Over three decades of scientific endeavors to realize programmable matter, a
substance that can change its physical properties based on user input or
responses to its environment, there have been many advances in both the
engineering of modular robotic systems and the corresponding algorithmic theory
of collective behavior. However, while the design of modular robots routinely
addresses the challenges of realistic three-dimensional (3D) space, algorithmic
theory remains largely focused on 2D abstractions such as planes and planar
graphs. In this work, we present the 3D geometric space variant for the
well-established amoebot model of programmable matter, using the face-centered
cubic (FCC) lattice to represent space and define local spatial orientations.
We then give a distributed algorithm for the classical problem of leader
election that can be applied to 2D or 3D geometric amoebot systems, proving
that it deterministically elects exactly one leader in O(n) rounds
under an unfair sequential adversary, where n is the number of amoebots in
the system. We conclude by demonstrating how this algorithm can be transformed
using the concurrency control framework for amoebot algorithms (DISC 2021) to
obtain the first known amoebot algorithm, both in 2D and 3D space, to solve
leader election under an unfair asynchronous adversary.Comment: 16 pages, 4 figures, 2 table