Rigid Tiles

Rigid Armed Tiles:
A depiction of the typical four-armed tile.
ranched junction molecules that have rigid arms (usually of unit length) where the arms are fixed with respect to the central vertex such that the following conditions hold true: (1) the positions of the arms are fixed with respect to one another, and (2) the arms are fixed with respect to the configuration such that the active areas of the cohesive ends do not change position by twisting (arms do not experience twist strain*)

* the compression or elongation of a DNA strand as a result of external forces [See04]. Thus the DNA strands are either too short to curve, or are reinforced for rigidity.

Additional Geometric Constraints

- No branched junction molecule can have fewer than 2 arms or more than 12 arms
- Final DNA structures must be complete
- The design must prevent the formation of structures smaller than the target structure

Platonic and Archimedean Solids in the Octet Truss

There are some solids that naturally occur in the octet truss: the tetrahedron, the octahedron, the truncated tetrahedron, the truncated octahedron and the cuboctahedron. These solids are contained as subgraphs within the octet truss. Thus, they automatically satisfy the design constraints of the branched junction molecule and are promising candidates for DNA self-assembly. Indeed the cube [CS91], the octahedron [ZS94], and the truncated octahedron [SQJ04] have already been built, albeit using other assembly methods. However, buckyballs use “dendrite” structures that are essentially three-armed branched junction molecules [H+08].

Shapes Found/Tiled in the Octet Truss
  • Three-armed structures
    Tetrahedron and Octahedron in the Octet Truss
    • Tetrahedron (see highlighted figure to the right)
    • Truncated Tetrahedron
    • Truncated Octahedron
  • Four-armed structures
    • Octahedron
    • Cuboctahedron
  • Non-convex structures with convex faces
    • Octahemioctahedron
    • Cuboctahemioctahedron
    • Tetrahemihexahedron

Why use the Octet Truss?

For rigid tiles, the octet truss provides a reasonable geometric model.
- Is highly symmetric and has a n appropriate number of edges for the application

Octet Truss
  Part of the Octet Truss
- Currently, branched junction molecules with 2, 4, 5, 6, 8, and 12 arms have been fabricated (see [See07] for an overview), so the 12-regular octet truss accurately reflects current capacity.
- All the edges are unit length, and the even spacing of the edges about the vertices prohibits any unattainable angles.
- The octet truss, being the only semi-regular tiling of 3-space is the next obvious 3D structure after the regular cubic lattice in which to draw graphs.

Shapes We Have Designed Which are not in the Octet Truss
  • Three-armed structures
    • Cube
    • Truncated Cube
    • Dodecahedron
    • Truncated Dodecahedron
    • Great Rhombicuboctahedron
    • Truncated Icosahedron
    • Great Rhombicosidodecahedron
  • Four-armed structures
    • Small Rhombicuboctahedron
    • Small Rhombicosidodecahedron
    • Icosidodecahedron
  • Five-armed structures
    • Icosahedron
    • Snub Cube
    • Snub Dodecahedron