相关论文: On the equivalence of Playfair's axiom to the para…
As a much later addition to the original Euclidean geometry, the parallel postulate distinguishes non-Euclidean geometries from Euclidean geometry. This paper will show that the parallel postulate is unnecessary because the 4th Euclidean…
Euclidean geometry consists of straightedge-and-compass constructions and reasoning about the results of those constructions. We show that Euclidean geometry can be developed using only intuitionistic logic. We consider three versions of…
We use Herbrand's theorem to give a new proof that Euclid's parallel axiom is not derivable from the other axioms of first-order Euclidean geometry. Previous proofs involve constructing models of non-Euclidean geometry. This proof uses a…
We present a new model of a non-Euclidean plane, in which angles in a triangle sum up to $\pi$. It is a subspace of the Cartesian plane over the field of hyperreal numbers $\mathbb{R}^*$. The model enables one to represent the negation of…
We trace the development of arguments for the consistency of non-Euclidean geometries and for the independence of the parallel postulate, showing how the arguments become more rigorous as a formal conception of geometry is introduced. We…
We prove that the Euclidean plane is the only Riemannian plane with total curvature and free of conjugate points that satisfies Playfair's version of the parallel postulate.
The article deals with the connection between the second postulate of Euclid and non-Euclidean geometry. It is shown that the violation of the second postulate of Euclid inevitably leads to hyperbolic geometry. This eliminates…
In this paper, we reconstruct Euclid's theory of similar triangles, as developed in Book VI of the \textit{Elements}, along with its 20th-century counterparts, formulated within the systems of Hilbert, Birkhoff, Borsuk and Szmielew, Millman…
We have found a (classical) competition between duality and gauge symmetries when trying to obtain an explicit dual to the non-nonabelian version of the self-dual model proposed by Townsend, Pilch and van Nieuwenhuizen\cite{TPvN} (NASD)…
Exploring selected reductio ad absurdum proofs in Book 1 of the Elements, we show they include figures that are not constructed. It is squarely at odds with Hartshorne's claim that "in Euclid's geometry, only those geometrical figures exist…
Some translations into non-euclidean geometry of classical theorems of planar projective geometry are explored. The existence of some common triangle centers is dedeuced from theorems of Pascal and Chasles. Desargues' Theorem allows to…
The initial techniques developed in Euclid's Elements, well before the use of the parallel postulate, are reexamined in order to clarify even the most obscure details, particularly those related to equality, superposition and angle…
Interesting connection has been established between two apparently unrelated concepts, namely, quantum nonlocality and Bayesian game theory. It has been shown that nonlocal correlations in the form of advice can outperform classical…
Euclid's reasoning is essentially constructive. Tarski's elegant and concise first-order theory of Euclidean geometry, on the other hand, is essentially non-constructive, even if we restrict attention (as we do here) to the theory with…
We describe a strictly classical dice game, which emulates the main features of the EPR experiment, including violation of Bell's inequalities. Therefore, the standard interpretation that Bell's theorem provides necessary conditions for…
We prove the existence of pairs of models of the same cardinality lambda which are very equivalent according to EF games, but not isomorphic. We continue the paper math.LO/0404222, but we don't rely on it.
The deformation principle admits one to obtain a very broad class of nonuniform geometries as a result of deformation of the proper Euclidean geometry. The Riemannian geometry is also obtained by means of a deformation of the Euclidean…
The study of comparison theorems in geometry has a rich history. In this paper, we establish a comparison theorem for polyhedra in 3-manifolds with nonnegative scalar curvature, answering affirmatively a dihedral rigidity conjecture by…
We ascribe to the Euclidean Fifth Postulate a genuine constructive role, which makes it absolutely necessary in the parallel construction. For that, we present a reconstruction of the general principles underlying the Euclidean construction…
We study Saccheri`s three hypotheses on a two right-angled isosceles quadrilateral, with a rectilinear summit side. We claim that in the Hilbert`s foundation of geometry the euclidean parallelism is a theorem, and in the h-plane the…