Calculation of the Radius of Gyration (Rg) of the Truncated Icosahedron, Cuboctahedron, Rhombicuboctahedron, Octahedron and of their variations.




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 or at Visual Polyhedra.

An animation of the Truncated Icosahedron can be visualized here.

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 Cuboctahedron (CO) and of two of its variations (Octahemioctahedron (OH) and Cubohemioctahedron (CH)) are being calculated. The polyhedral shape consists of 8 trigons and 6 tetragons (12 vertices, 14 faces, 24 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 or at Visual Polyhedra.

An animation of the Cuboctahedron can be visualized here.

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


Parameter (Cuboctahedron) Value
Volume 2.357
Surface 9.464
Perimeter 24.000
out-sphere radius ro 1.000
mid-sphere radius rm 0.866
in-sphere radius ri (trigons) 0.816
in-sphere radius ri (tetragons) 0.707
Rg (CO) 0.648
Rg (OH) 0.671
Rg (CH) 0.632


Table 2. Shape parameters and calculated values for Rg (rounded) of the Cuboctahedron and of two variations with unit length a=1 ( = edge length). Rg scales linearly with the edge-length a. Units are arbitrary.
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Next, 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 or at Visual Polyhedra.

An animation of the Rhombicuboctahedron can be visualized here.

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 3. Shape parameters and calculated values for Rg (rounded) of the Samll Rhombicuboctahedron and of two variations with unit length a=1 ( = edge length). Rg scales linearly with the edge-length a. Units are arbitrary.
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Last, the Radius of Gyration (Rg) of two variations of the Octahedron (Stella Octangula (SO) and Tetrahemihexahedron (THH)) are being calculated. The polyhedral shape (octahedron) consists of 8 trigons (6 vertices, 8 faces, 12 edges) with an unilateral edge-length a (here we take unit length = 1). In case of the Stalla Octangula all eight faces are stellated with tetrahedra of edge lengths equal to a, in case of the Tetrahemihexahedron four faces are removed (4 of the 8 trigonal pyramids are 'cut out'). In both cases the position of the centroid remains the same.

The parameters for the solid shape are entered as input into the Rg-calculator (and also here) as shown below:


Parameter (Octahedron) Value
Volume 0.471
Surface 3.464
Perimeter 12.000
out-sphere radius ro 0.707
mid-sphere radius rm 0.500
in-sphere radius ri (trigons) 0.408
Rg (O) 0.387
Rg (SO) 0.592
Rg (THH) 0.387


Table 3. Shape parameters and calculated values for Rg (rounded) of the Octahedron and of two variations with unit length a=1 ( = edge length). Note, Rg of the THH is the same as for the Octahedron, no matter how many trigonal pyramids are removed from the octagon (provided the location of the centroid remains the same). 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