Shape

A shape is a graphical representation of an object's form. It describes the external boundary, outline, or surface of an object.

Importantly, shape is distinct from other properties like color, texture, or material. In the field of geometry, a shape excludes information about an object's position, size, orientation, or chirality. While a shape focuses on form, a figure includes both shape and size. For example, a figure of the Earth provides specific dimensions that a general spherical shape does not.

Geometric shapes are often categorized by their dimensions. Plane shapes, or two-dimensional figures, are constrained to lie on a plane. However, a 2D shape may also exist on a curved surface. In contrast, three-dimensional shapes are solid objects that occupy space.

These 3D shapes can be polyhedra, which have flat faces, or ellipsoids, which are egg-shaped. Other common 3D forms include cylinders and cones. We often use these geometric terms to describe real-world objects. For instance, we say a manhole cover is shaped like a disk because it approximates that geometric object.

Simple shapes can be classified into several broad categories. Polygons are shapes made of straight lines that form a closed chain. We name polygons based on their number of edges. Triangles have three edges, quadrilaterals have four, pentagons have five, and hexagons have six. This naming convention continues with heptagons, octagons, nonagons, and decagons. Within these groups, there are smaller sub-categories. Triangles can be equilateral, isosceles, obtuse, acute, or scalene. Quadrilaterals include rectangles, rhombi, trapezoids, and squares.

Other fundamental shapes include points, lines, planes, and conic sections like circles, ellipses, and parabolas.

Mathematicians use specific terms to compare how objects relate to one another. Two objects are congruent if one can become the other through rotations, translations, or reflections. This means they have the same shape and the same size.

Two objects are similar if they can be transformed through uniform scaling along with rotations, translations, or reflections. Similarity means they share the same shape, even if their sizes differ. There is also a concept called isotopy, where objects are transformed through deformations. These deformations must not tear the object or create new holes. A more flexible version of this is a homeomorphism. A homeomorphism is a continuous stretching and bending. Under this rule, a square and a circle are considered homeomorphic.

Advanced mathematical study allows for even more precise comparisons. Procrustes analysis is a technique used in many sciences to determine if two objects have the same shape. It can also measure the difference between two shapes. Scientists use this to compare things like the bones of different animals. In higher mathematics, quasi-isometry serves as a criterion to state that shapes are approximately the same. For complex shapes like plant structures or coastlines, mathematicians use differential geometry or fractals. These methods help describe forms that defy traditional, simple mathematical descriptions.

In the study of polygons, researchers have developed ways to define shape using complex numbers. Mathematicians J.A. Lester and Rafael Artzy advanced methods to describe the shape of triangles and quadrilaterals this way. For a triangle, the shape can be expressed as a ratio of its vertices. For a quadrilateral, the shape is associated with two complex numbers. These mathematical tools can identify specific types of shapes. For example, if certain conditions are met, a quadrilateral is proven to be a parallelogram or a rhombus. A polygon is considered to bound a convex set if its shape components meet specific mathematical signs.

Shape is a fundamental concept that connects geometry to how we perceive the world. Human vision relies on many different ways of representing shape. Some psychologists believe humans mentally break down complex images into simple geometric shapes. They call these basic units geons, which include forms like cones and spheres. Whether we are looking at a simple square or a complex coastline, our brains are constantly processing these boundaries. Understanding the math behind these boundaries helps us understand the structure of the physical universe.