degenerate

Degenerate is an adjective used to describe something that has lost its normal or desirable function, form, or vitality. It can refer to a person, object, or situation. In moral or social language it may carry pejorative connotations, signaling decline or corruption. The term also appears in scientific usage to denote a limiting or special case.

In geometry and combinatorics, a degenerate case is a configuration that collapses into a simpler form, often of lower dimension. Examples include a degenerate triangle whose vertices are collinear (area zero); a degenerate conic that factors into lines or a point; a degenerate polygon with zero area.

In linear algebra, a degenerate matrix has less than full rank and typically zero determinant; degenerate eigenvalues occur when two or more eigenvectors share the same eigenvalue. In probability, a degenerate distribution is concentrated at a single value. These degenerate structures are important in describing singular behavior or symmetry.

In physics, degeneracy refers to distinct quantum states sharing the same energy level, as in degenerate energy levels of atoms. Degenerate matter describes highly compressed fermion systems where quantum statistics dominate, such as in white dwarfs and neutron stars; degeneracy pressure arises from the Pauli exclusion principle.

In biology and genetics, degeneracy occurs in the sense of the genetic code’s redundancy: multiple codons can encode the same amino acid. In broader evolutionary biology, degeneracy can describe functional redundancy within a system, contributing to robustness.

The term also appears in cultural and historical contexts, such as the label degenerate art used by certain regimes to condemn modern artistic movements.