Definitions copied from various sources or written by me to explain the different variants and flavors of things that are in the family of lattices/grids/tilings.
This topic is of interest because not only because of the behavior of grid cells, but because overlapping receptive fields of neurons create similar structures as tilings, coverings, or packings.
Lattice
- an infinite array of discrete points generated by a set of discrete translation operations described in three dimensional space
- an infinite set of points in this space with the properties that coordinate-wise addition or subtraction of two points in the lattice produces another lattice point, that the lattice points are all separated by some minimum distance, and that every point in the space is within some maximum distance of a lattice point
Grid
- a grid or mesh is a regular tessellation of a manifold or 2-D surface that divides it into a series of contiguous cells, which can then be assigned unique identifiers and used for spatial indexing purpose
- regular grid is a tessellation of n-dimensional Euclidean space by congruent parallelotopes (e.g. bricks).
- unstructured grid or irregular grid is a tessellation of a part of the Euclidean plane or Euclidean space by simple shapes, such as triangles or tetrahedra, in an irregular pattern.
Mesh
- a polygon mesh is a collection of vertices, edges and faces that defines the shape of a polyhedral object.
Tessellation
- A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called tiles, with no overlaps and no gaps.
Tiling
- Like tessellation. A periodic tiling has a repeating pattern. Some special kinds include regular tilings with regular polygonal tiles all of the same shape, and semiregular tilings with regular tiles of more than one shape and with every corner identically arranged. The patterns formed by periodic tilings can be categorized into 17 wallpaper groups. A tiling that lacks a repeating pattern is called “non-periodic”.
Partition
- a partition of a set is a grouping of its elements into non-empty subsets, in such a way that every element is included in exactly one subset.
Covering
- In combinatorics and computer science, covering problems are computational problems that ask whether a certain combinatorial structure ‘covers’ another, or how large the structure has to be to do that.
Packing
- Packing problems are a class of optimization problems in mathematics that involve attempting to pack objects together into containers. The goal is to either pack a single container as densely as possible or pack all objects using as few containers as possible.
Manifold decomposition
- In topology, a branch of mathematics, a manifold M may be decomposed or split by writing M as a combination of smaller pieces. When doing so, one must specify both what those pieces are and how they are put together to form M.