Lattices in Mathematics?

Question

Can someone explain lattices, and pentagonal lattices please, in a language I can understand? I read so much about them and I don't see one comprehensible example or definition.

Answer 1

There are two main notions of "lattice" in mathematics. The first (and not the one you are asking about) definition of a lattice is that it is a set with a "partial ordering", meaning that given two elements of the set, sometimes you can compare them and sometimes not. To be a lattice, you must have a notion of least common multiple and greatest common divisor. The precise definition is technical, and it's better to think of a simple (yet rather general) example:

Consider a cube. The cube has 8 vertices, 12 edges, 6 faces, and one "three dimensional face", namely the whole cube. It also has one "negative one dimensional fact", namely the empty set. Now we can define a partial order on the collection
{empty set together with vertices, edges, faces, and the cube} by saying that one is less than the other if it is a subset.

However, you probably mean the second definition, which is even more complicated. A lattice is a "finite co-volume discrete subgroup of a Lie group". Yikes. So, what's an example? Well, the "integer lattice in the reals is just the integers drawn on the real number line. Another example is the set of points with integer coordinates in the plane. The points look like they form a "square" lattice in the sense that if you join points to their nearest neighbors in the lattice, you get a square grid.

To get a pentagonal lattice, you should look in not the ordinary Euclidean plane but in the hyperbolic plane. You can tile the hyperbolic plane by petagons which have right angles at every one of their vertices. The vertices of this triangulation is an example of a pentagonal lattice.

Answer 2

this is just a bunch of lines in the planes
the easiest one is
the one made of vertical lines with distance 1 between any 2 consecutive lines, and of horizontal lines, also with distance 1 between any 2 consecutive lines.
then you get the plane divided into all these squares.
now i guess you could do it in such a way that you get pentagones instead of squares or paralelograms

Answer 3

http://mathworld.wolfram.com/Lattice.html