The Search for a Universal Tile
Feng Zhu · 2002
One of the most remarkable discoveries in the theory of tilings concerns the existence of sets of prototiles which admit infinitely many tilings of the plane without any of these tilings being periodic. Sets of prototiles with this property are called aperiodic. In 1966, R. Berger discovered a set of 20426 prototiles with corresponding matching rules to be aperiodic. Subsequently, the number of aperiodic prototiles was reduced to two by R. Penrose in 1974. It is still an open problem whether there is a single universal aperiodic two-dimensional prototile with corresponding matching rules. In 1996, Gummelt identified a single decagon covering that covered the plane aperiodically. The covering allows tiles to overlap in five different ways. Our research is motivated by her approach. We seek to develop a partitioning of the regions such that points in different partitions never collide in an overlap. Such partitioning is achieved by a special coloring process which assigns each point to be one of the five colors. The coloring process is based on infinite decomposition of Robinson tiles and a Cantor-like construction. We will show that each color set has positive measure