Clusters Produced by Placing Rhombic Triacontahedra at the Vertices of Polyhedra

To find the possible candidate polyhedra, let us summarize the angles of face centers relative to the center of a single face of RT, as seen from the origin.face radians degrees 1 0 0 °1 p 180 °4 p ê 2 90 °4 p ê 5 36 °4 2 p ê 5 72 °4 3 p ê 5 108 °4 4 p ê 5 154 °4 p ê 3 60 °4 2 p ê 3 120 °Thedimensions applicable in the clusters can be determined on the basis of the relationship of the cube and RT.For instance, the cube edge is equal to the longer diagonal of the face of the RT (a golden rhombus), 2 sinHarctanH1.618LL= 1.70129.The diagonal of the cube equals the distance between opposite threefold vertices of the RT.Additional information for finding possible candidate polyhedra comes from a chart of truncations prepared by Szaniszló Bérczi.Figure 1 shows regular (Platonic) solids projected on a sphere.Archimedean solids are deduced from the regular solids by the truncation operation.A Platonic or Archimedean solid can be identified by its vertex configuration, because it is uniform; this is given by the Steiner symbol, which lists the faces that meet at a vertex.For example, H4, 4, 4L is the Steiner symbol for the cube, because three squares (4-sided faces) meet at each vertex.The RT-related structures should be arranged according to the third row of the table: H5, 6, 6L, H3, 5, 3, 5L, H3, 10, 10L, H5, 5, 5L, H3, 4, 5, 4L, H4, 6, 10L.