Nfold

nfold is a mathematical and descriptive term used to indicate multiplicity or repetition by a positive integer n. It functions as an adjective describing a structure, operation, or construction that occurs n times or consists of n identical components. The precise meaning of nfold depends on the context, but common interpretations include repetition, replication, or a fiber of size n.

In geometry and symmetry, n-fold often refers to rotational symmetry of order n. An object with n-fold rotational symmetry is invariant under rotations by 360/n degrees and is typically associated with cyclic groups of order n. Regular n-gons are classic examples of shapes exhibiting n-fold rotational symmetry.

In topology and geometry, the concept appears in coverings. An n-fold covering map is a continuous surjection where each point in the base has exactly n preimages in the covering space (locally). A standard example is the map z ↦ z^n from the unit circle to itself, which is an n-fold covering.

In algebra, n-fold describes constructions that involve n factors. The n-fold direct product of a vector space V is V × V × ... × V (n times). The n-fold tensor product V ⊗ V ⊗ ... ⊗ V (n factors) is another common usage, forming higher-order tensor spaces.

In analysis and probability, n-fold often denotes operations applied repeatedly, such as the n-fold convolution f^{*n} of a function with itself, or the sum of n independent copies of a random variable, or n-fold integrals over n variables.

See also: k-fold, onefold.