A shape is a graphical representation of an object's form or its external boundary, outline, or external surface. It is distinct from other object properties, such as color, texture, or material type. In geometry, shape excludes information about the object's position, size, orientation and chirality. A figure is a representation including both shape and size (as in, e.g., figure of the Earth). 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).
Classification of simple shapes
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 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 three-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 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.
In geometry
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 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 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 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.
Properties There are multiple ways to compare the shapes of two objects:
Congruence: Two objects are congruent if one can be transformed into the other by a sequence of rotations, translations, and/or reflections. Similarity: Two objects are similar if one can be transformed into the other by a uniform scaling, together with a sequence of rotations, translations, and/or reflections. Isotopy: Two objects are isotopic if one can be transformed into the other by a sequence of deformations that do not tear the object or put holes in it.
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 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 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 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 the naming convention of the Greek derived prefix with '-gon' suffix: Pentagon, Hexagon, Heptagon, Octagon, Nonagon, Decagon... See polygon
Equivalence of shapes 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 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:
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