Definition

Noun: 1. Similarity in form or structure: In biology, the state of having the same or a very similar form, shape, or structure between different organisms or parts. 2. Equivalence in structure: In mathematics, a one-to-one correspondence between two mathematical structures (such as groups, rings, or vector spaces) that preserves all the relationships between the elements. It indicates the structures are essentially the same from a structural perspective.

Usage Examples

• In Biology:
  • The isomorphism between the wings of a bat and the flippers of a whale is an example of convergent evolution.
  • Scientists studied the isomorphism in the skeletal structures of different vertebrate species.
• In Mathematics:
  • The two groups are related by an isomorphism, meaning they have identical algebraic properties.
  • Proving the isomorphism between these two vector spaces was the key to solving the problem.

Advanced Usage

• "Up to isomorphism": A common phrase in mathematics meaning that two objects are considered the same if there exists an isomorphism between them, ignoring any superficial differences in labeling or presentation.
  • In group theory, there is only one cyclic group of order 5, up to isomorphism.

Variants and Related Words

• Isomorphic (adjective): Having the same form or structure; related by an isomorphism.
  • The two graphs are isomorphic.
  • The chemical compounds show isomorphic crystal forms.
• Isomorph (noun): One of two or more isomorphic substances or forms.
• Isomorphous (adjective): Synonym for isomorphic, often used in crystallography and chemistry.
  • The minerals are isomorphous and can form a solid solution series.

Synonyms

• Similarity: Resemblance in appearance or nature (general term).
• Equivalence: The condition of being equal or interchangeable in value, function, or meaning (closest in mathematics).
• Correspondence: A close similarity, connection, or equivalence.

Antonyms

• Polymorphism: The occurrence of something in several different forms.
• Heteromorphism: The condition of having different forms.
• Anisomorphism: Lack of isomorphism; dissimilarity in form.

Noun

1. (biology) similarity or identity of form or shape or structure