CURVE STITCHING...ELECTRONICALLY!
Ian Robertson shows you how to extend the use of graphing calculators
in your classroom
I recently read "Curve Stitching - the art of sewing
beautiful mathematical patterns" by Jon Millington, in which he discusses
curve stitching as an art form. (Millington, 1989. 3, 85) Curve
stitching, in two and three dimensions, has frequently been used in geometry
and art classrooms. At a lower grades, patterns are intriguing and relatively
easy to produce, while at higher grades the topic can reconcile geometric
and algebraic approaches to the conics.
One of the simplest curves to stitch is the parabola.
In Fig. 1a) OA and OB are each divided into an equal number of equal segments
( note: segments in OA may not necessarily be the same length as those
in OB ). If each segment on OA is numbered 1,2,... starting at O
and correspondingly each segment of OB is numbered starting at B
, then a1 to b1 will form one of the lines in the envelope. Similarly,
joining a2 to b2 etc., will complete the pattern. Point an will always
be the same number of divisions from O as bn is from B. The envelope of
all such lines form the parabola, or more properly, each line in the envelope
provides one point - the point of contact between the tangent and the parabola.
This is pattern is shown in Figure 1 a & 1b.
Fig. 1 a)
Fig. 1 b)
This article discusses using a graphing calculator - in
this case the Texas Instruments TI-82 - to transfer curve stitching to
the realm of technology. While the actual construction of a model with
card and coloured thread using curve stitching principles is a worthwhile
creative and educational exercise, using a graphing calculator has additional
advantages in that further patterns can be explored much more quickly and
by use of suitable "linking" software ( e.g. TI - Link provided
by Texas Instruments) any pictures created by students can be printed by
computer.
A variety of mathematical skills and principles are called
into play in curve stitching and they vary depending on the algebraic or
geometric emphasis desired - but the principles of curve stitching represent
an opportunity to unify the two approaches. From an algebraic standpoint
two of the skills utilized are working with equations of lines ( y=mx+b
or the Ax + By + C = 0 form) and the division of line segments ( usually
studied with ratio and proportion).
In his recent book " What if...? The Straight Line
" Bob Alexander discusses calculator approaches to varying the relationships
between m and b in the y = mx + b form: if b = m2 , the envelope
of lines forms a parabola (Alexander,1994. 29). Similarly if the pattern
between the coefficients A,B and C in Ax + By + C
= 0 is a multiplicative one, the envelope of lines is again a parabola,
albeit one with a different orientation (Alexander,1994. 54). In Ontario,
such investigations are appropriate at the Grade 11 level and with extensions,
could be applied to the study of conic sections in Grade 12.
Sample results of such investigations are shown in Figures
3 (a & b)and 4 (a & b) respectively. While these pictures were
simply produced using the y = function mode of the calculator, the appendix
contains a program listing ( Param1) for the TI-82 which could generate
similar figures.
Fig. 3 a & b
Fig. 4 a & b
From a geometric standpoint curve stitching can readily
be adapted to the graphing calculator. The principles of joining one point
on one ray to another point on the other ray are the same, so the advantage
to the students is that with the ease of use of a menu driven programming
language such as is used by the TI- 82, they can be actively involved in
the development of the program and the pencil and paper exercises of division
of line segments or alternately using the equations of lines approach,
have to be thoroughly understood in order to develop a successful program.
The program ("STITCH") to curvestitch "parabolas"
within a triangle is listed in the appendix and Figures 5 & 6 illustrate
two sample results.
Fig.5 and Fig. 6
The menu choice to allow only one of the vertices (as
opposed to the entire triangle) to be curve stitched is easily removed.
If the co-ordinate axes are desired, they will be drawn when the third
line of the program - "Axes Off" - is removed.
This relatively simple program can be modified to curve
stitching for other polygons (see STITCH2 in appendix)- an extension perhaps
suitable when students have been introduced to trigonometry.
In summary, the traditional approach of curve stitching
used to generate conics and to explore geometry can be readily adapted
to the more modern technology of the graphing calculator. Such a development
has relevance and utility by allowing a blending of both the algebraic
and geometric approaches to conics in particular.
Bibliography:
Alexander, Bob. What if....? The Straight Line,
Toronto: Bob Alexander Publishing, 1994.
Millington, Jon. Curve Stitching, Diss, Norfolk (U.K):
Tarquin Publications, 1989.
Appendix:
Program - PARAM1 - use a standard range setting
ClrDraw
:AxesOn
:FnOff
:Disp "START SLOPE"
:Prompt M
:0®T
:ClrList L1,L2
:For(I,-abs M,abs M,2abs M/10)
:T+1®T
:I®L1(T)
:I2®L2(T)
:End
:"L1X+L2"®Y1
:DispGraph
Program: STITCH -
set standard range, and keep the coordinates for the three vertices within
the window
ClrDraw
:FnOff
:AxesOff
:Disp "INPUT (X,Y)"
:Disp "OF VERTICES"
:Prompt A,B,C,D,E,F
:Disp "INPUT NO."
:Disp "OF STITCHES"
:Prompt N
:Menu("NO. OF VERTICES STITCHED","ONE",1,"ALL",2)
:Lbl 1
:1®Z
:Goto 3
:Lbl 2
:3®Z
:Lbl 3
:Line(A,B,E,F)
:Line(C,D,E,F)
:Line(A,B,C,D)
:For(I,1,Z)
:For(L,1,N-1)
:L/N*(A-E)+E®W
:L/N*(B-F)+F®Y
:L/N*(E-C)+C®U
:L/N*(F-D)+D®V
:Line(W,Y,U,V)
:End
:A®P:B®Q:C®A:D®B:E®C:F®D:P®E:Q®F
:End
Program STITCH2 - use with range settings x e [30 , 150
] and y e [ -10, 130 ]
ClrDraw
:FnOff
:AxesOff
:Disp "HOW MANY"
:Disp "SIDES?"
:Prompt V
:Disp "HOW MANY"
:Disp "STITCHES PER SIDE?"
:Prompt N
:For(B,0,2p,2p/V)
:B+2p/V®G
:B+4p/V®T
:For(H,60/N,61,60/N)
:(60-H)sin B+Hsin G+90®P
:(60-H)cos B+Hcos G+60®Q
:(60-H)sin G+Hsin T+90®X
:(60-H)cos G+Hcos T+60®Y
:Line(P,Q,X,Y)
:End
:End
Written by:
Ian Robertson (Mathematics teacher )
Appleby College
Oakville, Ontario
