Ad Quadratum Construction and Study of the Regular Polyhedra

C ˆ D ˆ O i
2
The cube dihedral angle C ˆ
D is equal to the octahedron internal angle O ˆ i
19
O ˆ D C ˆ i
But, as we have seen previously in connection with fig. 6,
So that we can also conclude
T ˆ i ˆ C i
O ˆ D T ˆ i
The octahedron dihedral angle is therefore equal to the supplement of the internal angle
of the cube C ˆ and to the internal angle of the tetrahedron T ˆ i
i .
The tetrahedron being its own dual, we consider the relation between T ˆ D and T ˆ i.
We see
that (fig. 18):
T ˆ D ˆ T i .
But we previously established that
ˆ T i ˆ C i T ˆ D ˆ C i
The tetrahedron dihedral angle T ˆ D is the supplement of its internal angle T ˆ i,
and is also
equal to the cube internal angle.
We can therefore establish the following table:
Form Trigonometric
Ratio
Tetrahedron
cosTi 1
3
Cube
cosCi 1
3
Octahedron sinOi 1
Dodecahedron
sinDi 2
3
Internal Angle Dihedral Angle
T i
109.
47
T
D
Ci
70.
53
C i 70.
53 CD Oi 90 o
Oi 90 o
OD Ti
Ci
109.
47
D i
41.
81
Icosahedron tanI i 2
I i 63.
44
DD I i
I D Di
116.
56
138.
19
Although the information regarding the dihedral angle is not required for the construction
of the polyhedra, it is included here for the sake of completeness and to serve as a
verification of their construction.