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DLOG. Note D27
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Extension from 2 to 4 Dimensions
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 soon reach 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.
o-------------------------------------------------------------------------------o
| |
| o o |
| / \ / \ |
| / \ / \ |
| / \ / \ |
| / \ o 1100 o |
| / \ / \ / \ |
| / \ / \ / \ |
| / \ !e! / \ / \ |
| o 1 1 o ----> o 1101 o 1110 o |
| / \ / \ / \ / \ / \ |
| / \ / \ / \ / \ / \ |
| / \ / \ / \ / \ / \ |
| / \ / \ o 1001 o 1111 o 0110 o |
| / \ / \ / \ / \ / \ / \ |
| / \ / \ / \ / \ / \ / \ |
| / \ / \ / \ / \ / \ / \ |
| o 1 0 o 0 1 o o 1000 o 1011 o 0111 o 0100 o |
| |\ / \ /| |\ / \ / \ / \ /| |
| | \ / \ / | | \ / \ / \ / \ / | |
| | \ / \ / | | \ / \ / \ / \ / | |
| | \ / \ / | | o 1010 o 0011 o 0101 o | |
| | \ / \ / | | |\ / \ / \ /| | |
| | \ / \ / | | | \ / \ / \ / | | |
| | x_1 \ / \ / x_2 | | x_1 \ / \ / \ / x_2 | |
| o-------o 0 0 o-------o o---+---o 0010 o 0001 o---+---o |
| \ / | \ / \ / | |
| \ / | \ / \ / | |
| \ / | x_3 \ / \ / x_4 | |
| \ / o-------o 0000 o-------o |
| \ / \ / |
| \ / \ / |
| \ / \ / |
| o o |
| |
o-------------------------------------------------------------------------------o
Figure 19-a. Extension from 2 to 4 Dimensions: Areal
Figure 19-b shows the differential extension
from U% = [u, v] to EU% = [u, v, du, dv] in
the "bundle of boxes" form of venn diagram.
o-----------------------------o
| o-----o o-----o |
| / \ / \ |
| / o \ |
| / / \ \ |
| o o o o |
@ | du | | dv | |
/| o o o o |
/ | \ \ / / |
/ | \ o / |
/ | \ / \ / |
/ | o-----o o-----o |
/ o-----------------------------o
/
o-----------------------------------------/---o o-----------------------------o
| / | | o-----o o-----o |
| @ | | / \ / \ |
| o---------o o---------o | | / o \ |
| / \ / \ | | / / \ \ |
| / o \ | | o o o o |
| / / \ @-------\-----------@ | du | | dv | |
| / / @ \ \ | | o o o o |
| / / \ \ \ | | \ \ / / |
| / / \ \ \ | | \ o / |
| o o \ o o | | \ / \ / |
| | | \| | | | o-----o o-----o |
| | | | | | o-----------------------------o
| | u | |\ v | |
| | | | \ | | o-----------------------------o
| | | | \ | | | o-----o o-----o |
| o o o \ o | | / \ / \ |
| \ \ / \ / | | / o \ |
| \ \ / \ / | | / / \ \ |
| \ \ / \ / | | o o o o |
| \ @-----\-/-----------\------------ @ | du | | dv | |
| \ o / | | o o o o |
| \ / \ / \ | | \ \ / / |
| o---------o o---------o \ | | \ o / |
| \ | | \ / \ / |
| \ | | o-----o o-----o |
o-----------------------------------------\---o o-----------------------------o
\
\ o-----------------------------o
\ | o-----o o-----o |
\ | / \ / \ |
\ | / o \ |
\ | / / \ \ |
\| o o o o |
@ | du | | dv | |
| o o o o |
| \ \ / / |
| \ o / |
| \ / \ / |
| o-----o o-----o |
o-----------------------------o
Figure 19-b. Extension from 2 to 4 Dimensions: Bundle
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.
o---------------------------------------------------------------------o
| |
| |
| o-------------------o o-------------------o |
| / \ / \ |
| / o \ |
| / / \ \ |
| / / \ \ |
| / / \ \ |
| / / \ \ |
| / / \ \ |
| o o (du).(dv) o o |
| | | -->-- | | |
| | | \ / | | |
| | dv .(du) | \ / | du .(dv) | |
| | u <---------------@---------------> v | |
| | | | | | |
| | | | | | |
| | | | | | |
| o o | o o |
| \ \ | / / |
| \ \ | / / |
| \ \ | / / |
| \ \ | / / |
| \ \|/ / |
| \ | / |
| \ /|\ / |
| o-------------------o | o-------------------o |
| | |
| du . dv |
| | |
| V |
| |
o---------------------------------------------------------------------o
Figure 19-c. Extension from 2 to 4 Dimensions: Compact
Figure 19-d gives the "digraph" form of representation for the
differential extension U% -> EU%, 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.
o-----------------------------------------------------------o
| |
| .->-. |
| | | |
| * * |
| \ / |
| .-->--@--<--. |
| / / \ \ |
| / / \ \ |
| / . . \ |
| / | | \ |
| / | | \ |
| / | | \ |
| . | | . |
| | | | | |
| v | | v |
| .--. | .---------->----------. | .--. |
| | \|/ | | \|/ | |
| ^ @ ^ v @ v |
| | /|\ | | /|\ | |
| *--* | *----------<----------* | *--* |
| ^ | | ^ |
| | | | | |
| * | | * |
| \ | | / |
| \ | | / |
| \ | | / |
| \ . . / |
| \ \ / / |
| \ \ / / |
| *-->--@--<--* |
| / \ |
| . . |
| | | |
| *-<-* |
| |
o-----------------------------------------------------------o
Figure 19-d. Extension from 2 to 4 Dimensions: Digraph
Jon Awbrey
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