Three-dimensional Constructive Coefficient

The three-dimensional constructive coefficient is a general quantity that can be computed for any polyhedron. Adrià Garcia defines it as the sum of the number of faces times the number of edges per face (which represents the constructive geometry of the polygon) over the fixed numer 12. This normalization is possible because the coefficient, at least in the Platonic, Catalan and Archimedean solids, always happens to be a multiple of 12. The three-dimensional constructive coefficient is the same for a solid and its dual. Therefore, it provides a means to know which solids can be dual of each other. The following table summarizes this coefficient for all the Sacred Solids presented in this site:

SOLID	THREE-DIMENSIONAL CONSTRUCTIVE COEFFICIENT	DUAL
TETRAHEDRON	1 ← 3.4/12 →	TETRAHEDRON
OCTAHEDRON	2 ← 3·8/12 4·6/12 →	CUBE
TRIAKIS TETRAHEDRON	3 ← 12·3/12 (4·3+4·6)/12 →	TRUNCATED TETRAHEDRON
RHOMBIC DODECAHEDRON	4 ← 12·4/12 (8·3+6·4)/12 →	CUBOCTAHEDRON
DODECAHEDRON	5 ← 12·5/12 3·20/12 →	ICOSAHEDRON
TRIAKIS OCTAHEDRON	6 ← 24·3/12 (8·3+6·8)/12 →	TRUNCATED CUBE
TETRAKIS HEXAHEDRON	6 ← 24·3/12 (8·6+6·4)/12 →	TRUNCATED OCTAHEDRON
DELTOIDAL ICOSITETRAHEDRON	8 ← 24·4/12 (8·3+18·4)/12 →	RHOMBICUBOCTAHEDRON
RHOMBIC TRIACONTAHEDRON	10 ← 30·4/12 (20·3+12·5)/12 →	ICOSIDODECAHEDRON
PENTAGONAL ICOSITETRAHEDRON	10 ← 24·5/12 (32·3+6·4)/12 →	SNUB CUBE
DISDYAKIS DODECAHEDRON	12 ← 48·3/12 (8·6+12·4+6·8)/12 →	TRUNCATED CUBOCTAHEDRON
TRIAKIS ICOSAHEDRON	15 ← 60·3/12 (20·3+12·10)/12 →	TRUNCATED DODECAHEDRON
PENTAKIS DODECAHEDRON	15 ← 60·3/12 (12·5+20·6)/12 →	TRUNCATED ICOSAHEDRON
DELTOIDAL HEXECONTAHEDRON	20 ← 60·4/12 (20·3+30·4+12·5)/12 →	RHOMBICOSIDODECAHEDRON
PENTAGONAL HEXECONTAHEDRON	25 ← 60·5/12 (80·3+12·5)/12 →	SNUB DODECAHEDRON
DISDYAKIS TRIACONTAHEDRON	30 ← 120·3/12 (30·4+20·6+12·10)/12 →	TRUNCATED ICOSIDODECAHEDRON

SOLID

THREE-DIMENSIONAL CONSTRUCTIVE

COEFFICIENT

DUAL

TETRAHEDRON