I see this on old movies all the time, usually used in tandem with a shot of a radar screen, control panels, etc. Its this monitoring device that has a screen like an oscilliscope but instead of the standard oscope line, its a circle that sort of rotates around on an axis speeding up quickly until it forms this rounded square pattern.
What is this device called and what purpose does it serve?
It sounds like an oscilloscpe displaying [url=http://www.egr.msu.edu/classes/ece482/Teams/99spr/design2/web/resources/lissajous/lissajous.htm]lissajous* patterns. This can be done on dual-input scopes, using one input for the X-axis and the other for the Y-axis.
Grrrrrrrr. [Fixed link.](It sounds like an oscilloscpe displaying [url=http://www.egr.msu.edu/classes/ece482/Teams/99spr/design2/web/resources/lissajous/lissajous.htm).
Yep, that’s exactly it. Thanks for the link.
I was going to reply earlier, but I couldn’t spell Lisijus, so I went for a walk on the beach instead. But now it’s raining, so I’m back at the computer. 
Here’s a neat site where you can draw Lissajous figures yourself and see how the various parameters change the pattern.
However, I think the reason they were used in old SciFi movies is just to amaze and awe the audience.
Although Musicat is probably right about the “amaze and awe” factor, these figures might actually have a purpose in radars. The X and Y channels of the oscilloscope are connected to two different signals that are supposed to be frequency- and phase-matched but may drift due to environmental changes. Any drift is then easy to see as a “rotation” of the figure in the display. This might be used to check the stability of two oscillators against one another, or maybe even to measure a Doppler shift.
I was able to find one reference to WWII use of a Lissajous display for precise measurements of the pulse repetition rates of hostile radars.
