Shuffle-greedy strategy for growing tensegrity surfaces

Stereogram of a zig-zag tensegrity dodecahedron trapped in a bowl-like, 'non-inflated' configuration.

Shuffle tends to make the corresponding structure sturdier, and more 'inflated'. (Usually we are seeking a maximum-volume configuration of the surface.) Even in a fairly simple structure, the maximum-volume configuration can be hard to reach when starting point from a random configuration (see the semi-stable configuration of the dodecahedron surface in the stereogram above.)

For this reason, it may be best to prioritize doing as many shuffle edits as early as possible. The plan would be to reach the final, sturdy, maximum-volume configuration through a growing sequence of structures that are themselves sturdy and maximum-volume.

Recall that an undip word is a shuffle of two Dyck words: an up-word and a down-word. Dyck words are formed by insertion edits alone—there is no shuffling of the parentheses past each other. Thus, in the generation of a Dyck word, when a parenthesis pair, '()', is inserted at the correct location for the open-parenthesis, the close-parenthesis is also, automatically, in the correct position.

The rule is the same for an undip word when the up-word/down-word is considered in isolation: if the just-inserted u/n is in the correct location, the adjacent d/p is also in the correct location. However, it might be necessary to translate the adjacent d/p to the right, shuffling past some characters of the other kind. This should be done immediately after insertion by following the insertion with a sequence of shuffles until the d/p is correctly interleaved with the characters of the other kind. Only then would more insertions be possible on that orbit of the Hamilton circuit.

Example:

|u|.ns = undp

The growth sequence above builds a ud 3-strut tensegrity in the first orbit, inserts an np 'ear' (a spliced-in triangular envelope) to form a unpd quad envelope during the second orbit, and immediately shuffles its p to the right, completing a undp tetrahedron in two orbits.

The shuffle-greedy strategy settles how all shuffle edits will be done: all letters present in the undip word will be  correctly interleaved at the conclusion of each orbit. We still need a strategy for choosing which insertion to prioritize when insertions are possible. Of course, within the two Dyck words we must follow the Dyck rules and insert the lowest-order parentheses first. But strategic  decisions may remain.

The simplest strategy would be to do the first possible Dyck-rule insertion that presents itself (be it up or down.) Though we need to maintain a reasonable balance of up-words and down-words to have sturdy intermediate structures, this may be something that can be left to chance.