/drawing.rel E.Lubkin  phy122 Spr 84 86 91 92 95 210Sp85&90 299Fall90 361Fall91
                                   4-92 9-94 4-95,6,7 9-99 feb2001, jan,sep2002
                      Draw a Lorentz Transformation

Draw it FIRST, then learn what it is good for. TO BE GRADED & RETURNED FOR MORE
                                               WORK: putting labels in, etc.
To draw four FAMILIES of parallel lines, on a sheet of graph paper.
Work in PENCIL, and by HAND:  you will learn more if you do NOT use a
ruler AT ALL.  (Since the graph paper already has two families of
faintly colored lines, there will be six families, altogether.)

Step1  Choose two unequal positive whole numbers p, q.  Let p be the larger.

Step2  Mark a point with a FAT pencil dot     near the middle of the page,
where two major graph-paper lines meet.  This will be your `origin',
and each FAMILY will have one line through that origin.

Step3  Starting from the origin, count  p  boxes to the right, and  q  boxes up
, and mark a dot.  Then go  p  MORE boxes to the right,  q  MORE up, and
mark a dot.  Keep doing that, and you will get dots along half a straight line.
Also mark  p  left and  q  down from the origin, and repeat THAT, to
complete the other half.  Connect the dots, using pencil lightly.
THAT IS YOUR FIRST LINE.

Step4  Make a SECOND line through the origin, by now counting  p  boxes UP
and  q  boxes to the LEFT between dots, and repeating THAT over and over.
(That gets you half a line; you finish the other half by working  p
boxes DOWN and  q  RIGHT.)  Connect THESE dots, TO GET YOUR SECOND LINE.
                                                   .
Pause:  The two lines you have                                   .
    drawn will have come out                               .
 at right angles                                     .    
   to each other,                              .      
       and so will form a leaning cross: .
                                                       .
Step5  Next, instead of starting at the origin, build MORE lines by
`walking between dots' as in steps 3 and 4, BUT STARTING AT THE OTHER
marked points.  Each marked dot on your FIRST line will, using box-counting
of step 4, give a new line parallel to the second line you drew.
Each marked dot on your SECOND line, using box-counting of step 3, will
give a new line parallel to the first line. 
As you `dot' in each new line, also pencil lightly connecting the dots.
You now have MANY lines.
Pause:  Your leaning cross has become a picture which looks like
  pencil-drawn graph paper with big boxes, tilted with respect to the
  finely boxed guide lines of the original, printed graph paper.
  You have drawn TWO of the four FAMILIES of lines.

Step6  Now repeat step 3, starting at the origin, EXCEPT INTERCHANGE
 p  AND  q .  That means, count  q  boxes to the right and  p  boxes up,
between dots.

Step7  Like step 4, but with  p  and  q  swopped:  your `step' is now
 q  up and  p  left.
Pause:  You now have a SECOND cross, tilted the OTHER WAY.

Step8  Complete this SECOND cross to a second crisscross of big-boxed
graph paper, tilted the OTHER WAY.  This means, work as in step 5 but
with  p  and  q  swopped, being careful to build new lines only from
the dots marked in steps 6 and 7.
Pause:  Both pencilled pictures of tilted graph paper are on your one
sheet of paper, making your FOUR FAMILIES of parallel lines.

THE IDEA:  These will be  contours  for  t , x , t' , x' , with each
FAMILY good for one of these four.  If you guess which is which before
I explain, I bet you guess wrong!                                           eof