摘要
It is well known that a triangle can be divided by mid-point refinement into four sub-triangles with the same shape. Similarly, a tetrahedron can be parted into eight subtetrahedra, which are generally not uniform in shape. This paper proves that there exist a set of tetrahedra, which is called isometrically subdivisible tetrahedra(IST) and can be divided into eight isometric subtetrahedra, including identical and reflection ones. And a new classification of tetrahedra is put forward, based on which all tetrahedra can be categorized into 26 classes according to both the number of maximum equal edges and topological relations. The IST belongs only to three of the classes. That result provides a new viewpoint of spatial structure and may be used to tile or subdivide space uniformly or isometrically.
It is well known that a triangle can be divided by mid-point refinement into four sub-triangles with the same shape. Similarly, a tetrahedron can be parted into eight subtetrahedra, which are generally not uniform in shape. This paper proves that there exist a set of tetrahedra, which is called isometrically subdivisible tetrahedra(IST) and can be divided into eight isometric subtetrahedra, including identical and reflection ones. And a new classification of tetrahedra is put forward, based on which all tetrahedra can be categorized into 26 classes according to both the number of maximum equal edges and topological relations. The IST belongs only to three of the classes. That result provides a new viewpoint of spatial structure and may be used to tile or subdivide space uniformly or isometrically.
基金
Supported by National Key Basic Research Program(2004CB318000)
National Natural Science Foundation of China(60573154,61227802 and 61379082)
