Tensegrity structures coded by undip words

Undip words directly code Hamiltonian cubic planar graphs, but any connected planar graph can be converted to a tensegrity structure using the method demonstrated in "Making Mrs. Stott's Expansion with Mrs. Bright's Knot."

These are direct-view stereograms intended for viewing at an image size about like the screen of an iphone. Click through if the image is cropped on your browser.

unndnununuppdpdpdp

uuuddd

uunddp

uunpdd

uundpd

unnndnunuuupppdpdpdd

uuundddp

uunnddpp

undpnp

undupd

unpdnp

ududud

Straight undip words as lattice walks

To construct a straight word via its lattice walk:

Construct a diagonal walk beginning at the origin, staying in the first quadrant, and returning to the origin. This will serve as the diagonal skeleton. There are no other rules for the diagonal skeleton: it can retrace itself, or move ahead in a straight line.

Now trace the diagonal skeleton using the normal orthogonal moves (u/d for up/down; n/p for out/back). Each diagonal move in the skeleton needs two orthogonal moves to trace it, and there are always two choices in how this is done. Choose a way that does not result in an immediate retrace or two successive moves in the same direction.

Upon returning to the origin, you have the lattice walk of a straight undip word.

Straight undip words

The straight-loop word undp in one-step truchet tiles.

A straight undip word is a word in which up-letters and down-letters strictly alternate. The resulting triangle loop is just a straight belt of triangles These words generally correspond to various assemblies of tetrahedra, often connected only at edges. These are in effect mechanisms. Straight words offer a way to encode a robot skeleton in just a few letters.

A second try at First Ground

This is a denser First Ground than yesterday's. It has the same number of stitches between crochets as in de Dillmont's. (4 stitches = 4 steps = 2 strides.)  No longer is there an attempt to have two loops engaged at the dismounts. The undip formula is unchanged:

u (nu)^a [uu (up)^a nn (nd)^a]^x.

At two steps (one stride) per triangle, this is the smallest truchet that can be used with an arbitrary undip code. (One step per triangle loses its walking rhythm when the code does not alternate left and right.)

"First ground" in walking stitch crochet

This is a walking stitch pattern for a crocheted fabric similar to Thérèse de Dillmont's first ground for Irish crochet lace. The undip code is:

u (nu)^a [uu (up)^a nn (nd)^a]^x.

The scale is twice that of de Dillmont's because the interval between crochets here is four strides rather than four steps.


Image of "first ground" for irish crochet lace from "Irish Crochet Lace" by Therese de Dillmont.


de Dillmont's instructions for First Ground.

Walking crochet patterns

Here are walking crochet patterns on all the shapes that can be made with undip words of 2, 4, or 6 letters.


 The particular codes used:

ud                     undp                      uunddp

                         uudd                      uundpd
                   
                                                       uududd

                                                       uuuddd

Truchet tiles for walking crochet

Assembled according to an undip code word, these triangular truchet tiles show the tracks of the two-legged horse and the clowns.