Golden Truncated Icosahedron

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A truncated icosahedron with golden hexagonal faces is defined here as a golden truncated icosahedron (GTICO). When 30 squares fitted with vertex connections on a rhombic triacontahedron (RT) are expanded by the golden ratio, then the vertices meet again. These meeting points correspond to the vertices of a GTICO. In this Demonstration, the GTICO is related to a number of shapes. The most notable features include: (1) in a ring of 10 tetrahedra, the vertices of the tetrahedra are on the vertices of the GTICO; (2) the edges of the rings of 10 icosahedra coincide with the shorter edge of the golden hexagons; (3) icosahedra in the ring meet the tips of pentagrams; and (4) the edges of expanded squares divide the faces of the GTICO and the large icosahedron in various proportions related to the golden ratio.

Contributed by: Sándor Kabai (August 2022)
Open content licensed under CC BY-NC-SA



Some observations:

• The 30 squares correspond to the square faces of a small rhombicosidodecahedron (SRD).

• A compound of five cubes could be obtained if the expansion was continued beyond the meeting points.

• The lower edges of the tetrahedra lie across the initial squares (faces of RT and SRD).

• The edges of expanded squares trace out pentagrams, which could be best seen on the faces of the dodecahedron and GTICO.

• The tips of RT fit into the central pentagons of the pentagrams.

• The 10 tetrahedra are part of that cluster of 30 (not shown). The occurrences of the golden ratio are abundant (it could be an exciting challenge to count them).

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