Changes

A blank sheet of paper can be represented in linear text as a blank space, but that way of doing it tends to be confusing unless the logical expression under consideration is set off in a separate display.

A blank sheet of paper can be represented in linear text as a blank space, but that way of doing it tends to be confusing unless the logical expression under consideration is set off in a separate display.

For example, consider the axiom drawn below:

For example, consider the axiom or formal initial that is shown below:

[[Image:Logical_Graph_(()).jpg|center]]

[[Image:Logical_Graph_(()).jpg|center]]

This can be written in linear text as "(( )) = ", or set off in the following way:

This can be written inline as “&nbsp;<math>(~(~)~)~=</math>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;” or set off in a text display:

::::: (( )) =

(~(~)~) & = & & . \\

When we turn to representing the corresponding expressions in computer memory, where they can be manipulated with utmost facility, we begin by transforming the planar graphs into their [[duality (mathematics)|topological dual]]s.  The planar regions of the original graph correspond to nodes (or points) of the [[dual graph]], and the boundaries between planar regions in the original graph correspond to edges (or lines) between the nodes of the [[dual graph]].

When we turn to representing the corresponding expressions in computer memory, where they can be manipulated with utmost facility, we begin by transforming the planar graphs into their [[duality (mathematics)|topological dual]]s.  The planar regions of the original graph correspond to nodes (or points) of the [[dual graph]], and the boundaries between planar regions in the original graph correspond to edges (or lines) between the nodes of the [[dual graph]].