Optimize printing massive strings to System.out with respect to time
Are you sure the bottleneck is the printing, as opposed to whatever new ClassThatBuildsOnBuilder(builder); is doing? 
Is coordinates always going to have a width of four?
It appears you're building the whole string in memory, and then printing all your output at once at the end. While printing each line to standard-out individually may be slower, it also may take better advantages of the built-in output buffering. (Printing each word individually as you describe having originally done probably isn't a great idea.


Would something like this work?

  private void printRow(){
    System.out.format("#%.3f #%.3f #%.3f #%.3f%n",
      coordinates[0],
      coordinates[1],
      coordinates[2],
      coordinates[3]);


You could also try a Formatter, which would let you use a StringBuilder as you already are, but also use printf syntax. 

Finally, I don't know how to take manual control of System.out's buffering behavior, but it's probably possible and may help.