Program assists in exploring Archimedean spirals and related plant structures.
Documentation includes classroom notes.




GENERAL INTRODUCTION
 The flowers and fruits of certain plants -such as daisy, sunflower, pine and
sequoia- show remarkable spiral patterns. The florets of the flower or the
scales of the cone form a primary spiral; more directly visible to the naked
eye, however, are two secondary spirals or rays, one radiating clockwise, the
other anticlockwise.
 The primary spiral has the characteristics of an Archimedean point spiral.
These characteristics are best expressed in terms of the properties of radial
lines from the centre of the spiral to its points. First, the angle between
successive radial lines takes a fixed value. Second, the length of the radial
line to the Nth point is proportional to the square root of N.
OPTIONS AND OPERATION
 This program helps the user to construct and analyse Archimedean point
spirals; in particular to model plant spirals. The program is menu-driven with
the following options:
1: SET ANGLE sets the size (in degrees) of the fixed 	angle value generating
the spiral.
2: DRAW BLOBS draws the spiral slowly, representing each  point by a blob. This
takes 150 seconds.
3: DRAW DOTS draws the spiral quickly, representing each point by a dot. This
takes 5 seconds.
4: DRAW RAYS superimposes a pattern of rays onto the spiral. The number of rays
is set, then each is drawn in turn. (Press ENTER to cue the next one). Best
used on the dot spiral.
5: SHOW SKETCH shows the current spiral sketch.
6: DEMO PLANT shows a typical plant spiral.
7: QUIT terminates the program.
To return to the menu from the current sketch, press ENTER.
CLASSROOM NOTES
 I have used this program with success with lower and upper secondary students
and with teachers in initial and inservice training. The classroom activity is
divided into three phases, each of which starts with an organising
introduction; continues with an extended period of pair or small group work by
students; and concludes with class reporting and discussion of results.
 I usually start by getting students to work with real pine and sequoia cones,
and with photocopies of photographs of daisy and sunflower heads: looking for
spiral patterns; marking them out with coloured pens; counting the number of
spiral rays making up each visible set. When students have reported and
discussed their results, I mention that pine cones usually have 8 rays one way,
13 the other, sometimes 5 with 8, while daisies and sunflowers commonly have 13
with 21, although larger numbers are found.
 In the next phase, I introduce the calculator program to students. Taking an
example, such as the 61 degree spiral, I demonstrate how to set the angle to
61, then how to draw the blob spiral. As this takes place slowly, it can be
talked through, focusing attention on the 61 degree turn between successive
blobs, and the way in which blobs lie at an increasing distance from the
centre. I then show the quicker dot spiral option, and use the ray option to
superimpose the visible 6 rays on this example. Finally, I explain how to move
each way between menu and sketch.
 Having introduced students to the program, I present them with two open
questions to tackle in their pairs or groups.
* Why do some angle values (such as 61) produce spirals with rays which
bend clockwise or anticlockwise, while others (such as 60) give spirals
with straight rays?
* Is there any relationship between the size of the angle generating the
spiral and the number and pattern of rays? In particular, is it possible to
predict these features of the spiral from the angle size?
 In the final phase, I introduce the demonstration option in the calculator
program, showing the plant spiral. I emphasise that it is well packed, tightly
but evenly spread, and without overlaps. The students are asked to find angle
values which produce such a spiral.
USEFUL REFERENCES
 * Peter Stevens, Patterns in Nature (Penguin, 1976).
 * any reference to spiral phyllotaxis.
AUTHOR
 Kenneth Ruthven, University of Cambridge
 Post: 17 Trumpington Street, Cambridge, CB2 1QA, UK
 Email: KR18@UK.AC.CAM.PHX