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→Duality : logical and topological: del unnecessary specs
For example, overlaying the corresponding [[dual graph]]s on the plane-embedded graphs shown above, we get the following composite picture:
For example, overlaying the corresponding [[dual graph]]s on the plane-embedded graphs shown above, we get the following composite picture:
{| align="center" border="0" cellpadding="10" cellspacing="0"
{| align="center" cellpadding="10"
| [[Image:Logical_Graph_Figure_4_Visible_Frame.jpg|500px]] || (4)
| [[Image:Logical_Graph_Figure_4_Visible_Frame.jpg|500px]] || (4)
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Extracting the dual graphs from their composite matrices, we get this picture:
Extracting the dual graphs from their composite matrices, we get this picture:
{| align="center" border="0" cellpadding="10" cellspacing="0"
{| align="center" cellpadding="10"
| [[Image:Logical_Graph_Figure_5_Visible_Frame.jpg|500px]] || (5)
| [[Image:Logical_Graph_Figure_5_Visible_Frame.jpg|500px]] || (5)
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In the case of our last example, a moment's contemplation of the following picture will lead us to see that we can get the corresponding parenthesis string by starting at the root of the tree, climbing up the left side of the tree until we reach the top, then climbing back down the right side of the tree until we return to the root, all the while reading off the symbols, in this case either "<math>(\!</math>" or "<math>)\!</math>", that we happen to encounter in our travels.
In the case of our last example, a moment's contemplation of the following picture will lead us to see that we can get the corresponding parenthesis string by starting at the root of the tree, climbing up the left side of the tree until we reach the top, then climbing back down the right side of the tree until we return to the root, all the while reading off the symbols, in this case either "<math>(\!</math>" or "<math>)\!</math>", that we happen to encounter in our travels.
{| align="center" border="0" cellpadding="10" cellspacing="0"
{| align="center" cellpadding="10"
| [[Image:Logical_Graph_Figure_6_Visible_Frame.jpg|500px]] || (6)
| [[Image:Logical_Graph_Figure_6_Visible_Frame.jpg|500px]] || (6)
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First the plane-embedded maps:
First the plane-embedded maps:
{| align="center" border="0" cellpadding="10" cellspacing="0"
{| align="center" cellpadding="10"
| [[Image:Logical_Graph_Figure_7_Visible_Frame.jpg|500px]] || (7)
| [[Image:Logical_Graph_Figure_7_Visible_Frame.jpg|500px]] || (7)
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Next the plane maps and their dual trees superimposed:
Next the plane maps and their dual trees superimposed:
{| align="center" border="0" cellpadding="10" cellspacing="0"
{| align="center" cellpadding="10"
| [[Image:Logical_Graph_Figure_8_Visible_Frame.jpg|500px]] || (8)
| [[Image:Logical_Graph_Figure_8_Visible_Frame.jpg|500px]] || (8)
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Finally the dual trees by themselves:
Finally the dual trees by themselves:
{| align="center" border="0" cellpadding="10" cellspacing="0"
{| align="center" cellpadding="10"
| [[Image:Logical_Graph_Figure_9_Visible_Frame.jpg|500px]] || (9)
| [[Image:Logical_Graph_Figure_9_Visible_Frame.jpg|500px]] || (9)
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And here are the parse trees with their traversal strings indicated:
And here are the parse trees with their traversal strings indicated:
{| align="center" border="0" cellpadding="10" cellspacing="0"
{| align="center" cellpadding="10"
| [[Image:Logical_Graph_Figure_10_Visible_Frame.jpg|500px]] || (10)
| [[Image:Logical_Graph_Figure_10_Visible_Frame.jpg|500px]] || (10)
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