## DEMODISK1 - © 1992,1994, R.J.K. Stowasser, Techn.Univ.Berlin

DEMODISK1 offers examples of the art of visual communication which have been created by R.J. K. Stowasser as a by-product of a European joint project called New Approaches of the Teaching of Engineering Mathematics (1991-1994).

The philosophy behind the animations can be described in the following way:

Not long ago, envelopes of curves, involutes, caustics, and parallels were standard topics for freshmen. The rich concept 'curve' served as an assembler of the isolated parts of school mathematics: Geometry, Algebra, Trigonometry, Analytical Geometry, Calculus. As a result of the trend towards generalization and rigour, in math education, the 'special curves' and most of the 'vital geometric spark which has ignited so many minds in the past', have been left out. Even hyperbolas and parabolas are not necessarily objects of geometric or kinematic interest any more. Their study has all too often degenerated into treating them as mere graphs of functions. Analysis courses at school or university maintain this unsatisfactory view. Even in non-obligatory courses such as Differential Geometry 'special curves' are mostly used for illustrative purposes only.

As a result of this fault in their training mathematics teachers are not aware of the educational potency of 'special curves' as a fruitful field for exploration. They have no competence to use geometric, kinematic, algebraic and other precalculus tools. Little is thought of anyone who can draw cardioids and other beautiful curves using compasses and ruler.

By all means, the fascinating shapes showing the very essence of a curve can usually not be brought out by paper and pencil work. Beauty does not show up when we are drawing rough sketches or plotting a few points in a Cartesian coordinate system. Computers fundamentally change the situation. They give us the powerful drawing tools which are necessary to produce aesthetically appealing curves. And most importantly: the very dynamic nature of the curves becomes manifest when step by step generation of the curve takes place on the screen.

Because we now have graphics and animations, geometrical methods -- particularly precalculus methods of the 17th century, neclected today -- gain new educational importance. The rich material on curves in Newton's Lucasian Lectures on Algebra, or the problems in the Calculus textbooks of Johann Bernoulli can in fact be exploited properly using kinematic computer representations.

Today's technology allows for efficient revitalising of geometric and analytic problem solving (model building) and makes a new kind of visual communication possible in contrast to the common verbal problem posing found in the textbooks. Like no ready-made sets of exercises the DEMODISK1 offers a starting point to rich fields of activity to engage the curiosity of 'open-eyed minds'. DEMODISK1 is an invitation to aks questions and to look around for answers making use of all available sources of information including such as 'Mathematica', 'Maple', Dictionaries,.... common sense and friends.

If you arrive at some 'formulas' which make computable what you see on the screen, you have certainly practiced some mathematical modelling. We hope that the aesthetic quality of DEMODISK1 is a strong motivator and we would be glad to hear from you.

### Literature:

1. Bruce, J.W., Giblin, P.J., and Rippon, P.J. Microcomputers and Mathematics. Cambridge University Press 1990.
2. Brieskorn, E., Knörrer, H. Ebene algebraische Kurven. Birkhäuser Boston 1981 Plane Algebraic Curves, Birkhäuser Boston 1986.
3. Stowasser, R.J.K. Unifying Ideas for the Mathematics Curriculum under Control of Aesthetics. Beiträge zum Mathematikunterricht 1994. Verlag Barbara Franzbecker, D-31162 Bad Salzdetfurth.
4. Haapasalo, L., Stowasser, R.J.K. Computeranimationen - Wiederbelebung der Geometrie. Mathematik Lehren, Heft 65, 1994. Erhard Friedrich Verlag, D-315917 Seelze.