2nfold

2nfold is a term used in mathematics and related fields as an adjective to indicate multiplicity by a factor of 2n. It describes objects that consist of 2n equivalent parts, copies, or preimages, or structures that exhibit a symmetry or repetition with period 360/(2n) degrees. The phrase is common in discussions of coverings, tilings, and combinatorial constructions, where the numeric factor 2n specifies the degree or order of repetition.

In topology, a 2n-fold cover refers to a covering map p: E -> B in which every point of B has exactly 2n preimages in E. The degree of the covering is 2n. Connected 2n-fold covers correspond to subgroups of the fundamental group of B with index 2n, reflecting a standard relationship between covering spaces and group theory. This framework is used to study how spaces branch and how their fundamental group encodes possible multi-sheeted unwinding.

In geometry and symmetry, 2n-fold often describes rotational symmetry of order 2n. Objects with 2n-fold rotational symmetry are invariant under rotation by 360 degrees divided by 2n. For example, a regular 2n-gon has a rotational symmetry group of order 2n, and patterns or tessellations with 2n-fold symmetry repeat every 360/(2n) degrees. The full dihedral symmetry of such a polygon includes reflections as well, yielding broader symmetry groups.

In combinatorics and enumeration, 2n-fold may describe constructions built from 2n identical components or processes that are replicated n times in a doubling fashion. The exact meaning can vary with context, but the core idea is multiplicity or symmetry by the factor 2n. See also k-fold, 2-fold, and covering space.