Geometric shape

A geometric figure is a term used in geometry, which is used inconsistently and often remains undefined. Often it refers to certain subsets of the plane or three-dimensional space. Sometimes just figures are meant that are composed of simple parts such as lines and circles, sometimes complicated subsets like fractals are also included. The term is used both in the Euclidean geometry as well as in the non-Euclidean geometry.

Definition and delimitation

Are not covered by the definition as a subset further structuring such as an ordered pair of points, because the quantities and are the same for two points.

As another example, a segment on with a dot. Two different choices for lead in the same subset of the plane, namely the distance are therefore identical as figures in the sense defined above.

Overview and Examples

Plane geometric figures

In addition to individual points in the plane and all the plain, even the simplest figures are the straights. In affine geometry is called points and lines as affine subspaces and assigns a dimension to. Points are then zero-dimensional and one-dimensional subspaces of the two-dimensional straight line affine plane. Play an important role in geometry, certain subsets of straight lines, namely the routes between two points and half-lines.

The character class of the polygons obtained by connecting at least three points by stretching. Even the simplest of polygons, triangles, allow rich geometric definitions and theorems ( see also triangle geometry, trigonometry ). Triangles and therefore play an important role because it is so let polygons with more than three vertices, ie quadrilaterals, pentagons, hexagons, always decomposed into triangles.

Often by additional conditions on distances and angles can be regarded define special cases of polygons. At the regular polygons, all sides are equal and also all angles between adjacent sides equal. In three corners arise equilateral triangles, squares at the four corners. About Beaten regular polygons, such as the pentagram are also called stars. More special types of triangles are isosceles with two equal sides and right angles with a right angle. A quadrilateral with four equal (and then needed right ) angles is called a rectangle, a rectangle with four equal sides rhombus. A parallelogram is a quadrilateral with the respective opposite sides are parallel.

Also with the help of the concept of distance can be defined circles, namely the set of all points that have a fixed distance from a given point. As in the classical geometry structures play a very important ruler and compass, count circles in addition to the straight lines of the fundamental figures in geometric problems. As the circle can also be the other conic sections, namely, ellipses, parabolas and hyperbolas defined by elementary geometric distance conditions. For example, the ellipse, the set of points for which the sum of distances from two given points is the same.

The conics in coordinates by polynomial equations describe the second degree: they are called quadrics. Examples of curves that are defined by equations of higher degree, are the Cartesian leaf or the Cassini curves. Alternatively, curves can also be described by means of parameters as ways. This representation can be used for example to investigate different types of spirals or cycloid. The latter are formed geometrically by rolling circles on straight lines or other circuits.

Spatial geometric figures

As in the plane are also in three-dimensional Euclidean space the affine subspaces (points, lines and planes ) together with distances and half-line the simplest geometric figures. As subsets of planes in space, all plane figures can also be regarded as figures in space. Routes can be assembled in closed or open spatial polygons. Generally, one can also consider curves in three-dimensional space, such as the helix or nodes.

The two-dimensional space, the polygons corresponding to the polyhedron are geometric shapes that are limited only by planar side surfaces. The most regular polyhedra are the Platonic solids, which are characterized by the fact that all its faces are congruent regular polygons. Already mathematicians in ancient Greece it was known that there are exactly five Platonic solids: tetrahedron, cube, octahedron, icosahedron and dodecahedron. Another class of regular polyhedra with high symmetry are the Archimedean body, such as the cuboctahedron. The complete classification of all strictly convex body with only regular polygons as faces was not until the 20th century with the Johnson - bodies.

Other commonly regarded types of polyhedra are the pyramids and prisms. A right prism with a square as a base side is called a cuboid. An oblique prism with a parallelogram as base side is called a parallelepiped or Spat.

Generalizations of pyramids and prisms on non- polygonal base sides are cones and cylinders. The right circular cone and right circular cylinders are examples of a more important character class, the rotation of the body. Among them is the torus generated by rotating a circle about an axis located at the county level.

The three-dimensional analogue of the circle, that is the set of all points in space that are from a given point at the same distance, the ball. You can also generate a rotational body, namely by rotating a circle about a diameter. The sphere is the most important case of a quadric in three-dimensional space. More quadrics are the ellipsoids, paraboloids and hyperboloids, which are also called second-order surfaces. The geometrical properties, in particular the curvature properties, common areas are examined differential geometry in the mathematical subfield of ( elementary ). This space can be specified as a set of solutions of equations or by parametric equations.

Non-Euclidean geometric figures

• Lune
• Spherical triangle

Fractal geometric figures

• Koch curve
• Dragon curve
• Sierpinski triangle
• Mandelbrot set
• Gosper curve
• Menger Sponge