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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