  In this work, the relativistic phenomena of Lorentz contraction and time
dilation are derived using a modified distance formula appropriate for discrete
space. This new distance formula is different than Pythagoras's theorem but
converges to it for distances large relative to the Planck length. First, four
candidate formulas developed by different people over the last 70 years will be
considered. Three of the formulas are shown to be identical for conditions that
best describe discrete space; this equation is then used in the rest of the
paper. It is shown that this new distance formula is applicable to all
size-scales, from the Planck length upwards, and solves two major historical
problems associated with a discrete space-time model. One problem it solves is
maintaining isotropy in discrete space. The second problem it solves is the
commonly perceived incompatibility of the model's concept of an immutable
"atom" of space and the Lorentz contraction of this atom required by special
relativity. With the new distance formula, it is shown that the Lorentz
contraction of the atom of space does not occur regardless of the relative
velocities of two reference frames. It is also shown that time dilation of the
atom of time does not occur. Also discussed is the possibility of any object
being able to travel at the speed of light for specific temporal durations
derived in this work.
