  In this work, we show the geometric properties of a family of polyhedra
obtained by folding a regular tetrahedron along regular triangular grids. Each
polyhedron is identified by a pair of nonnegative integers. The polyhedron can
be cut along a geodesic strip of triangles to be decomposed and unfolded into
one or multiple bands (homeomorphic to a cylinder). The number of bands is the
greatest common divisor of the two numbers. By a proper choice of pairs of
numbers, we can create a common triangular band that folds into different
multiple polyhedra that belongs to the family.
