I am hoping someone can tell me this can be done and how to do it.
Every time I log into linux I open three terminal windows and one gedit window. I arrange the terminals one atop the other on one side of the screen and the gedit window takes up the other half of the screen.

I would like the desktop arrangement to come up automatically when I log in. Does anyone know how to do this?

Thanks

Don't have a clear answer for you, but I think the best solution will depend on what desktop environment & window manager you are using.
I think some Desktop Environments also have options to restart applications that were open at the time of the last shutdown.

For the terminal emulator windows you could put a section into your autostart script the opens the windows. I have never used it, but there seem to be a couple of useful options, which again might also depend on what terminal emulator you are using.
E.g. this looks like something that might help you.

There is an Autostart folder in the /home/user hidden .kde folder where you can put a script. Since you are using gedit, you probably aren't using kde. Take a look at the link below for info on Unity or Gnome, second post:

http://askubuntu.com/questions/20991...login-on-12-10

We're just guessing here because you neglected to indicate which Desktop you are using as well as the distribution of Linux,