"an identical angle-angle-angle (AAA) sequence, they are congruent (unlike for plane triangles). The plane-triangle congruence theorem angle-angle-side (AAS) does not hold for spherical triangles. As in plane geometry, side-side-angle (SSA) does not imply congruence. A symbol commonly used for congruence is an equals symbol with a tilde above it, ≅, corresponding to the Unicode character 'approximately equal to' (U+2245). In the UK, the three-bar equal sign ""≡"" (U+2261) is sometimes used. Congruence (geometry) In geometry, two figures or objects are congruent if they have the same shape and size, or if one has the same shape and size as the"
"Similarity (geometry) Two geometrical objects are called similar if they both have the same shape, or one has the same shape as the mirror image of the other. More precisely, one can be obtained from the other by uniformly scaling (enlarging or reducing), possibly with additional translation, rotation and reflection. This means that either object can be rescaled, repositioned, and reflected, so as to coincide precisely with the other object. If two objects are similar, each is congruent to the result of a particular uniform scaling of the other. A modern and novel perspective of similarity is to consider geometrical"
"drawings. Similarity (geometry) Two geometrical objects are called similar if they both have the same shape, or one has the same shape as the mirror image of the other. More precisely, one can be obtained from the other by uniformly scaling (enlarging or reducing), possibly with additional translation, rotation and reflection. This means that either object can be rescaled, repositioned, and reflected, so as to coincide precisely with the other object. If two objects are similar, each is congruent to the result of a particular uniform scaling of the other. A modern and novel perspective of similarity is to consider"
"Congruence (geometry) In geometry, two figures or objects are congruent if they have the same shape and size, or if one has the same shape and size as the mirror image of the other. More formally, two sets of points are called congruent if, and only if, one can be transformed into the other by an isometry, i.e., a combination of rigid motions, namely a translation, a rotation, and a reflection. This means that either object can be repositioned and reflected (but not resized) so as to coincide precisely with the other object. So two distinct plane figures on a"
"into the other by an isometry, i.e., a combination of rigid motions, namely a translation, a rotation, and a reflection. This means that either object can be repositioned and reflected (but not resized) so as to coincide precisely with the other object. So two distinct plane figures on a piece of paper are congruent if we can cut them out and then match them up completely. Turning the paper over is permitted. Two geometrical objects are called similar if they both have the same shape, or one has the same shape as the mirror image of the other. More precisely,"
"mathematics, quasi-isometry can be used as a criterion to state that two shapes are approximately the same. Simple shapes can often be classified into basic geometric objects such as a point, a line, a curve, a plane, a plane figure (e.g. square or circle), or a solid figure (e.g. cube or sphere). However, most shapes occurring in the physical world are complex. Some, such as plant structures and coastlines, may be so complicated as to defy traditional mathematical description – in which case they may be analyzed by differential geometry, or as fractals. In geometry, two subsets of a Euclidean"
"objects always have either the same shape or mirror image shapes, and have the same size. Objects that have the same shape or mirror image shapes are called geometrically similar, whether or not they have the same size. Thus, objects that can be transformed into each other by rigid transformations, mirroring, and uniform scaling are similar. Similarity is preserved when one of the objects is uniformly scaled, while congruence is not. Thus, congruent objects are always geometrically similar, but similar objects may not be congruent, as they may have different size. A more flexible definition of shape takes into consideration"
"Homomorphism In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ""homomorphism"" comes from the ancient Greek language: ""ὁμός (homos)"" meaning ""same"" and ""μορφή (morphe)"" meaning ""form"" or ""shape"". However, the word was apparently introduced to mathematics due to a (mis)translation of German ""ähnlich"" meaning ""similar"" to ""ὁμός"" meaning ""same"". Homomorphisms of vector spaces are also called linear maps, and their study is the object of linear algebra. The concept of homomorphism has been generalized, under the name of morphism, to many"
"Isomorphism In mathematics, an isomorphism (from the Ancient Greek: ἴσος ""isos"" ""equal"", and μορφή ""morphe"" ""form"" or ""shape"") is a homomorphism or morphism (i.e. a mathematical mapping) that can be reversed by an inverse morphism. Two mathematical objects are isomorphic if an isomorphism exists between them. An ""automorphism"" is an isomorphism whose source and target coincide. The interest of isomorphisms lies in the fact that two isomorphic objects cannot be distinguished by using only the properties used to define morphisms; thus isomorphic objects may be considered the same as long as one considers only these properties and their consequences. For"
"space have the same shape if one can be transformed to the other by a combination of translations, rotations (together also called rigid transformations), and uniform scalings. In other words, the ""shape"" of a set of points is all the geometrical information that is invariant to translations, rotations, and size changes. Having the same shape is an equivalence relation, and accordingly a precise mathematical definition of the notion of shape can be given as being an equivalence class of subsets of a Euclidean space having the same shape. Mathematician and statistician David George Kendall writes: In this paper ‘shape’ is"
"one can be obtained from the other by uniformly scaling (enlarging or shrinking), possibly with additional translation, rotation and reflection. This means that either object can be rescaled, repositioned, and reflected, so as to coincide precisely with the other object. If two objects are similar, each is congruent to the result of a uniform scaling of the other. Solid geometry was the traditional name for the geometry of three-dimensional Euclidean space. Stereometry deals with the measurements of volumes of various solid figures (three-dimensional figures) including pyramids, cylinders, cones, truncated cones, spheres, and prisms. Rational number is any number that can"
"Homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bi continuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are the mappings that preserve all the topological properties of a given space. Two spaces with a homeomorphism between them are called homeomorphic, and from a topological viewpoint they are the same. The word ""homeomorphism"" comes from the Greek words ""ὅμοιος"" (""homoios"") = similar or same and ""μορφή"" (""morphē"") = shape, form, introduced to mathematics by Henri Poincaré in 1895."
"similar. Several types of curves have the property that all examples of that type are similar to each other. These include: A similarity (also called a similarity transformation or similitude) of a Euclidean space is a bijection from the space onto itself that multiplies all distances by the same positive real number , so that for any two points and we have where """" is the Euclidean distance from to . The scalar has many names in the literature including; the ""ratio of similarity"", the ""stretching factor"" and the ""similarity coefficient"". When = 1 a similarity is called an isometry"
"Equidimensional Equidimensional is an adjective applied to objects that have nearly the same size or spread in multiple directions. As a mathematical concept, it may be applied to objects that extend across any number of dimensions. More specifically, it's also used to characterize the shape of three-dimensional solids. The word ""equidimensional"" is sometimes used by geologists to describe the shape of three-dimensional objects. In that case it is a synonym for equant. Deviations from equidimensional are used to classify the shape of convex objects like rocks or particles. For instance Th. Zingg in 1935 pointed out that if ""a"", ""b"""
"different lengths. Cuboid In geometry, a cuboid is a convex polyhedron bounded by six quadrilateral faces, whose polyhedral graph is the same as that of a cube. While mathematical literature refers to any such polyhedron as a cuboid, other sources use ""cuboid"" to refer to a shape of this type in which each of the faces is a rectangle (and so each pair of adjacent faces meets in a right angle); this more restrictive type of cuboid is also known as a rectangular cuboid, right cuboid, rectangular box, rectangular hexahedron, right rectangular prism, or rectangular parallelepiped. By Euler's formula the"
"Isothetic polygon An isothetic polygon is a polygon whose alternate sides belong to two parametric families of straight lines which are pencils of lines with centers at two points (possibly the point at infinity). The most well-known example of isothetic polygons are rectilinear polygons, and the former term is commonly used as a synonym for the latter one. The term is produced from Greek roots: ""iso-"" for ""equal, same, similar"" and ""thetos"" (position, placement), i,e., the term is supposed to mean ""polygon with similarly placed sides"". The term was suggested during the early years of the computational geometry. Much emphasis"
"Equivalent definitions of mathematical structures In mathematics, equivalent definitions are used in two somewhat different ways. First, within a particular mathematical theory (for example, Euclidean geometry), a notion (for example, ellipse or minimal surface) may have more than one definition. These definitions are equivalent in the context of a given mathematical structure (Euclidean space, in this case). Second, a mathematical structure may have more than one definition (for example, topological space has at least seven definitions; ordered field has at least two definitions). In the former case, equivalence of two definitions means that a mathematical object (for example, geometric body)"
"aA""n"". Bicupola (geometry) In geometry, a bicupola is a solid formed by connecting two cupolae on their bases. There are two classes of bicupola because each cupola half is bordered by alternating triangles and squares. If similar faces are attached together the result is an ""orthobicupola""; if squares are attached to triangles it is a ""gyrobicupola"". Cupolae and bicupolae categorically exist as infinite sets of polyhedra, just like the pyramids, bipyramids, prisms, and trapezohedra. Six bicupolae have regular polygon faces: triangular, square and pentagonal ortho- and gyrobicupolae. The triangular gyrobicupola is an Archimedean solid, the cuboctahedron; the other five are"
"find out exactly which two circles are the solution. Homothetic center In geometry, a homothetic center (also called a center of similarity or a center of similitude) is a point from which at least two geometrically similar figures can be seen as a dilation or contraction of one another. If the center is ""external"", the two figures are directly similar to one another; their angles have the same rotational sense. If the center is ""internal"", the two figures are scaled mirror images of one another; their angles have the opposite sense. If two geometric figures possess a homothetic center, they"
"object becomes ""equidimensional"". Separate ""a"" and ""c"" further from ""b"" and it becomes ""bladed"". For example, the convex envelope for some humans might plot near the black dot in the upper left of the figure. Theodor Zingg PhD thesis: Equidimensional Equidimensional is an adjective applied to objects that have nearly the same size or spread in multiple directions. As a mathematical concept, it may be applied to objects that extend across any number of dimensions. More specifically, it's also used to characterize the shape of three-dimensional solids. The word ""equidimensional"" is sometimes used by geologists to describe the shape of"
"(rigid motion). Two sets are called similar if one is the image of the other under a similarity. As a map , a similarity of ratio takes the form where is an orthogonal matrix and is a translation vector. Similarities preserve planes, lines, perpendicularity, parallelism, midpoints, inequalities between distances and line segments. Similarities preserve angles but do not necessarily preserve orientation, ""direct similitudes"" preserve orientation and ""opposite similitudes"" change it. The similarities of Euclidean space form a group under the operation of composition called the ""similarities group "". The direct similitudes form a normal subgroup of and the Euclidean group"
"is also called a quadratic form. Two ""n""-dimensional quadratic spaces and are isometric if there exists an invertible linear transformation (isometry) such that The isometry classes of ""n""-dimensional quadratic spaces over ""K"" correspond to the equivalence classes of ""n""-ary quadratic forms over ""K"". Let ""R"" be a commutative ring, ""M"" be an ""R""-module and be an ""R""-bilinear form. A mapping is the ""associated quadratic form"" of ""b"", and is the ""polar form"" of ""Q"". Alternatively, a quadratic form may be characterized as follows: Two elements ""v"" and ""w"" of ""V"" are called orthogonal if . The kernel of a bilinear"
"Geometric shape A geometric shape is the geometric information which remains when location, scale, orientation and reflection are removed from the description of a geometric object. That is, the result of moving a shape around, enlarging it, rotating it, or reflecting it in a mirror is the same shape as the original, and not a distinct shape. Objects that have the same shape as each other are said to be similar. If they also have the same scale as each other, they are said to be congruent. Many two-dimensional geometric shapes can be defined by a set of points or"
"Bicupola (geometry) In geometry, a bicupola is a solid formed by connecting two cupolae on their bases. There are two classes of bicupola because each cupola half is bordered by alternating triangles and squares. If similar faces are attached together the result is an ""orthobicupola""; if squares are attached to triangles it is a ""gyrobicupola"". Cupolae and bicupolae categorically exist as infinite sets of polyhedra, just like the pyramids, bipyramids, prisms, and trapezohedra. Six bicupolae have regular polygon faces: triangular, square and pentagonal ortho- and gyrobicupolae. The triangular gyrobicupola is an Archimedean solid, the cuboctahedron; the other five are Johnson"
"most algebraic structures, including groups and rings, a homomorphism is an isomorphism if and only if it is bijective. In topology, where the morphisms are continuous functions, isomorphisms are also called ""homeomorphisms"" or ""bicontinuous functions"". In mathematical analysis, where the morphisms are differentiable functions, isomorphisms are also called ""diffeomorphisms"". A canonical isomorphism is a canonical map that is an isomorphism. Two objects are said to be canonically isomorphic if there is a canonical isomorphism between them. For example, the canonical map from a finite-dimensional vector space ""V"" to its second dual space is a canonical isomorphism; on the other hand,"
"plane is called mirror symmetric (see mirror image). The axis of symmetry of a two-dimensional figure is a line such that, if a perpendicular is constructed, any two points lying on the perpendicular at equal distances from the axis of symmetry are identical. Another way to think about it is that if the shape were to be folded in half over the axis, the two halves would be identical: the two halves are each other's mirror image. Thus a square has four axes of symmetry, because there are four different ways to fold it and have the edges all match."
"Arc (geometry) In Euclidean geometry, an arc (symbol: ⌒) is a closed segment of a differentiable curve. A common example in the plane (a two-dimensional manifold), is a segment of a circle called a circular arc. In space, if the arc is part of a great circle (or great ellipse), it is called a great arc. Every pair of distinct points on a circle determines two arcs. If the two points are not directly opposite each other, one of these arcs, the minor arc, will subtend an angle at the centre of the circle that is less than radians (180"
"Dual polygon In geometry, polygons are associated into pairs called duals, where the vertices of one correspond to the edges of the other. Regular polygons are self-dual. The dual of an isogonal (vertex-transitive) polygon is an isotoxal (edge-transitive) polygon. For example, the (isogonal) rectangle and (isotoxal) rhombus are duals. In a cyclic polygon, longer sides correspond to larger exterior angles in the dual (a tangential polygon), and shorter sides to smaller angles. Further, congruent sides in the original polygon yields congruent angles in the dual, and conversely. For example, the dual of a highly acute isosceles triangle is an obtuse"
"is a continuous function of these lines. Linear geometries are incidence structures in which any two distinct points formula_6 and formula_7 are joined by a unique line formula_8. Such geometries are called ""topological"" if formula_8 depends continuously on the pair formula_10 with respect to given topologies on the point set and the line set. The ""dual"" of a linear geometry is obtained by interchanging the rôles of points and lines. A survey of linear topological geometries is given in Chapter 23 of the ""Handbook of incidence geometry"". The most extensively investigated topological linear geometries are those which are also dual"
"Intersection In mathematics, the intersection of two or more objects is another, usually ""smaller"" object. All objects are presumed to lie in a certain common space except in set theory, where the intersection of arbitrary sets is defined. The intersection is one of basic concepts of geometry. Intuitively, the intersection of two or more objects is a new object that lies in each of original objects. An intersection can have various geometric shapes, but a point is the most common in a plane geometry. Definitions vary in different contexts: set theory formalizes the idea that a smaller object lies in"
"of abstract objects, theoretic models, language, music and even knowledge itself. This article describes symmetry from three perspectives: in mathematics, including geometry, the most familiar type of symmetry for many people; in science and nature; and in the arts, covering architecture, art and music. The opposite of symmetry is asymmetry. A geometric shape or object is symmetric if it can be divided into two or more identical pieces that are arranged in an organized fashion. This means that an object is symmetric if there is a transformation that moves individual pieces of the object but doesn't change the overall shape."
"meaning. In geometry, juxtaposition of names of points represents lines or line segments. In physics, juxtaposition is also used for ""multiplication"" of a numerical value and a physical quantity, and of two physical quantities, for example, three times π would be written as 3π and area equals length times width as ""A"" = ""lw"". Juxtaposition Juxtaposition is an act or instance of placing two elements close together or side by side. This is often done in order to compare/contrast the two, to show similarities or differences, etc. Juxtaposition in literary terms is the showing contrast by concepts placed side by"
"Cuboid In geometry, a cuboid is a convex polyhedron bounded by six quadrilateral faces, whose polyhedral graph is the same as that of a cube. While mathematical literature refers to any such polyhedron as a cuboid, other sources use ""cuboid"" to refer to a shape of this type in which each of the faces is a rectangle (and so each pair of adjacent faces meets in a right angle); this more restrictive type of cuboid is also known as a rectangular cuboid, right cuboid, rectangular box, rectangular hexahedron, right rectangular prism, or rectangular parallelepiped. By Euler's formula the numbers of"