ATCMA 2001, 
Melbourne, Au

Computer Screens and Toys.

Ivan Cnop, Vrije Universiteit Brussel, Brussels, Belgium
icnop@vub.ac.be

Abstract.

In this paper we show that experimenting with simple motion on a computer screen leads to a surprising variety of visual objects in the plane, and allows to describe natural objects in space as well. Using commercial toys for building moving objects makes these constructions accessible to pupils from lower secondary on. Co-ordinate systems in the plane as well as trigonometry can be introduced rather early in this way. 
Using a CAS we can generate much faster the orbits of moving objects, together with their animation, and it gives the student a good visual impression of 3D objects. A CAS moreover provides a link to the underlying theories. Advanced theorems and applications can be motivated before they arise in the tertiary education curriculum. 
Part of this work was written in Mathematica under the 
"Exploot" (Experimental Learning Environment using CAS) 
project of the Flemish Ministry of Education at the Vrije Universiteit Brussels, Belgium and part of this material is visible on the projects' web site

Introduction.

Is "Computer Screens and Toys" the right title for addressing an international conference?
Let us start by some questions concerning

1 + (∑_(j = 1)^(n - 1))cos(((2 π j  )/n))

Q1: what is this sum?
      Q2: can you prove this result?
      Q3: can you explain the meaning of this result in secondary school?
      
      If we do not know the answer to the first question,our CAS will immediately provide the answer:

1 + (∑_(j = 1)^(n - 1))cos(((2 π j  )/n))

For the second question, we can use trigonometric identities or reformulate the problem in terms of de Moivre’s formula and points on the unit circle.
	Do we know properties of a regular n-gon? 
	Do we know what the sum of the n-th roots of unity is?
	Do we know facts about geometric series?

Answering positively to the third question seems impossible. Nobody will ever think of teaching trigonometry to pupils age twelve. The reason is that the formulation in terms of trigonometry hides the facts from elementary geometry that are behind most trigonometric identities.
These elementary facts remain hidden because the mathematics curriculum imposes complicated formulations and many mathematicians seem to enjoy this because it gives them an aura of importance.

Multiplication of points in the plane.

Let us look at a computer screen.
It is a plane and it is therefore natural to start computer-assisted learning in the plane instead of accumulating numbers on a line. Successful packages such as Cabri II and Geometer’s Sketchpad do exactly this. 
	Everything in trigonometry has to do with a point moving along the unit circle and the way the angle, over which it moves, is measured. Writing cosines is like looking at the picture through glasses completely covered with the exception of an horizontal slit, writing sines is looking at the picture through glasses with only a vertical slit. One can expect that the vision is impaired by imposing such restrictions on ourselves and on the pupils. Looking at the plane gives a lot more information.
	Select a point on the computer screen as the origin,and a horizontal axis on which we choose a unit and put real numbers. Each other point is completely determined by a direction and its distance to the origin. Directions are measured counter-clockwise from the positive horizontal axis. A simple way to measure the (polar) angle with arbitrary precision is to measure the length it cuts off along the unit circle. Navigation chooses orientation clockwise from North (or the magnetic north pole), and measures angles in degrees, minutes and seconds (with precision limited to .1" ). Both systems have evolved through history and there is no reason to replace one of these standards by the other. If we need a shorthand notation we can write
	cis(t) , unit(t) 
or the Euler notation

(E^(I t))

for the direction making an angle t from the positive axis.

We know how to add vectors in the plane by the parallelogram rule and use this to add points. We usually draw only two sides of the parallelogram to get the sum. We know how to multiply a vector in the plane by some real number and use this to multiply a point by a real number. The only thing we do not yet know to perform is the multiplication of two points, both not on the horizontal axis. Let us state that in this case we add their angles and multiply their distances to the origin.

This does not conflict with earlier material. This product gives the corresponding images under similarities and rotations. The theorem of Thales can be reformulated using products and many results from plane geometry can be obtained in this way. Let students experiment with multiplication of a (all points in a) geometric figure by a point, and see what properties are preserved.
	What we have defined under product of points is in fact the product of complex numbers in polar form. I want to avoid the word "complex" since I am only talking about points in the plane, and these points are very real. It is unfortunate that multiplying points or vectors in the plane is forbidden in classical curriculum. It is indeed a very natural operation. It is a special case of the matrix multiplication in the plane.
	Doubling angles, or reversing angles is even simpler. A double angle means that we square directions, a third power is given by three times the angle, and so on. A reversed angle is obtained by the division operation, and we can continue in this way. Each of the directions   unit(2π j/n)   in the original question becomes 1 when you take its nth power. We have found all solutions  z =  unit(2π j/n)   of the equation

(z^n) = 1

They are the vertices of a regular n-gon inscribed in the unit circle, with one vertex at 1.

A balancing act.

Let us build a wheel with spokes at multiples of 2π/n. 
This model will prove the answer to the question in the introduction.Since multiplying angles corresponds to powers of points,adding all directions forms a simple geometric series that adds op to zero.
In class,a geometric proof is much simpler:adding one by one all line segments (starting with the spoke at the vertex 1) gives a regular polygon that closes at the origin.

We now only have to project all points on the horizontal axis to get cosine expressions. We have proved the sum is indeed 0.

       	A spoke wheel model is in equilibrium when suspended in its centre.This is no longer the case if the spokes do not have the same length,since the irregular polygon will not close.
       	However,if the length of the spokes is given by some random number (such as the outcomes of rolling dice) the sum result will end up close to zero.A drunk person stumbling over some random distance (with distances averaging to 1) in each of the n directions 2πj/n will not get far:

Show[GraphicsArray[{%%, Show[%, %%]}], 
      PlotLabel -> StyleForm["fig 3", TextStyle]]

This is a good point to start introducing probability. The main advantage of doing this in two dimensions is that drawing vectors in the plane is much more obvious to the eye than drawing vectors on the real line or adding numbers on a calculator. Instead of cutting each spoke to a given length we can put weights
        		w0, w1, w2,…, w(n-1)
at the end   unit(2πj/n)   of each spoke. We obtain a new type of scale and the result of adding all vectors gives the direction in which the scale tilts; how badly it will tilt depends on the length of the sum vector. Using only this sum vector or point in the plane we cannot determine the values of the n initial weights. More information is needed. Even in the real case,we cannot find two values if we only know their sum. We can if,together with the sum,we also know their difference. This corresponds to using directions 1 and –1, weighing the difference on a traditional scale.
	Let us rearrange the weights by attaching them to spokes (initially numbered from 0 to n -1) skipping every other spoke as we count around the circle. We have to go round the circle twice to attach all weights in order. We do this again taking multiples of 3, next 4, and so on. If the number of spokes is a prime number, we never get two weights on the same spoke. Let the pupils verify this fact.
	Placing the weights on spokes numbered
        0  1  2  3  4 … up to  n-1,    for the initial sum, followed by
        0  2  4  6  8 … until all spokes are occupied, and again on
        0  3  6  9  12 …
        continuing until
        0  n-1  2n-2  3n-3  4n-4 …    and finally
        0  0  0  0 …
        	Adding each time the weights multiplied by the corresponding units on each line we get n    totals s1, s2,…, sn-1 and s.
        	The last one is just the arithmetic sum of all weights. This is the same result as when adding spokes of length wj in the corresponding direction given by the numbering in each row. Now the original weights can be computed by solving the system of these n linear equations with right-hand sides s, s1, s2,…, sn-1. This is possible because the determinant with entries consisting of all powers of directions is not zero. The CAS can compute this determinant. The CAS also solves the system of linear equations for us in no time. The method is simple and faithful if we take a prime number of spokes, but we do not like to solve 13 x 13 systems by hand. Here we get another bonus.	
        	It is not necessary to know how to solve linear systems to obtain the unknown weights. If we have a prime number of spokes, making new sums for the n right-hand sides s, s1, s2,…, sn-1 in the same way but counting spokes in reverse order (clockwise from the horizontal spoke) will return the initial weights w0, w1,w2, …, wn-1 after dividing the new sums by n  . This result can be proved instantly by using once more the above reasoning about adding directions to zero.
        	Suppose now that same drunk person did modify the totals s, s1, s2, …, sn-1 somewhat in a random fashion. Since making new totals will reduce the effect of such modifications, we obtain a much better approximation for the original weights if we compute the weights by this method. If the original weights are whole numbers, it will yield the right values by rounding if the differences between exact and computed values are less than  .5  . We have found an error-correcting method.
        	It is nice that we can explain this material to kids early in secondary school. We have just explained how the Discrete Fourier Transform DFT and its reciprocal work. It is the basis of many applications in image manipulation and signal processing.Image enhancing and medical imaging use DFT. Scanners in medicine, radar images, speech technology, image compression and CD records would not exist without this technique.

One-armed robots.

A robot arm can assume all positions on a circle, not only those at the vertices of a regular n-gon. Not much is to be said about a robot whose arm describes a circular motion around the origin. More interesting is a robot arm in two sections joined in such a way that the angle the second section makes (with the horizontal half-axis direction) is always a given multiple of the angle with the horizontal half-axis where the first arm is attached. This can be manufactured using well-known mechanical toys, such as Meccano, Fischer, K’nex and Lego. Every building set with gear systems will allow building prototypes and drawings can be realised using the Spirograph drawing set. Young pupils in elementary school will enjoy building their own in the classroom or sketching the curves involved. Next photograph shows a model built in Meccano:

The left arm describes a circle centred at the point where parts are fixed to the base plate, and the sectioned arm to the right describes some cycloid figure. Let us draw the movement of such a robot arm in two sections as the main section describes a half circle. In the following graphic the gear ratio between the angles is 2, and the second section is half the length of the first. We obtain the upper half of a curve called "limaçon" since it looks like part of a snail.

z +  .5 (z^2)

This does not seem to be an efficient robot: it starts off moving too rapidly and gets very slow at the left end of its sway. Other gear ratios may give a better result. In the case of gear ratios that are whole numbers, we always get a closed curve if we complete a full circle for the main angle. Some of these curves seem familiar. Some of these can also be sketched with the plastic Spirograph discs. Since in the Spirograph all disc ratios are rational (fractions), a Spirograph drawing will always close up after a sufficient number of turns. 		
	The main benefit of making such models on a computer screen instead of making them in toy hardware, or on paper, is that they are generated much faster in any CAS software. Teachers can build large numbers of experiments and models for themselves. Figures such as the deltoid and other cycloids are easy to obtain while it is not possible to draw cusps with a Spirograph. We make full turns in the following examples.

z  +    .5 (z^(-2))

z +  .5 (z^(-1))

z +  .5

z  +   .5 z

z  +   .5 (z^2)

z  +   .5 (z^3)

z  +   .5 (z^4)

A well-known sketch is obtained for gear ratio –1. 
It was already known in the XVIIth century that this is a tool for drawing an ellipse. The large axis is twice the sum of the arm sections and its small axis is twice the positive difference of the arm sections. We do not immediately find the focal points as in the construction of an ellipse with two pins and a piece of rope. The Pythagorean theorem yields that focal points are at  –2  and  +2  if the product of the arm section lengths is 1. Therefore the following are confocal ellipses:

a z + (1/a) (1/z)       ,  
    different values of  a

The family of hyperbolas with same focal points and orthogonal to these ellipses is obtained by computing the same action on points which are on rays, i.e. all points in the same direction from the origin. 		
	Next, we generalise this construction of the ellipse to get a surprising result. We replace points on the unit circle (which indicate directions) by other points in the plane. Instead of taking the image of a unit circle we map a shifted image of the unit circle (adding the same vector to all points on the unit circle in fig 1) with the same gear ratio – 1. If the shift is not horizontal, the result is no longer symmetric and for some values of the translation vector we get a section of an air plane wing. Let the pupils bring their scale models of planes into the classroom to check this.

z + (1/z)          on a shifted circle

We are not a playing around any more. Since the work of Joukowski, we know it is possible to model the laminar flow of liquids and gases around such objects by using the same mapping on circles that are slightly bigger than the unit circle. This is the reason why planes can fly and sailboats can sail. Other forms, whose equations are sometimes difficult to write in full in terms of cosine and sine expressions, are easily obtained. The deltoid obtained above is only one of them.

z  +    .5 (z^(-2))

Spirals are obtained by multiplying the motion by the angle parameter. Many natural designs can be created. Teachers can build their own graphics gallery.


    	Did we get rid of trigonometry completely? Not really. In order to draw a point on the screen it is necessary to compute co-ordinates of the points and these are expressions involving cosine and sine. But we do not have to do this by hand any more. The CAS does this for us and it therefore can be considered part of the computer graphics package, just as we do not bother about counting pixels on the screen to position each point precisely.

Multiply hinged robot arms.

Adding more sections to the robot arm, each with its own gear ratio, will generate more figures. Gear ratios that are natural numbers correspond to polynomial mappings with coefficients the respective arm section lengths. Adding sections with negative (whole number) gear ratios leads to rational mappings. For a CAS this extension is no problem even if the coefficients are negative. It gets increasingly difficult however to manufacture toy models in hardware because of friction and strain and it is no longer possible to sketch models using three or more plastic Spirograph disks.Rational gear ratios will give curves which, just as in the Spirograph case, close only after a sufficiently large number of turns. Irrational gear ratios will never close and may even fill parts of the plane if we continue running the main angle around the unit circle. Let us therefore stick to ratios that are whole numbers. A list-oriented program for making such models is in [H.Uscka] and this author refers to the web-site of Frank Farris for symmetry properties of the resulting pictures.

Up to now we have mapped the (complex) plane by (complex) polynomials and (complex) rational functions.Over the last three centuries a very powerful (complex) analysis has been developed and these results are readily available for us. In particular,orthogonal trajectories (and other angles as well) are preserved under such maps (excluding a finite number of exceptional points) and the theory of conformal mappings describes flows around these objects. Since Gauss, we all know that polynomials have as many roots in the plane as indicated by their degree. Images winding around the origin may give information about the number and position of the zeroes and poles of the functions involved. Time dependence in the robot arm motion is easily obained by plotting the curve depending on the time parameter, and can be visualised also by adding a third axis.
        	Let us look at the motion of the robot arm adding a third t axis (or time axis), sticking out of the computer screen, to the picture. Simple motion around a circle gives a helix, and more complicated maps will give some other spring-like curve in space.

What we see is the three-dimensional graph of the (complex) Fourier series for the given sequence of coefficients,and its projections in terms of cosines and sines. Many properties can be derived from this. The horizontal projection is always an even function, since it consists only of cosines. The vertical projection is always an odd function since it consists only of sines. If the sequence of coefficients is symmetric around zero the result will be real, which means it stays in the horizontal co-ordinate plane. When approximating a simple discontinuity, all values at that point are exactly halfway between one-sided limits and near the discontinuity one sees some overshoot called Gibb’s phenomenon. The overshoot comes from the fact that all sine terms take off in the same direction. The harmonic conjugate functions show surprisingly different behavior.
    	It is interesting that such material can be shown to students early in the curriculum, even long before the traditional calculus sequence is completed. They learn to approximate periodic functions by Fourier series. Image compression and sound are important fields of application. If students have at their disposal a CAS that includes playing sound samplings or some computer interface that performs sampling, students will enjoy experimenting and listening to their electronic compositions.

Generating bodies by Spirograph images.

Above we have described motion as mapping the unit circle by polynomials and rational maps or other maps. This replaces the simple circular revolution having as co-ordinates cosine and sine of the angle. Each time we encounter a pair consisting of the cosine and the sine of an angle,we can replace it by such motion. We know how to rotate a graph around a chosen axis. It creates a surface of revolution. If we replace the circular rotation by another motion, we may get some surprising results. A surface may be self-intersecting or fold in itself. A similar gallery showing strange forms was created by[H.Ebisui] using (complicated) trigonometric expressions. It is interesting to investigate which ones can be obtained by using only rational maps acting on directions on the unit circle. A torus is obtained by rotating a circle around an axis that does not intersect the circle. A new family of spheres, tori and knots can now be investigated.

Shell-like forms can be obtained from a torus if we multiply the size of the section by the angle (or some function of the angle) over which is rotated.
	The general ellipsoid is not a surface of revolution. Combining two ellipse constructions we obtain any ellipsoid. Many more forms can be obtained combining symmetric rational maps, all straightforward generalisations of the construction of the ellipse.

The umbilical torus is the result of rotating a deltoid, and was made into a sculpture by H. Ferguson.
http://www.hmco.com/college/mathematics.

It can be generalised and inspire more artists and designers.In the past artists have been hampered by the amount of trigonometry involved in the mathematical description of objects.

We can introduce new co-ordinate systems in space using homothetical images of these bodies. Finally, since we have the same mapping properties, we can start studying exact flows in three dimensions around such objects in the same way as flows around the airfoil are studied. It opens new perspectives in industrial design.

Conclusion

We are proposing a new education strategy,where education delivers the appropriate material (on computer screens) bridging the gap between child's toys and professional careers:

Toys -> Computer screens -> (({{Math concepts}, {Technology}, {Innovative design}}))

Such material is covering all fields: mathematics, engineering, natural sciences, design, art, ...
But this is quite different from letting them play with commercial computer games.

References

Ivan CNOP: "New Insight in Mathematics by live CAS documents",  Proceedings of the ACA Conference, Albuquerque, NM, 2001.
Hirotaka EBISUI: "Ovoids", ATCM Proceedings, Guangzhou, PRC, 1999.
Hania USCKA: "Complex Polynomials", Exploot CD#1 (in Dutch), Exploot Project, 2001.

Acknowledgement:
All brand names of toys and software packages mentioned above belong to their respective manufacturers.

http://we.vub.ac.be/exploot/
      

icnop@vub.ac.be