In Chainmail, units have movement penalties for changing formation or facing. The rules give the penalties for a unit changing from a column to a line formation and vice versa, from a column to a square, and from a line to a square. Only foot troops can form a square. Essentially, you could chart out the formations like so:
Line – Column – Square
Any change from one to the next incurs a penalty of one move. So it takes one movement to reform for each step across the range. As a unit reforms or turns, it loses movement in order to maintain its organization. That makes sense. A unit has to exercise some discipline and energy to reform itself.
The interesting bit in this section, though, is the reliance again on some former knowledge or assumptions. Specifically, none of the formations are described or specified in any way. This caused me to exercise the extent of my stupidity until I realized some of the assumptions inherent in these particular rules.
Initially, I was looking at the straight geometric arrangement of the troops as the determining factor for their arrangement. I assumed that a line is a single file formation of troops, a column is a rectangular body of troops that is significantly wider or longer in one direction than the other, and that a square is a body of troops that approximates a square as closely as possible. So, if you have 24 models in a unit, a line would have them arranged in a 1 x 24 formation of figures. A column would be either 2 x 12 or 3 x 8. A square would be 5 x 5 with a missing model in the formation, possibly in the back rank or the very center of the formation. A 4 x 6 formation would either be a column or a square, depending on how literal you wanted to be about a square.
Then I started thinking in terms of historical formations. Thinking in these terms, a line is the same, a single rank or file of troops. A column is a block of troops, arranged in ranks, facing in a single direction. A square is a block of troops arranged in equal ranks around a central command, facing outward in all directions. The important thing to consider in determining the unit's formation is its facing. A column faces one direction, and a square faces in all four outward directions.
This facing also determines how it can move and how much it is penalized for doing so. Any move in other than a straight line is penalized. Oblique moves incur a 1/4 move penalty, left or right face incur a half move penalty, and turning about takes a full move. So, units in lines and columns move in a particular direction with the appropriate movement penalty. Most troops formed into a square formation, though, would need to reform into a column before moving. The only exceptions would be the troops that are allowed to form a hedgehog as described later in the rules. Square formations are not subject to flanking, but they are immobile until they reform for movement.