Non-Desarguesian Projective Plane Geometries Which Satisfy The Harmonic Point Axiom

Canadian Journal of Mathematics ◽

10.4153/cjm-1956-060-1 ◽

1956 ◽

Vol 8 ◽

pp. 532-562 ◽

Cited By ~ 2

Author(s):

N. S. Mendelsohn

Keyword(s):

Projective Plane ◽

Desarguesian Projective Plane ◽

Fourth Harmonic ◽

Straight Lines

1. Introduction and summary. In her papers (12) and (13) R. Moufang discusses projective plane geometries which satisfy the axiom of the uniqueness of the fourth harmonic point. Her main result is that in such geometries non-homogeneous co-ordinates may be assigned to the points of the plane (except for the “line at infinity”) in such a way that straight lines have equations of the forms aαx + y + β = 0, or x + γ − 0.

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Resolving Sets and Semi-Resolving Sets in Finite Projective Planes

The Electronic Journal of Combinatorics ◽

10.37236/2582 ◽

2012 ◽

Vol 19 (4) ◽

Cited By ~ 8

Author(s):

Tamás Héger ◽

Marcella Takáts

Keyword(s):

Projective Plane ◽

Finite Projective Plane ◽

Projective Planes ◽

The Other ◽

Blocking Set ◽

Metric Dimension ◽

Desarguesian Projective Plane ◽

Resolving Set ◽

Incidence Graph ◽

Vertex Set

In a graph $\Gamma=(V,E)$ a vertex $v$ is resolved by a vertex-set $S=\{v_1,\ldots,v_n\}$ if its (ordered) distance list with respect to $S$, $(d(v,v_1),\ldots,d(v,v_n))$, is unique. A set $A\subset V$ is resolved by $S$ if all its elements are resolved by $S$. $S$ is a resolving set in $\Gamma$ if it resolves $V$. The metric dimension of $\Gamma$ is the size of the smallest resolving set in it. In a bipartite graph a semi-resolving set is a set of vertices in one of the vertex classes that resolves the other class.We show that the metric dimension of the incidence graph of a finite projective plane of order $q\geq 23$ is $4q-4$, and describe all resolving sets of that size. Let $\tau_2$ denote the size of the smallest double blocking set in PG$(2,q)$, the Desarguesian projective plane of order $q$. We prove that for a semi-resolving set $S$ in the incidence graph of PG$(2,q)$, $|S|\geq \min \{2q+q/4-3, \tau_2-2\}$ holds. In particular, if $q\geq9$ is a square, then the smallest semi-resolving set in PG$(2,q)$ has size $2q+2\sqrt{q}$. As a corollary, we get that a blocking semioval in PG$(2, q)$, $q\geq 4$, has at least $9q/4-3$ points. A corrigendum was added to this paper on March 3, 2017.

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New upper bounds on the smallest size of a complete arc in a finite Desarguesian projective plane

Journal of Geometry ◽

10.1007/s00022-013-0154-6 ◽

2013 ◽

Vol 104 (1) ◽

pp. 11-43 ◽

Cited By ~ 10

Author(s):

Daniele Bartoli ◽

Alexander A. Davydov ◽

Giorgio Faina ◽

Stefano Marcugini ◽

Fernanda Pambianco

Keyword(s):

Projective Plane ◽

Upper Bounds ◽

Desarguesian Projective Plane ◽

Complete Arc ◽

Finite Desarguesian Projective Plane

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An Easy Proof for Some Classical Theorems in Plane Geometry

Canadian Mathematical Bulletin ◽

10.4153/cmb-1992-073-8 ◽

1992 ◽

Vol 35 (4) ◽

pp. 560-568 ◽

Cited By ~ 1

Author(s):

C. Thas

Keyword(s):

Projective Plane ◽

Projective Geometry ◽

The Other ◽

Plane Geometry ◽

The Real ◽

Easy Proof ◽

Special Cases ◽

Euclidean Planes ◽

Straight Lines ◽

Short Method

AbstractThe main result of this paper is a theorem about three conies in the complex or the real complexified projective plane. Is this theorem new? We have never seen it anywhere before. But since the golden age of projective geometry so much has been published about conies that it is unlikely that no one noticed this result. On the other hand, why does it not appear in the literature? Anyway, it seems interesting to "repeat" this property, because several theorems in connection with straight lines and (or) conies in projective, affine or euclidean planes are in fact special cases of this theorem. We give a few classical examples: the theorems of Pappus-Pascal, Desargues, Pascal (or its converse), the Brocard points, the point of Miquel. Finally, we have never seen in the literature a proof of these theorems using the same short method see the proof of the main theorem).

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Finite groups as automorphism groups of orthocomplemented projective planes

Journal of the Australian Mathematical Society ◽

10.1017/s144678870003888x ◽

1978 ◽

Vol 25 (1) ◽

pp. 19-24 ◽

Cited By ~ 3

Author(s):

Richard J. Greechie

Keyword(s):

Finite Group ◽

Projective Plane ◽

Finite Groups ◽

Automorphism Groups ◽

Projective Planes ◽

Desarguesian Projective Plane ◽

Absolute Point

AbstractA construction is given for a non-desarguesian projective plane P and an absolute-point free polarity on P such that the group of collineations of P which commute with the polarity is isomorphic to an arbitrary preassigned finite group.

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A Class of Non-Desarguesian Projective Planes

Canadian Journal of Mathematics ◽

10.4153/cjm-1957-045-0 ◽

1957 ◽

Vol 9 ◽

pp. 378-388 ◽

Cited By ~ 32

Author(s):

D. R. Hughes

Keyword(s):

Positive Integer ◽

Projective Plane ◽

Collineation Group ◽

Difference Sets ◽

Projective Planes ◽

Desarguesian Projective Plane ◽

Desarguesian Plane ◽

Desarguesian Projective Planes

In (7), Veblen and Wedclerburn gave an example of a non-Desarguesian projective plane of order 9; we shall show that this plane is self-dual and can be characterized by a collineation group of order 78, somewhat like the planes associated with difference sets. Furthermore, the technique used in (7) will be generalized and we will construct a new non-Desarguesian plane of order p2n for every positive integer n and every odd prime p.

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Projective Geometry in the One-Dimensional Affine Group

Canadian Journal of Mathematics ◽

10.4153/cjm-1964-066-9 ◽

1964 ◽

Vol 16 ◽

pp. 683-700 ◽

Cited By ~ 1

Author(s):

Hans Schwerdtfeger

Keyword(s):

Projective Space ◽

Projective Plane ◽

Simple Group ◽

Final Section ◽

Theoretical Interpretation ◽

Special Group ◽

One Dimensional ◽

The One ◽

Straight Lines ◽

Geometrical Investigation

The idea of considering the set of the elements of a group as a space, provided with a topology, measure, or metric, connected somehow with the group operation, has been used often in the work of E. Cartan and others. In the present paper we shall study a very special group whose space can be embedded naturally into a projective plane and where the straight lines have a very simple group-theoretical interpretation. It remains to be seen whether this geometrical embedding in a projective space can be extended to other classes of groups and whether the method could become an instrument of geometrical investigation, like co-ordinates or reflections. In the final section it is shown how a geometrical theorem may lead to relations within the group.

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