Thursday, October 9, 2014

Tensegrities from children's drawings

Many tensegrities can be generated from the operation of the map operation snub on planar graphs.

A connected drawing (for example, one drawn by not lifting the pencil off the paper between the start and finish, retracing over the same line being allowed) when adorned with dots every place lines cross, branch, or terminate, describes a planar map. Map operations exist that can translate any orientable map into a design for a tensegrity. Via such a map operation, any connected drawing drawn on a sheet of paper describes a tensegrity structure in the broad sense: a self-prestressed system that is not necessarily rigid. In particular, the tensegrity the drawing describes may be an articulated space mechanism rather than a rigid 3D structure.

The simplest procedure to describe uses map operation snub: replace every edge with a rectangle (more properly speaking, a topological quadrilateral) and exploit the orientability of the plane to consistently place struts along the all the 'Z' or all the 'S' diagonals of the rectangles.

When the map operation is applied, some of the edges in the resultant may have the same vertex at both ends (a degenerate edge,) or some edges may connect the same two vertices as another edge (a parallel, or redundant edge,) in other cases an edge may entirely coincide with a strut (a bow edge), in this way the the tensegrity may be simpler than the intermediate resultant map that describes it. For example, applying the map operation snub to a triangle produces a the famous three-strut tensegrity though the intermediate map is rather more complicated.