q and a circle c through p there exists a uniquely determined circle through q that touches c at p, i.e., intersects c only in the point p, or coincides with c. Received by the editors October 3, 1994 and, in revised form, July 20, 1996. 1991 Mathematics Subject Classification. Primary 41A05, 51H15; Secondary 05B15, 51B15.

Toroidal circle planes that are not Minkowski planes

We construct first examples of circle planes on the torus that are no Minkowski planes, but satisfy the same axiom of joining as flat Minkowski planes. The circle planes constructed by us form a special class ofhyperbola structures (see [4]) or(B*)-Geometrien (see [2]).

One-Glance(ish) Proofs of Pythagoras' Theorem for 60-Degree and 120-Degree Triangles

Summary In this article we present some elementary proofs of the 60-degree and 120-degree counterparts of Pythagoras′ theorem that mimic the two (most) famous one-glance proofs of Pythagoras′ theorem.