3D Dimer Constant Computed
March 1999
Since 1938, physicists and mathematicians around the world have
been trying to calculate a fundamental quantity known as the
dimer constant. This year, the threedimensional constant was
obtained using a unique computational approach by I. Beichl
of ITL's Mathematical and Computational Sciences Division in
collaboration with F. Sullivan of the IDA Center for Computing
Sciences. The dimer constant describes the rate of growth of the
number of ways to arrange dominos (in twodimensions) or bricks
(in three dimensions) on a lattice of increasing size. Dimer
constants provide key information related to fundamental models
of materials and are factors in computing the partition function
for the monomerdimer system. From the partition function, all
thermodynamic properties such as specific heat of a material can
be computed from first principles. The twodimensional constant
was obtained analytically in 1961, while the threedimensional
problem has defied exact solution. The approximate solution by
Beichl and Sullivan is based on the use of importance sampling
techniques to estimate the permanent of a related matrix efficiently.
The computed value, which comes with rigorous error bars, far exceeds
the accuracy of any previously obtained result. A paper describing
this work recently appeared in the Journal of Computational Physics
(volume 149, number 1, February 1999).
