Synthetic geometry

Last updated

Synthetic geometry (sometimes referred to as axiomatic geometry or even pure geometry) is geometry without the use of coordinates. It relies on the axiomatic method for proving all results from a few basic properties initially called postulate, and at present called axioms.

Contents

The term "synthetic geometry" was coined only after the 17th century, and the introduction by René Descartes of the coordinate method, which was called analytic geometry. So the term "synthetic geometry" was introduced to refer to the older methods that were, before Descartes, the only known ones.

According to Felix Klein

Synthetic geometry is that which studies figures as such, without recourse to formulae, whereas analytic geometry consistently makes use of such formulae as can be written down after the adoption of an appropriate system of coordinates. [1]

The first systematic approach for synthetic geometry is Euclid's Elements. However, it appeared at the end of the 19th century that Euclid's postulates were not sufficient for characterizing geometry. The first complete axiom system for geometry was given only at the end of the 19th century by David Hilbert. At the same time, it appeared that both synthetic methods and analytic methods can be used to build geometry. The fact that the two approches are equivalent has been proved by Emil Artin in his book Geometric Algebra .

Because of this equivalence, the distinction between synthetic and analytic geometry is no more in use, except at elementary level, or for geometries that are not related to any sort of numbers, such as some finite geometries and non-Desarguesian geometry.[ citation needed ]

Logical synthesis

The process of logical synthesis begins with some arbitrary but definite starting point. This starting point is the introduction of primitive notions or primitives and axioms about these primitives:

From a given set of axioms, synthesis proceeds as a carefully constructed logical argument. When a significant result is proved rigorously, it becomes a theorem .

Properties of axiom sets

There is no fixed axiom set for geometry, as more than one consistent set can be chosen. Each such set may lead to a different geometry, while there are also examples of different sets giving the same geometry. With this plethora of possibilities, it is no longer appropriate to speak of "geometry" in the singular.

Historically, Euclid's parallel postulate has turned out to be independent of the other axioms. Simply discarding it gives absolute geometry, while negating it yields hyperbolic geometry. Other consistent axiom sets can yield other geometries, such as projective, elliptic, spherical or affine geometry.

Axioms of continuity and "betweenness" are also optional, for example, discrete geometries may be created by discarding or modifying them.

Following the Erlangen program of Klein, the nature of any given geometry can be seen as the connection between symmetry and the content of the propositions, rather than the style of development.

History

Euclid's original treatment remained unchallenged for over two thousand years, until the simultaneous discoveries of the non-Euclidean geometries by Gauss, Bolyai, Lobachevsky and Riemann in the 19th century led mathematicians to question Euclid's underlying assumptions. [3]

One of the early French analysts summarized synthetic geometry this way:

The Elements of Euclid are treated by the synthetic method. This author, after having posed the axioms, and formed the requisites, established the propositions which he proves successively being supported by that which preceded, proceeding always from the simple to compound, which is the essential character of synthesis. [4]

The heyday of synthetic geometry can be considered to have been the 19th century, when analytic methods based on coordinates and calculus were ignored by some geometers such as Jakob Steiner, in favor of a purely synthetic development of projective geometry. For example, the treatment of the projective plane starting from axioms of incidence is actually a broader theory (with more models) than is found by starting with a vector space of dimension three. Projective geometry has in fact the simplest and most elegant synthetic expression of any geometry. [5]

In his Erlangen program, Felix Klein played down the tension between synthetic and analytic methods:

On the Antithesis between the Synthetic and the Analytic Method in Modern Geometry:
The distinction between modern synthesis and modern analytic geometry must no longer be regarded as essential, inasmuch as both subject-matter and methods of reasoning have gradually taken a similar form in both. We choose therefore in the text as common designation of them both the term projective geometry. Although the synthetic method has more to do with space-perception and thereby imparts a rare charm to its first simple developments, the realm of space-perception is nevertheless not closed to the analytic method, and the formulae of analytic geometry can be looked upon as a precise and perspicuous statement of geometrical relations. On the other hand, the advantage to original research of a well formulated analysis should not be underestimated, - an advantage due to its moving, so to speak, in advance of the thought. But it should always be insisted that a mathematical subject is not to be considered exhausted until it has become intuitively evident, and the progress made by the aid of analysis is only a first, though a very important, step. [6]

The close axiomatic study of Euclidean geometry led to the construction of the Lambert quadrilateral and the Saccheri quadrilateral. These structures introduced the field of non-Euclidean geometry where Euclid's parallel axiom is denied. Gauss, Bolyai and Lobachevski independently constructed hyperbolic geometry, where parallel lines have an angle of parallelism that depends on their separation. This study became widely accessible through the Poincaré disc model where motions are given by Möbius transformations. Similarly, Riemann, a student of Gauss's, constructed Riemannian geometry, of which elliptic geometry is a particular case.

Another example concerns inversive geometry as advanced by Ludwig Immanuel Magnus, which can be considered synthetic in spirit. The closely related operation of reciprocation expresses analysis of the plane.

Karl von Staudt showed that algebraic axioms, such as commutativity and associativity of addition and multiplication, were in fact consequences of incidence of lines in geometric configurations. David Hilbert showed [7] that the Desargues configuration played a special role. Further work was done by Ruth Moufang and her students. The concepts have been one of the motivators of incidence geometry.

When parallel lines are taken as primary, synthesis produces affine geometry. Though Euclidean geometry is both an affine and metric geometry, in general affine spaces may be missing a metric. The extra flexibility thus afforded makes affine geometry appropriate for the study of spacetime, as discussed in the history of affine geometry.

In 1955 Herbert Busemann and Paul J. Kelley sounded a nostalgic note for synthetic geometry:

Although reluctantly, geometers must admit that the beauty of synthetic geometry has lost its appeal for the new generation. The reasons are clear: not so long ago synthetic geometry was the only field in which the reasoning proceeded strictly from axioms, whereas this appeal — so fundamental to many mathematically interested people — is now made by many other fields. [5]

For example, college studies now include linear algebra, topology, and graph theory where the subject is developed from first principles, and propositions are deduced by elementary proofs. Expecting to replace synthetic with analytic geometry leads to loss of geometric content. [8]

Today's student of geometry has axioms other than Euclid's available: see Hilbert's axioms and Tarski's axioms.

Ernst Kötter published a (German) report in 1901 on "The development of synthetic geometry from Monge to Staudt (1847)"; [9]

Proofs using synthetic geometry

Synthetic proofs of geometric theorems make use of auxiliary constructs (such as helping lines) and concepts such as equality of sides or angles and similarity and congruence of triangles. Examples of such proofs can be found in the articles Butterfly theorem, Angle bisector theorem, Apollonius' theorem, British flag theorem, Ceva's theorem, Equal incircles theorem, Geometric mean theorem, Heron's formula, Isosceles triangle theorem, Law of cosines, and others that are linked to here.

Computational synthetic geometry

In conjunction with computational geometry, a computational synthetic geometry has been founded, having close connection, for example, with matroid theory. Synthetic differential geometry is an application of topos theory to the foundations of differentiable manifold theory.

See also

Notes

  1. Klein 1948 , p. 55
  2. Greenberg 1974 , p. 59
  3. Mlodinow 2001, Part III The Story of Gauss
  4. S. F. Lacroix (1816) Essais sur L'Enseignement en Général, et sur celui des Mathématiques en Particulier, page 207, Libraire pur les Mathématiques.
  5. 1 2 Herbert Busemann and Paul J. Kelly (1953) Projective Geometry and Projective Metrics, Preface, page v, Academic Press
  6. Felix Klein (1872) Ralf Stephan translator (2006) "A comparative review of researches in geometry"
  7. David Hilbert, 1980 (1899). The Foundations of Geometry , 2nd edition, §22 Desargues Theorem, Chicago: Open Court
  8. Pambuccian, Victor; Schacht, Celia (2021), "The Case for the Irreducibility of Geometry to Algebra", Philosophia Mathematica, 29 (4), doi:10.1093/philmat/nkab022
  9. Ernst Kötter (1901). Die Entwickelung der Synthetischen Geometrie von Monge bis auf Staudt (1847). (2012 Reprint as ISBN   1275932649)

Related Research Articles

<span class="mw-page-title-main">Euclidean geometry</span> Mathematical model of the physical space

Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry, Elements. Euclid's approach consists in assuming a small set of intuitively appealing axioms (postulates) and deducing many other propositions (theorems) from these. Although many of Euclid's results had been stated earlier, Euclid was the first to organize these propositions into a logical system in which each result is proved from axioms and previously proved theorems.

<span class="mw-page-title-main">Euclidean space</span> Fundamental space of geometry

Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's Elements, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension n, which are called Euclidean n-spaces when one wants to specify their dimension. For n equal to one or two, they are commonly called respectively Euclidean lines and Euclidean planes. The qualifier "Euclidean" is used to distinguish Euclidean spaces from other spaces that were later considered in physics and modern mathematics.

<span class="mw-page-title-main">History of geometry</span> Historical development of geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers (arithmetic).

In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises by either replacing the parallel postulate with an alternative, or relaxing the metric requirement. In the former case, one obtains hyperbolic geometry and elliptic geometry, the traditional non-Euclidean geometries. When the metric requirement is relaxed, then there are affine planes associated with the planar algebras, which give rise to kinematic geometries that have also been called non-Euclidean geometry.

Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the nature of mathematics. In this latter sense, the distinction between foundations of mathematics and philosophy of mathematics turns out to be vague. Foundations of mathematics can be conceived as the study of the basic mathematical concepts and how they form hierarchies of more complex structures and concepts, especially the fundamentally important structures that form the language of mathematics also called metamathematical concepts, with an eye to the philosophical aspects and the unity of mathematics. The search for foundations of mathematics is a central question of the philosophy of mathematics; the abstract nature of mathematical objects presents special philosophical challenges.

<span class="mw-page-title-main">Projective space</span> Completion of the usual space with "points at infinity"

In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet at infinity. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally, an affine space with points at infinity, in such a way that there is one point at infinity of each direction of parallel lines.

In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts. The basic intuitions are that projective space has more points than Euclidean space, for a given dimension, and that geometric transformations are permitted that transform the extra points to Euclidean points, and vice versa.

<span class="mw-page-title-main">Affine geometry</span> Euclidean geometry without distance and angles

In mathematics, affine geometry is what remains of Euclidean geometry when ignoring the metric notions of distance and angle.

<span class="mw-page-title-main">Finite geometry</span> Geometric system with a finite number of points

A finite geometry is any geometric system that has only a finite number of points. The familiar Euclidean geometry is not finite, because a Euclidean line contains infinitely many points. A geometry based on the graphics displayed on a computer screen, where the pixels are considered to be the points, would be a finite geometry. While there are many systems that could be called finite geometries, attention is mostly paid to the finite projective and affine spaces because of their regularity and simplicity. Other significant types of finite geometry are finite Möbius or inversive planes and Laguerre planes, which are examples of a general type called Benz planes, and their higher-dimensional analogs such as higher finite inversive geometries.

Absolute geometry is a geometry based on an axiom system for Euclidean geometry without the parallel postulate or any of its alternatives. Traditionally, this has meant using only the first four of Euclid's postulates. The term was introduced by János Bolyai in 1832. It is sometimes referred to as neutral geometry, as it is neutral with respect to the parallel postulate. The first four of Euclid's postulates are now considered insufficient as a basis of Euclidean geometry, so other systems are used instead.

<span class="mw-page-title-main">Line (geometry)</span> Straight figure with zero width and depth

In geometry, a straight line, usually abbreviated line, is an infinitely long object with no width, depth, or curvature, an idealization of such physical objects as a straightedge, a taut string, or a ray of light. Lines are spaces of dimension one, which may be embedded in spaces of dimension two, three, or higher. The word line may also refer, in everyday life, to a line segment, which is a part of a line delimited by two points.

<span class="mw-page-title-main">Karl Georg Christian von Staudt</span> German mathematician

Karl Georg Christian von Staudt was a German mathematician who used synthetic geometry to provide a foundation for arithmetic.

<span class="mw-page-title-main">Space (mathematics)</span> Mathematical set with some added structure

In mathematics, a space is a set with some added structure. While modern mathematics uses many types of spaces, such as Euclidean spaces, linear spaces, topological spaces, Hilbert spaces, or probability spaces, it does not define the notion of "space" itself.

In geometry, Pasch's axiom is a statement in plane geometry, used implicitly by Euclid, which cannot be derived from the postulates as Euclid gave them. Its essential role was discovered by Moritz Pasch in 1882.

Ordered geometry is a form of geometry featuring the concept of intermediacy but, like projective geometry, omitting the basic notion of measurement. Ordered geometry is a fundamental geometry forming a common framework for affine, Euclidean, absolute, and hyperbolic geometry.

Geometry is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a geometer. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts.

Foundations of geometry is the study of geometries as axiomatic systems. There are several sets of axioms which give rise to Euclidean geometry or to non-Euclidean geometries. These are fundamental to the study and of historical importance, but there are a great many modern geometries that are not Euclidean which can be studied from this viewpoint. The term axiomatic geometry can be applied to any geometry that is developed from an axiom system, but is often used to mean Euclidean geometry studied from this point of view. The completeness and independence of general axiomatic systems are important mathematical considerations, but there are also issues to do with the teaching of geometry which come into play.

Geometric Algebra is a book written by Emil Artin and published by Interscience Publishers, New York, in 1957. It was republished in 1988 in the Wiley Classics series (ISBN 0-471-60839-4).

Mathematics is a broad subject that is commonly divided in many areas that may be defined by their objects of study, by the used methods, or by both. For example, analytic number theory is a subarea of number theory devoted to the use of methods of analysis for the study of natural numbers.

References