Operation

These two programs can create figures generated by straight lines. Points around the circumference of a circle (Rose 1) or a square (Rose 2) are joined by straight lines according to a rule chosen by the user.

When the number of points is large, certain rules give a curve stitching pattern; when this occurs each of the straight lines is a tangent to a generated figure; this figure is the envelope of the lines. In mathematical terms, this process is the dual of a figure generated by a series of points, a locus. However, such an envelope is not always apparent and the attention is then focused on the pattern of lines created.

The user can create figures by controlling the size of the circle or square, within prescribed limits; the number of points on the circumference, again within prescribed limits; the rule for joining up points; the speed at which the drawing is done and the colour.

Three figures can be superimposed on the same diagram and monochrome copies of the figures can be printed out.

After the initial setting up procedure at the start of the program, there is an optional instruction and demonstration procedure. The program then operates from commands indicated on the screen. The instructions can be accessed by pressing I.

The program is suitable for 10 year olds up to sixth formers. Pupils of differing ability and mathematical maturity can use the program provided they can understand a function rule such as P→P+20. The function rule uses modulo arithmetic; if there are 80 points on the circle or square, P→P+20 maps 60→80; 61→1, 62→2 and so on.

Educational Notes

The programs can be used in an instructional mode, but they are designed for individual or group use in an exploratory mode. The following objectives may be achieved:

1. exploring a sequence of rules and explaining the results obtained;
2. discovering how to produce a given figure;
3. exploring the relation between the number of points and the rule employed;
4. illustrating aspects of coordinate geometry and number theory.

Suggestions For Use

The program can either be related to specific topics within the mathematics curriculum or be used in the context of investigational work. The following suggestions are ways in which the program could be used in either context.

Rose 1: a series of rules of the form P→P+K, where K=0, 1, 2, 3 ... give concentric circular envelopes; for this purpose, the number of points on the circle should be give the same circle; the first two rules gie the same circle if drawn with the same radius but with 50 and 100 points respectively. The pupil could try to work out the reason for this.

Rose 1: a series of rules of the form P→P*K, where K=2, 3, 4 ... and a large number of points give a series of cusped figures starting with the cardioid (K=2) and nephroid (K=3).

Make the program produce a predefined figure. For example:

1. Rose 2: 3 points on each side; P→P+2 gives a square within a square.
2. Rose 1: 5 points; both P→P+2 and P→P+3 give the same pentagram
3. Rose 2: Can a tessellation of squares be produced?

Rose 1 and 2 with a 72 point circle and a 19 point square, using the overlay option and two colours, the following observations could be investigated.

1. P→P*19 and P→P+18; the same overall circle or square is apparent.
2. P→P*23 and P→P+47 produce identical patterns.
3. P→P*29 produces the same 4 cusp pattern as P→P*5.

Other rules which can be explored with both programs are P→K, P→K/P (fractional parts are ignored), compound rules such as P→2*P+10 and its relation to P→2*P.

Screen Dump

The screen dump routine is used by permission of Beebug Publications Ltd and remains the copyright of Beebug Publications Ltd.

In The Program:

* is the symbol used for multiplication
/ is the symbol used for division
+ is the symbol used for addition
- is the symbol used for substraction

Loading

To load, type CHAIN"ROSE1" or CHAIN"ROSE2" and press RETURN

There are two programs on the cassette. Mathematical Roses 1 is on Side 1 and Mathematical Roses 2 is on Side 2.

The screen dump program follows both programs. It loads automatically when the option to print is accepted. The dump only works with Epson printers.