Transformability and Reversibility of Unfoldings of Doubly-Covered Polyhedra - Discrete and Computational Geometry and Graphs - page 80

(1) (2)

(3)

( (

Fig. 3. The pair of a rhombic dodecahedron and a rectangular parallelepiped which is
reversible.

edge lengths √ 3 , √ 3 and 2, and whose d ih ed r al angle of the edge with length

2is ˀ/ 2. We call such tetrahedron a ( √ 3 , √ 3 , 2)- tetrahedron (or Sommerville
tetrahedron, [ 9 , 10 ]) and denote it by ST (see Fig. 4 ).
By dissecting a ST into halves by the plane orthogonal to an edge with
length 2, we obtain a tetrahedron Q which is also a reflective space-filler, and
by dissecting Q into halves again by the plane orthogonal to the edge with
length 2, we obtain a tetrahedron R which is also a reflective space-filler (see
Fig. 4 ). We call Q and R respectively a half-ST and a quarter-ST respectively.
The quarter-ST plays an important role in [ 2 ] where they call it a tetradron.
Note that the mirror image of a half- ST or a quarter- ST is not strictly
congruent to the original.

3

3 3

2

f

2 2

2

2 e

1

1

e

1

1

3

2 2

3 3 3

Fig. 4. Reflective space-filling tetrahedra: (1) a ( √ 3 , √ 3 , 2)-tetrahedron which is
denoted by ST : (2) a half- ST : (3) a quarter- ST .