Changes

A blank sheet of paper can be represented as a blank space in a line of text, 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 as a blank space in a line of text, 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 in box form below:

For example, consider an equation of the following form:

{| align="center" cellpadding="10"

| [[Image:Logical_Graph_Figure_3_Visible_Frame.jpg|500px]]

|}

` ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` `

` ` ` ` ` | | ` ` ` | | ` ` ` = ` ` ` ` ` ` ` ` ` `

` ` ` ` ` | | ` ` ` | | ` ` ` ` ` ` ` ` ` ` ` ` ` `

` ` ` ` ` | o-------o | ` ` ` ` ` ` ` ` ` ` ` ` ` `

We can write this in line as "(()) = " or set it off as follows:

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

: (( )) =

| width="33%" | <math>(~(~)~)</math>

| width="33%" | &nbsp;

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 topological duals.  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 topological duals.  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.