Tetrahedra With Right Angle Faces

Prepared by:

Joseph Malkevitch
Department of Mathematics and Computer Studies
York College (CUNY)
Jamaica, New York



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It is an interesting fact that a cube can be decomposed into tetrahedra whose faces all have right angles. Sometimes such a tetrahedron is called an orthoscheme. Of course, one can have tetrahedra with fewer than 4 right angles. The purpose of this microworld is to determine if interesting new insights into tetrahedra can be obtained by combining a partitions approach to tetrahedra which also considers the right angles in the faces of the tetrahedron. Partition types take into account that the 6 edges of a tetrahedron might be such that 4 of them have one length, and two of them have a different length. In this case we have partition type {4, 2}. Also graph theoretical properties of the equal length edges can be used as well (see references). Depending on one's point of view there are either 11 or 25 partition types.

Question 1

a. For each of the 11 partition types of tetrahedra determine the number of right angles that tetrahedra of this partition type can have so as to further refine classifying tetrahedra.

b. For each of the 25 types of tetrahedra (using graph theory information) determine the number of right angles that tetrahedra of this partition can have so to further refine classifying tetrahedra.

Note: One can look at whether tetrahedra can have 0, 1, 2, 3 or 4 right angles. One is especially interested in examples where the edge lengths of the tetrahedron if possible are integers. Right triangles such as 3, 4, 5 triangles have integer side lengths. Such integer triples are known as Pythagorean Triples.


Alonso, O., and J. Malkevitch, J., Classifying Triangles and Quadrilaterals. Mathematics Teacher, 106(7)(2013) 541-548.

Malkevitch, J., and D. Mussa, The transition from two dimensions to three dimensions- some geometry of the tetrahedron, Consortium Number 105, Fall/Winter 2013, p. 1-5.