similarities

Similarities refer to attributes or relations that make two or more entities resemble each other in some respect. They are central in science, mathematics, and everyday judgment, and they arise whenever comparisons are made across objects, ideas, or data.

In mathematics, two figures are similar if one can be obtained from the other by a similarity transformation, which includes resizing, rotation, reflection, and translation. In such cases corresponding angles are equal and corresponding sides are proportional, with a scale factor describing size change. Congruence implies equality of size, while similarity allows proportionality without identical size.

In data analysis and computing, similarity is quantified by measures such as cosine similarity between vectors, the Jaccard index for sets, the Dice coefficient, and Pearson correlation. These metrics underpin clustering, information retrieval, and recommender systems. Similarity can be derived from distance; many tools convert a distance measure into a similarity score.

In perception and cognition, similarity judgments guide categorization and recognition. Gestalt principles describe grouping by shared features like color, shape, or texture, while psychological theories examine how people assess likeness under different contexts and scales of observation.

Linguistics and biology also use similarity: lexical or phonetic similarity among words, and genetic sequence similarity used in phylogenetics and functional prediction. These assessments help identify relatedness, approximate function, or evolutionary relationships.

Because similarity is context-dependent, a high degree of likeness in one feature does not guarantee identity or equivalence in another. Caution is warranted when drawing conclusions from similarity alone, and it is common to consider both similarities and differences.