Definition

1. Adjective:
   • Having identical or similar form, shape, or structure. In mathematics and sciences, it describes a one-to-one correspondence between two structures that preserves their essential properties and relationships.
   • Having similar appearance but genetically different. In biology, it can refer to organisms or structures that look alike but have different origins or genetic makeup.

Usage Examples

• Adjective:
  • The two chemical compounds are isomorphic, sharing the same crystalline structure.
  • In graph theory, the two networks are isomorphic; you can map every node from one to the other while preserving all connections.
  • The biologist noted that the wings of bats and birds are isomorphic, serving similar functions but evolving from different ancestral structures.

Advanced Usage

• "Isomorphic to": Used to state that one mathematical object is isomorphic to another.
  • The group of rotations of a square is isomorphic to the cyclic group of order four.
• "Isomorphic embedding": In mathematics, an injective homomorphism that preserves structure.
  • The paper discusses an isomorphic embedding of one vector space into another.

Variants and Related Words

• Isomorphism (noun): The condition or property of being isomorphic; the specific mapping that demonstrates the correspondence.
  • The isomorphism between the two algebraic systems was proven in the theorem.
• Isomorph (noun): A substance or organism that is isomorphic with another.
  • Quartz and cristobalite are isomorphs of silicon dioxide.

Synonyms

• Structurally identical
• Congruent (in specific mathematical contexts)
• Homomorphic (similar but not necessarily one-to-one; a broader term)

Antonyms

• Heteromorphic (differing in form or structure)
• Non-isomorphic

Related Phrases and Concepts

• "Isomorphic graphs": In discrete mathematics, graphs that are identical in their connectivity when vertex labels are ignored.
  • The algorithm checks if the two input graphs are isomorphic graphs.
• "Isomorphic representation": In computer science and mathematics, a way of representing data or a system that maintains its structural relationships.
  • The software uses an isomorphic representation to convert data between different formats without loss of information.

Adjective

1. having similar appearance but genetically different