Synthetic geometry

# Synthetic geometry

Discussion

Encyclopedia
Synthetic or axiomatic geometry is the branch of geometry
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 ....

which makes use of axiom
Axiom
In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...

s, theorem
Theorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...

s and logical arguments to draw conclusions, as opposed to analytic
Analytic geometry
Analytic geometry, or analytical geometry has two different meanings in mathematics. The modern and advanced meaning refers to the geometry of analytic varieties...

and algebraic
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...

geometries which use analysis
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...

and algebra
Algebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...

to perform geometric computations and solve problems.

## Logical synthesis

The process of logical synthesis begins with some arbitrary but defined starting point.
• Primitives
Primitive notion
In mathematics, logic, and formal systems, a primitive notion is an undefined concept. In particular, a primitive notion is not defined in terms of previously defined concepts, but is only motivated informally, usually by an appeal to intuition and everyday experience. In an axiomatic theory or...

are the most basic ideas. Typically they include objects and relationships. In geometry, the objects are things like points, lines and planes while the fundamental relationship is that of incidence – of one object meeting or joining with another.
• Axiom
Axiom
In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...

s
are statements about these primitives, for example that any two points are together incident with just one line (i.e. that for any two points, there is just one line which passes through both of them).

From a given set of axioms, synthesis proceeds as a carefully constructed logical argument. Where a significant result is proved rigorously, it becomes a theorem
Theorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...

.

Any given set of axioms leads to a different logical system. In the case of geometry, each distinct set of axioms leads to a different geometry.

### Properties of axiom sets

If the axiom set is not categorical
Morley's categoricity theorem
In model theory, a branch of mathematical logic, a theory is κ-categorical if it has exactly one model of cardinality κ up to isomorphism....

(so that there is more than one model) one has the geometry/geometries debate to settle. That's not a serious issue for a modern axiomatic mathematician, since the implication of axiom
Axiom
In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...

is now a starting point for theory rather than a self-evident plank in a platform based on intuition. And since the Erlangen program
Erlangen program
An influential research program and manifesto was published in 1872 by Felix Klein, under the title Vergleichende Betrachtungen über neuere geometrische Forschungen...

of Klein the nature of any given geometry has been seen as the connection of symmetry
Symmetry
Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...

and the content of propositions, rather than the style of development.

## History

The geometry of Euclid
Euclid
Euclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...

was synthetic, though not all of his books covered topics of pure geometry.

The close axiomatic study of Euclidean geometry
Euclidean geometry
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...

in the 19th Century led to the discovery of non-Euclidean geometries
Non-Euclidean geometry
Non-Euclidean geometry is the term used to refer to two specific geometries which are, loosely speaking, obtained by negating the Euclidean parallel postulate, namely hyperbolic and elliptic geometry. This is one term which, for historical reasons, has a meaning in mathematics which is much...

having different axioms. Gauss
Gauss
Gauss may refer to:*Carl Friedrich Gauss, German mathematician and physicist*Gauss , a unit of magnetic flux density or magnetic induction*GAUSS , a software package*Gauss , a crater on the moon...

, Bolyai and Lobachevski independently discovered hyperbolic geometry
Hyperbolic geometry
In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...

, in which the Euclidean axiom of parallelism is replaced by an alternative. Poincaré
Poincaré
Several members of the French Poincaré family have been successful in public and scientific life:* Henri Poincaré , physicist, mathematician and philosopher of science* Lucien Poincaré , physicist, brother of Raymond and cousin of Henri...

soon discovered the first physical geometric model of hyperbolic geometry, in a form known as the Poincaré disc. Eventually, hyperbolic geometry would become accessible to analysis using Mobius transformations.

The heyday of synthetic geometry can be considered to have been the 19th century, when analytic methods based on coordinates
Coordinate system
In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric element. The order of the coordinates is significant and they are sometimes identified by their position in an ordered tuple and sometimes by...

and calculus
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...

were ignored by some geometers such as Jakob Steiner
Jakob Steiner
Jakob Steiner was a Swiss mathematician who worked primarily in geometry.-Personal and professional life:...

, in favour of a purely synthetic development of projective geometry
Projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant under projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts...

. For example, the treatment of the projective plane
Projective plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines that do not intersect...

starting from axioms of incidence is actually a broader theory (with more models) than is found by starting with a vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

of dimension three. Projective geometry has in fact the simplest and most elegant synthetic expression of any geometry.

Another example concerns inversive geometry as advanced by Ludwig Immanuel Magnus
Ludwig Immanuel Magnus
Ludwig Immanuel Magnus was a German Jewish mathematician who, in 1831, published a paper about the inversion transformation, which leads to inversive geometry....

, which can be considered synthetic in spirit. The closely related operation of reciprocation
Multiplicative inverse
In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x−1, is a number which when multiplied by x yields the multiplicative identity, 1. The multiplicative inverse of a fraction a/b is b/a. For the multiplicative inverse of a real number, divide 1 by the...

expresses analysis of the plane.

On the other hand, the theory of special relativity
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...

was originally developed analytically via the linear algebra
Linear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...

of the Lorentz transformation
Lorentz transformation
In physics, the Lorentz transformation or Lorentz-Fitzgerald transformation describes how, according to the theory of special relativity, two observers' varying measurements of space and time can be converted into each other's frames of reference. It is named after the Dutch physicist Hendrik...

; then in 1912 Lewis and Wilson developed a synthetic approach (see reference), bringing greater confidence in the foundations of spacetime theory.

## Computational synthetic geometry

In conjunction with computational geometry
Computational geometry
Computational geometry is a branch of computer science devoted to the study of algorithms which can be stated in terms of geometry. Some purely geometrical problems arise out of the study of computational geometric algorithms, and such problems are also considered to be part of computational...

, a computational synthetic geometry has been founded, having close connection, for example, with matroid
Matroid
In combinatorics, a branch of mathematics, a matroid or independence structure is a structure that captures the essence of a notion of "independence" that generalizes linear independence in vector spaces....

theory. Synthetic differential geometry
Synthetic differential geometry
In mathematics, synthetic differential geometry is a reformulation of differential geometry in the language of topos theory, in the context of an intuitionistic logic characterized by a rejection of the law of excluded middle. There are several insights that allow for such a reformulation...

is an application of topos
Topos
In mathematics, a topos is a type of category that behaves like the category of sheaves of sets on a topological space...

theory to the foundations of differentiable manifold
Differentiable manifold
A differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since...

theory.