Changes

Figure 19-a lays out the ''areal view'' or the ''angular form'' of venn diagram for universes of 2 and 4 dimensions, indicating the embedding map of type '''B'''² → '''B'''⁴.  In many ways these pictures are the best kind there is, giving full canvass to an ideal vista.  Their style allows the clearest, the fairest, and the plainest view that we can form of a universe of discourse, affording equal representation to all dispositions and maintaining a balance with respect to ordinary and differential features.  If only we could extend this view!  Unluckily, an obvious difficulty beclouds this prospect, and that is how precipitately we run into the limits of our plane and visual intuitions.  Even within the scope of the spare few dimensions that we have scanned up to this point subtle discrepancies have crept in already.  The circumstances that bind us and the frameworks that block us, the flat distortion of the planar projection and the inevitable ineffability that precludes us from wrapping its rhomb figure into rings around a torus, all of these factors disguise the underlying but true connectivity of the universe of discourse.

Figure 19-a lays out the ''areal view'' or the ''angular form'' of venn diagram for universes of 2 and 4 dimensions, indicating the embedding map of type '''B'''² → '''B'''⁴.  In many ways these pictures are the best kind there is, giving full canvass to an ideal vista.  Their style allows the clearest, the fairest, and the plainest view that we can form of a universe of discourse, affording equal representation to all dispositions and maintaining a balance with respect to ordinary and differential features.  If only we could extend this view!  Unluckily, an obvious difficulty beclouds this prospect, and that is how precipitately we run into the limits of our plane and visual intuitions.  Even within the scope of the spare few dimensions that we have scanned up to this point subtle discrepancies have crept in already.  The circumstances that bind us and the frameworks that block us, the flat distortion of the planar projection and the inevitable ineffability that precludes us from wrapping its rhomb figure into rings around a torus, all of these factors disguise the underlying but true connectivity of the universe of discourse.

<pre>

<br>

o-------------------------------------------------------------------------------o

<p>[[Image:Diff Log Dyn Sys -- Figure 19-a -- Extension from 2 to 4 Dimensions.gif|center]]</p>

|                                                                               |

<p><center><font size="+1">'''Figure 19-a.  Extension from 2 to 4 Dimensions:  Areal'''</font></center></p>

|                  / \                                    / \                  |

|          o      1 1     o        ---->         o 1101  o 1110  o          |

Figure 19-a.  Extension from 2 to 4 Dimensions:  Areal

</pre>

Figure 19-b shows the differential extension from ''U''^&nbsp;&bull;&nbsp;=&nbsp;[''u'',&nbsp;''v''] to E''U''^&nbsp;&bull;&nbsp;=&nbsp;[''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v''] in the ''bundle of boxes'' form of venn diagram.

Figure 19-b shows the differential extension from ''U''^&nbsp;&bull;&nbsp;=&nbsp;[''u'',&nbsp;''v''] to E''U''^&nbsp;&bull;&nbsp;=&nbsp;[''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v''] in the ''bundle of boxes'' form of venn diagram.

<pre>

<br>

                                                  o-----------------------------o

<p>[[Image:Diff Log Dyn Sys -- Figure 19-b -- Extension from 2 to 4 Dimensions.gif|center]]</p>

                                                  |     o-----o  o-----o      |

<p><center><font size="+1">'''Figure 19-b.  Extension from 2 to 4 Dimensions:  Bundle'''</font></center></p>

                                                  |    /       \ /      \    |

Figure 19-b.  Extension from 2 to 4 Dimensions:  Bundle

</pre>

As dimensions increase, this factorization of the extended universe along the lines that are marked out by the bundle picture begins to look more and more like a practical necessity.  But whenever we use a propositional model to address a real situation in the context of nature we need to remain aware that this articulation into factors, affecting our description, may be wholly artificial in nature and cleave to nothing, no joint in nature, nor any juncture in time to be in or out of joint.

As dimensions increase, this factorization of the extended universe along the lines that are marked out by the bundle picture begins to look more and more like a practical necessity.  But whenever we use a propositional model to address a real situation in the context of nature we need to remain aware that this articulation into factors, affecting our description, may be wholly artificial in nature and cleave to nothing, no joint in nature, nor any juncture in time to be in or out of joint.

Figure 19-c illustrates the extension from 2 to 4 dimensions in the ''compact'' style of venn diagram.  Here, just the changes with respect to the center cell are shown.

Figure 19-c illustrates the extension from 2 to 4 dimensions in the ''compact'' style of venn diagram.  Here, just the changes with respect to the center cell are shown.

<pre>

<br>

o---------------------------------------------------------------------o

<p>[[Image:Diff Log Dyn Sys -- Figure 19-c -- Extension from 2 to 4 Dimensions.gif|center]]</p>

|                                                                     |

<p><center><font size="+1">'''Figure 19-c.  Extension from 2 to 4 Dimensions:  Compact'''</font></center></p>

|          /                     \ /                    \          |

|    |                      |  -->--  |                      |    |

|    |      u      <---------------@--------------->     v      |    |

Figure 19-c.  Extension from 2 to 4 Dimensions:  Compact

</pre>

Figure 19-d gives the ''digraph'' form of representation for the differential extension ''U''^&nbsp;&bull;&nbsp;&rarr;&nbsp;E''U''^&nbsp;&bull;, where the 4 nodes marked "@" are the cells ''uv'', ''u''(''v''), (''u'')''v'', (''u'')(''v''), respectively, and where a 2-headed arc counts as two arcs of the differential digraph.

Figure 19-d gives the ''digraph'' form of representation for the differential extension ''U''^&nbsp;&bull;&nbsp;&rarr;&nbsp;E''U''^&nbsp;&bull;, where the 4 nodes marked "@" are the cells ''uv'', ''u''(''v''), (''u'')''v'', (''u'')(''v''), respectively, and where a 2-headed arc counts as two arcs of the differential digraph.

<pre>

<br>

o-----------------------------------------------------------o

<p>[[Image:Diff Log Dyn Sys -- Figure 19-d -- Extension from 2 to 4 Dimensions.gif|center]]</p>

<p><center><font size="+1">'''Figure 19-d.  Extension from 2 to 4 Dimensions:  Digraph'''</font></center></p>

|                          .->-.                           |

|                      .-->--@--<--.                      |

|                  /       |    |      \                  |

|          .--. | .---------->----------. | .--.          |

|          *--* | *----------<----------* | *--*          |

|                      *-->--@--<--*                      |

|                          *-<-*                          |

Figure 19-d.  Extension from 2 to 4 Dimensions:  Digraph

</pre>

===Thematization of Functions : And a Declaration of Independence for Variables===

===Thematization of Functions : And a Declaration of Independence for Variables===