Parallel Postulate Research Articles

How Hyperbolic Geometry Became Respectable

(1994). How Hyperbolic Geometry Became Respectable. The American Mathematical Monthly: Vol. 101, No. 5, pp. 464-470.

La ?Filosofia delle Matematiche? di P. Galluppi

This paper provides the first presentation together with a historical and critical analysis of the Filosofia delle Matematiche of the philosopher (and mathematician?) P. Galluppi. The work, which until now escaped the attention of scholars of the Studi Galluppiani, is preserved in a manuscript in the archives of the National Library in Naples. The manuscript represents one of the first independent works explicitly and exclusively dedicated to the “philosophy of mathematics”, considered as an autonomous field of research, and is closely related to Galluppi's well known treatise Su l'analisi e su la sintesi (1807, 1935). Galluppi's philosophy of mathematics is analysed in its contextual relations with the philosophy of Kant and of Condillac. Of special interest are his relevant remarks concerning the axiomatic foundations of Euclid's Elements, where a particular emphasis is given to the postulate of parallels.

Gersonide, le pseudo-Tūsi, et le postulat des paralléles.

Euclid's Elements were translated into Hebrew from Arabic in the 13th century; but precious few of the Arabic commentaries have come down to us in a Hebrew version (al-Fārābī, Ibn al-Hayṯam). Nonetheless, a study of several texts dealing with the Fifth Postulate (the Parallel Postulate) of Book I reveals that the Hebrew authors are greatly indebted to Arabic sources.We shall examine three attempted proofs of the Parallel Postulate. The two attempts by Moses ha-Levi of Seville (13th century) and Alfonso of Valladolid (14th century) are mathematically unconvincing. Nevertheless they are interesting historically: Moses ha-Levi exploits the movement of lines which are infinite in actu; and Alfonso, starting from a critique of Ibn al-Hayṯam and al-Nayrīzī, claims to innovate in the use of the method of superposition.In contrast, Gersonides' attempt (14th century) is a well-articulated series of premises and proofs, including several arguments which we have traced back to the Taḥrīr Uṣūl Uqlīdis of Pseudo-Ṭūsī. We feel it is important to emphasize this relationship, even though it is impossible to establish the route by which these arguments found their way to Gersonides.

The remarkable Ibn al-Haytham

‘I saw that I can reach the truth only through concepts whose matter are sensible things and whose form is rational.’The achievements in experimental and theoretical science of the Arab scholar al-Haytham (also known as Alhazen, from his latinized first name al-Hasan) make him as much a figure of the sixteenth and seventeenth centuries as of his own tenth and eleventh centuries.When his writings become known in the West the importance of his contribution to optics was widely recognized and he was studied by Galileo, Kepler, Fermat, Snell and Descartes. Mathematicians remember al-Haytham chiefly for Alhazen's Problem on the reflection of light from a circular mirror, which he solved by the method of conic sections; Huygens, Gregory, l'Hospital, Barrow (and many others) later took up the problem with the new analytical methods of geometry. Al-Haytham also wrote a commentary on the postulates of Euclid, and his attempted proof of the parallel postulate has similarities to Lambert's quadrilateral and Playfair's axiom in the eighteenth century. His theory of cognition may produce yet further interest in his work.

Parallel axiom in convexity lattices

The convexity lattices, introduced by Bennett and Birkhoff, generalize the lattices of convex sets. We present three forms of Parallel Axiom in such lattices and define Euclidean and two classes of non-Euclidean lattices via the number of parallel lines through a point. The paper deals with these three classes of lattices.

Automatic Generation of Hyperbolic Tilings

than Euclid's other assumptions The author discusses the (such as the one that states fundamentals of plane hyperbolic that two points determine exgeometry and describes a method for generating hyperbolic tilings by actly one line), and for a long computer, based on the recently detime people tried to prove its veloped theory of automatic groups. redundancy within the Euclidean framework. However, by the 1820s at least three people had independently discovered that a selfconsistent geometry that does not satisfy the parallel axiom could be built: Janos Bolyai in Hungary, Carl Friedrich Gauss in Germany and Nikolai Ivanovich Lobachevskii in Russia [4]. Their creation became known as Lobachevskiian geometry until Felix Klein, at the turn of this century, introduced the now more common term hyperbolic geometry. (The term non-Euclidean geometry is also commonly used, but it encompasses geometries other than the hyperbolic.) The denial of one of Euclid's axioms remained a profoundly disturbing idea, and it was not until the second half of the century that the general mathematical public started to get a clear grasp of the nature of hyperbolic geometry. This understanding was much aided by the construction by Eugenio Beltrami [5], in 1866, of an explicit model of hyperbolic geometry-something like a map of hyperbolic space that can be drawn on a piece of Euclidean paper. This model, now known as the projective, or Klein model, after its popularizer, represents n-dimensional hyperbolic space by an open ball {(Xl.,'Xn)ERn X+ 2 <* I+<} in Euclidean space, and hyperbolic lines by Euclidean straight-line segments in the ball. Figure 1 shows that, given a line L and a point P outside of L, there are infinitely many Silvio Levy (mathematician, editor), Geometry Center, University of Minnesota, 1300 S. Second St., Minneapolis, MN 55454, U.S.A. Received 13 March 1991. LEONARDO, Vol. 25, No. 3, pp. 349-354, 1992 349 This content downloaded from 157.55.39.255 on Mon, 01 Aug 2016 05:43:04 UTC All use subject to http://about.jstor.org/terms Fig. 2. A tiling of the hyperbolic plane, shown in the Klein model. All the pentagons shown here are congruent and rightangled. The sum of the angles of a hyperbolic polygon is always less than what it would be for a Euclidean polygon of the same number of sides, and the difference is equal to the polygon's hyperbolic area. Fig. 3. The same tiling of Fig. 2, shown in the Poincare disk model. parallels to L through P, that is, lines that are coplanar with L but do not intersect it. (Sometimes only lines that meet at infinity, that is, on the circle that bounds the Klein model, are called parallel, while lines that meet outside infinity are called ultraparallel. I will not enforce this distinction.) Although the Klein model is projectively correct-that is, it represents straight lines by straight lines-it distorts shapes drastically. As one moves away from the center of the disk, an object of constant hyperbolic size quickly starts looking very small in the model and also gets flattened against the edge. Figure 2 shows a number of identical regular pentagons, which tile the hyperbolic plane much as squares tile the Euclidean plane. The tile at the center appears much bigger than its closest neighbors, and just a few layers out the triangles are already too small to make out. The figure also makes clear how the Klein model Fig. 4. Circles centered at Pare mapped to Fig. 5. The locus of themselves by rotations around P. Notice from a fixed hyperb that a hyperbolic circle in the Poincare appears as an arc of model appears round, although other model, but it does ni shapes are preserved only approximately. orthogonally, therefo poi1 olic

Theory of a Spinor Field as a Part of the Space-Time Structure

It is shown that in order to describe the motion of a particle with space-time extension in conformity with the theory of general relativity where the axiom of parallels is violated, it is necessary to introduce a new spinor field as a part of the space-time structure which is the angular part of a tetrad field preparing at every world point a tetrad to be used as the reference frame for the description of the rotational motion of such a particle. Some of its characteristic features are discussed on the one particle level on the basis of the Lagrangian formalism. Possible relation to supersymmetry is also noted, which arises through the quantization of this field.

Beltrami's model and the independence of the parallel postulate

E. Beltrami in 1868 did not intend to prove the consistency of non-euclidean plane geometry nor the independence of the euclidean parallel postulate. His approach would have been unsuccessful if so intended. J. Houel in 1870 described the relevance of Beltrami's work to the issue of the independence of the euclidean parallel postulate. Houel's method is different from the independence proofs using reinterpretation of terms deployed by Peano about 1890, chiefly in using a fixed interpretation for non-logical terms. Comparing the work of Beltrami and Houel with the treatment of non-euclidean geometry after the development of the axiomatic method in the 1890s indicates an important shift in mathematicians’ attitudes towards mathematical theories.

Gauss's geodesy and the axiom of parallels

It is a myth that Gauss measured a certain large triangle specifically to determine its angle sum; he did so in order to link his triangulation of Hanover with contiguous ones. The sum of the angles differed from 180° by less than two thirds of a second; he is known to have mentioned in conversation that this constituted an approximate verification of the axiom of parallels (which he regarded as an empirical matter because his studies of hyperbolic trigonometry had led him to recognize the possibility of logical alternatives to Kant and Euclid). However, he never doubted Euclidean geometry in his geodetic work. On the contrary, he continually used 180° angle sums as a powerful check for observational errors, which helped him to achieve standards of precision equivalent to today's. Nor did he ever plan an empirical investigation of the geometrical structure of space.

Ansätze zur begründung theoretischer terme in der mathematik—Die theorie des unendlichen bei Johann Schultz

This article discusses a publication by the Königsberg mathematician Johann Schultz (1739–1805) on a theory of the infinite. Rather than develop the infinite as a useful heuristic fiction, Schultz establishes the reality of the mathematical infinite and develops this concept to prove the parallel postulate. Despite the inconsistencies of his theory, which appeared particularly in applications to a “Geometry of the Infinite,” Schultz achieved important advances in his theory, including the distinction between the concepts of number and magnitude, the introduction of the concept of set as a generalization of the number concept for the infinite, and the separation of the infinitely-small from the infinitely-large. Since contemporary mathematicians did not recognize the inconsistencies of Schultz' theories and their faulty application in geometry, they assumed the errors to lie in his theoretical foundations, which therefore, with but few exceptions, were not developed further.

Central collineations and the parallel axiom in stable planes

Central collineations and the parallel axiom in stable planes

Minkowskian geometry, convexity conditions and the parallel axiom

The problem of characterising Minkowskian spaces is an important problem of that branch of differential geometry in which spaces more general than the complete Riemann and Finsler spaces are studied axiomatically using synthetic geometric methods. The fundamental theorem in this field is the result that a Desarguesian straightG-space in which the parallel axiom holds and the spheres are convex is Minkowskian. However the question as to whether the hypothesis of the space being Desarguesian is necessary or not has remained unsolved for over forty years. It is therefore natural to investigate conditions stronger than the mere convexity of spheres. In this paper such geometric conditions derived from functions which measure the distance between lines and points on lines are studied. Besides characterising the Minkowskian spaces these investigations also bring out the interplay between the parallel axiom and the convexity and linearity conditions.

A thirteenth century ‘proof’ of the parallel postulate

Proposition I.14 of Witelo's Perspectiva purports to provide a proof of the claim contained in Euclid's fifth postulate. The Latin text of the proposition is presented and translated into English; a commentary on the nature of the ‘proof’ is also provided.

Near affine Hjelmslev planes

A study is made of two generalizations of affine Hjelmslev planes in which the parallel axiom is not required to hold. Integer invariants are obtained for the finite planes in these new classes. Formulas are derived which enable one to compute the cardinalities of certain subsets of points and lines in terms of the invariants, and results are obtained on the nonexistence of planes with certain sets of invariants.

The Triangle Area Formula Implies the Parallel Postulate

The Triangle Area Formula Implies the Parallel Postulate

The Triangle Area Formula Implies the Parallel Postulate

The Triangle Area Formula Implies the Parallel Postulate

The Dependence of Mathematics on Reality

MOST mathematicians today regard mathematics as an abstract structure built up from axioms by the use of logical principles. By altering these axioms, different types of mathematics can be developed; for example, Euclidean and non-Euclidean geometries differ by only one axiom—the parallel postulate—but this difference is quite sufficient to lead to very different conclusions in each system. This modern concept of mathematics stresses the idea that man invents his mathematics (and hence controls it to some extent.) rather than discovers it (the assumption here being that mathematics exists apart from and independent of man). Mathematics becomes an intellectual game. You select your axioms according to taste and work out the corresponding mathematics, using logical principles. If the results are aesthetically pleasing or elegant, this new system becomes a part of mathematics. Notice here that for the modern mathematician there is no mention of applicability to reality. The system stands alone on its consistency and beauty. If, for some reason, the system can be used in explaining certain properties of the real world, this is of no import to the mathematician. G. H. Hardy, in his book A Mathematician's Apology, expresses this view quite forcefully. It has been said by Huxley that six monkeys pecking unintelligently on typewriters would, after millions of millions of years, eventually write all the books in the British Museum (Sir James Jeans quotes Huxley's statement on page 4 of The Mysterious Universe). Similarly, mathematicians randomly selecting various axioms would eventually produce significant mathematical systems, but undoubtedly only after an enormous amount of time had been wasted.

A Geometric Approach to the Heine-Borel Theorem

In a topological plane with strong enough topological properties one can use [6] open triangular regions to define a base for the topology. Similarly, one can use these regions to define boundedness of a set. In this setting we show that in the absolute plane geometry, the holding of the Heine-Borel theorem is equivalent to every four points being contained in some such region and that this second condition is equivalent to the parallel postulate. Thus we give two new conditions equivalent to the parallel postulate.

Historically Speaking— The Parallel Postulate

Euclid's famous parallel postulate was responsible for an enormons amount of mathematical activity over a period of more than twenty centuries. The failure of mathematicians to prove Euclid's statement, from his other postulates contributed to Euclid's fame and eventually led to the invention of non-Euclidean geometries.

Independence of the Incidence Postulates

CONSISTENCY is an essential property of any mathematical system. Without it two conflicting statements may be deduced within the system. Independence, on the other hand, is not considered essential, and, indeed, it is sometimes not desirable. If the set of postulates in a high school geometry text were independent, then the first few basic theorems would be so difficult that virtually no high school student could understand their proofs. At the foundations level, however, it is considered aesthetically desirable that the postulates of a system be independent. It is generally not an easy matter to prove that a set of postulates is independent.1 It is believed that when Euclid stated his five postulates for geometry he thought that his parallel postulate was demonstrable from the other postulates, but he was unable to give a valid proof of it. Mathematicians sought a proof for two thousand years, but it has been only within the last century or so that the search has been shown to be futile. It took two thousand years and the development of non-Euclidean geometries to show that the parallel postulate is independent.