Pappian Plane Research Articles

Harmonic ovals

We show that a central oval in a Moufang Fano plane is a Moufang oval if and only if the oval is harmonic. This shows, for the first time, that harmonic ovals that are not conics exist. In addition, we show that, if a strongly harmonic oval in a Moufang Fano plane exists, then the plane is Pappian and the oval is a conic and that if a harmonic oval in a Moufang plane that is not a Fano plane exists, then the plane is Pappian and the oval is a conic. We also show local versions of these results, characterising conics by the property of: being harmonic at external points on a fixed secant line, being harmonic at external points on a fixed external line, being harmonic at points on a fixed tangent line and being harmonic at all secant lines on a fixed point of the oval. Finally, we give some related characterisation of conics in terms of degenerations of Pascal’s theorem, including the theorem that an oval in a Moufang plane with the four point Pascal property is a conic in a Pappian plane.

Nonexistence of certain Fano subplanes of Figueroa planes

Using Grundhöfer's construction of the Figueroa planes from Pappian planes which have an order three planar collineation ${\widehat \alpha }$, we show that any Figueroa plane (finite or infinite) of characteristic not two cannot have a Fano subplane which includes an ${\widehat \alpha }$ invariant triangle.

On some classes of linear spaces embedded in a Pappian plane

We deal with the following problem. Let L be a suitable finite linear space embedded in a Pappian plane P and suppose that L is embeddable in a finite projective plane π′ of order n. It is true that a finite subplane π of P isomorphic to π′ containing L exists?

The Collineation Groups of Figueroa Planes

AbstractWe describe the structure of the collineation groups of Figueroa planes, giving examples and explanations that show why the description in Dempwolff [3] is not completely accurate. We also give criteria for when Figueroa planes are isomorphic, and show that certain subplanes are Figueroa or Pappian planes.

Ovale in p-adischen Ebenen

Continuously differentiable ovals in compact disconnected Pappian planes are characterized by their finite homomorphic images.

The ternary rings of Desarguesian and Pappian planes ? A simple proof using perspectivities

The aim of this paper is to give a direct, simple proof of the well-known theorems — that the Hall ternary ring (R, T) of a Pappian projective plane is a linear ternary ring over a field, and that of a Desarguesian plane is a linear one over a skew field — by making repeated application of the perspectivity theorem in a Pappian plane and the characterization of Desarguesian planes in terms of perspectivities.

A characterization of Thalesian orthogonality in Pappian planes of characteristic 2

In an affine plane over a field K, any Thalesian orthogonality relation is equivalent to ⊥ d for some d∈K∖{0}, where ⊥ d denotes the relation with constant of orthogonality d/it (i.e., after suitable coordinatization the slopes m, m*∈K∖{0} of orthogonal lines satisfy m·m*=d) (cf. [1], [5]). In the present paper we show that in a Pappian plane of characteristic two any orthogonality relation admitting the same group as ⊥1 is equivalent to ⊥1. This gives a characterization of Thalesian orthogonality over perfect fields of characteristic two.

Eine projektive Spiegelungsgruppe, welche S-Gruppe ist

Let π be a projective plane with the flag (A,b). Assume that there exist all involutory homologies with centres on b and axis through A. Let G be the collineation group generated by the set S of all these involutory homologies. It is shown that π is a pappian plane if (G,S) is an S - group.

Ternary rings with zero associated to Desarguesian and Pappian planes

Ternary rings with zero associated to Desarguesian and Pappian planes

On an application of the ‘square’ functional equation to a geometric characterization of quadratic functions from the standpoint of conformal-mapping properties

J. Steiner defined a conic in a pappian plane as the set of points obtained from the intersection of corresponding lines from two projectively related line pencils. This definition is not applicable to non-pappian planes; hence the following specialization: In a general incidence plane let A, B, P be non-collinear points, c~ a line containing neither A nor P, and fl a line containing none of A, B, P; moreover, let AB not contain c~n ft. Then there is a map * from the lines through A onto the lines through B by means of perspectivities as expressed by A ~ P ~ B but not A ~ B . The set of points {0nO* ] ~ line through A} is then a special conic in Steiner's sense. In Moufang planes 2~ is such a conic.

Products of Reflections in an Affine Moufang Plane

Let be a Moufang plane. By specializing one line ω, the line at infinity, weobtain an affine Moufang plane . The group generated by the shears of is called the equiaffine group. Veblen [9, § 52] asked whether every equiaffinity is a product of two affine reflections. He gave a proof which will work in an affine Pappian plane, using the following two properties.Property 1. If an equiaffinity fixes two distinct proper points of , it fixes every point collinear with them.Property 2. Let e be an equiaffinity and P a point such that ppe2pe3 is a triangle. Then the lines PePe2 and PPe3 are parallel.Without using these properties, it will be proved that the answer to Veblen's question is “yes“ if and only if the Moufang plane is Pappian.