by Dr. Bruce McLaughlin
This article summarizes the hypothesis that the universe comprises a countably infinite array of cubes.
Introduction
Following George Stoney’s work in the 1870’s, Max Plank, in 1899, invented a system of units which, as he argued, “would not depend on the choice of special bodies or substances and would be valid for all epochs and all cultures including extraterrestrial and extrahuman ones and could therefore serve as natural units of measurements.” These units are:
PARAMETER 
EXPRESSION

VALUE (SI) 
Planck Length 
l=[(hG)/c^{3}]^{1/2} 
1.61625×10^{35} m 
Planck Mass 
m=[(hc)/G]^{1/2} 
2.17644×10^{8} kg 
Planck Time 
t=[(hG)/c^{5}]^{1/2} 
5.39124×10^{44} s 
Planck Charge 
q=(4πε_{0}hc)^{1/2} 
1.87554×10^{18} C 
Planck Temperature 
T=[(hc^{5})/(Gk_{B}^{2})]^{1/2} 
1.41678×10^{32} K 
where h is the reduced Plank’s Constant, c is the velocity of light, G is the gravitational constant and k_{B} is Boltzmann’s constant; the Plank Charge was added later. These units have largely been ignored by the engineering community because they are not particularly useful for engineering computations. But the community of scientists has not completely abandoned them. In string theory, for example, the Planck Length roughly corresponds to the size of strings which comprise all particles in the Standard Model. The Planck Length is also a crucial component of the equation that describes entropy in Black Holes. How odd that a century old curiosity should keep popping up in cosmology and particle physics.
Background
In our everyday experience space and time seem very different. We are forced to progress through time in a specific way yet we can move through space at will. Even so, all contemporary models of fundamental physics claim that reality comprises a spacetime continuum. Time appears merely as a fourth dimension just like the three ordinary dimensions of space. But some scientists have proposed that this view of reality is intrinsically false. They claim space is not a continuum and neither is time.
Analysis
Hundreds of pages have been written on this subject most recently by Stephen Wolfram in A New Kind of Science. The basic building blocks of space, which cannot be subdivided, may be called pixels, voxels, cells, nodes or something else. Voxel, for volume element, will be used in this paper.
A filling of space or tessellation can be accomplished with many polyhedra but none are simpler than a cube. Selecting a cubic voxel and incorporating the Plank Length suggests the universe is constructed around a cubic lattice such that the smallest identifiable physical entity is bounded by eight vertices, twelve edges each equal to the Plank Length and six flat faces. The universe would thereby be divided into an ordered (numbered), countably infinite array of volumes. The volume enclosed by an undeformed voxel would be [1.6 x10^{35}]^{3} m^{3}. If a voxel can be deformed by some stimulus, the eight vertices would retain their identities but may slightly change their positions, the twelve edges would remain straight but each may slightly change its length and each of the six flat faces would either remain flat or possibly deform into a minimal surface with zero mean curvature (e.g. soap film); in general, four points do not define a plane. 13.7 billion years after the beginning, the deformable cubic lattice may no be longer perfect but may contain line and surface defects analogous to dislocations and stacking faults.
The “state” of a particular voxel would not be a continuous property; the number of allowable discrete states would probably be small with many values of energy, spin, etc. associated with a given state. The state of every voxel in the universe would be updated at successive time intervals equal to the Plank Time (5.4x10^{44} sec). Other scenarios of updating are possible but this is the simplest. What are some justifications and consequences for this hypothetical universe of voxels?
 Motion, as we understand it, would not exist. Nothing would move through the cubic lattice from one voxel to another. Individual voxels would simply change their state at Planck time intervals. Think of watching a gymnast on a television screen. The gymnast appears to be moving on the screen. But no part of the screen is moving!
 The maximum speed would be the speed of light. (Plank Length)/(Plank Time) = c. This would apply to all directions and not just the lattice axes. For example, even a simple rule of time transformation specifying that a voxel should transform if exactly 2 of its 26 neighbors had transformed on the previous step yields a nearly spherical wave front.
 The EPR paradox, for a quantum entangled pair of particles, questions the idea that two distant, noninteracting systems can be correlated in such a way that a change in the state of one results in an immediate response of the other. But, in a universe of voxels, all things are connected. Countably infinite strings of voxels connect any two particles. These strings constitute ‘hidden variables.’ As we are informed by scripture, God is before all things and in Him all things hold together. (Col 1:17)
 Because any scientific observer and associated detection instrumentation would be transforming in sync with the property being observed, no evidence of voxels or voxels being updated at successive time intervals could be detected.
 Why would the one true God, existing as three distinct, transcendent, immanent, infinite, eternal and immutable persons, create such a system? One reason might be that it would give God the ability to fast forward and slow motion the universe without leaving any evidence detectible by us. For example, if God decided to increase the velocity of light by a factor of say 100, the length of the basic time interval would be smaller by a factor of 100,000. To a transcendent observer, the universe would be changing 100,000 times faster but no entity inside the universe would sense a difference because the other Planck Units would have changed accordingly.
 Small groups of voxels (i.e. three groups of three contiguous voxels) might be the fundamental particles of String Theory.
Conclusion
The concept of spacetime continuum is an integral component of General Relativity which examines the universe on a grand scale. Quantum Mechanics, Bohmian Mechanics, and String Theory study the universe on much smaller scales of space and time which are, nevertheless, many orders of magnitude larger than the Plank Length and Planck Time. A definitive experiment has not been devised to determine whether space and time are continuous or discrete.