Geometriclikeness

Geometriclikeness is a concept in geometry that describes when two figures share the same shape, regardless of size. More formally, two nondegenerate figures are geometrically alike, or similar, if there exists a similarity transformation that maps one onto the other.

A similarity transformation in the plane combines translation, rotation, reflection, and uniform scaling. Such transformations preserve angles and multiply all lengths by a common factor. Because of this, they preserve the shape of figures while allowing change of size. Geometriclikeness thus partitions figures into similarity classes, where every member is a scaled, rotated, translated, or reflected version of any other member.

For polygons, two polygons are similar if their corresponding angles are equal and the ratios of corresponding side lengths are constant. The correspondence of vertices must be maintained, and the orientation may be preserved or reversed. Circles are a notable exception: every circle is similar to every other circle, since all have the same shape under appropriate scaling.

Geometriclikeness is distinct from congruence. Congruent figures are identical in shape and size (distance-preserving), while similar figures may differ in size but retain the same form. In higher dimensions, the same idea applies to solids and manifolds, with similarity transformations including uniform scaling in multiple directions.

Applications of geometriclikeness appear in geometry education, computer graphics, pattern recognition, architecture, and design, where recognizing or generating shapes up to scale is important. It provides a framework for classifying shapes by form rather than by size.