Visualizing 3-hyperplanes

[ICN5D]

Wow, those diagrams are very cool, Wendy. They encourage me to make them for the 4 and 5D rotopes. I'm fuelled now, ready to experiment. Maybe even adapt them to my linear construction diagrams, showing the entire sequence from a point in an incmat display. Have you ever thought about using any shape symbols in place of the v,e, h, c, hd, hw ? Like what I see on Richard's incmat website, those matrices have shape notations in them. It's probably what I would want to do, for the construction diagrams.

[ICN5D]

Take something like this for example:

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Cylinder

||O = [ |O-2 , |(O) ]


3D       [ ||O ]
         /  |  \
2D    |O  |(O)  |O
        \ /   \ /
1D      (O)   (O)
          \    /
0D         \  /
            \/
-1D        n

|O = circle    |(O) = line torus   (O) = circle edge


However, for the cyltrianglinder |>|O, the diagonal lines that I can make with text aren't good enough to connect the lacings. The more graphical version that Keiji makes would work better.

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Cyltrianglinder

|>|O = [ ||O^2 , ||O-|O , |>(O) ]

4D                  |>|O     
                                           
3D       ||O   ||O  ||O      |>(O)                                 
                                                                                                         
2D     |O  |O  |O        |(O) |(O) |(O)                           
           
1D             (O)    (O)    (O)                                                                                   
0D                                                                                                             
-1D                    n


||O = cylinder      |>(O) = triangle torus



[ICN5D]

And here's one for the cylhemoctahedrinder. It's cool because you can see the square pyramid nature in it.

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Cylhemoctahedrinder

||>|O = [ |>|O^2 , |>|O^2 , |||O-|O , ||>(O) ]



5D                                ||>|O                                      1  x ||>|O                     

4D           |>|O--|>|O--|>|O--|>|O--|||O         ||>(O)                    4x |>|O , 1x |||O , 1x ||>(O)   

3D     ||O-||O-||O-||O-||O-||O-||O-||O   |>(O)-|>(O)-|>(O)-|>(O)-||(O)      8x ||O  , 4x |>(O) , 1x ||(O)

2D          |O-|O-|O-|O-|O      |(O)-|(O)-|(O)-|(O)-|(O)-|(O)-|(O)-|(O)     5x |O , 8x |(O)

1D                        (O)-(O)-(O)-(O)-(O)                               5x (O)     

0D                                                                       

-1D                                  n                                                     


||>(O) = square pyramid torus


[wendy]

One might suppose that the incmats for lace-prisms could be directly derived from the symbol itself. The theory is already complete, the current main obsticle is that given a set of nodes, to be able to recognise the underlying symmetry. Since the number of surtopes X in Y is simply a ratio of symmetries, and the nodes in X being a subset of those in Y, then it is a matter of recognising what sort of symmetry might be represented by some exotic arrangement of, say {3,3,5}.

ASCII art has its limitations, so it is more useful to highlight particular points, than to create a general all-purpose figure. The series i gave highlight how the pyramid and comb products work.

[ICN5D]

After studying that graph of the ||>|O, I notice some very peculiar symmetry with it. The diagonal division separating the flat from round cells is like a mirror. Both sequences of cells are reflected, but skewed by one dimension. The upper left is from the square-pyramid*disk. The lower right is from the square-pyramid*disk-edge. The edge of the disk is n-1 dimensions from the solid disk, which is what creates this pattern. Every rotope has this symmetry, be it any polytope*n-sphere will have this reflected mirror cartesian product. It's always the product with a solid n-sphere plus the n-sphere's edge. I never noticed this before, because I never made this kind of graph, that includes all surtope cells. I knew that n*circle is with both a n*disk + n*edge, but I never saw the full pattern with every surtope displayed. It shows how the curved surtopes lace together to make the square pyramid torus ||>(O).