A tensor which has the same components in all rotated coordinate systems. All rank-0 tensors (scalars) are isotropic, but no rank-1 tensors (vectors) are. The unique rank-2 isotropic tensor is the Kronecker delta, and the unique rank-3 isotropic tensor is the permutation symbol. The number of isotropic tensors of rank 0, 1, 2, ... are 1, 0, 1, 1, 3, 6, 15, 36, 91, 232, ... (OEIS A005043). These numbers are called the Motzkin sum numbers and are given by the recurrence relation a_n = (n - 1)/(n + 1)(2a_(n - 1) + 3a_(n - 2)) with a(0) = 1 and a(1) = 0.

Kronecker delta | permutation tensor | scalar | syzygy | tensor | vector