Classifying Tetrahedra Inscribed in a Sphere

Joseph Malkevitch
Department of Mathematics
York College (CUNY)
Jamaica, New York 11451


web page:

A tetrahedron is a convex n-dimensional solid formed from 4 points in three dimensional space which don't lie in a plane. Such a convex polyhedron has 4 vertices, 4 faces and 6 edges. We can classify tetrahedra by partition type. There are 11 partitions of 6 (for example, {5,1} or {3, 1, 1, 1}} and we will interpret, say, the partition {4, 2} as meaning a tetrahedron with 4 edges of equal length and two edges of equal length, but different from the length of the other 4 edges. It turns out there exist tetrahedra of all 11 partition types. One can refine partition type further by noting the "graph theory" properties of the edges of the different lengths. Thus, partition type {4,2} can be "subdivided" into those where the 4 equal edges form a simple circuit or not. One can see if the two equal edges can be in a path, or be disjoint. When the equal edges form a circuit, the other two equal length edges must be disjoint. Thus, {4,2} subdivides into two partition types. It can be shown that the 11 partition types refine to a total of 25 partition types in this way.

Some tetrahedra are such that all 4 of the vertices of the tetrahedron lie on a sphere.

Research Problem

Determine what are the possible partition types of tetrahedra whose vertices lie on a sphere. What happens if one wants integer edge lengths?


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