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74-465: A plane shape or plane figure is constrained to lie on a plane , in contrast to solid 3D shapes. A two-dimensional shape or two-dimensional figure (also: 2D shape or 2D figure ) may lie on a more general curved surface (a two-dimensional space ). Some simple shapes can be put into broad categories. For instance, polygons are classified according to their number of edges as triangles , quadrilaterals , pentagons , etc. Each of these

148-431: A finitely generated group are defined to be the ends of the corresponding Cayley graph ; this definition is independent of the choice of a finite generating set. Every finitely-generated infinite group has either 0,1, 2, or infinitely many ends, and Stallings theorem about ends of groups provides a decomposition for groups with more than one end. If two connected locally finite graphs are quasi-isometric then they have

222-427: A group with respect to a symmetric generating set describes the size of balls in the group. Every element in the group can be written as a product of generators, and the growth rate counts the number of elements that can be written as a product of length n . According to Gromov's theorem , a group of polynomial growth is virtually nilpotent , i.e. it has a nilpotent subgroup of finite index . In particular,

296-541: A plane is a flat two- dimensional surface that extends indefinitely. Euclidean planes often arise as subspaces of three-dimensional space R 3 {\displaystyle \mathbb {R} ^{3}} . A prototypical example is one of a room's walls, infinitely extended and assumed infinitesimal thin. In two dimensions, there are infinitely many polytopes: the polygons. The first few regular ones are shown below: The Schläfli symbol { n } {\displaystyle \{n\}} represents

370-444: A quasi-isometry is a function between two metric spaces that respects large-scale geometry of these spaces and ignores their small-scale details. Two metric spaces are quasi-isometric if there exists a quasi-isometry between them. The property of being quasi-isometric behaves like an equivalence relation on the class of metric spaces. The concept of quasi-isometry is especially important in geometric group theory , following

444-419: A regular n -gon . The regular monogon (or henagon) {1} and regular digon {2} can be considered degenerate regular polygons and exist nondegenerately in non-Euclidean spaces like a 2-sphere , 2-torus , or right circular cylinder . There exist infinitely many non-convex regular polytopes in two dimensions, whose Schläfli symbols consist of rational numbers {n/m}. They are called star polygons and share

518-416: A subexponential growth rate . Any such group is amenable . The ends of a topological space are, roughly speaking, the connected components of the “ideal boundary” of the space. That is, each end represents a topologically distinct way to move to infinity within the space. Adding a point at each end yields a compactification of the original space, known as the end compactification . The ends of

592-450: A " p " have a different shape, at least when they are constrained to move within a two-dimensional space like the page on which they are written. Even though they have the same size, there's no way to perfectly superimpose them by translating and rotating them along the page. Similarly, within a three-dimensional space, a right hand and a left hand have a different shape, even if they are the mirror images of each other. Shapes may change if

666-410: A coffee cup by creating a dimple and progressively enlarging it, while preserving the donut hole in a cup's handle. A described shape has external lines that you can see and make up the shape. If you were putting your coordinates on a coordinate graph you could draw lines to show where you can see a shape, however not every time you put coordinates in a graph as such you can make a shape. This shape has

740-412: A different finite generating set T results in a different graph and a different metric space, however the two spaces are quasi-isometric. This quasi-isometry class is thus an invariant of the group G . Any property of metric spaces that only depends on a space's quasi-isometry class immediately yields another invariant of groups, opening the field of group theory to geometric methods. More generally,

814-458: A group has a Følner sequence then it is automatically amenable. An ultralimit is a geometric construction that assigns to a sequence of metric spaces X n a limiting metric space. An important class of ultralimits are the so-called asymptotic cones of metric spaces. Let ( X , d ) be a metric space, let ω be a non-principal ultrafilter on N {\displaystyle \mathbb {N} } and let p n  ∈  X be

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888-419: A map ϕ : R → X {\displaystyle \phi :\mathbb {R} \to X} such that there exists C , K > 0 {\displaystyle C,K>0} so that is called a ( C , K ) {\displaystyle (C,K)} -quasi-geodesic. Obviously geodesics (parametrised by arclength) are quasi-geodesics. The fact that in some spaces

962-405: A mirror is the same shape as the original, and not a distinct shape. Many two-dimensional geometric shapes can be defined by a set of points or vertices and lines connecting the points in a closed chain, as well as the resulting interior points. Such shapes are called polygons and include triangles , squares , and pentagons . Other shapes may be bounded by curves such as the circle or

1036-599: A outline and boundary so you can see it and is not just regular dots on a regular paper. The above-mentioned mathematical definitions of rigid and non-rigid shape have arisen in the field of statistical shape analysis . In particular, Procrustes analysis is a technique used for comparing shapes of similar objects (e.g. bones of different animals), or measuring the deformation of a deformable object. Other methods are designed to work with non-rigid (bendable) objects, e.g. for posture independent shape retrieval (see for example Spectral shape analysis ). All similar triangles have

1110-558: A quasi-isometry f : M 1 → M 2 {\displaystyle f:M_{1}\to M_{2}} . The map between the Euclidean plane and the plane with the Manhattan distance that sends every point to itself is a quasi-isometry: in it, distances are multiplied by a factor of at most 2 {\displaystyle {\sqrt {2}}} . Note that there can be no isometry, since, for example,

1184-421: A reflection of each other, and hence they are congruent and similar, but in some contexts they are not regarded as having the same shape. Sometimes, only the outline or external boundary of the object is considered to determine its shape. For instance, a hollow sphere may be considered to have the same shape as a solid sphere. Procrustes analysis is used in many sciences to determine whether or not two objects have

1258-533: A region D in R of a function f ( x , y ) , {\displaystyle f(x,y),} and is usually written as: The fundamental theorem of line integrals says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve. Let φ : U ⊆ R 2 → R {\displaystyle \varphi :U\subseteq \mathbb {R} ^{2}\to \mathbb {R} } . Then with p , q

1332-583: A sequence of base-points. Then the ω –ultralimit of the sequence ( X , d n , p n ) {\displaystyle (X,{\frac {d}{n}},p_{n})} is called the asymptotic cone of X with respect to ω and ( p n ) n {\displaystyle (p_{n})_{n}\,} and is denoted C o n e ω ( X , d , ( p n ) n ) {\displaystyle Cone_{\omega }(X,d,(p_{n})_{n})\,} . One often takes

1406-427: 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’

1480-434: A shape defined by n − 2 complex numbers S ( z j , z j + 1 , z j + 2 ) ,   j = 1 , . . . , n − 2. {\displaystyle S(z_{j},z_{j+1},z_{j+2}),\ j=1,...,n-2.} The polygon bounds a convex set when all these shape components have imaginary components of the same sign. Human vision relies on

1554-440: A triangle. The shape of a quadrilateral is associated with two complex numbers p , q . If the quadrilateral has vertices u , v , w , x , then p = S( u , v , w ) and q = S( v , w , x ) . Artzy proves these propositions about quadrilateral shapes: A polygon ( z 1 , z 2 , . . . z n ) {\displaystyle (z_{1},z_{2},...z_{n})} has

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1628-417: A vector A by itself is which gives the formula for the Euclidean length of the vector. In a rectangular coordinate system, the gradient is given by For some scalar field f  : U ⊆ R → R , the line integral along a piecewise smooth curve C ⊂ U is defined as where r : [a, b] → C is an arbitrary bijective parametrization of the curve C such that r ( a ) and r ( b ) give

1702-460: A way that no edges cross each other. Such a drawing is called a plane graph or planar embedding of the graph . A plane graph can be defined as a planar graph with a mapping from every node to a point on a plane, and from every edge to a plane curve on that plane, such that the extreme points of each curve are the points mapped from its end nodes, and all curves are disjoint except on their extreme points. Quasi-isometry In mathematics ,

1776-472: A wide range of shape representations. Some psychologists have theorized that humans mentally break down images into simple geometric shapes (e.g., cones and spheres) called geons . Meanwhile, others have suggested shapes are decomposed into features or dimensions that describe the way shapes tend to vary, like their segmentability , compactness and spikiness . When comparing shape similarity, however, at least 22 independent dimensions are needed to account for

1850-408: Is a one-dimensional manifold . In a Euclidean plane, it has the length 2π r and the area of its interior is where r {\displaystyle r} is the radius. There are an infinitude of other curved shapes in two dimensions, notably including the conic sections : the ellipse , the parabola , and the hyperbola . Another mathematical way of viewing two-dimensional space

1924-402: Is by homeomorphisms . Roughly speaking, a homeomorphism is a continuous stretching and bending of an object into a new shape. Thus, a square and a circle are homeomorphic to each other, but a sphere and a donut are not. An often-repeated mathematical joke is that topologists cannot tell their coffee cup from their donut, since a sufficiently pliable donut could be reshaped to the form of

1998-718: Is called a quasi-isometric embedding if it satisfies the first condition but not necessarily the second (i.e. it is coarsely Lipschitz but may fail to be coarsely surjective). In other words, if through the map, ( M 1 , d 1 ) {\displaystyle (M_{1},d_{1})} is quasi-isometric to a subspace of ( M 2 , d 2 ) {\displaystyle (M_{2},d_{2})} . Two metric spaces M 1 and M 2 are said to be quasi-isometric , denoted M 1 ∼ q . i . M 2 {\displaystyle M_{1}{\underset {q.i.}{\sim }}M_{2}} , if there exists

2072-495: Is called a quasi-isometry from ( M 1 , d 1 ) {\displaystyle (M_{1},d_{1})} to ( M 2 , d 2 ) {\displaystyle (M_{2},d_{2})} if there exist constants A ≥ 1 {\displaystyle A\geq 1} , B ≥ 0 {\displaystyle B\geq 0} , and C ≥ 0 {\displaystyle C\geq 0} such that

2146-437: Is characterized as being the unique contractible 2-manifold . Its dimension is characterized by the fact that removing a point from the plane leaves a space that is connected, but not simply connected . In graph theory , a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such

2220-430: Is defined as: A vector can be pictured as an arrow. Its magnitude is its length, and its direction is the direction the arrow points. The magnitude of a vector A is denoted by ‖ A ‖ {\displaystyle \|\mathbf {A} \|} . In this viewpoint, the dot product of two Euclidean vectors A and B is defined by where θ is the angle between A and B . The dot product of

2294-611: Is divided into smaller categories; triangles can be equilateral , isosceles , obtuse , acute , scalene , etc. while quadrilaterals can be rectangles , rhombi , trapezoids , squares , etc. Other common shapes are points , lines , planes , and conic sections such as ellipses , circles , and parabolas . Among the most common 3-dimensional shapes are polyhedra , which are shapes with flat faces; ellipsoids , which are egg-shaped or sphere-shaped objects; cylinders ; and cones . If an object falls into one of these categories exactly or even approximately, we can use it to describe

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2368-440: Is found in linear algebra , where the idea of independence is crucial. The plane has two dimensions because the length of a rectangle is independent of its width. In the technical language of linear algebra, the plane is two-dimensional because every point in the plane can be described by a linear combination of two independent vectors . The dot product of two vectors A = [ A 1 , A 2 ] and B = [ B 1 , B 2 ]

2442-424: Is given by an ordered pair of real numbers, each number giving the distance of that point from the origin measured along the given axis, which is equal to the distance of that point from the other axis. Another widely used coordinate system is the polar coordinate system , which specifies a point in terms of its distance from the origin and its angle relative to a rightward reference ray. In Euclidean geometry ,

2516-480: 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 the fact that realistic shapes are often deformable, e.g. a person in different postures, a tree bending in the wind or a hand with different finger positions. One way of modeling non-rigid movements

2590-557: Is quasi-isometric. In this case, every function from one space to the other is a quasi-isometry. If f : M 1 ↦ M 2 {\displaystyle f:M_{1}\mapsto M_{2}} is a quasi-isometry, then there exists a quasi-isometry g : M 2 ↦ M 1 {\displaystyle g:M_{2}\mapsto M_{1}} . Indeed, g ( x ) {\displaystyle g(x)} may be defined by letting y {\displaystyle y} be any point in

2664-516: Is the first step in the proof of the Mostow rigidity theorem . Furthermore, this result has found utility in analyzing user interaction design in applications similar to Google Maps . The following are some examples of properties of group Cayley graphs that are invariant under quasi-isometry: A group is called hyperbolic if one of its Cayley graphs is a δ-hyperbolic space for some δ. When translating between different definitions of hyperbolicity,

2738-477: Is therefore congruent to its mirror image (even if it is not symmetric), but not to a scaled version. Two congruent 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

2812-405: Is used in the vulgar sense, and means what one would normally expect it to mean. [...] We here define ‘shape’ informally as ‘all the geometrical information that remains when location, scale and rotational effects are filtered out from an object.’ Shapes of physical objects are equal if the subsets of space these objects occupy satisfy the definition above. In particular, the shape does not depend on

2886-554: The Švarc–Milnor lemma states that if a group G acts properly discontinuously with compact quotient on a proper geodesic space X then G is quasi-isometric to X (meaning that any Cayley graph for G is). This gives new examples of groups quasi-isometric to each other: A quasi-geodesic in a metric space ( X , d ) {\displaystyle (X,d)} is a quasi-isometric embedding of R {\displaystyle \mathbb {R} } into X {\displaystyle X} . More precisely

2960-405: The Euclidean metric ) that sends every n {\displaystyle n} -tuple of integers to itself is a quasi-isometry: distances are preserved exactly, and every real tuple is within distance n / 4 {\displaystyle {\sqrt {n/4}}} of an integer tuple. In the other direction, the discontinuous function that rounds every tuple of real numbers to

3034-1596: The complex plane , z ↦ a z + b , a ≠ 0 , {\displaystyle z\mapsto az+b,\quad a\neq 0,}   a triangle is transformed but does not change its shape. Hence shape is an invariant of affine geometry . The shape p = S( u , v , w ) depends on the order of the arguments of function S, but permutations lead to related values. For instance, 1 − p = 1 − u − w u − v = w − v u − v = v − w v − u = S ( v , u , w ) . {\displaystyle 1-p=1-{\frac {u-w}{u-v}}={\frac {w-v}{u-v}}={\frac {v-w}{v-u}}=S(v,u,w).} Also p − 1 = S ( u , w , v ) . {\displaystyle p^{-1}=S(u,w,v).} Combining these permutations gives S ( v , w , u ) = ( 1 − p ) − 1 . {\displaystyle S(v,w,u)=(1-p)^{-1}.} Furthermore, p ( 1 − p ) − 1 = S ( u , v , w ) S ( v , w , u ) = u − w v − w = S ( w , v , u ) . {\displaystyle p(1-p)^{-1}=S(u,v,w)S(v,w,u)={\frac {u-w}{v-w}}=S(w,v,u).} These relations are "conversion rules" for shape of

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3108-410: The composition of two quasi-isometries is a quasi-isometry, it follows that the property of being quasi-isometric behaves like an equivalence relation on the class of metric spaces. Given a finite generating set S of a finitely generated group G , we can form the corresponding Cayley graph of S and G . This graph becomes a metric space if we declare the length of each edge to be 1. Taking

3182-415: The ellipse . Many three-dimensional geometric shapes can be defined by a set of vertices, lines connecting the vertices, and two-dimensional faces enclosed by those lines, as well as the resulting interior points. Such shapes are called polyhedrons and include cubes as well as pyramids such as tetrahedrons . Other three-dimensional shapes may be bounded by curved surfaces, such as the ellipsoid and

3256-422: The position of each point . It is an affine space , which includes in particular the concept of parallel lines . It has also metrical properties induced by a distance , which allows to define circles , and angle measurement . A Euclidean plane with a chosen Cartesian coordinate system is called a Cartesian plane . The set R 2 {\displaystyle \mathbb {R} ^{2}} of

3330-620: The shape of triangle ( u , v , w ) . Then the shape of the equilateral triangle is 0 − 1 + i 3 2 0 − 1 = 1 + i 3 2 = cos ⁡ ( 60 ∘ ) + i sin ⁡ ( 60 ∘ ) = e i π / 3 . {\displaystyle {\frac {0-{\frac {1+i{\sqrt {3}}}{2}}}{0-1}}={\frac {1+i{\sqrt {3}}}{2}}=\cos(60^{\circ })+i\sin(60^{\circ })=e^{i\pi /3}.} For any affine transformation of

3404-404: The sphere . A shape is said to be convex if all of the points on a line segment between any two of its points are also part of the shape. There are multiple ways to compare the shapes of two objects: Sometimes, two similar or congruent objects may be regarded as having a different shape if a reflection is required to transform one into the other. For instance, the letters " b " and " d " are

3478-763: The base-point sequence to be constant, p n = p for some p ∈ X ; in this case the asymptotic cone does not depend on the choice of p ∈ X and is denoted by C o n e ω ( X , d ) {\displaystyle Cone_{\omega }(X,d)\,} or just C o n e ω ( X ) {\displaystyle Cone_{\omega }(X)\,} . The notion of an asymptotic cone plays an important role in geometric group theory since asymptotic cones (or, more precisely, their topological types and bi-Lipschitz types ) provide quasi-isometry invariants of metric spaces in general and of finitely generated groups in particular. Asymptotic cones also turn out to be

3552-546: The converse is coarsely true, i.e. that every quasi-geodesic stays within bounded distance of a true geodesic, is called the Morse Lemma (not to be confused with the Morse lemma in differential topology). Formally the statement is: It is an important tool in geometric group theory. An immediate application is that any quasi-isometry between proper hyperbolic spaces induces a homeomorphism between their boundaries. This result

3626-425: The discovery. Both authors used a single ( abscissa ) axis in their treatments, with the lengths of ordinates measured along lines not-necessarily-perpendicular to that axis. The concept of using a pair of fixed axes was introduced later, after Descartes' La Géométrie was translated into Latin in 1649 by Frans van Schooten and his students. These commentators introduced several concepts while trying to clarify

3700-462: The endpoints of C and a < b {\displaystyle a<b} . For a vector field F  : U ⊆ R → R , the line integral along a piecewise smooth curve C ⊂ U , in the direction of r , is defined as where · is the dot product and r : [a, b] → C is a bijective parametrization of the curve C such that r ( a ) and r ( b ) give the endpoints of C . A double integral refers to an integral within

3774-399: The endpoints of the curve γ. Let C be a positively oriented , piecewise smooth , simple closed curve in a plane , and let D be the region bounded by C . If L and M are functions of ( x , y ) defined on an open region containing D and have continuous partial derivatives there, then where the path of integration along C is counterclockwise . In topology , the plane

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3848-609: The following two properties both hold: The two metric spaces ( M 1 , d 1 ) {\displaystyle (M_{1},d_{1})} and ( M 2 , d 2 ) {\displaystyle (M_{2},d_{2})} are called quasi-isometric if there exists a quasi-isometry f {\displaystyle f} from ( M 1 , d 1 ) {\displaystyle (M_{1},d_{1})} to ( M 2 , d 2 ) {\displaystyle (M_{2},d_{2})} . A map

3922-617: The ideas contained in Descartes' work. Later, the plane was thought of as a field , where any two points could be multiplied and, except for 0, divided. This was known as the complex plane . The complex plane is sometimes called the Argand plane because it is used in Argand diagrams. These are named after Jean-Robert Argand (1768–1822), although they were first described by Danish-Norwegian land surveyor and mathematician Caspar Wessel (1745–1818). Argand diagrams are frequently used to plot

3996-418: The image of f {\displaystyle f} that is within distance C {\displaystyle C} of x {\displaystyle x} , and letting g ( x ) {\displaystyle g(x)} be any point in f − 1 ( y ) {\displaystyle f^{-1}(y)} . Since the identity map is a quasi-isometry, and

4070-454: The naming convention of the Greek derived prefix with '-gon' suffix: Pentagon, Hexagon, Heptagon, Octagon, Nonagon, Decagon... See polygon In geometry, two subsets of a Euclidean 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

4144-422: The nearest integer tuple is also a quasi-isometry: each point is taken by this map to a point within distance n / 4 {\displaystyle {\sqrt {n/4}}} of it, so rounding changes the distance between pairs of points by adding or subtracting at most 2 n / 4 {\displaystyle 2{\sqrt {n/4}}} . Every pair of finite or bounded metric spaces

4218-448: The object is scaled non-uniformly. For example, a sphere becomes an ellipsoid when scaled differently in the vertical and horizontal directions. In other words, preserving axes of symmetry (if they exist) is important for preserving shapes. Also, shape is determined by only the outer boundary of an object. Objects that can be transformed into each other by rigid transformations and mirroring (but not scaling) are congruent . An object

4292-401: The order of polynomial growth k 0 {\displaystyle k_{0}} has to be a natural number and in fact # ( n ) ∼ n k 0 {\displaystyle \#(n)\sim n^{k_{0}}} . If # ( n ) {\displaystyle \#(n)} grows more slowly than any exponential function, G has

4366-502: The ordered pairs of real numbers (the real coordinate plane ), equipped with the dot product , is often called the Euclidean plane or standard Euclidean plane , since every Euclidean plane is isomorphic to it. Books I through IV and VI of Euclid's Elements dealt with two-dimensional geometry, developing such notions as similarity of shapes, the Pythagorean theorem (Proposition 47), equality of angles and areas , parallelism,

4440-419: The particular value of δ may change, but the resulting notions of a hyperbolic group turn out to be equivalent. Hyperbolic groups have a solvable word problem . They are biautomatic and automatic .: indeed, they are strongly geodesically automatic , that is, there is an automatic structure on the group, where the language accepted by the word acceptor is the set of all geodesic words. The growth rate of

4514-407: 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 . Some common shapes include: Circle , Square , Triangle , Rectangle , Oval , Star (polygon) , Rhombus , Semicircle . Regular polygons starting at pentagon follow

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4588-647: The points ( 1 , 0 ) , ( − 1 , 0 ) , ( 0 , 1 ) , ( 0 , − 1 ) {\displaystyle (1,0),(-1,0),(0,1),(0,-1)} are of equal distance to each other in Manhattan distance, but in the Euclidean plane, there are no 4 points that are of equal distance to each other. The map f : Z n → R n {\displaystyle f:\mathbb {Z} ^{n}\to \mathbb {R} ^{n}} (both with

4662-427: The positions of the poles and zeroes of a function in the complex plane. In mathematics, analytic geometry (also called Cartesian geometry) describes every point in two-dimensional space by means of two coordinates. Two perpendicular coordinate axes are given which cross each other at the origin . They are usually labeled x and y . Relative to these axes, the position of any point in two-dimensional space

4736-542: The same unit of length . Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin , usually at ordered pair (0, 0). The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin. The idea of this system was developed in 1637 in writings by Descartes and independently by Pierre de Fermat , although Fermat also worked in three dimensions, and did not publish

4810-404: The same vertex arrangements of the convex regular polygons. In general, for any natural number n, there are n-pointed non-convex regular polygonal stars with Schläfli symbols { n / m } for all m such that m < n /2 (strictly speaking { n / m } = { n /( n − m )}) and m and n are coprime . The hypersphere in 2 dimensions is a circle , sometimes called a 1-sphere ( S ) because it

4884-406: The same number of ends. In particular, two quasi-isometric finitely generated groups have the same number of ends. An amenable group is a locally compact topological group G carrying a kind of averaging operation on bounded functions that is invariant under translation by group elements. The original definition, in terms of a finitely additive invariant measure (or mean) on subsets of G ,

4958-422: The same shape, or to measure the difference between two shapes. In advanced 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 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

5032-481: The same shape. These shapes can be classified using complex numbers u , v , w for the vertices, in a method advanced by J.A. Lester and Rafael Artzy . For example, an equilateral triangle can be expressed by the complex numbers 0, 1, (1 + i√3)/2 representing its vertices. Lester and Artzy call the ratio S ( u , v , w ) = u − w u − v {\displaystyle S(u,v,w)={\frac {u-w}{u-v}}}

5106-453: The shape of the object. Thus, we say that the shape of a manhole cover is a disk , because it is approximately the same geometric object as an actual geometric disk. A geometric shape consists of 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

5180-403: The size and placement in space of the object. For instance, a " d " and a " p " have the same shape, as they can be perfectly superimposed if the " d " is translated to the right by a given distance, rotated upside down and magnified by a given factor (see Procrustes superimposition for details). However, a mirror image could be called a different shape. For instance, a " b " and

5254-432: The sum of the angles in a triangle, and the three cases in which triangles are "equal" (have the same area), among many other topics. Later, the plane was described in a so-called Cartesian coordinate system , a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates , which are the signed distances from the point to two fixed perpendicular directed lines, measured in

5328-460: The way natural shapes vary. There is also clear evidence that shapes guide human attention . Plane (geometry) In mathematics , a Euclidean plane is a Euclidean space of dimension two , denoted E 2 {\displaystyle {\textbf {E}}^{2}} or E 2 {\displaystyle \mathbb {E} ^{2}} . It is a geometric space in which two real numbers are required to determine

5402-425: The work of Gromov . Suppose that f {\displaystyle f} is a (not necessarily continuous) function from one metric space ( M 1 , d 1 ) {\displaystyle (M_{1},d_{1})} to a second metric space ( M 2 , d 2 ) {\displaystyle (M_{2},d_{2})} . Then f {\displaystyle f}

5476-663: Was introduced by John von Neumann in 1929 under the German name "messbar" ("measurable" in English) in response to the Banach–Tarski paradox . In 1949 Mahlon M. Day introduced the English translation "amenable", apparently as a pun. In discrete group theory , where G has the discrete topology , a simpler definition is used. In this setting, a group is amenable if one can say what proportion of G any given subset takes up. If

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