Mathematical objects cannot be directly perceived through our senses. Therefore problems arise when mathematics is to be taught or disseminated. However, students (and mathematicians) usually use external representations (as for example words, schemas, figures, symbols, gestures, tools) when dealing with mathematical objects. A blackboard covered in formulae is a commonly shared image reminding of a mathematics lesson. However mathematical activity has always included manipulation of instruments, as for example ruler and compass. The Laboratory of Mathematics of the University of Modena owns a collection of about 200 geometrical mechanisms (mathematical machines) that have been reconstructed on the basis of original historical documents. The majority of the machines date back to the 17th-19th century (e.g. perspectographs, curvigraphs pantographs for transformations), but there are also some mechanisms dating back to antiquity, concerning conic sections and the solution of problems.
Research activities concerning these mechanisms sit within a project of a Thematic network, funded by the European Community within the 5th Framework (1999/2003). The aim of this project is to build an exhibition of mathematical experiments showing the impact and importance of mathematics in everyday life. The co-ordinator of the project is Prof. Albrecht Beutelspacher (University of Giessen, De). The exhibition is divided into different modules, coordinated by the different partners. The University of Modena works with Perspective. The research groups participating in the project belong to the following institutions: the University of Giessen, the University of Milan, the University of Florence, the Atractor group in Porto, Middlesex University (London).
The work of the Laboratory includes dissemination activities and education research.

Dissemination activities

The Laboratory in Modena has already prepared a number of exhibitions (Modena 1992 e 1998; Torino 1996; Cesenatico 2000; Treviso e Cesena 2002, Firenze 2002) and some limited thematic exhibitions during the week for the Scientific Culture and during seminars and conferences for teachers. In particular, in the year 2001 the section Perspective and Anamorphosis was presented in Modena within the Apriti Sesamo exhibition, at the Foro Boario. Different parts of the collection have been exhibited in European (France, Germany, Spain, The Netherlands) and non-European countries (USA, Canada, Mexico, Brasil, Japan). Complex models were constructed for Italian (e.g. the Citta' della Scienza in Naples) and foreign (Amsterdam, Grenoble, Bordeaux and Mexico City) Scientific Institutions.

The collection of machines is called Theatrum Machinarum. The catalogue (updated until 1999) is on the web ( and is available also on a CD (that can be requested to the Laboratory). A CD about Perspective, drawing on the exhibition Perspectiva Artificialis, is in preparation and will be available in April 2003.
The collection Theatrum Machinarum can be shown in exhibitions (for more information, see Contacts).

Education research

The machines have been reconstructed with a didactical aim in mind.
The Laboratory:
a) belongs to the project SCUOLA CITTA' ITINERARI in Modena and is opened twice a week for schools
b) hosts the sub-project 'Mathematical Mechanism' carried out in collaboration by the MPI and the Department of Mathematics, University of Modena and Reggio Emilia focused on INNOVAZIONE IN DIDATTICA DELLA MATEMATICA:LA FUNZIONE DEGLI ‘STRUMENTI’ (innovation in mathematics education: the role of tools)
c) hosts undergraduate mathematics students
d) developed on-line activities about transformations and curvigraphs within the SeT project (, unità N e O) that are published by the BDP (
Schools interested in implementing these activities in the classroom may buy already made kits containing five copies of each machine, in order to have five groups of students working with the same machine (the worksheets are also available on paper at the exhibition - other machines can be constructed if required)
e) is available for secondary school classrooms' and university students' visits and scientific consultancy is provided in order to choose the appropriate programme for the visit.
Besides producing on-line and paper publications, the Laboratory has established a permanent collaboration with the journal Alice.

The research activity of the Laboratory has been presented in many national and international conferences. They were invited to two international conferences, presenting work about curvigraphs and pantographs and about perspectographs:
· International Congress of Mathematicians, Berlin 1998
· 4th Int. Design Thinking Res. Symp. on Design Representation (MIT - Cambridge MA - 1999).

The activity of the laboratory is co-ordinated by Prof. Mariolina. Bartolini Bussi, and organised by four teachers (Annalisa Martinez, Marcello Pergola, Marco Turrini and Carla Zanoli) and a contract researcher (Michela Maschietto).


Prof. Mariolina Bartolini Bussi (
Dott. Michela Maschietto (

Università di Modena e Reggio Emilia
Dipartimento di Matematica
Via G.Campi, 213/b
41100 Modena

Tel. 059 205 5176



translation: F.Olivero