
1. Observing Spiral
Patterns > 2.
Spiral Applet > 3. Dynamical Model Applet
> 4. Cylindrical Spirals > 5. Cylindrical
Spirals Applet > 6. The Farey Tree
and the Golden Mean
4. Cylindrical Spirals
In the three previous sections, you explored
spiral structures that are observed when looking at a flower from
above, or a pine cone from below. These patterns often continue
into spirals wrapping around the stem of the flower, the body of
a cactus or of a pine cone. The geometry is not that of a plane
with a center anymore, but that of a cylinder. In this section,
we explore the kind of spiral leaves make around stems, scales and
prickers around cylindrical pine cones or cacti. To the right is
a view of such patterns on Leanne's Robertson's cactus that some
of you pricked your fingers on a little while ago ( by the way,
it has parastechy numbers 16 , 26 = 2 (8, 13), a kind of double
spiral configuration we may talk about later) We have two reasons
to study this new type of spirals: the first one is raise the awareness
that the cyndrical and planar spiral are generated by the same mechanism.
The planar spirals evolve, as the plant grows, into cylindrical
ones. The second reason, more mathematical, is that cylindrical
spirals, although less stricking to the eyes, turn out to be easier
mathematical objects to study.
In this section, we go over steps similar
to Section 1, but instead of planar spiral, we observe and create
planar ones.
Task
20: Using plants that you
have available (eg. cylindrical pine cones, cacti, pineapples, palm
trees, flowers stems from which you may want to cleanly cut the
leaves), observe and count spirals winding up in opposite direction
up the cylindrical surface.
Task
21: Using the loose
cylinder circles sheet, find a way to place one dot on each line
in such a way that, as you close up the sheet into a cylinder, the
dots form a regular spiral. Use the bottom black dot as your starting
point. You may want to cut one white margin of the sheet to have
the printed lines nicely close up into circles around the cylinder.
Task
22: What kind of rule similar
to that of the Remark in Task 2 are
you applying to make your construction? What kind of geometric objects
do you get when the sheet of paper is flat and you join the dots
of your spiral in the order you have drawn them?
Task
23: Now use the tight
and tighter
cylindrical circle sheets to experiment with different types of
cylindrical spiral. The way your spiral changes should be determined
by the position of the first dot you draw above the line with the
printed dot. Play with the position of this point until you observe
interesting patterns. An efficient way to do this is to experiment
in groups, each person choosing different starting position, and
comparing notes at the end.
Task
24: Connect the dots using
the same closest neighbour rule as you did (or should have!) in
Task 5. Try to find families of them
winding in opposite directions, and count them.
Congratulations! You have now found cylindrical
generative spirals and parastechies. As before, the apparent spirals,
especially in dense structures, are usually the parastechies, which
join points with their nearest, or second nearest neighbour. And
as before, the number of parastechies are usually successors in
the Fibonacci sequence.
1. Observing
Spiral Patterns > 2.
Spiral Applet > 3. Dynamical Model Applet>
4. Cylindrical Spirals > 5. Cylindrical
Spirals Applet > 6. The Farey Tree
and the Golden Mean

