Irreducible componentIn algebraic geometry, an irreducible algebraic set or irreducible variety is an algebraic set that cannot be written as the union of two proper algebraic subsets. An irreducible component is an algebraic subset that is irreducible and maximal (for set inclusion) for this property. For example, the set of solutions of the equation xy = 0 is not irreducible, and its irreducible components are the two lines of equations x = 0 and y =0.
Tropical forestTropical forests (a.k.a. jungle) are forested landscapes in tropical regions: i.e. land areas approximately bounded by the tropic of Cancer and Capricorn, but possibly affected by other factors such as prevailing winds. Some tropical forest types are difficult to categorize. While forests in temperate areas are readily categorized on the basis of tree canopy density, such schemes do not work well in tropical forests. There is no single scheme that defines what a forest is, in tropical regions or elsewhere.
Combinatorial mapA combinatorial map is a combinatorial representation of a graph on an orientable surface. A combinatorial map may also be called a combinatorial embedding, a rotation system, an orientable ribbon graph, a fat graph, or a cyclic graph. More generally, an -dimensional combinatorial map is a combinatorial representation of a graph on an -dimensional orientable manifold. Combinatorial maps are used as efficient data structures in image representation and , in geometrical modeling.
Szilassi polyhedronIn geometry, the Szilassi polyhedron is a nonconvex polyhedron, topologically a torus, with seven hexagonal faces. The 14 vertices and 21 edges of the Szilassi polyhedron form an embedding of the Heawood graph onto the surface of a torus. Each face of this polyhedron shares an edge with each other face. As a result, it requires seven colours to colour all adjacent faces. This example shows that, on surfaces topologically equivalent to a torus, some subdivisions require seven colors, providing the lower bound for the seven colour theorem.
Tropical cyclones and climate changeClimate change can affect tropical cyclones in a variety of ways: an intensification of rainfall and wind speed, a decrease in overall frequency, an increase in the frequency of very intense storms and a poleward extension of where the cyclones reach maximum intensity are among the possible consequences of human-induced climate change. Tropical cyclones use warm, moist air as their source of energy or "fuel". As climate change is warming ocean temperatures, there is potentially more of this fuel available.
Regular map (graph theory)In mathematics, a regular map is a symmetric tessellation of a closed surface. More precisely, a regular map is a decomposition of a two-dimensional manifold (such as a sphere, torus, or real projective plane) into topological disks such that every flag (an incident vertex-edge-face triple) can be transformed into any other flag by a symmetry of the decomposition. Regular maps are, in a sense, topological generalizations of Platonic solids. The theory of maps and their classification is related to the theory of Riemann surfaces, hyperbolic geometry, and Galois theory.
TorusIn geometry, a torus (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanar with the circle. The main types of toruses include ring toruses, horn toruses, and spindle toruses. A ring torus is sometimes colloquially referred to as a donut or doughnut. If the axis of revolution does not touch the circle, the surface has a ring shape and is called a torus of revolution, also known as a ring torus.
Császár polyhedronIn geometry, the Császár polyhedron (ˈt͡ʃaːsaːr) is a nonconvex toroidal polyhedron with 14 triangular faces. This polyhedron has no diagonals; every pair of vertices is connected by an edge. The seven vertices and 21 edges of the Császár polyhedron form an embedding of the complete graph K_7 onto a topological torus. Of the 35 possible triangles from vertices of the polyhedron, only 14 are faces.
Transformation geometryIn mathematics, transformation geometry (or transformational geometry) is the name of a mathematical and pedagogic take on the study of geometry by focusing on groups of geometric transformations, and properties that are invariant under them. It is opposed to the classical synthetic geometry approach of Euclidean geometry, that focuses on proving theorems. For example, within transformation geometry, the properties of an isosceles triangle are deduced from the fact that it is mapped to itself by a reflection about a certain line.
TessellationA tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety of geometries. A periodic tiling has a repeating pattern. Some special kinds include regular tilings with regular polygonal tiles all of the same shape, and semiregular tilings with regular tiles of more than one shape and with every corner identically arranged.
Position (geometry)In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents the position of a point P in space in relation to an arbitrary reference origin O. Usually denoted x, r, or s, it corresponds to the straight line segment from O to P. In other words, it is the displacement or translation that maps the origin to P: The term position vector is used mostly in the fields of differential geometry, mechanics and occasionally vector calculus.
Generalized flag varietyIn mathematics, a generalized flag variety (or simply flag variety) is a homogeneous space whose points are flags in a finite-dimensional vector space V over a field F. When F is the real or complex numbers, a generalized flag variety is a smooth or complex manifold, called a real or complex flag manifold. Flag varieties are naturally projective varieties. Flag varieties can be defined in various degrees of generality. A prototype is the variety of complete flags in a vector space V over a field F, which is a flag variety for the special linear group over F.
