# MATHEMATICAL LABORATORY

In the Mathematical Laboratory of the University Museum of Natural History and Scientific Instruments there are about 160 mathematical machines built by teachers of Liceo A.Tassoni in the scope of an innovation project for the teaching of geometry developed by the Nucleo di Ricerca in Storia e Didattica della Matematica (University of Modena) . The project , coordinated by prof. Mariolina Bartolini Bussi (bartolini@unimo.it) has produced mathematical machines, didactical itineraries, animation films and simulations by computer. With these teaching aids we introduce a historical dimension and a manipulative and visual one in our classrooms. Our didactical research purpose is the historical contextualization of problems, of theories and of methods.Many "machines" realize projects or ideas of mathematicians, from the Ancient Greek up to now. For the use of these "machines" in the classroom, students have to elaborate abstract themes and proofs.

Classroom activities have been developed on curricular themes using machines and also some seminars have been organized with teachers of different schools . Two exibitions were organized , in Modena in 1992 and in Torino in 1996. We are collaborating with Vierkant Foundation -Amsterdam (http://www.cs.vu.nl./~vierkant/) and with the group Cabri-Geometre of Grenoble (http//www-cabri.imag.fr/).

The research project on Mathematical Machines had been financed by the Municipality of Modena, by CNR, by MURST and by University of Modena.

Mathematical machines of our laboratory can be divided in the following groups:

1) Pantographs,i.e. linkages that allow to draw the image of a figure under a given transformation. Geometrical transformations considered are: symmetry to an axis, simmetry to a point, translation, rotation, glide reflection, homothety, similarity, affinity, inversion, perspectivity. In some machines, two or more linkages are connected : so that it is possible to visualize the result of the composition of two transformations.

2) Curve drawers, i.e. machines (built by wood, plexiglas, metal bars and stretched threads) which can force a point or a straight line to move along a given trajectory. They are: machines drawing a straight line (Kempe and Hart), Descartes machines, Cavalieri machines and DeL'Hospital machines (drawing conics), ellipsographs (Van Schooten), Newton square drawing a strophoid and a cissoid, mecanical linkages for lemniscate, for conchoid, for limancon,machines drawing envelopes.

3) Models (built by wood, plexiglas, metal bars and stretch threads) by which it is possible to show many important properties and theorems; i.e. cones, to derive the symptom of conic sections, cones and cylinders to show Dandelin theorems, models to show conics as metamoorphoses of circles, Durer perspectograph, models to illustrate the generation of geometrical transformations in the space.

4) Machines that were built to solve very important problems in the history of mathematics, like mesolabon, trisectors, the squares of Bombelli ecc.

DIDACTICAL ITINERARIES.

The didactical research studies of N.R.S.D.M.have concerned particular themes like :the concept of function, complex numbers, probability, geometrical transformations, conics. (References)

VIDEOTAPES
Videotapes have been realized with the contribution of the Calculus Centre of the University and the Ufficio Cinema of Municipality of Modena.
• Conics families.
• Complex numbers: intersections of a parabola with a line.
• Complex numbers: intersections of a cubic curve with a line.
• Pithagoras theorem and affinities.
• Carnot theorem and affinities.