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Tarski Geometry Axioms – Part II

Roland Coghetto, Adam Grabowski (2016)

Formalized Mathematics

In our earlier article [12], the first part of axioms of geometry proposed by Alfred Tarski [14] was formally introduced by means of Mizar proof assistant [9]. We defined a structure TarskiPlane with the following predicates: of betweenness between (a ternary relation), of congruence of segments equiv (quarternary relation), which satisfy the following properties: congruence symmetry (A1), congruence equivalence relation (A2), congruence identity (A3), segment construction (A4), SAS (A5), betweenness...

Teilmengengraphen und projektive Räume.

Günter Pickert (1985)

Aequationes mathematicae

Ternärringe ohne Planaritätsbedingung

Václav Havel, Ivan Studnička (1979)

Časopis pro pěstování matematiky

Ternary rings with a left quasi-zero belonging to translation planes

Dalibor Klucký, Libuše Marková (1980)

Sborník prací Přírodovědecké fakulty University Palackého v Olomouci. Matematika

The Betten-Walker spread and Cayley's ruled cubic surface.

Havlicek, Hans, Riesinger, Rolf (2006)

Beiträge zur Algebra und Geometrie

The classification of SPG-systems of index 2.

De Winter, St. (2003)

Advances in Geometry

The classification of spreads in $P G (3, q)$ admitting linear groups of order $q (q + 1)$ . II: Even order.

Jha, V., Johnson, N. L. (2003)

Advances in Geometry

The classification of subplane covered nets.

Johnson, Norman L. (1995)

Bulletin of the Belgian Mathematical Society - Simon Stevin

The Conic y = x2 in Moufang Planes

R. ARTZY (1971)

Aequationes mathematicae

The contributions of Hilbert and Dehn to non-archimedean geometries and their impact on the italian school

Cinzia Cerroni (2007)

Revue d'histoire des mathématiques

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 degrees of certain strata of the dual variety

Israel Vainsencher (1979)

Compositio Mathematica

The Desargues configuration and inversive geometry

H. Struve, R. Struve (1986)

Beiträge zur Algebra und Geometrie = Contributions to algebra and geometry

The elementary geometry of a triangular world with hexagonal circles.

Pambuccian, Victor (2008)

Beiträge zur Algebra und Geometrie

The envelope of the Wallace-Simpson lines of a triangle. A simple proof of the Steiner problem on the deltoid.

Miguel de Guzmán (2001)

RACSAM

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 generating rank of the symplectic line-Grassmannian.

Blok, Rieuwert J. (2003)

Beiträge zur Algebra und Geometrie

The geometry of $G L (2, q)$ in translation planes of even order $q^{2}$ .

Johnson, N.L. (1978)

International Journal of Mathematics and Mathematical Sciences

The geometry of GL(2,q) in translation planes of even order $q^{2}$ (Corrigendum).

Johnson, N.L. (1981)

International Journal of Mathematics and Mathematical Sciences

The geometry of null systems, Jordan algebras and von Staudt's theorem

Wolfgang Bertram (2003)

Annales de l’institut Fourier

We characterize an important class of generalized projective geometries $(X, X^{'})$ by the following essentially equivalent properties: (1) $(X, X^{'})$ admits a central null-system; (2) $(X, X^{'})$ admits inner polarities: (3) $(X, X^{'})$ is associated to a unital Jordan algebra. These geometries, called of the first kind, play in the category of generalized projective geometries a rôle comparable to the one of the projective line in the category of ordinary projective geometries. In this general set-up, we prove an analogue of von Staudt’s...

The group of projectivities in free-like geometries

A. Barlotti, E. Schreiber, K. Strambach (1978)

Rendiconti del Seminario Matematico della Università di Padova

The hyperplanes of $DW (5, 2)$ .

Pralle, Harm (2005)

Experimental Mathematics

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