Great Rhombicuboctahedron

This structure has 48 vertices and 72 edges. Because each vertex in this structure has a degree of three, augmenting edges must be added so that all vertices are of even degree. The diagram below shows an Euler circuit with 3-way rotational symmetry. One augmenting edge (light blue) bisects each of the squares, while there are two augmenting edges (green) in each octagonal face; original edges of the structure are represented by the dark blue lines. The black arrows show the direction of the scaffolding strand as it traverses the Euler circuit during threading. The staple strands are absent from the image for clarity, but the staples would clearly follow in the opposite direction to the scaffolding strand, holding the DNA in place at the vertices.

Threading sequence: 1, 9, 10, 26, 27, 44, 43, 48, 40, 43, 41, 25, 26, 9, 8, 7, 24, 25, 42, 24, 23, 42, 41, 40, 39, 22, 23, 7, 6, 8, 1, 6, 5, 21, 22, 38, 39, 48, 47, 46, 34, 47, 35, 37, 38, 21, 20, 19, 18, 37, 36, 18, 17, 36, 35, 34, 33, 16, 17, 19, 4, 20, 5, 4, 3, 15, 16, 32, 33, 46, 45, 44, 28, 45, 29, 31, 32, 15, 14, 13, 12, 31, 30, 12, 11, 30, 29, 28, 27, 10, 11, 13, 2, 14, 3, 2, 1

The same Schlegel diagram is presented below with different colors to highlight the three identical paths that form as a result of this graph's 3-way rotational symmetry. These paths connect to complete the Euler circuit:

With this symmetric circuit, there are only 6 unique vertex configurations necessary to self-assemble the Great Rhombicuboctahedron. The directions of the staple strands (red) are shown along with the scaffolding strand (black) in the closer look at these configurations: