Kaplan's research work focuses on the application of computer graphics and mathematics in art and design. He is an expert on computational applications of
tiling theory.
Exotic geometries in protein assembly In 2019, Kaplan helped to apply the concepts of
Archimedean solids to
protein assembly, and together with an experimental team at
RIKEN demonstrated that these exotic geometries lead to ultra-stable
macromolecular cages. These new systems could have applications in targeted
drug delivery systems or the design of new materials at the
nanoscale.
Einstein problem In 2023, Kaplan was part of the team that solved the
einstein problem, a major open problem in tiling theory and
Euclidean geometry. The problem is to find an "aperiodic monotile", a single
geometric shape which can
tesselate the plane
aperiodically (without
translational symmetry) but which cannot do so periodically. The discovery is under professional review and, upon confirmation, will be credited as solving a longstanding mathematical problem. In 2022, hobbyist
David Smith discovered a shape constructed by gluing together eight
kites (in this case, each kite is a sixth of a regular hexagon) which seemed from Smith's experiments to tile the plane but would not do so periodically. He contacted Kaplan for help analyzing the shape, which the two named the "hat". After Kaplan's computational tools also found the tiling to continue indefinitely, Kaplan and Smith recruited two other mathematicians, Joseph Samuel Myers and
Chaim Goodman-Strauss to help prove they had found an aperiodic monotile. Smith also found a second tile, dubbed the "turtle", which seemed to have the same properties. In March 2023, the team of four announced their proof that the hat and turtle tiles, as well as an infinite family of other tiles interpolating the two, are aperiodic monotiles. Both the hat and turtle tiles require some reflected copies to tile the plane. After the initial preprint, Smith noticed that a tile related to the hat tile could tile the plane either periodically or aperiodically, with the aperiodic tiling not requiring reflections. A suitable manipulation of the edge prevents the periodic tiling. In May 2023 the team of Smith, Kaplan, Myers, and Goodman-Strauss posted a new preprint proving that the new shape, which Smith called a "spectre", is a strictly chiral aperiodic monotile: even if reflections are allowed, every tiling is non-periodic and uses only one chirality of the spectre. This new shape tiles a plane in a pattern that never repeats without the use of mirror images of the shape, hence been called a "vampire einstein". ==Honors and awards==