An undip word specifies a step-by-step procedure that reassembles a spherical map from its edge-rhombs (the quadrilateral faces into which the map operation radial, Ra(), would dissect it.) If a map is a triangulation of the sphere, and its dual is hamiltonian, then there is a least one undip word that reassembles the primal map from its edge-rhombs. (In the special case that we obtained the triangulation as the meta, a.k.a. barycentric subdivision, Mt(), of another map, we are guaranteed that the dual is hamiltonian.)

The assembly procedure does not care what we might have drawn on the faces of the edge-rhombs, and, since many structures and machines can be represented as truchet tiles drawn on edge rhombs, an undip word is also a step-by-step guide to the assembly of any such structure or machine.

In the following I'll assume the reader is familiar with the procedure of reading an undip word to weave twogs into baskets as described in Make a Basket from a Word.

Edge-rhombs are topological objects that can be stretched in any way we wish, but it will forestall confusion if we idealize edge-rhombs as the rhombic shape created by joining two equilateral triangles at a common edge, and further assume that an edge of our primal map (the triangulation) forms the longer diagonal. Therefore, an edge of the dual map (a cubic or trivalent map) forms the short diagonal.

TWOGS through the state of play "uun—"

In order to represent the "assembly" of a twog weaving, I have made a set of paper edge-rhombs with the short diagonal marked in each of three colors. In the photographs below, green is the "photon color" and blue and pink are the "energy colors."

An undip word gains a half edge for free at the beginning of the word, and another half edge for free at the end of the word (the last being the other half of the first.) These moves are encoded, not by a letter, but simply by the fact that the word begins and ends. In reassembling edge-rhombs, we show this peculiar opening move by playing just half of an edge-rhomb (i.e., from an edge-rhomb sliced in two along its longer diagonal.)

The first letter is either a u or n, let's say u. That says, "emit a photon to the left, and then build on to the right." The photon edge will not be completed until later when it gets "absorbed." We show the state of incompletion by placing just half an edge-rhomb on the photon side while building on with a full edge-rhomb on the electron side. If our edge-rhombs have been decorated as truchet tiles, we can now physically build everything that is fully drawn at this state of play. All else must wait until more tiles are added or joined.

Below, edge-rhombs decorated as twogs are carried through to the state-of-play uun—.

Suppose the next letter is d. One can visualize how a half a photon rhomb will be added on the left side of the last played rhomb and then joined to another half photon rhomb already in the work.

That's pretty much all there is to reassembling a map from an undip word that codes it.

Below are four tensegrity patterns also taken through to the state of play uun—.

ZIG-ZAG TENSEGRITY through "uun—"

CIRCUIT TENSEGRITY through "uun—"

STAR TENSEGRITY through "uun—"

DIAMOND TENSEGRITY through "uun—"

## Friday, October 7, 2011

## Wednesday, October 5, 2011

### Tensegrities from Maps

Anthony Pugh in his book An Introduction to Tensegrity, identifies three categories of tensegrity patterns: diamond (also known as rhombic), zig-zag, and circuit. Lawrence Pendred found one more: star. Tensegrities conforming to these patterns form stable membranes that can generally be curved to any shape. All four are associated with map operations. The table above shows these tensegrity patterns with their truchet tiles (the truchet tiles replace the edge-rhomb associated with each edge in the base map.)

The diamond (or rhombic, see Xi-Qiao Feng et al.) tensegrity pattern is isomorphic to the map operation snub, Sn(). The compression elements, or struts, of the tensegrity structure, correspond to the chiral edge of snub; the remaining non-chiral edges—which, by themselves, could just as well have been generated by the map operation expand, Ex()—correspond to the tension elements, or tendons, of the tensegrity. When the base map is a triangle on the sphere (as shown above,) this map operation yields the famous 3-strut, 9-tendon, T-prism. When the base map is a tetrahedron, this map operation yields the equally well-known 6-strut, 24-tendon, expanded octahedron.

Pendred's star tensegrity pattern is isomorphic to the map operation capra, Ca(). The compression struts of the tensegrity correspond to the chiral edge of capra; the remaining non-chiral edges—which, by themselves, could just as well have been generated by the map operation chamfer, Ch()—correspond to the tendons of the tensegrity. This arrangement is less rigid than the diamond pattern, but sometimes has useable strength.

The situation is a bit messier for the zig-zag and circuit tensegrity patterns. These patterns have elements that crossover each other without actually touching—thus they cannot fully correspond to the result of a map operation. A stratagem is to use truchet tiles to draw these patterns (as in the table above) with narrow tendons overlying wide struts—not to indicate that they pass in front, but rather that they may pass either in front or behind as the curvature of the surface dictates. Such “floating” tendons and struts are non-physical, but the problem of deciding whether tendons or struts pass in front is fully determined in practice by the spatial coordinates of their endpoints. If the surface the tensegrity structure conforms to is sufficiently curved, strut-tendon and strut-strut collisions are avoided.

The zig-zag tensegrity pattern has tendons that align with the edges of truncate, Tr(), (see Yue Li et al.,) while the struts align with an additional chiral edge which, at its midpoint, crosses over a tendon.

The circuit tensegrity pattern has tendons and struts all of which align with the edges of medial, Me(). The struts join in the common tendon vertex only on alternate passes.

Clearly, there exist many more tensegrity patterns since any consistent way to add struts to a tension network has a chance of working like Snelson's bag-of-bars. In particular, it would be interesting to see if the other chiral map operations correspond to tensegrity patterns.

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