Geometry’s shape ignores size, place, and mirror flips
Shape sketches an object’s form or outline, apart from colour, texture, or material. In geometry it also strips away position, size, orientation, and chirality—move, enlarge, rotate, or reflect and the shape stays. A figure keeps size with shape, as in “figure of the Earth.”
Plane figures are confined to a flat plane, whereas a two-dimensional figure in general may sit on a curved surface. Everyday classification starts with polygons counted by edges, then splits triangles into equilateral, isosceles, scalene, acute or obtuse and quadrilaterals into rectangles, rhombi and trapezoids. Regular polygons from five sides up take a Greek prefix and the suffix -gon, from pentagon to decagon. Points, lines, planes and conic sections such as circles, ellipses and parabolas round out the basic kit, and common solids include polyhedra, ellipsoids, cylinders and cones.
Comparisons come in grades. Congruent objects match after rotations, translations and reflections, similar ones after uniform scaling as well, and isotopic ones after any deformation that neither tears nor punches holes. Context can matter: b and d are mirror images and therefore congruent, yet readers may not treat them as the same shape. A shape is convex when the segment joining any two of its points stays inside it.
Statistician David George Kendall defined shape informally as the geometric information left once location, scale and rotation are filtered out. By that rule a d and a p share one shape, since sliding, flipping upside down and enlarging the d lays it perfectly over the p. Procrustes analysis uses such superimposition to compare, say, the bones of different animals.
Real objects often bend, like a person changing posture or a tree in wind, so flexible definitions use homeomorphisms, continuous stretching without tearing. Under them a square and a circle count as equivalent but a sphere and a doughnut do not, hence the joke that topologists cannot tell a coffee cup from a doughnut. Coastlines and plant structures can be too intricate for classical description and are studied with differential geometry or as fractals.
Source: Shape
