Folding Tetrahedra and Four-Dimensional Origamis

By pure analogy with a usual three-dimensional origami, we can fold a regular octahedral material along flat faces in 4-space. The octahedron comprises four congruent tetrahedra. We will show a procedure to fold a tetrahedron along bisectors of the dihedral angles. This procedure demonstrates that each two of four surfaces of the given tetrahedron coincide with each other and that the point of intersection of those bisectors is the center of the incircle. We will prove that there is a kind of a folded tetrahedron whose flaps swing freely. Consistently joining such folded tetrahedra which construct the regular octahedron, we obtain a four-dimensional bird-base. As its cross-section in 3-space, we can see a traditional bird base. Concerning the three-incircle theorem on a triangle found by HUSIMI and HUSIMI (1979), we will prove the similar theorem for one of four congruent tetrahedra which consist of an octahedron having the 4-fold symmetry.