Calculation of the Radius of Gyration (Rg) of the Truncated Icosahedron and of the Rhombicuboctahedron



As an example, the Radius of Gyration (Rg) of a Truncated Icosahedron (the shape of the C60-molecule) is being calculated. The polyhedral shape consists of 20 hexagons and 12 pentagons (60 vertices, 32 faces, 90 edges) with an unilateral edge-length a (here we take unit length = 1). All parameters for the calculation of Rg, such as out-, mid- and in-sphere radii (ro, rm, ri), respectively, can be found at Wolfram MathWorld .


C60 C60 C60




The parameters for the solid shape are entered as input into the Rg-calculator as shown below:
Parameter (Truncated Icosahedron) Value
Volume 55.288
Surface 72.607
Perimeter 90.000
out-sphere radius ro 2.478
mid-sphere radius rm 2.427
in-sphere radius ri (pentagons) 2.327
in-sphere radius ri (hexagons) 2.267
Rg of solid 1.833
Rg of faces 2.366
Rg of edges 2.444
Rg of vertices ( = ro) 2.478


Table 1. Shape parameters and calculated values for Rg (rounded) of the regular truncated icosahedron (C60) of unit length a=1 ( = edge length). Rg scales linearly with the edge-length a. Units are arbitrary.
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As another example, the Radius of Gyration (Rg) of the Small Rhombicuboctahedron (RCO) and of two of its variations (Small Cubicuboctahedron (CCO) and Small Rhombihexahedron (RH)) are being calculated. The polyhedral shape consists of 8 trigons and 18 tetragons (24 vertices, 26 faces, 48 edges) with an unilateral edge-length a (here we take unit length = 1). All parameters for the calculation of Rg, such as out-, mid- and in-sphere radii (ro, rm, ri), respectively, can be found at Wolfram MathWorld .

The parameters for the solid shape are entered as input into the Rg-calculator as shown below:


Parameter (Small Rhombicuboctahedron) Value
Volume 8.714
Surface 21.464
Perimeter 48.000
out-sphere radius ro 1.399
mid-sphere radius rm 1.307
in-sphere radius ri (trigons) 1.274
in-sphere radius ri (tetragons) 1.207
Rg (RCO) 0.991
Rg (CCO) 0.926
Rg (RH) 0.989


Table 2. Shape parameters and calculated values for Rg (rounded) of the Samll Rhombicuboctahedron of unit length a=1 ( = edge length). Rg scales linearly with the edge-length a. Units are arbitrary.
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Author: M.Kriechbaum, TU-Graz (2013), e-mail: manfred.kriechbaum@tugraz.at