Rhombic Triacontahedron

The rhombic triacontahedron has 32 vertices, 60 edges (all the same length), 30 identical rhombic faces. The rhombic triacontahedron is the dual to the icosidodecahedron, as such it is one of only 13 so called Catalan solids.

Equirectangular projection.

Vertex and face numbering.

The vertex indices are provided below. Values of 2 translates to 1/φ and a value of 3 translates to φ, where φ is the golden ratio of (1+sqrt(5))/2. All (x,y,z) values are combinations of 0, +/- 1, +/- φ and +/- 1/φ.

// Vertices
#define NV 32
XYZ v[NV] = {
   {-1,0,3},{-1,1,1},{0,2,3},{0,-2,3},{-1,-1,1},{3,1,0},
   {1,1,1},{0,3,1},{1,0,3},{3,0,2},{1,-1,1},{0,-3,1},
   {-3,0,2},{-3,1,0},{-2,3,0},{2,3,0},{3,-1,0},{2,-3,0},
   {3,0,-2},{0,-3,-1},{-2,-3,0},{-3,-1,0},{-3,0,-2},{-1,-1,-1},
   {1,-1,-1},{0,-2,-3},{-1,0,-3},{1,0,-3},{1,1,-1},{0,2,-3},{-1,1,-1},{0,3,-1}
};

Faces in terms of the vertex indices above.

// Faces
#define NF 30
FACE f[NF] = {
   {0,2,8,3},{25,26,29,27},{2,7,6,8},{3,8,10,11},{0,1,7,2},{27,28,31,29},
   {0,3,11,4},{19,24,27,25},{26,29,31,30},{5,6,7,15},{10,11,17,16},{5,15,31,28},
   {1,7,14,13},{4,11,20,21},{13,14,31,30},{16,17,19,24},{19,20,21,23},{5,6,8,9},
   {0,1,13,12},{19,23,26,25},{8,9,16,10},{5,18,27,28},{0,4,21,12},{13,22,26,30},
   {16,18,27,24},{21,22,26,23},{7,14,31,15},{11,17,19,20},{5,9,16,18},{12,13,22,21}
};

Edges in terms of vertex indices above.

// Edges
#define NE 60
EDGE e[NE] = {
   {4,21},{4,11},{11,20},{20,21},{11,17},{17,19},{19,20},{19,23},
   {23,21},{19,24},{24,16},{16,17},{16,10},{10,11},{23,26},{26,25},
   {25,19},{25,27},{27,24},{27,28},{28,5},{5,18},{18,27},{18,16},
   {16,9},{9,5},{10,8},{8,9},{8,6},{6,5},{11,3},{3,8},{3,0},{0,4},
   {8,2},{2,7},{6,7},{7,15},{15,5},{15,31},{31,28},{31,29},{29,27},
   {2,0},{0,1},{1,7},{1,13},{13,14},{14,7},{14,31},{13,30},{30,31},
   {30,26},{26,29},{21,22},{22,26},{12,13},{13,22},{21,12},{12,0}
};

The short diagonals of the faces of the rhombic triacontahedron give the edges of a dodecahedron. The long diagonals give the edges of the icosahedron.

The ratio of the long diagonal of each rhombic face to the short diagonal is the golden ratio, φ, known as "golden rhombi".

There are at least two nets (flattened) layout in common usage.

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