Truncated Icosahedron

This structure has 60 vertices and 90 edges. Two different types of augmenting edges are used here, resulting in a lower bound of 8 vertex configurations when a directional field is applied to the graph. By using a pentagon as the central face in our projection of the Schlegel diagram, 5-way symmetry can be achieved with the augmenting edge arrangement and Euler circuit shown below. The direction that the scaffolding strand traverses during threading is represented by the black arrows. The staple strands, which attach at the vertices and traverse the circuit in the opposite direction from the scaffolding strand, are left out of the diagram for clarity.

Threading sequence: 1, 9, 10, 26, 9, 8, 25, 24, 44, 43, 23, 42, 43, 57, 58, 44, 45, 25, 26, 27, 28, 10, 11, 1, 2, 12, 13, 30, 12, 11, 29, 28, 47, 46, 27, 45, 46, 58, 59, 47, 48, 29, 30, 31, 32, 13, 14, 2, 3, 15, 16, 34, 15, 14, 33, 32, 50, 49, 31, 48, 49, 59, 60, 50, 51, 33, 34, 35, 36, 16, 17, 3, 4, 18, 19, 38, 18, 17, 37, 36, 53, 52, 35, 51, 52, 60, 56, 53, 54, 37, 38, 39, 40, 19, 20, 4, 5, 6, 7, 22, 6, 20, 21, 40, 41, 55, 39, 54, 55, 56, 57, 41, 42, 21, 22, 23, 24, 7, 8, 5, 1

The diagram below uses different colors to show the five identical paths that connect to form the 5-way symmetrical Euler circuit for the truncated icosahedron. This rotational symmetry helps to lower the number of vertex configurations required for the threading process.

This threading sequence yields 8 unique vertex configurations, which matches the lower bound indicated by the directional field. In the illustration below of these 8 configurations, the black arrows represent the direction of the scaffolding strand while the red arrows represent the direction of the staple strands.