## 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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