Geomancy is the most mathematically transparent of the divinatory systems, and the number sixteen is not a choice anyone made. It is forced.
Generating a figure
The practitioner makes a line of marks, traditionally poked into sand or earth with a stick, later dots on paper, without counting them. The line is then counted, and only one thing about it is retained: whether the total is odd or even.
Odd gives a single point. Even gives a double point.
Four such lines produce a figure of four rows, each row either single or double. The four rows are read as Fire, Air, Water and Earth, from the top down.
The arithmetic
Four rows, two possible states each, gives two to the fourth power: sixteen. That is every figure that can exist. There is no seventeenth, and no figure has been left out of the tradition by oversight.
This is why Via and Populus are the two extremes of the set, all singles and all doubles, and why every other figure is a mixture. It is also why each figure has exactly one inverse, obtained by flipping every row, and why the sixteen fall naturally into eight inverse pairs.
The system is a four-bit binary code with an interpretation attached to each of its sixteen values, arrived at several centuries before anyone described it in those terms.
Where the figures came from
The names in use here are Latin (Via, Populus, Acquisitio, Carcer), because the system reached Europe through Latin translation. The Arabic tradition had its own names, and behind those the individual figures are older than any of the naming systems.
What the attributions are not
Each figure carries an element, a planet and a zodiacal sign, and this atlas records them. Those attributions were added to the figures, not derived from them, and the sources do not entirely agree with one another. Agrippa, Cattan and the later English handbooks differ on several assignments.
The point structure is intrinsic and can be checked. The astrological correspondences are a tradition, and a contested one in its details.