In elementary geometry, a polytope is a geometric object with flat sides, which exists in any general number of dimensions.

In convex geometry, a face of a polytope P is the intersection of any supporting hyperplane of P and P. From this definition it follows that the set of faces of a polytope includes the polytope itself and the empty set.

* E₆ polytope in geometry

Corresponding to each tiling of the quartic ( partition of the quartic variety into subsets ) is an abstract polyhedron, which abstracts from the geometry and only reflects the combinatorics of the tiling ( this is a general way of obtaining an abstract polytope from a tiling ) – the vertices, edges, and faces of the polyhedron are equal as sets to the vertices, edges, and faces of the tiling, with the same incidence relations, and the ( combinatorial ) automorphism group of the abstract polyhedron equals the ( geometric ) automorphism group of the quartic.

In geometry, a uniform tessellation is a vertex-transitive tessellation made from uniform polytope facets.

In geometry, the barycentric subdivision is a standard way of dividing an arbitrary convex polygon into triangles, a convex polyhedron into tetrahedra, or, in general, a convex polytope into simplices with the same dimension, by connecting the barycenters of their faces in a specific way.

In geometry, a cross-polytope, orthoplex, hyperoctahedron, or cocube is a regular, convex polytope that exists in any number of dimensions.

* Density ( polytope ) in geometry

* Rectification ( geometry ) – truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points

* Flag ( geometry ), part of a polygon, polyhedron or higher polytope

* E5 polytope in geometry

* E₇ polytope in geometry

# Cauchy's theorem on geometry of convex polytopes states that a convex polytope is uniquely determined by the geometry of its faces and combinatorial adjacency rules.

In Euclidean geometry, rectification is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points.

* Peak ( geometry ), an ( n-3 )- dimensional element of a polytope

A value of 0 means that the pixel does not have any coverage information and is transparent ; i. e. there was no color contribution from any geometry because the geometry did not overlap this pixel.

A value of 1 means that the pixel is opaque because the geometry completely overlapped the pixel.

In terms of analytic geometry, the restriction of classical geometry to compass and straightedge constructions means a restriction to first-and second-order equations, e. g., y

As well as statistics, means are often used in geometry and analysis ; a wide range of means have been developed for these purposes, which are not much used in statistics.

He manages to find some rational points on these curves – elliptic curves, as it happens, in what seems to be their first known occurrence — by means of what amounts to a tangent construction: translated into coordinate geometry

This performance was attained by means of the highly optimized yet platform independent LIBRT ray-tracing engine in BRL-CAD and by using solid implicit CSG geometry on several shared memory parallel machines over a commodity network.

In differential geometry, volume is expressed by means of the volume form, and is an important global Riemannian invariant.

In geometry, confocal means having the same foci.

This wheel supercharges the engine air intake to a degree that can be controlled by means of a wastegate or by dynamically modifying the turbine housing's geometry ( as in a VGT turbocharger ).

According to Plutarch, Plato gave the problem to Eudoxus and Archytas and Menaechmus, who solved the problem using mechanical means, earning a rebuke from Plato for not solving the problem using pure geometry ( Plut., Quaestiones convivales VIII. ii, 718ef ).

This means that although the local geometries of spacetime are generated by the theory of relativity based on spacetime intervals, we can approximate 3-space by the familiar Euclidean geometry.

Similar formulas in plane geometry can be proven with more elementary means.

In other contexts, such as in Euclidean geometry and informal use, sphere sometimes means ball.

In the Posterior Analytics, Aristotle ( 384 BC – 322 BC ) laid down the axiomatic method, to organize a field of knowledge logically by means of primitive concepts, axioms, postulates, definitions, and theorems, taking a majority of his examples from arithmetic and geometry.

René Descartes published La Géométrie ( 1637 ) aimed to reduce geometry to algebra by means of coordinate systems, giving algebra a more foundational role ( while the Greeks embedded arithmetic into geometry by identifying whole numbers with evenly spaced points on a line ).

In general, this means that only part of the geometry of the slice can be given by the scientist, while the geometry everywhere else will then be dictated by Einstein's equations on the slice.

An xlink: href attribute on a GML geometry property means that the value of the property is the resource referenced in the link.

In the Principia he outlined his philosophical method, which incorporated experience, geometry ( the means whereby the inner order of the world can be known ), and the power of reason ; and he presented his cosmology, which included the first presentation of his Nebular hypothesis.

This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts.

Euler discussed a generalization of Euclidean geometry called affine geometry, which retains the fifth postulate unmodified while weakening postulates three and four in a way that eliminates the notions of angle ( whence right triangles become meaningless ) and of equality of length of line segments in general ( whence circles become meaningless ) while retaining the notions of parallelism as an equivalence relation between lines, and equality of length of parallel line segments ( so line segments continue to have a midpoint ).

The metric tensor that defines the geometry — in particular, how lengths and angles are measured — is not the Minkowski metric of special relativity, it is a generalization known as a semi-or pseudo-Riemannian metric.

In geometry, a simplex ( plural simplexes or simplices ) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension.

Taylor's theorem also generalizes to multivariate and vector valued functions on any dimensions n and m. This generalization of Taylor's theorem is the basis for the definition of so-called jets which appear in differential geometry and partial differential equations.

In mathematics, particularly differential geometry, a geodesic ( or ) is a generalization of the notion of a " straight line " to " curved spaces ".

The Langlands program seeks to attach an automorphic form or automorphic representation ( a suitable generalization of a modular form ) to more general objects of arithmetic algebraic geometry, such as to every elliptic curve over a number field.

It is a very broad and abstract generalization of the differential geometry of surfaces in R < sup > 3 </ sup >.

Only after Felix Klein's Erlangen program was affine geometry recognized for being a generalization of Euclidean geometry.

Affine geometry can be viewed as the geometry of affine space, of a given dimension n, coordinatized over a field K. There is also ( in two dimensions ) a combinatorial generalization of coordinatized affine space, as developed in synthetic finite geometry.

In Riemannian geometry for example, ramification is a generalization of the notion of covering maps.

In differential geometry, the exponential map is a generalization of the ordinary exponential function of mathematical analysis to all differentiable manifolds with an affine connection.

In differential geometry, a pseudo-Riemannian manifold ( also called a semi-Riemannian manifold ) is a generalization of a Riemannian manifold.

One natural generalization in differential geometry is hyperbolic n-space H < sup > n </ sup >, the maximally symmetric, simply connected, n-dimensional Riemannian manifold with constant sectional curvature − 1.

In geometry, inversive geometry is the study of those properties of figures that are preserved by a generalization of a type of transformation of the Euclidean plane, called inversion.

In the spirit of generalization to higher dimensions, inversive geometry is the study of transformations generated by the Euclidean transformations together with inversion in an n-sphere:

The generalization of the time-independent perturbation theory to the multi-parameter case can be formulated more systematically using the language of differential geometry, which basically defines the derivatives of the quantum states and calculate the perturbative corrections by taking derivatives iteratively at the unperturbed point.

In computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basis is a particular kind of generating subset of an ideal I in a polynomial ring R. One can view it as a multivariate, non-linear generalization of:

The sheaf of rational functions K < sub > X </ sub > of a scheme X is the generalization to scheme theory of the notion of function field of an algebraic variety in classical algebraic geometry.

The generalization using elements of a mathematical ring requires methods of inversive ring geometry.

In the mathematical field of differential geometry, a Cartan connection is a flexible generalization of the notion of an affine connection.