Important Concepts in Geometry: Axioms

Important concepts in geometry
The following are some of the most important concepts in geometry. [7][38][39]
Axioms
An illustration of Euclid's parallel postulate

See also: Euclidean geometry and Axiom
Euclid took an abstract approach to geometry in his Elements,[40] one of the most influential books ever
written.[41] Euclid introduced certain axioms, or postulates, expressing primary or self-evident properties of
points, lines, and planes.[42] He proceeded to rigorously deduce other properties by mathematical
reasoning. The characteristic feature of Euclid's approach to geometry was its rigor, and it has come to be
known as axiomatic or synthetic geometry.[43] At the start of the 19th century, the discovery of non-
Euclidean geometries by Nikolai Ivanovich Lobachevsky (1792–1856), János Bolyai (1802–1860), Carl
Friedrich Gauss (1777–1855) and others[44] led to a revival of interest in this discipline, and in the 20th
century, David Hilbert (1862–1943) employed axiomatic reasoning in an attempt to provide a modern
foundation of geometry.[45]
Points
Main article: Point (geometry)
Points are considered fundamental objects in Euclidean geometry. They have been defined in a variety of
ways, including Euclid's definition as 'that which has no part' [46] and through the use of algebra or nested
sets.[47] In many areas of geometry, such as analytic geometry, differential geometry, and topology, all
objects are considered to be built up from points. However, there has been some study of geometry without
reference to points.[48]
Lines
Main article: Line (geometry)
Euclid described a line as "breadthless length" which "lies equally with respect to the points on itself". [46] In
modern mathematics, given the multitude of geometries, the concept of a line is closely tied to the way the
geometry is described. For instance, in analytic geometry, a line in the plane is often defined as the set of
points whose coordinates satisfy a given linear equation,[49] but in a more abstract setting, such
as incidence geometry, a line may be an independent object, distinct from the set of points which lie on it.
[50] In differential geometry, a geodesic is a generalization of the notion of a line to curved spaces.[51]
Planes
Main article: Plane (geometry)
A plane is a flat, two-dimensional surface that extends infinitely far. [46] Planes are used in every area of
geometry. For instance, planes can be studied as a topological surface without reference to distances or
angles;[52] it can be studied as an affine space, where collinearity and ratios can be studied but not
distances;[53] it can be studied as the complex plane using techniques of complex analysis;[54] and so on.
Angles
Main article: Angle
Euclid defines a plane angle as the inclination to each other, in a plane, of two lines which meet each other,
and do not lie straight with respect to each other. [46] In modern terms, an angle is the figure formed by
two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle.[55]
Acute (a), obtuse (b), and straight (c) angles. The acute and obtuse angles are also known as oblique angles.
In Euclidean geometry, angles are used to study polygons and triangles, as well as forming an object of

study in their own right.[46] The study of the angles of a triangle or of angles in a unit circle forms the basis
of trigonometry.[56]
In differential geometry and calculus, the angles between plane curves or space curves or surfaces can be
calculated using the derivative.[57][58]
Curves
Main article: Curve (geometry)
A curve is a 1-dimensional object that may be straight (like a line) or not; curves in 2-dimensional space are
called plane curves and those in 3-dimensional space are called space curves.[59]
In topology, a curve is defined by a function from an interval of the real numbers to another space. [52] In
differential geometry, the same definition is used, but the defining function is required to be
differentiable [60] Algebraic geometry studies algebraic curves, which are defined as algebraic
varieties of dimension one.[61]
Surfaces
Main article: Surface (mathematics)
A sphere is a surface that can be defined parametrically (by x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ) or implicitly

(by x2 + y2 + z2 − r2 = 0.)
A surface is a two-dimensional object, such as a sphere or paraboloid.[62] In differential
geometry[60] and topology,[52] surfaces are described by two-dimensional 'patches' (or neighborhoods) that
are assembled by diffeomorphisms or homeomorphisms, respectively. In algebraic geometry, surfaces are
described by polynomial equations.[61]
Manifolds
Main article: Manifold
A manifold is a generalization of the concepts of curve and surface. In topology, a manifold is a topological
space where every point has a neighborhood that is homeomorphic to Euclidean space.[52] In differential
geometry, a differentiable manifold is a space where each neighborhood is diffeomorphic to Euclidean
space.[60]
Manifolds are used extensively in physics, including in general relativity and string theory.[63]
Length, area, and volume

Main articles: Length, Area, and Volume
See also: Area § List of formulas, and Volume § Volume formulas
Length, area, and volume describe the size or extent of an object in one dimension, two dimension, and
three dimensions respectively.[64]
In Euclidean geometry and analytic geometry, the length of a line segment can often be calculated by
the Pythagorean theorem.[65]
Area and volume can be defined as fundamental quantities separate from length, or they can be described
and calculated in terms of lengths in a plane or 3-dimensional space. [64] Mathematicians have found many
explicit formulas for area and formulas for volume of various geometric objects. In calculus, area and
volume can be defined in terms of integrals, such as the Riemann integral[66] or the Lebesgue integral.[67]
Metrics and measures
Main articles: Metric (mathematics) and Measure (mathematics)
Visual checking of the Pythagorean theorem for the (3, 4, 5) triangle as in the Zhoubi Suanjing 500–200 BC. The
Pythagorean theorem is a consequence of the Euclidean metric.
The concept of length or distance can be generalized, leading to the idea of metrics.[68] For instance,
the Euclidean metric measures the distance between points in the Euclidean plane, while the hyperbolic
metric measures the distance in the hyperbolic plane. Other important examples of metrics include
the Lorentz metric of special relativity and the semi-Riemannian metrics of general relativity.[69]
In a different direction, the concepts of length, area and volume are extended by measure theory, which
studies methods of assigning a size or measure to sets, where the measures follow rules similar to those of
classical area and volume.[70]
Congruence and similarity

Main articles: Congruence (geometry) and Similarity (geometry)
Congruence and similarity are concepts that describe when two shapes have similar characteristics. [71] In
Euclidean geometry, similarity is used to describe objects that have the same shape, while congruence is
used to describe objects that are the same in both size and shape. [72] Hilbert, in his work on creating a
more rigorous foundation for geometry, treated congruence as an undefined term whose properties are
defined by axioms.
Congruence and similarity are generalized in transformation geometry, which studies the properties of
geometric objects that are preserved by different kinds of transformations. [73]
Compass and straightedge constructions

Main article: Compass and straightedge constructions
Classical geometers paid special attention to constructing geometric objects that had been described in
some other way. Classically, the only instruments allowed in geometric constructions are
the compass and straightedge. Also, every construction had to be complete in a finite number of steps.
However, some problems turned out to be difficult or impossible to solve by these means alone, and
ingenious constructions using parabolas and other curves, as well as mechanical devices, were found.
Dimension
Main article: Dimension
The Koch snowflake, with fractal dimension=log4/log3 and topological dimension=1
Where the traditional geometry allowed dimensions 1 (a line), 2 (a plane) and 3 (our ambient world
conceived of as three-dimensional space), mathematicians and physicists have used higher dimensions for
nearly two centuries.[74] One example of a mathematical use for higher dimensions is the configuration
space of a physical system, which has a dimension equal to the system's degrees of freedom. For
instance, the configuration of a screw can be described by five coordinates. [75]
In general topology, the concept of dimension has been extended from natural numbers, to infinite
dimension (Hilbert spaces, for example) and positive real numbers (in fractal geometry).[76] In algebraic
geometry, the dimension of an algebraic variety has received a number of apparently different definitions,
which are all equivalent in the most common cases.[77]
Symmetry
Main article: Symmetry
A tiling of the hyperbolic plane
The theme of symmetry in geometry is nearly as old as the science of geometry itself.[78] Symmetric
shapes such as the circle, regular polygons and platonic solids held deep significance for many ancient
philosophers[79] and were investigated in detail before the time of Euclid.[42] Symmetric patterns occur in
nature and were artistically rendered in a multitude of forms, including the graphics of da Vinci, M.C.
Escher, and others.[80] In the second half of the 19th century, the relationship between symmetry and
geometry came under intense scrutiny. Felix Klein's Erlangen program proclaimed that, in a very precise
sense, symmetry, expressed via the notion of a transformation group, determines what geometry is.
[81] Symmetry in classical Euclidean geometry is represented by congruences and rigid motions, whereas
in projective geometry an analogous role is played by collineations, geometric transformations that take
straight lines into straight lines.[82] However it was in the new geometries of Bolyai and Lobachevsky,
Riemann, Clifford and Klein, and Sophus Lie that Klein's idea to 'define a geometry via its symmetry group'
found its inspiration.[83] Both discrete and continuous symmetries play prominent roles in geometry, the
former in topology and geometric group theory,[84][85] the latter in Lie theory and Riemannian geometry.
[86][87]
A different type of symmetry is the principle of duality in projective geometry, among other fields. This meta-
phenomenon can roughly be described as follows: in any theorem,
exchange point with plane, join with meet, lies in with contains, and the result is an equally true theorem.
[88] A similar and closely related form of duality exists between a vector space and its dual space.[89]

Important Concepts in Geometry: Axioms

Uploaded by

Copyright:

Available Formats

Important Concepts in Geometry: Axioms

Uploaded by

Document Information

Original Description:

Original Title

Copyright

Available Formats

Share this document

Share or Embed Document

Sharing Options

Did you find this document useful?

Is this content inappropriate?

Copyright:

Available Formats

Important Concepts in Geometry: Axioms

Uploaded by

Copyright:

Available Formats

Important concepts in geometry

An illustration of Euclid's parallel postulate

In Euclidean geometry, angles are used to study polygons and triangles, as well as forming an object of

A sphere is a surface that can be defined parametrically (by x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ) or implicitly

Length, area, and volume

Congruence and similarity

Compass and straightedge constructions

The Koch snowflake, with fractal dimension=log4/log3 and topological dimension=1

You might also like