**Taxicab Geometry**

Imagine a rectangular lattice and the only way to move around is to go 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

We define a circle as usual - a set of point equidistant from a given point. But remember, we now measure distance using our metric. So a circle centered on the origin is a diamond. We define pi as usual as periphery/diameter and we get pi=4 exactly.

In this grid each node is connected to exactly 4 others and pi=4. Is this a coincidence?

Is there a grid where each node is connected to exactly 3 others? Yes, a hexagonal grid has this, and pi=3 exactly.

Is there a grid where each node is connected to exactly 6 others? Yes, a triangular grid has this, and pi=6 exactly.

So the first thing we learn is that the value of pi is dependent on the metric used. In this metric pi=connectivity.

Physics uses the metric d(m,n)=sqrt(m^2+n^2). What does this mean for pi?

**The General Case**This geometry is not constrained to grids (rectangular, hexagonal, triangular 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 metric is d(m,n)=m+n.

It's very general. It's topological. It's all about connectivity.

A fun way to imagine this geometry: Think of this space as a fishing net. The knots in the net are the points of space 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.

Thought: Could we build physics on such a space and if so what would it look like?

Content written and posted by Ken Abbott abbottsystems@gmail.com