The spiropath contains a pattern which is repeated over and over
  again with different starting points and in different initial
  directions. Imagine replacing each pattern by a single line
  segment joining the starting point and end point of that pattern
  (or equivalently by the single vector which is the sum of all the
  vectors forming a single example of the pattern). Then the
  repeated pattern reduces to repeated vectors of the same length
  end to end with a given angle of turn between the vectors, so the
  question of whether the path will close up reduces to considering
  the result of <a href=
  "http://nrich.maths.org/public/viewer.php?obj_id=5432&amp;part=index">
  the Spirostars problem</a> .<br />
  <br />
  The diagrams given contain a small triangle which is the Logo
  'turtle'. As the path is drawn by the Logo software you can see
  the turtle moving around. It can be hidden but we have chosen to
  show it to indicate astarting point and initial direction for the
  spiropath.<br />
  <br />
  



 
  If we define a 'turtle' as a point and a direction, we can use the notation
  (<i>x</i>, <i>y</i>, <font face="symbol">q</font
>) or (<i>z</i>, <font face="symbol">q</font
>) or <i>ze</i><sup><i>i</i><font face="symbol">q</font
></sup>. The motif which forms the 
  pattern in the spiropath
  is repeated but from a different starting point each time and, in general, 
  with a different initial direction. Each repetition of the set of instructions 
  which draws the motif has the effect of mapping 
  turtles to turtles where the turtle gives the initial point and the initial 
  direction. These motifs have the same form but different starting points and 
  initial directions. When the motif is repeated over and over again it may 
  return to the same initial 'turtle' and repeat a cyclic pattern as in the 
  first three examples, or it may never return to a previous starting point and
  path may go on to infinity. 


            
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