Animal Testing

A configuration of unit cubes in three dimensions with integer coordinates is called an animal if the boundary of their union is homeomorphic to a sphere. Shermer discovered several animals from which no single cube may be removed such that the resulting configurations are also animals [16]. Here we obtain a dual result: we give an example of an animal to which no cube may be added within its minimal bounding box such that the resulting configuration is also an animal. We also present two O(n)-time algorithms for determining whether a given configuration of n unit cubes is an animal.