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Galileo’s odd-number sequence, $1, 3, 5, \ldots$, is not itself the physical motion of a uniformly accelerating object; rather, it is a dimensionless discrete representation of that motion. Continuous uniformly accelerated motion is recovered by assigning physical distance and time scales to this mathematical pattern and examining how those scales transform as the observation interval is progressively refined.
As the observation frequency increases by a factor of $n$, the elementary time interval scales as $1/n$, while the corresponding elementary distance scales as $1/n^2$. The dimensionless odd-number sequence remains unchanged under this refinement, while the physical scale associated with each term changes systematically. The resulting $n^2$ scaling follows from the quadratic dependence of displacement on time and provides a concrete example of a more general finite-difference principle: for a function whose relevant increment scales as a homogeneous quantity of degree $p$, the corresponding discrete sequence exhibits an $n^p$ scaling under refinement.
This framework also suggests accessible extensions for students. Uniform motion, for example, produces the sequence $1, 1, 1, \ldots$, with linear scaling by $n$, while a cubic position law produces the sequence $1, 7, 19, 37, \ldots$, with scaling by $n^3$. These examples allow students to investigate how different continuous functional relationships are encoded in discrete numerical patterns.
By distinguishing among the abstract numerical pattern, its geometric representation, physical dimensions, measurement units, and the numerical values assigned to those quantities, students can see more clearly how continuous physical behavior is represented through discrete mathematics. In this framework, the limit $\Delta t \rightarrow 0$ is not a change to the underlying mathematical pattern; rather, it is a refinement of the physical scale used to map that enduring discrete structure onto continuous motion.
[1] J. Fiala, “From Discrete Pattern to Continuous Motion: Galileo’s Law of Odd Numbers,” The Physics Teacher, under review.
