Tuesday, March 6, 2018

Working backwards from an undip word to a growth sequence

Stereogram of a zig-zag tensegrity dodecahedron. The Hamilton circuit follows the orange struts.

Here's an undip word for the dodecahedron (12 faces, 20 vertices, 30 edges):


What would be a practical growth sequence for this surface?

A bad idea would be to do all the insertions, say the 'up' word first and then the 'down' word:


and then do all the shuffles to get things in the proper interleaved order. The first stage is a formless loop:

Stereogram of an un-shuffled dodecahedron.

Seems more sensible to start with just the lowest-order (outermost) parenthesis pairs in the order they appear, and then do shuffles as soon as they become possible (since shuffle edits tend to increase the connectedness and stability of the structure.)

Here is the same word with the letters spaced vertically according to their parenthesis order.

          u   d
         u      d
        u         d
u   d u            d
 n               p
  n            p
   n         p
     n      p
       n   p

If we dedicate the first orbit of the Hamilton circuit to the zeroth-order insertions and perform the pair insertion when we encounter the letter of the pair:

|unu = udnpud

Stereogram of zig-zag tensegrity udnpud, the first-order parenthesis pairs in the dodecahedron word.
We then have two shuffles we can do: switch d-n and p-u

|unu|..s.s = undupd, which results in a stable shape:

Stereogram of zig-zag tensegrity undupd, a triangular prism.

The next orbit of the Hamilton circuit does the 2nd-order insertions, of which there are two:

|unu|..s.s.|..np..ud = unnpduudpd, which is a bowl-like shape.

The next orbit of the Hamilton circuit can shuffle the first p past duu and the second d past the second p.


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