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The basic idea is that we build up the spiral in the lower half of the playfield, keeping tracking of the min and max coordinates as we go along. Once all the input has been processed, we iterate over the range of coordinates and write out the content that was saved.
Note that the dx and dy values are in the range 0 to 3, corresponding to the relative offsets -1, 0, 1, 0. Updating the direction is thus done by adding 1, modulo 4. Converting to a relative offset is done by subtracting 1, modulo 2.
"'"::00p09p19p Set x, minx, and maxx to 39.
35*::10p29p39p Set y, miny, and maxy to 15.
020p130p Set dx to 0 (relative offset -1) and dy to 1 (relative offset 0).
140p Set the segment length to 1.
0 Set the index to 0 (on the stack).
-- The main input loop start here --
~ Read a character from stdin.
:48*`_ If it's greater than 32 (i.e. not whitespace or EOL)...
00g: Get the current value of x and make a duplicate.
::09g\`8+0\p If it's less than minx, update minx.
::19g`8+1\p If it's greater than maxx, update maxx.
20g1-2% Calculate (dx-1)%2, which converts dx into the range -1 to 1.
+00p Add that to x and save (leaving the original x on the stack).
10g: Get the currrent value of y and make a duplicate.
::29g\`8+2\p If it's less than miny, update miny.
::39g`8+3\p If it's greater than maxy, update maxy.
30g1-2% Calculate (dy-1)%2, which converts dy into the range -1 to 1.
+10p Add that to y and save (leaving the original y on the stack).
p Store the character we read at the original x,y coordinates.
1+ Increment the index by 1.
:40g%! Mod that with the current segment len; if zero it's time to turn.
:20g+4%20p Add the turn condition to the dx value, thus turning if needed.
:30g+4%30p Similarly add the turn condition to the dy value.
:40g+40p Also add it to the segment length, so it gets longer every turn.
!* Multiply the index by !turn, so it's reset to zero on every turn.
^ Repeat the loop, reading another character.
0`!#v_^ If the char <= 32, we test for EOF and repeat the loop if not.
> If it is EOF, we start the output routine.
29g Initialise the y coordinate with miny.
:39g\- Calculate maxy - miny to use as the height countdown.
-- The outer output loop starts here --
\:50p Swap y to the top and save in a temporary variable.
09g Initialise the x coordinate with minx.
:19g\- Calculate maxx - miny to use as our width countdown.
\ Swap x to the top of the stack.
-- The inner output loop starts here --
:50g Take a copy of x and the y save in the temporary variable.
g, Read the character at those coordinate and write to stdout.
\:_ Swap the width countdown to the top of the stack and break if zero.
1- Otherwise decrement the width countdown.
\1+ Swap the x value back to the top and increment it.
< Repeat the inner loop.
$$ Clear the width countdown and x value from the stack.
1+ Increment the y value.
\ Swap the height countdown to the top of the stack.
:¬#@_ If it's zero, exit.
1- Otherwise decrement it.
55+, Output a line break.
^ Repeat the outer loop.