  Unstable nonchaotic solutions embedded in the chaotic attractor can provide
significant new insight into chaotic dynamics of both low- and high-dimensional
systems. In particular, in turbulent fluid flows, such unstable solutions are
referred to as exact coherent structures (ECS) and play an important role in
both initiating and sustaining turbulence. The nature of ECS and their role in
organizing spatiotemporally chaotic dynamics, however, is reasonably well
understood only for systems on relatively small spatial domains lacking
continuous Euclidean symmetries. Construction of ECS on large domains and in
the presence of continuous translational and/or rotational symmetries remains a
challenge. This is especially true for models of excitable media which display
spiral turbulence and for which the standard approach to computing ECS
completely breaks down. This paper uses the Karma model of cardiac tissue to
illustrate a potential approach that could allow computing a new class of ECS
on large domains of arbitrary shape by decomposing them into a patchwork of
solutions on smaller domains, or tiles, which retain Euclidean symmetries
locally.
