For a rod at rest, lay a ruler against it and read both ends. Even if the two ends are read at slightly different times the answer is the same, because the rod does not move.
With a moving rod, mark the rear end first and the front end afterwards, and the rod has moved on in between, so the result comes out too long. Mark the front end first and it comes out too short.
The length of a moving rod is found by reading the positions of both ends at the same time. By the relativity of simultaneity the same time differs between frames, so the length can come out differently depending on the frame that measures it.
A rod of length lies on the ground, and someone runs from one end to the other at speed . The length , measured in the frame where the rod is at rest, is called the proper length.
Measured by clocks on the ground, the run takes .
The runner is present both when passing one end and when passing the other. The time on the runner's own clock is therefore a proper time, and by time dilation it is .
In the runner's frame the runner is at rest, and the rod slides past to the left at speed .
From the moment one end of the rod passes the runner until the other end passes, the runner's clock reads . Both passings happen at one place, so no simultaneity has to be decided here.
This time multiplied by the rod's speed is the length of the rod for the runner.
The length of a moving body is of its length at rest. This is Lorentz contraction, or length contraction.
The point in the figure stops at , , , and in turn, and falls through , , , and . As approaches , the length of the moving body approaches .
What contracts is the other person's ruler, not your own. From their side, it is your ruler that is short. As with time dilation, the same holds from both frames.
No force has squeezed the rod. Measured by someone moving with it, its length is always .
Only lengths along the direction of motion contract. Widths across the motion are the same in every frame.
Stand two rods A and B of equal length upright, across the direction of motion, and let them pass each other. With a brush on the tip of each, each rod leaves a mark on the other as they pass.
If B's rod were shorter in A's frame, B's brush would mark A's rod partway up and A's brush would pass over the top of B's rod. In B's frame it would be the other way round, with A's brush marking B's rod.
Which rod carries the mark is a single fact, so both cannot be true. Across the motion, neither rod contracts.
Measure a moving ball by reading both ends at once, and it is contracted along its motion only. What the eye sees is different again, because the time light takes to reach the eye adds its own effect.