Great Rhombicosidodecahedron

This structure has 120 vertices and 180 edges. Just like the other Platonic and Archimedean structures examined in this research, the Great Rhombicosidodecahedron has all vertices of odd degree, and so augmenting edges must be added for an Euler circuit to be possible. The graph below shows the three different types of augmenting edges bisecting every face except the top and bottom decagon bases; the resulting Euler circuit has 5-way symmetry.

Because of the complexity and size of this structure's Schlegel diagram, it is more productive to show the threading sequence for only 1/5 of the Euler circuit, since this path is simply repeated five times around the center of rotation to complete the structure's assembly. Illustrated in the diagram below, the direction of the scaffolding strand is indicated by the black arrows.

Threading sequence: 1, 2, 12, 1, 11, 20, 19, 18, 17, 72, 18, 71, 72, 73, 74, 75, 80, 79, 78, 77, 76, 75, 118, 76, 117, 118, 119, 74, 67, 73, 68, 17, 16, 11, 12, 13, 14, 15, 16, 69, 15, 70, 61, 14, 58, 13, 59, 2, 3, 4, 51, 3, 60, 59, 58, 57, 56, 62, 57, 61, 62, 63, 64, 65, 70, 69, 68, 67, 66, 65, 120, 66, 119, 120, 111, 64, 107, 63, 108, 56, 55, 51, 52, 53, 54, 55, 109, 54, 110, 101, 53, 47, 52, 48, 4, 5, 6, 50, 5, 49, 48, 47, 46, 45, 102, 46, 101, 102, 103, 104, 105, 110, 109, 108, 107, 106, 105, 112, 106, 111, 112, 113, 104, 97, 103, 98, 45, 44, 49, 50, 41, 42, 43, 44, 99, 43, 100, 91, 42, 37, 41, 38, 6, 7, 8, 40, 7, 39, 38, 37, 36, 35, 92, 36, 91, 92, 93, 94, 95, 100, 99, 98, 97, 96, 95, 114, 96, 113, 114, 115, 94, 87, 93, 88, 35, 34, 39, 40, 31, 32, 33, 34, 89, 33, 90, 81, 32, 27, 26, 25, 82, 26, 81, 82, 83, 84, 85, 90, 89, 88, 87, 86, 85, 116, 86, 115, 116, 117, 84, 77, 83, 78, 25, 24, 29, 30, 21, 22, 23, 24, 79, 23, 80, 71, 22, 19, 21, 20, 10, 30, 9, 29, 28, 27, 31, 28, 8, 9, 10, 1

With this 5-way symmetrical Euler circuit, there are only 12 unique vertex configurations necessary to self-assemble the complex. This minimum matches the lower bound of 12 found with the graph's directional diagram. The directions of the staple strands (red) are included along with the scaffolding strand (black) in this illustration of the different configurations: