The DNA self-assembly techniques of tiling and threading are used in conjunction with graph theory concepts to model the construction of the three-arm polytopes that are not found in the octet truss.
Tiling:
The first method used was branched junction molecule to tile the three-arm polytopes. These structures have two different types of angles when they are created by tiling with rigid armed tiles. The first is the angle between adjacent arms of the tile; this angle is determined by the shape of the face that the two arms form. The second is the dihedral angle; this is determined by the angle between two adjacent faces. The angles are unique for each polytope, therefore the tiles for our target structure will not be able to form smaller structures.
When tiling our structures, our goal was to use two tile types, the minimum number needed to construct polytopes with odd degree vertices. When tiling a polytope, the number of unhatted letters must equal the number of hatted ones. If the tile arms have odd degree, using only one tile type will result in an imbalance between the number of hatted and unhatted letters. Therefore, the smallest number of tiles that can be used is two.
When tiling these polytopes with the minimum number of tiles, a pattern was developed: the number of unique dihedral angles is equal to the number of different bond edge types. Using this equality, all of our three-armed structures are able to be built with only two tile types.
Threading:
The origami method is the second method used to build designs for these polytopes. A long single strand of DNA called the scaffolding strand is bent and curved to form the desired structure. This strand follows the Euler cycle that we have created. We then diagrammed the stapling and vertex configurations for each polytope.