Portal:Geometry

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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.

In modern times, geometric concepts have been extended. They sometimes show a high level of abstraction and complexity. Geometry now uses methods of calculus and abstract algebra, so that many modern branches of the field are not easily recognizable as the descendants of early geometry. (See areas of mathematics.)

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A dodecahedron, one of the five Platonic solids

A regular polytope is a geometric figure with a high degree of symmetry. Examples in two dimensions include the square, the regular pentagon and hexagon, and so on. In three dimensions the regular polytopes include the cube, the dodecahedron, and all other Platonic solids. other platonic solids include: the terahedron, the octahedron, the icosahedron, and the5th dimensional hendecatope. There exist examples in higher dimensions also. Circles and spheres, although highly symmetric, are not considered polytopes because they do not have flat faces. The strong symmetry of the regular polytopes gives them an aesthetic quality that interests both non-mathematicians and mathematicians.

Many regular polytopes, at least in two and three dimensions, exist in nature and have been known since prehistory. The earliest surviving mathematical treatment of these objects comes to us from ancient Greek mathematicians such as Euclid. Indeed, Euclid wrote a systematic study of mathematics, publishing it under the title Elements, which built up a logical theory of geometry and number theory. His work concluded with mathematical descriptions of the five Platonic solids.

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Euclid of Alexandria Euclid of Alexandria

Euclid (also referred to as Euclid of Alexandria) (Greek: Εὐκλείδης) (c. 325–c. 265 BC), a Greek mathematician, who lived in Alexandria, Hellenistic Egypt, almost certainly during the reign of Ptolemy I (323 BC–283 BC), is often considered to be the "father of geometry". His most popular work, Elements, is thought to be one of the most successful textbooks in the history of mathematics. Within it, the properties of geometrical objects are deduced from a small set of axioms, thereby founding the axiomatic method of mathematics.

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Construction of an astroid

One of way on constructing an astroid, by tracking the path a point on the smaller circle follows as it rolled round within the larger cricle. Hence, an astroid is a hypocycloid.

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Categories

Algebraic geometry • Classical geometry
Conformal geometry • Convex geometry
Coordinate systems • Differential geometry
Digital geometry • Dimension • Discrete geometry
Duality theories • Figurate numbers
Frames of reference • Geometers
Geometric algorithms • Geometric graph theory
Geometric group theory • Geometric shapes
Homogeneous spaces • Incidence geometry
Integral geometry • Metric geometry
Symmetry • Trigonometry

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Topics in Geometry

Basic topics Trigonometry Euclidean geometry Non-Euclidean geometry
  • point
  • line
    • line segment
  • plane
  • Hyperplane
  • Geometric shape
  • Polygon
  • Polyhedron
  • List of basic geometry topics
  • Trigonometry
  • Trigonometric functions
  • Law of sines
  • Law of cosines
  • Law of tangents
  • Trigonometric identity
  • Pythagorean theorem
  • Euler's formula
  • De Moivre's formula
  • List of trigonometric identities
  • List of basic trigonometry topics
  • List of trigonometry topics
Differential geometry Riemannian geometry Algebraic geometry Other
  • Differential geometry
  • Differential geometry of curves
  • Differential geometry of surfaces
  • Manifold
  • Tensor analysis
  • Jets
  • Differential topology
  • Fiber bundle
  • connection
  • List of differential geometry topics
  • Riemannian geometry
  • Riemannian manifold
  • Levi-Civita connection
  • Geodesic
  • Riemannian symmetric space
  • Riemannian submanifold
  • Curvature of Riemannian manifolds
  • Isometry
  • Laplace-Beltrami operator
  • List of formulas in Riemannian geometry
  • Algebraic geometry
  • Algebraic curve
  • Algebraic surface (list)
  • Algebraic group
  • Complex manifold
  • List of algebraic geometry topics
  • Lie group
  • Index theory
  • Homogeneous space
  • List of mathematical shapes
  • List of geometry topics
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