nonsimilarity

Nonsimilarity is the property of two objects or figures that are not similar, meaning there is no similarity transformation that can map one onto the other. In Euclidean geometry, a similarity transformation consists of translation, rotation, reflection, and uniform scaling.

For polygons, two polygons are similar if their corresponding angles are equal and their corresponding sides are in proportion. If these conditions fail, the polygons are nonsimilar. In particular, if at least one corresponding angle differs, the polygons cannot be similar. Circles are an exception in a sense: all circles are similar to each other because any circle can be obtained from another by scaling; thus any two circles are not described as nonsimilar.

Examples illustrate the concept. A 30-60-90 triangle and a 45-45-90 triangle are not similar. A rectangle and a triangle are not similar. Among polygons, two pentagons with different angle measures and side ratios are nonsimilar unless their corresponding features match in the proportional way required by similarity.

Beyond strict geometry, nonsimilarity is used descriptively in classification, pattern recognition, and education to denote a lack of shape equivalence under the standard similarity transformations. It serves to distinguish objects that share no shape-preserving correspondence from those that do.

Relation to other ideas: nonsimilarity is the negation of the similarity relation. It complements concepts such as congruence (exact equality of size and shape) and general shape descriptors used in analysis and computer vision. See also similarity (geometry), transformation, congruence, and shape descriptor.