Imagine an integer lattice and the only way to move around is to jump from node to node by horizontal and vertical movements. Diagonal movements are not allowed.
Think of a Taxicab on the Manhattan street grid. It certainly can't drive diagonally through a block!
So the node point (m,n) is at a distance m+n from the origin (m and n are integers of course) and the metric on the space is therefore d(m,n)=m+n
A "circle" centered on the origin is a diamond and pi=4 exactly.
In this grid each node is connected to exactly 4 others and pi=4. Is this just a coincidence or a fundamental result?
Is there a grid where each node is connected to exactly 3 others? Yes, a hexagonal grid has this. And pi=3.
The General Case This geometry is not constrained to grids (rectangular, hexagonal or otherwise). So long as the connectivity is correct the whole thing could be a mesh piled in a giant heap on the floor!
There are just two rules: You can only move between nodes along a connection, and the (minimum) number of connections between 2 nodes is the distance between the nodes.
It's incredibly general. It's topological. It's all about connectivity. Examples: the "rectangular grid" is a mesh with connectivity=4 and the "hexagonal grid" is a mesh with connectivity=3.
A fun way to imagine this geometry: Think of space as a fishing net. The knots in the net are the points of space, the snippets of reality. And the rope between the knots are the connections. It does not matter how you handle the net - throw it on the ground in a heap if you wish - the topology is unchanged. So reality is unchanged.
Could we build physics on such a mesh and if so what would it look like?
Content written and posted by Ken Abbott firstname.lastname@example.org