Representations of B3

B3 is a group that consists of 48 elements. The action of these elements can be represented as reflexions and rotations in the three dimensional space. Therefore, this group can transform one single point of space into 48 different points. The result is a geometrical object that consists of 48 vertices.

However, if the point that you start with exactly lies on such plane of reflexion, it won't be transformed into another point. It turns out that in such a case, exactly half of the 48 elements remain passive and one ends up with a geometrical object that consists of 24 vertices. Similarly, geometrical objects with 12, 8 and 6 vertices can be obtained. Starting from the most general representation with 48 vertices, all other representations are obtained by identifying vertices.

Truncated Cuboctahedron

48 vertices
72 lines
26 faces

rhomicuboctahedron

24 vertices
48 lines
26 faces

Truncated HexaHedron

24 vertices
36 lines
14 faces

Truncated OctaHedron

24 vertices
36 lines
14 faces

Cuboctahedron

12 vertices
24 lines
14 faces

Hexahedron

8 vertices
12 lines
6 faces

Octahedron

6 vertices
12 lines
8 faces

There are transformations that allow to generate new polyhedra from these basic polyhedra. The first of such transformations is the duality operation. The center of each face consitutes a vertex to the new polyhedra. The following table shows the duals. The number of faces and vertices interchange, the number of edges stays the same.

Dual Truncated Cuboctahedron

26 vertices
72 edges
48 faces

Dual Rhombicuboctahedron

26 vertices
48 edges
24 faces

Dual Truncated HexaHedron

14 vertices
36 edges
24 faces

Dual Truncated Octahedron

14 vertices
36 edges
24 faces

DUAl cuboctahedron

14 vertices
24 edges
12 faces

Another interesting transformation consists in merging the vertices of a polyhedron with the vertices of its dual and construct from this set of points a new polyhedron. The mergers are visualized in the following animations.