Mathematician  Peter  Maurer  made   public   the  simple
       algorithm that is used to  produce these  pictures. (Ref.
       The  American Mathematical Monthly, Volume 94,pp631-645.)
       For obvious reasons the algorithm referred to is known as
       "The Rose".
                Here, in  terms  that  can easily be transcribed
       to any standard computer language, is what you have to do
       to produce such pictures on your own computer.
                The program uses whole number variables N, D and
       A, and real number varibles X, R, oldX, oldY, newX, newY.
                At the start of the program, set the variables N
       and  D equal to two  whole  numbers  between  1  and  359
       (inclusive), and set A, oldX, and oldY equal to zero.
                Now  you   set  up  a  loop  consisting  of  the
       following steps.
                1. SetA  equal to A+D. If A>360 replace A by the
       remainder obtained when  you  divide  A  by  360 (that is
       compute A mod 360 and set  A equal to the result).
                2. Calculate N*A, then reduce  it  mod  360(i.e.
       take  the  remainder on dividing N*A by 360), convert the
       result from degrees  to  radians,  and set X equal to the
       final  result.  (To  convert  from  degrees  to  radians,
       multiply by 0.01745.)
                3. Set R equal to the sine of X.
                4. Convert A from degrees to  radians, and set T
       equal to the result.
                5. Set newX equal to R*sin(T) and set newY equal
       to  R*cos(T).  (This converts the polar coordinates (R,T)
       to rectangular coordinates (newX,newY).)
                6. Draw a  line  from  the  point(oldX,oldY)  to
       (newX,newY).
                7.  If  A  is equal to zero, then stop, else set
       oldX equal to newX,  oldY  equal  to newY, and go back to
       step one.
                The  only  input the program requires are values
       for the two  integer variables N and D. Different choices
       of these two numbers  produce  often strikingly different
       patterns. For some choices the pattern  turns  out  to be
       fairly  rudimentary,  sometimes  just  a  single dot. For
       other  values  you  will  obtain pictures  every  bit  as
       attractive as these:                                                            Enter N=4, D=43 or  N=5, D=97.

                They  may  not smell as sweet, but the roses you
       produce on your  computer  screen  can bring every bit as
       much pleasure as the real thing.

                p.s. Keith Devlin is a professor of mathematics.
       He formerly worked in this country  but  now lives in the
       United States.

                  Margaret Wright. K3X