A Solution to the Animal Problem

Let A be an animal. We consider magnifications of A in x-, y-, or z-direction. But this is not always possible because there may be voxels that block the magnification. To overcome this difficulty, we consider an AD-movement. This is performed by considering the magnification in two directions (e.g., y-direction and z-direction) such that any series of blocking voxels never forms a cycle. If a magnification is possible for two directions then it is also possible for the third direction. After all undilatable voxels have been erased, we can apply a previously shown result for showing that the animal problem is positively solvable. This paper is a continuation of [4]. Let B be a well-composed picture [3] that does not contain any cavity or any tunnel. Is B SD-equivalent (i.e., equivalent by repeated simple deformations) to a single voxel? This problem was named B-problem by the late Azriel Rosenfeld. In [4], we have given a positive solution to this problem. The animal problem proposed by Janos Pach was the question whether every animal can be reduced to a single unit cube by a finite sequence of moves, each consisting of either adding or deleting a cube provided the result is an animal again at each move. Azriel Rosenfeld called this problem the A-problem. In this animal problem, the deformation is not based on SD but must preserve animality of the picture at each step. This means that the B-problem is weaker than the A-problem. In this paper, we give a positive solution to the animal problem. We assume that readers are familiar with basic definitions in digital topology, such as provided in [1, 2], and, in particular, with the discussion in [4]. An animal is defined as a topological 3-ball in R, consisting of unit cubes (i.e., a subcomplex in the 3D grid cell space). In accordance with terms of digital topology, hereafter, we call a unit cube also a voxel. Let A be an animal consisting of black voxels (black means that a unit cube is present, i.e., in a given “object”) and p be a black voxel. 1 Professor emeritus at Hiroshima University, Japan; email: [email protected]