Skew Dodecahedron (D12), Regularoid

Skew Dices can have surprising shapes while being perfectly isohedral (ie all faces are equivalents).
The best example is the dodecahedron that has 12 pentagonal faces.
The usual D12 is a regular dodecahedron (regular pentagons) but it is possible to build an infinite family of D12 by playing with 2 parameters and few constraints.

In this family, the dodecahedron has 3 kinds of vertices leading to a pentagon with 2 couples of adjacent edges having the same length.

It can be noted that a dual isogonal family exists with skewed icosahedra having three kind of faces : two with equilateral triangles and the last one with an isosceles triangle.

Illustrations are showing 2 different ways to build the skewed D12 family :
- Method 1 : by defining a point on a cube face (with its symmetry by the face center)
- Method 2 : by picking two nested cubes in a larger one

In method 1, the twelve points will lead to twelve triangles defining the planes of the 12 faces.
A family of isohedral solids can be defined by adding triangular pyramids on the axis of the cube vertices.
A family of isogonal solids can be defined by adding triangular faces instead of pyramids (adding 8 faces, thus defining a skew icosahedron).

In method 2, the larger cube is used to define the face centers and the two other cubes to define the vertices of two tetrahedra. These 14 points defines the 12 planes of the isohedral solid.

Note : the faces are not numbered since the purpose of the model is to show the geometry of a skew D12 and to have Ardechoedres !

7 versions are presented  :
- A skew D12 close to a regular dodecahedron (between the regular pentagonal D12 and the rhombic dodecahedron), with the internal shapes of method 2
- A skew D12 close to a cube, with the internal shapes of method 2
- A skew D12 close to a cube, with the internal shapes of method 2 and faces close to Ardèche shape
- A skew D12 close to a tetrahedron, with the internal shapes of method 2
- A skew D12 close to a tetrahedron, with the internal shapes of method 2 and faces close to Ardèche shape
- A skew D12 close to a cube and faces close to Ardèche shape (empty)
- A skew D12  close to a tetrahedron and faces close to Ardèche shape (empty)