Truncated Dodecahedron

This structure has 60 vertices and 90 edges. To create the complex, the vertices of the dodecahedron are "sliced off" to create equilateral triangles. Then, the faces of the dodecahedron become decagons.Because each vertex has degree of three, augmenting edges were added to the structure so then all the vertices are of even degree for the Euler circuit.

By projecting this Schlegel diagram through the center octagon face, it is possible to put exactly 3 augmenting edges through each octagon face except for the inner and outer ones. This placement yields the Euler circuit shown below, with the black arrows indicating the direction of the scaffolding strand. The original edges of the structure are represented by the dark blue lines, while the augmenting edges are light blue and green.

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

The different colors in the diagram below shows the 5-way rotational symmetry that can be achieved with these augmenting edges. This rotational symmetry allows the connection of five identical paths so that the number of unique vertex configurations needed during the threading process is minimized.

With this 5-way symmetrical circuit, only 8 unique vertex configurations are required to complete the threading process. These configurations are shown in the illustration below; the directions of the staple strands (red) are included along with the scaffolding strand (black).