Dynamic Geometry Software (DGS) includes packages such as Cabri Géomètre (also ... of the Apple II, famously developed by Steve Jobs in his garage.
Dynamic Geometry Software a powerful tool for teaching mathematics, not just geometry! Adrian Oldknow Emeritus Professor of Mathematics and Computing Education Mathematics Centre, Chichester Institute of Higher Education, W. Sussex PO21 1HR, UK Abstract Dynamic Geometry Software (DGS) includes packages such as Cabri Géomètre (also available in the Texas Instruments' TI-92), the Geometer's Sketchpad and Thales, for PC and Mac. These enable users to define objects such as points, lines (or segments, rays and vectors) and circles (or arcs) and to construct geometric objects depending on them by classical constructions (straight edge and compass), linear transformations (translation, rotation, dilation, reflection) and/or algebraic relationships using coordinates (Cartesian and polar). Measurements may be made, on which computations can be performed and from which other objects can be defined. Loci of moving points and envelopes of moving lines can be defined and drawn, and such curves change shape instantly as adjustments are made to their defining objects (parameters). The talk will give examples of how such software can be used to (a) develop mathematical concepts such as variable, dependency, domain and parameter, (b) solve a variety of mathematical problems in novel ways and (c) provide a powerful means of illustrating a variety of topics in undergraduate mathematics from areas such as calculus, numerical analysis, linear algebra, complex number, dynamical systems, mathematical modelling, mechanics, statistics, and computer graphics.
1. Historical developments It can be argued that the key breakthrough in educational computing came with the launch of the Apple II, famously developed by Steve Jobs in his garage. This was the first lowprice, desk-top machine with "high resolution" colour graphics and analog input. It sold widely, both to the education and the domestic market. Given the strong place of classical geometry in the US school curriculum software it was not surprising that software was soon developed to exploit the facilities offered by this hardware. In 1985 Judah Schwartz and Michal Yerushalmy at MIT's Educational Development Center produced The Geometric Supposer for the Apple II. This allowed students to become geometric experimenters, and to make their own discoveries (albeit usually ones that someone else had already found!). Many other countries also place considerable emphasis on classical geometry in their school curricula, and so several other geometric packages have been developed such as Cabri Géomètre (France), the Geometer's Sketchpad (USA), Geometry Inventor (Israel) and Thales (Austria). As the hardware platforms have rapidly improved, so have the range of facilities of the software - which I shall refer to, collectively, as DGS (for Dynamic Geometry Software). A further breakthrough came a couple of years' ago with the launch of Texas Instruments' TI-92 - a hand-held computer containing a virtually full implementation of Cabri II in its ROM. The launch of Windows CE as a common operating system for palm-top computers should ensure that DGS has a very healthy future - but to do what? Before addressing that question I shall first describe a few properties of the software. 2. Technical considerations The DGS packages named above have much more in common, than they have differences between them - so they form a relatively new type of generic software. They enable users to define objects (sometimes called primitives - i.e. building blocks) such as points, straight objects (segments, lines, rays and vectors) and circles (or arcs). From these you can construct further geometric objects (which are dependent on them) by classical constructions of the "straight edge and compass" type (midpoints, bisectors, parallels, 1
perpendiculars etc.), and/or linear "affine" transformations (translation, rotation, dilation, reflection) and/or algebraic relationships using coordinates (Cartesian and polar). An ordered set of points can be designated to define a polygon. Measurements may be made of objects, such as the length of a segment, the size of an angle, the area of a polygon, the circumference of a circle, or the coordinates of a point. As a "free" object, such as a point, is dragged around the screen so all objects and measurements which depend on it change dynamically. Thus the reader must bear in mind that this paper consists of static text and illustrations - and must always try to imagine what dynamic possibilities are presented! Thus, for example, Figs 2.1 and 2.2 can be considered as "stills" from a film in which, as point A is dragged around the screen, the triangle ABC appears to change shape smoothly, and the numerals in the area measurement are constantly changing.
Area ABC = 2.75 cm2
Area ABC = 2.45 cm2
A
A
B
C
B
Fig. 2.1
C Fig. 2.2
Computations can be performed using any of the measurements, and these results used to define other objects. As an object is dragged, such as a point P on a given segment AB, the path taken by a dependent object, such as point P', can be traced. The path can itself become an object: the locus of P' with P. Similarly the envelopes of moving lines can be defined and drawn. Such curves change shape instantly as adjustments are made to their defining objects (parameters). As an example both of technique for using a DGS and for how it can applied outside classical geometry consider the following screen, Fig. 2.3, obtained from Sketchpad. L t
AB = 2.24 cm A
f
B
O
F
N
G h C
I EG = 1.05 cm
t E
H
D
Area IGH = 2.08 cm2
J (Area IGH) 1cm
M
h
K
= 2.08 cm
Fig. 2.3 The problem is well-known. AB represents a length of fence, and CD a long straight wall. The picture on the left is a view from above of a triangular area IGH enclosed by the wall 2
and two identical pieces of fence IG and GH. The problem is to find how to choose G so that the area IGH is maximal. Here is a brief record of the steps needed to create this image. "Cr." is short for "Create", "Co." for "Construct", "Int." for "Intersection", "Perp." for "Perpendicular", "Tran." for "Translate", "Rot." for "Rotate": 01 Cr. segments AB, CD 02 Co. midpoint E of CD 03 Co. line through E perp. to CD 04 Co circle centre E, radius AB 05 Co. F as int. of circle and line 06 Hide circle and line 07 Cr. segment EF, and point G on EF 08 Co. circle centre G, radius AB 09 Co. int H,I of circle and CD 10 Hide circle 11 Measure distances AB, EG 12 Co. polygon IGH, measure area IGH 13 Calculate area divided by 1 cm. 14 Cr. segment JK 15 Rot. K, centre J, through 90°, to L 16 Tran. J to M on JK by distance E 17 Tran. J to N on JL by result of step 13 18 Co. line through M perp. to JK 19 Co. line through N perp. to JL 20 Co. O as int. of these lines 21 Hide the lines. Cr. segment JL 22 Cr. the locus of O with G 23 Tidy up the labelling 24 Select all objects 25 Copy them to clipboard 26 Paste image in Word. 3. Mathematical and Educational considerations So that gives a flavour of how to use a DGS - but also suggests that a nice compact notation system for such geometric constructions would be a big advantage! Now consider how the image in Fig. 2.3 can be manipulated dynamically to solve the problem. On the right of the picture we now have the graph of the area of the triangle t against its height h. We can drag G on the segment EF until O appears to be at the top of the graph, and the value shown as Area IGH is the biggest yet seen. Thus we can obtain a numerical solution to the problem, graphically, without use of either algebra or calculus! What is more, the methodology is quite general. We could easily have divided the area IGH by a different length - so it is simple to scale either axis. The graph (locus) is now a function of the parameter f - the length of the defining fence AB. As, say, B is dragged so the whole graph is deformed dynamically. This power alone makes a strong case for having DGS available for mathematics teaching, but we have more to look at than just spotting the maximum point on the graph tf (h) of t as function of h with parameter f . We can see the geometry of the corresponding triangle. Lo and behold, no matter what the value of f, the triangle IGH always seems to have greatest area when the angle at G is a right-angle! So we come to another feature of using DGS - it can help us discover invariant properties; but it only provides evidence - not proof! However the insights gained often enable a proof to be arrived at. As mathematicians we know that the area of triangle IGH is given by t = 1/2 f 2 sin ∠IGH , and we can deduce that, for constant f this occurs when sin ∠IGH is maximum, i.e. when ∠IGH is a right-angle. But this does not give much insight into the solution. Now consider reflecting the isosceles triangle IGH in CD as mirror line to get the rhombus IGHG'. Of course we all know that the rhombus of greatest area is the square, so IGH must be half a square! Perhaps the reader would like to consider this style of problem-solving, deduction and proof in terms of the van Hiele levels for geometric thinking: Visualisation, Analysis, Informal Deduction, Formal Deduction and Rigour. We could also argue that we have not avoided algebra, but instead used a rather subtle form of geometric algebra. I have been careful to use words like function and parameter, which are usually associated with algebraic 3
functions, not with geometry. But here we have a clear visualisation of these concepts. The point G is the independent variable, whose domain is the segment EF. The point O is the dependent variable, and its position is a function of length h of segment GE. The length f of the segment AB is the parameter for this function. The locus of O with G is the graph of this function. The algebraic notation tf (h) is a sophisticated way for mathematicians to express this relationship. The dynamic version of the image above, which changes as G or B are dragged about, is a concrete embodiment of this relationship. These are powerful ideas which can be developed through the use of DGS, but the teacher needs to work on making them explicit and getting students to articulate them. There are many who are critical of the overuse of computers in education, particularly in mathematics - especially in a climate in which so many students stare at TV and computer screens at home! As teachers we have a duty to develop written and oral communication in mathematics - as well as to create motivating images. The rest of the paper gives some examples of, and references to, interesting applications of DGS in geometry, in problem-solving and in teaching other areas of mathematics. 4. Solving geometric problems The first example is a real-life discovery prompted by a simple problem. Choose 3 points A,B,C. Can you always construct 3 circles with these as centres so that they just touch each other in pairs? This is not a well defined problem since circles can touch (i.e. have a common tangent) either internally or externally - so we will specialise the problem to one where all contacts are external. Fig. 4.1 shows how the problem can be set up in a DGS using ideas of variable, domain etc. which were introduced above. The triangle ABC is constructed, and D is a point (variable) on segment AC (domain). Circles through D are constructed with centres A and C. These meet sides AB and CB respectively in points E and F. The circle centre B through E is constructed. We can now slide D on AC until this circle (dependent variable) appears to pass through point F. Without intending to be rude I call this approximate, exploratory, technique an "engineering approach"! With the resulting drawing we can see if any insights arise.
A
A
E B
E
D
F
B
C
Fig. 4.1
D G F
C
Fig. 4.2
Fig. 4.2 shows one way in which they can. Here perpendiculars (i.e. common tangents) are drawn through D and E to meet in G. (Note: there is no geometric significance in the labels used - they just represent the order in which the points were defined.) The reader might like to see if s/he can identify the shaded shape and derive a property which fixes G as one of the well-known centres of the triangle ABC. Using this can you derive a construction for 4
the points D,E,F ? As you now deform the triangle ABC do the circles maintain their tangency? So experimentation can lead to discoveries which can be proved by arguments assisted by identifying key features in a fairly accurate drawing. Of course very few such "discoveries" will be ones not yet known to be known! In this case the points D,E,F have a nice property shown in Fig 4.3, namely that AF, BD and CE are concurrent in the point H , called the "Gergonne point" of the triangle, after the French mathematician Joseph Diez Gergonne (1771-1859). Such an active approach to classical geometry can easily reveal other known objects, such as the Euler line, in a living way which constrasts with the "dead language" approach to geometry which caused its demise in the English school curriculum. It can also motivate students, and teachers, to find out more about periods of historical development of their subject - such as the arguments about geometry in the early 1800's in which Gergonne was a key player! Note that the triangle ABC in Fig. 4.3 had to be deformed considerably to show that H and G are, in general, distinct points. Of course it is also very easy to enlarge the diagram on the screen and to "zoom in" on areas of interest. None of which could be readily done with paper and pencil.
A K
N
D E H B
F
M
P
G
G H O
C L
Fig. 4.3
Fig. 4.4
Fig. 4.4 starts to show some of the remarkable features of the line GH (i.e. through in-centre G and Gergonne point H). Points O and M are slideable on the line GH (shortened to the segment KL), and points N and P are free points. Circles centre M through N, and centre O through P are constructed and these 4 points are dragged about using the "engineering approach" until one circle appears to touch the original 3 circles internally, and the other touches them externally. These circles are known as the inner- and outer-Soddy circles after the Nobel-prizewinning English chemist Frederick Soddy (1877-1956) who derived their radii (a feat already achieved by Descartes 300 years earlier!). Actually the points HOGM form a harmonic range, i.e. HO:OG = HM:MG. (As do the centroid, circumcircle, orthocentre and nine-point centre on the Euler line!) We shall call the line GH the "Soddy line" of the triangle. Now, technically, the triangles ABC and DEF are "in perspective" with the Gergonne point H as the centre. The theorem of Desargues tells us that the intersections P,Q,R of corresponding sides of these triangles (where P = AB ∩ DF etc.) are collinear in a line called the "axis of perspective" - so we shall call this line PQR the "Gergonne line" of the triangle. My own little contribution (if indeed it is a "new discovery") concerns 5
properties of these two lines - of which a key one is that the Gergonne and Soddy lines are always orthogonal! (Oldknow 1995, Oldknow 1996) The purpose of this example was not personal glorification, but to illustrate how DGS can become the geometer's equivalent of the astronomer's Hubble telescope, and reveal sights never before visible. This section concludes with a puzzle taken from a leading UK newspaper, The Sunday Times, which the reader is invited to tackle with DGS. During an exam Johnny amused himself by sliding his set square around the edge of his exam paper. The set square, which had an area of 60 cm2, slid around on the paper (30cm by 20 cm) with the two acuteangled vertices always touching the perimeter of the paper (indeed with both of them touching the same side for stretches along each of the sides of the paper). The right-angled vertex never touched the perimeter. In one complete loop the right-angled vertex covered a distance of 76cm. How long is the set square's longest side? (Schofield 1997)
X''
Y
Z Z'' Z'
X X'
Fig. 4.5 Fig. 4.5 shows an accurate scale drawing of the rectangular paper. XYZ is the "prototype" set square. X' and X'' are two possible (slideable) positions of X on the sides. We really want to be able identify the locus of the right-angled vertex Z' as X' makes a full tour of the perimeter of the paper. I guess many readers will be tempted to dive into coordinates, and algebra - but I hope you can resist the temptation, at least for a while! Try the problem as a "thought experiment" first, and then see if you can construct it in a DGS. The locus is indeed surprising. Once you have discovered it, can you use the diagram to give you some insights that might lead to proof? (Hint: e.g. look at the quadrilateral formed by the two right angled triangles at the bottom left corner of Fig. 4.5 - what property does this have?) This problem, though, is a special case of what we could call the general "falling triangle" problem. What is the locus of Z' if XYZ is a general triangle (i.e. not necessarily rightangled at Z)? What if the paper was a parallelogram, rather than a rectangle? Leonardo da Vinci discovered something about these generalisations (Wells 1991). In fact I frequently use the simpler "falling ladder" problem to motivate the parametric equations for the loci which arise from points on the hypotenuse X'Y'. 5. Mathematical Modelling and Simulation The problem in section 2 was a simple example of an optimisation problem of the form which would often be tackled analytically using differential calculus. The following problem appears in the first textbook published on differential calculus: Analyse des infiniment petits in Paris in 1696 by a French nobleman, the Marquis de L’Hospital (or L’Hôpital) (1661-1704). He was tutored by Jean Bernoulli (1667-1748) who was in turn a major proponent of the calculus developed by Gottfried Wilhelm Leibniz's (1646-1716). 6
One of the problems L’Hospital used to show the power of the calculus in solving physical problems is one known as “L’Hospital’s weight problem”. (Drijvers 1996, Oldknow 1998b) AB is a horizontal bar of unit length. A rope AC of length a tied to A carries a pulley at C. A rope of length b attached at one end to B passes over the pulley and carries a mass at its free end D. The problem is to determine the equilibrium position of D. [Assuming, that is, all the usual idealisations of weightless and inextensible strings, smooth and small pulleys, point masses etc.]
1 A
a
C
c
B
b-c D
L’Hospital took a = 0.4 and b = 1 . He found the result using differential calculus.
Fig. 5.1
In order to model this in a DGS we need a construction to enable 2 lengths to be subtracted in order to form (b - c), and this can easily be achieved with the use of compasses.
C
A
B
a B'
F D
AC = 2.004 cm DF = 5.007 cm
b
EG = 5.006 cm J E
G a
H
JK = 2.600 cm
I H'
K
Fig. 5.2 Fig. 5.2 is another static image from which the reader will need some imagination to add the dynamism. Again the labelling of points has not been tampered with; it reflects the order in which they were created. Here we start with a key idea, that of a "slider" to set a parameter. Segment AB is constructed, with C as a point on it. The perpendicular is drawn to AB at A, and points D,E constructed on it. The vector AD is marked and use to translate AB to DB', and point F is constructed on it. The line parallel to AB through E is drawn, and 7
point G taken on it. The lines are hidden and the segment EG drawn as the bar. Distance EG is measured, and G slid on its (hidden) domain until a (nearly) round number is found. This gives a scale of 5cm to 1unit. Distances AC and DF are measured, then C and F are each slid on their domains AB and DB' until suitable scaled values for the parameters a and b are set. This completes the setting up of the initial conditions. Now to the construction. A circle is drawn with centre E and radius AC. A point H is taken on this to represent the pulley, and segment EH drawn to represent the shorter rope. We want to construct a ray through H along GH, so first the vector GH is marked and then H is translated by this to give point H', and the ray HH' is drawn. Now we can draw a circle centre G radius DF, repesenting the stretched rope along GH (produced). If H is near enough to G then b-c will be positive and this circle will intersect the ray HH', at point I. The ray HH' is hidden and the segment HH' shown instead. We now just need to turn the surplus rope HI "round the corner" at H. First draw a circle centre H through I, then the perpendicular to EG at H. This intersects the lower part of the circle in K, and the bar EG in J. Now GH + HK represents the longer rope, and K the weight. The dependent variable is now the distance JK, which we want to maximise. The locus of K with H is also shown, and we can slide the independent variable H on its circular domain until K is at the lowest point of the locus, and JK shows the greatest distance. [Note: the portion of the locus above EG only has physical significance if EH is a rigid bar rather than a rope.] Again the solution obtained this way has avoided the use of algebra and calculus. It is quite general - just slide C and F to set up different initial conditions for the parameters a and b. With b large enough the locus will be a closed curve. How does that critical length depend upon a? Analysts might think that the closed curve has a simple description - but it's not as simple as it looks! But now we can take a step further. Does the image provided by the "solution picture" conjure up any ideas of the geometric relationships between the objects? Again different people will focus on different aspects of such an image, so it is not guaranteed to deliver (in the way algebra usually does for problems that are already known to be solvable!). But just look at the segments EH, GH and HK. Imagine producing EH through H, when it looks as if it is the angle bisector of ∠GHK . Using simple ideas of forces (tensions) in balance, can you justify physically why this must be the case if K is in equlibrium? With this information can you now construct the exact solution geometrically? I wonder what Newton would have thought of l'Hospital's choice of problem to justify the use of calculus? The other example in this section is given in less detail but concerns the possible modes of motion which can be taken by a mechanism well known in mechanical engineering as a "4bar linkage". The particular form of the problem arose in the study of the oscillations of a common type of garden sprinkler! (Oldknow 1997a).
8
Here AB is the frame, AL is the driver crank, LM the coupler and MB the follower crank. As AL is turned with steady speed about A by a water-driven turbine, so the loosely jointed coupler LM forces the water sprinkler bar at M to oscillate round B. E
b
F
J
H K
C M b
c
c
L I a
a D
A
B
d
G
Fig. 5.3 Fig. 5.3 should now be sufficient to reveal the construction of the geometric model, where the parameters a,b,c are set by sliders I,J,K with domains DC, EF, GH. The investigator can now find what conditions must be satisfied by a,b,c,d for the mechanism to be welldefined for all positions of the independent variable L on its circular domain. What will be critical values for M to perform oscillations about B, rather than circular motion? The sprinkler designer might be interested not just in finding the endpoints of the arc NO which forms the range of the dependent variable M on its circular codomain but also exploring the speed of M at various parts of its path. The point L can be animated to traverse its domain with constant speed, and the resulting changes in the mechanism observed "in real time". In order to have this information in tangible form, can you figure out a way to draw a "response graph" of the angle swept out by bar BM against that by AL as a locus using the technique of section 2? 6. Enhancing Mathematics Teaching with DGS Obviously DGS has a strong claim as a tool to help learn, teach, explore and discover plane geometry; particularly in its classical, transformation and coordinate forms. I hope that I have made the case that it is also a very versatile and instructive tool for problem-solving and modelling. The paper concludes with some examples of the use of DGS in teaching other areas of mathematics. The first one is from the area of iterated functions, or "dynamic systems" and shows how the analytic functions of a DGS can be used to produce dynamic "cobweb- and stair-diagrams". We start by showing how to use locus to produce a graph defined by a conventional algebraic function such as: y = k x (1-x) .
9
G K
B: (1.00, 0.00) A
C
F
E
x
B
D
E: (0.28, 0.00) x E =0.28
H
k
I
J k = 2.51 cm
k xE (1 - Ex) 1cm
=0.51
Fig. 6.1 The built-in coordinate system has been used to define the origin A and unit point B. Points C and D are taken on the x-axis which is hidden, and segment CD drawn. The independent variable E is taken as a point on the domain CD. Points F and G are taken on the y-axis, which is hidden and the segment FG drawn. A slider HJI is constructed for the parameter k, which is measured. The coordinates of E are measured, and its x-coordinate isolated using the built-in calculator. The value of k x (1 - x) - without units - is then calculated as the corresponding y-coordinate. The x- and y-coordinates are then used to define the position of the dependent variable K. The locus of K with E gives the required graph. As J slides on HI so the parameter k is varied and the family of graphs is visible dynamically. We now need to find a way of producing graphically an iteration where the y-coordinate of K becomes the x-coordinate of a new point L, whose ycoordinate is recomputed from the function y = k x (1 - x) . This process can then be turned into a little program of steps called a "macro", or "script", and so be repeatedly applied to successive points to form a chain of iterates. The result is illustrated in Fig. 6.2 . Here the graphs of y = f(x) = k x ( 1- x), and of y = f(f(x)) are both drawn, as well as the line y = x .
H
F G x12
O x0
k = 3.23 k
Fig. 6.2
10
The picture shows the first 12 iterates, with the last 2 joined by 4 bold segments. The actual image also uses colour to help sort out the various features. The whole image can be made to flow dynamically in one of two ways. The starting point for the iteration marked x0 can be slid on the x-axis to examine the effect of changing the initial condition. Also the value of the parameter k can be adjusted with the slider. The point F is the "fixed point" (other then O) for the function f, and the points F,G,H are the fixed points for the function f 2 . For the settings shown we see that F is a repellor, while G,H form an attracting 2-cycle. The tangent at F is also shown as a broken line. (Griffiths & Oldknow, 1993) The next example comes from an introduction to eigenvalues and eigenvectors for a 2 by 2 a b matrix in a first course of Linear Algebra. The matrix A = is set up on sliders c d (which now need to be able to go negative as well). The point P traverses the unit circle, centre O, and the image P' of P under the matrix transformation defined by A is calculated, and its locus drawn. The value of the ratio k = OP'/OP is displayed, and an eigenvector v will be encountered when P' lies on OP (possibly produced in either direction), when k will be the modulus of the corresponding eigenvalue. The slopes of OP and OP' are also shown.
c a = 1.00
a
b
d
b = -0.25
c = -1.00 d = 1.00
P: (0.60, 0.80) P
a x P + byP =0.39 c x P + dyP =0.20
P': (0.39, 0.20)
P' O m P'O m PO
Slope PO = 1.34 Slope P'O = 0.51
= 0.44
Fig. 6.3 Again the reader must use some imagination to make this image appear dynamic as P traverses the circle, stopping when the two slope measurements are equal e.g. when the slopes are -2 the ratio of lengths is 1.5 , from which we can deduce one eigenvector (invariant line) and the corresponding eigenvalue. Is there another pair? Changing any of the parameters a,b,c,d with the sliders immediately allows a different matrix to be explored. The next examples use a similar idea, but this time defining an affine transformation of a point P by a rotation through a given angle α centre O to give P', followed by a dilation with factor k centre O to give P'' and finally a translation by a vector GH to give P''' .
11
F
P''' α = 80° E
D
P' P
C
A
B
m CA = 0.70 m OU
P''
H O
U OU = 1.00 cm
q = 2.12 cm G p = 2.75 cm Fig. 6.4 Fig. 6.4 illustrates how the various parameters for this particular affine transformation may be set up. Then dragging F on its circle, or sliding C on its segment, or dragging H round the screen will alter the transformation. Now the transformation from P to P''' can be defined as a macro or script, ready for repeated application. Fig. 6.5 shows the result of such a sequence of transformations, which leads to a fixed point provided the mapping is a "contraction mapping" - one with |k| < 1. Again you must use imagination to see how such a figure will deform fluidly as the initial condition or any of the parameters are altered. R m ECR = 45.2° C
E V
V: (0.2, 0.5)
F
O
X m OF m OX
P1 P0
Fig. 6.5
12
= 0.79
Now the Mandlebrot iteration, which is usually defined in terms of complex numbers, can similarly be regarded as succesive applications of affine transformations in the plane. If a given complex number c is represented by the point C, and a variable complex number z by point Z in the Argand diagram, then the image z' = z2 + c of z can be represented by a point Z' obtained as an affine transform of Z. By de Moivre's theorem, squaring a complex number is equivalent to squaring its modulus and doubling its argument (angle) - and hence is the product of a dilation centre O factor |z| and rotation centre O through angle arg(z). Similarly the addition of the complex number c is equivalent to a translation by the vector OC. Fig. 6.6 shows the effect of such a transformation on a point Z whose domain is an arc of a circle centre O by giving the locus of its image Z' with Z. The image of the circular arc AB is the circular arc A'B'.
Z'
D
R
C
Z
B
A'
B' O
U
A
Fig. 6.6
Points D and R are the succesive dilation and rotation of Z before the translation takes R to Z' . So the construction steps which take Z to Z' can be defined as a macro or script, and again applied repeatedly to produce a chain of points representing the "orbit" of Z. Fig. 6.7 shows how this can reveal some of the structure of the Mandelbrot set. C
C: (-0.350, 0.585) Z': (-0.434, 0.246)
Z'
O
Fig. 6.7 Here the cardioid H and circle K are superimposed as guidelines. H is constructed by taking a point P on the circle centre O, radius 1/2, forming its complex square Q and forming OR as the vector difference OP-OQ. The locus of R is H. K is the circle centre (1,0) and radius 1/4. As C is dragged round the plane, so the first 20 or so points of its orbits 13
are displayed, where the iteration starts from C itself. The 20-th point is labelled Z', and the last couple of segments are shown in bold. In the position shown C is just inside the cardioid H, and the orbit is contracting (slowly) towards a fixed point. Again the reader must imagine how the star-shape changes dynamically as C is slowly dragged radially across the boundary of H. There is a small "bump" outside H in which region the orbit of C settles to an 8-pointed starlike limit cycle, so we have an attracting 8-cycle. Then as C is dragged further outside H so the successive stars grow and the orbit diverges. Tracking round the boundary of H reveals many such "bumps" with limit cycles of different periods. We leave the reader to locate a region of period 3 - a key theorem states that chaos is an inevitable consequence! The orbits for C inside the circle K reveal 2-cycles, and again there are patches of odd behaviour just outside its boundary. (Griffiths & Oldknow 1993, Oldknow 1997b). Well, this Odyssey is just about finished. The reader might like to think about constructing other teaching and learning images where the dynamic changes reveal insights. For example I have not touched on Statistics, but it would be easy to think of ways of illustrating linear regression, say. Similarly for numerical integration techniques in Numerical Analysis. I shall finish, though, with a final example from mathematics applied to CAD (Computer Aided Design). This starts by recognising that the family of Bézier curves can be generated geometrically by a series of dilations as in Fig. 6.8. t O
U
T
t = 0.40 P2
B1,1
P1 B0,2
B1,2
B2,1
B0,3
B0,1
P3
P0 Fig.6.8 Here 4 points P0, P1, P2, P3 , known as "control points", are used to define an arc of a cubic curve. The independent variable T has segment OU as domain, and the ratio OT:OU defines a parameter t in [0,1]. The point B0,1 is obtained by dilating P1 with centre P0 and factor t. The points B1,1 and B2,1 are similarly defined. The loci of each of these points with T are, of course, the (linear) segments P0P1 , P1P2 and P2P3 . Now we have a recursive process of dilations by which the points B0,2 and B1,2 are first defined - these have quadratic loci (not shown) - and then used to define B0,3 . As T slides on OU then B0,3 traverses the cubic arc from P0 to P3 , but it does not, in general pass through P1 or P2 . The ideas can be easily extended to polynomial arcs of degree n-1 defined by n control points. This has the snag that moving any one control point changes the whole shape of the curve. An alternative method in more common use is called "parametric B-spline cubics". Their derivation is usually given in terms of the convolution of functions, but they have a less well known, though nice, geometric relationship with the cubic and linear Bézier points. The cubic B-spline point C0,3 corresponding to the cubic Bézier point B0,3 on the 4 points P0, P1, P2, P3 is formed by first finding the centroid G0 of the triangle P1 P2 B0,3 and then 14
forming the midpoint of B1,1 G0 . The locus of C0,3 with T gives the required arc, shown in Fig. 6.9. When there are a fairly large number of points, successive sets of 4 points, like P1, P2, P3, P4 etc. are used to define further arcs, each of which has continuity of degree 2 at its joins with neighbouring arcs. If the sets of 4 points are continued cyclically then the arcs will constitute a closed curve as in Fig. 6.10 where alternate arcs are drawn thickly. t O
U
T B1,1
P3
t = 0.40
P4
P2
P1
P1
P6
B0,3
P5 P0 P9
P0
P2 P7
P3
P8
Fig. 6.9
Fig. 6.10
Dragging any control point will only change the shape of the 4 adjoining arcs which depend upon it, and this feature, known as "local control" is very useful in CAD. (Oldknow 1998a) 7. Conclusion I hope this extended paper has shown just some of the potential for the use of Dynamic Geometry Systems, and whetted the readers' appetite to try the ideas out themselves. The relatively low cost of the software, widespread availability and ease of use makes it an accessible, motivating and exciting tool for the learner, researcher and teacher alike. 8. References Drijvers, P. (1996). Old mathematics and new technology: L'Hospital's weight problem and the TI-92. The SAC Newsletter, 1 No 1, Amsterdam University. Griffiths, H.B. and Oldknow, A.J.(1993). Mathematics of Models. Ellis Horwood, Chichester. Oldknow, A.J. (1995). Computer Aided Research in the Geometry of the Triangle. Mathematical Gazette, 79 No 485. Oldknow, A.J. (1996). The Euler-Soddy-Gergonne Triangle of a Triangle. American Mathematical Monthly, 103 No 4. Oldknow, A.J.(1997a). Modelling a Garden Sprinkler. International Journal of Computer Algebra in Mathematics Education, 4 No 3. Oldknow, A.J. (1997b). Some Complex Issues Raised by Dearing. Teaching Mathematics and its Applications, 16 No 3. Oldknow, A.J. (1998a). Dilations, splines and CAD - as easy as ABC. Teaching Mathematics and its Applications, 17 No 1. Oldknow, A.J. (1998b). Using IT to support mathematics learning. submitted to Teaching Mathematics and its Applications. Schofield, R. (1997). Brainteaser 1824. The Sunday Times Colour Supplement, 30 Aug. Wells, D. (1991). The Penguin Dictionary of Curious and Interesting Geometry. Penguin, London.
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