Lab · Groundwork

From relations between nodes to a geometry

A graph is only a list of who is linked to whom. Here is how you get distance, dimension, curvature and area from that list, by counting. You need no maths beyond counting: every number is computed live from the links, and no coordinates are used.

Use Next. This explains the method on spaces whose answer is known, then applies it to our graph. It does not claim that our graph is a space.

What to take away

Geometry can be counted. Distance is hops, dimension is how fast a ball fills up, curvature is the gap from the flat count.
Entanglement gives the area. If links are entangled pairs, the links on a border count its entropy.
A number is not a space. A value of d means something only if it stays the same at larger sizes and does not depend on details of the rule.
Time is a separate step. Everything up to curvature is space. Time needs events and their order.

Did we make these rules up?

Mostly no, and that is what makes them testable. Each one has a simple reason, and each can give a wrong answer.

Distance: the fewest steps. Distance is what it takes to get somewhere. If influence moves along links, the fewest links is the natural measure. It also obeys the properties any distance must have, by mathematics and not by choice.
Dimension: how fast a ball fills. On a line, doubling the radius doubles what you enclose; on a plane it multiplies by four; in a volume by eight. This idea is old mathematics, used well before graphs.
Curvature: the gap from the flat count. A classical result: on a curved surface, small circles enclose less or more than on a plane, and that gap measures the curvature. You can tell the Earth is curved without leaving it.
Area: the links on a border. For a pair of entangled qubits this is a theorem. Saying that this area also behaves like an area in gravity is a hypothesis.

Some choices are ours, and we keep them in sight: what counts as a link when the state is not made of simple pairs, the thresholds of the tests, and the choice of ignoring link strengths. Other reasonable choices could give different numbers. So we test them first on spaces whose answer is known: if the ruler fails on a grid, we do not use it on our graph.