Tiling space with tetrahedra - a partitions perspective

Prepared by:

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

email:

malkevitch@york.cuny.edu

web page:

http://york.cuny.edu/~malk

It is well known that it is not possible to tile 3-dimensional space with congruent regular tetrahedra. A regular tetrahedron has 4 vertices, 4 faces, and 6 edges, and all of its faces are congruent equilateral triangles. A recent strand of research has to been to determine exactly how densely on can pack pack 3-space with regular tetrahedra. It is also known that there are some tetrahedra T with which one can pack (tile) 3-space without overlaps or gaps with congruent copies of T. The purpose of this microworld is to adopt a partitions point of view for these questions in the hope of obtaining richer insight.

The partitions of 6 are 11 in number. Given a partition of 6, say, {4, 2} we can associate this partition with a tetrahedron which has 4 edges of equal length and two other edges with equal length but different from the length of the other 4 edges. One can can refine the 11 partition types that can occur for tetrahedra from this point of view with 25 partition types using graph theory ideas, as explained in Malkevitch and Musaa, in the references.

Question 1

For each of the 11 partition types of tetrahedra based on partitions of the number 6, the number of edges of a tetrahedron, determine if there is a tetrahedron of this partition type which will tile 3-space. As noted above for the partition {6} the answer is no.

Question 2

For each of the 25 partition types of tetrahedra based on partitions of the number 6, the number of edges of a tetrahedron, and graph theoretical aspects of the relative position of edges determine if there is a tetrahedron of this partition type which will tile 3-space. As noted above for the partition {6} the answer is no.

Comment: The paper of Marjorie Senechal discusses some of these matters but does not adopt a partitions point of view.

Comment: One can also approach this problem from a compositions point of view instead of a partitions point of view. A composition of n is an ordered collection of positive integers whose sum is n. For n equal to 6 their are 32 partitions of 6.

References:

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.

Senechal, Marjorie. "Which tetrahedra fill space?." Mathematics Magazine 54, no. 5 (1981): 227-243.