# Helical boundary conditions

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In mathematics, **helical boundary conditions** are a variation on periodic boundary conditions. Helical boundary conditions provide a method for determining the index of a lattice site's neighbours when each lattice site is indexed by just a single coordinate. On a lattice of dimension *d* where the lattice sites are numbered from 1 to *N* and *L* is the width (i.e. number of elements per row) of the lattice in all but the last dimension, the neighbors of site *i* are:

It is not necessary that *N* = *L*^{d}. Helical boundary conditions make it possible to use only one coordinate to describe arbitrary-dimensional lattices.

## References

*Monte Carlo Methods in Statistical Physics*, Mark E. J. Newman, G. T. Barkema