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'Spiroflowers' printed from https://nrich.maths.org/
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
the Spirostars problem .
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.
If we define a 'turtle' as a point and a direction, we can use the
notation $(x, y, \theta)$ or $(z, \theta)$ or $ze^{i\theta}$. 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.