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# Shape

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 Title: Shape Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### Shape

An example of the different definitions of shape. The two triangles on the left are congruent, while the third is similar to them. The last triangle is neither similar nor congruent to any of the others, but it is homeomorphic.

A shape is the form of an object or its external boundary, outline, or external surface, as opposed to other properties such as color, texture, or material composition.

Psychologists have theorized that humans mentally break down images into simple geometric shapes called geons.[1] Examples of geons include cones and spheres.

## Contents

• Classification of simple shapes 1
• Shape in geometry 2
• Equivalence of shapes 2.1
• Congruence and similarity 2.2
• Homeomorphism 2.3
• Shape analysis 3
• Similarity classes 4
• References 6

## Classification of simple shapes

A variety of polygonal 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 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 the shape of the object. Thus, we say that the shape of a manhole cover is a circle, because it is approximately the same geometric object as an actual geometric circle.

## Shape in geometry

There are several ways to compare the shape of two objects:

• Congruence: Two objects are congruent if one can be 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, an 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 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 arbitrary as to defy traditional mathematical description – in which case they may be analyzed by differential geometry, or as fractals.

### 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