Tuesday, March 6, 2018

Growing tensegrity surfaces: insertions and shuffles

A triangulation with a Hamiltonian dual can be built up or 'grown' in a systematic way starting from a single triangular envelope. As determined by the route of the Hamilton circuit (traced in red in the photos,) the first triangle is either ud or np. A new triangular envelope can be spliced into the same Hamilton circuit, by slitting open the first envelope along an edge crossed by the Hamilton circuit, slitting open the second triangular envelope similarly, and taping them together to form a quadrilateral envelope.

Two triangular envelopes, ud and np, with marked and oriented Hamilton circuits (Hamilton circuit marked in red: these triangles are also marked with a zig-zag tensegrity pattern) are slit open and taped together to form a unpd quadrilateral envelope with a marked and oriented Hamilton circuit.

Each time the triangulation grows, it grows by this type of insertion—it is always a triangular envelope that is being spliced-in at an edge crossed by the Hamilton circuit.

One other local mutation is needed. What is described as a shuffle of two letters in the undip word, is physically a switching of the diagonal in a triangulated quadrilateral. To do this the triangle edge separating the two letters to be shuffled must be slit open, as well as the other two triangle edges bordering the letters. The three slits necessarily form a 'Z' (or 'S' ) shape because one letter is 'up' letter and the other 'down'. The two triangles freed by these slits can be manipulated to change places in the Hamilton circuit and taped back together in that altered position.

Shuffling of up and down letters corresponds to slitting three edges in a 'Z' (or 'S',) switching the order of the two freed triangles in the Hamilton circuit, an taping things back up. In the photo a shuffle edit turns a unpd quadrilateral envelope into a nupd tetrahedron.

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