1 Video-based Hypermedia for Communicating Mathematics Tom Apostol, Teresa Chambel, José Francisco Rodrigues and Lara Santos A project in progress at the CMAF, University of Lisbon Applying Hyper-Video techniques for communicating mathematical concepts

2 Video-based Hypermedia for Communicating Mathematics The combi Combination of video and computer animation is a powerful visualization tool It portrays a large quantity and diversity of information in a brief period of time With text it conveys mathematics in a richer and more stimulating cultural context We show with examples how video based hypermedia technology greatly enhances communication of mathematics, by providing flexible interactive mechanisms that integrate and navigate different types of media, video, animation, and text in ways that can be adapted to a great variety of learning styles and situations. We shall use films from the virtual exhibition BEYOND THE THIRD DIMENSION and the video and workbook from "The History of Pi", produced by Project MATHEMATICS!

3 Video-based Hypermedia for Communicating Mathematics The combi WORLD MATHEMATICAL YEAR 2000 LAUNCHED BY THE INTERNATIONAL MATHEMATICAL UNION SUPPORTED BY UNESCO On the 6 th May of 1992, in Rio de Janeiro, following a proposal of its president J.-L. Lions, the International Mathematical Union declared that the Year 2000 would be the World Mathematical Year. The Declaration of Rio set three aims: - The great challenges of 21 st Century - Mathematics, a key for Development - The Image of Mathematics

4 Video-based Hypermedia for Communicating Mathematics Mathematics as an Educational Resource 2000 – 2001 A programatic activity within the research center CMAF-University of Lisbon, Portugal Communicating Mathematics for the future and present generations: 1.Mathematical contents for univeristy and pre-university teaching 2.Raising public awareness in Mathematics, with the SPM and EMS 3.Using multimedia and computational tools for visualization Support:Departamento do Ensino Secundário – Ministério da Educação Fundação para a Ciência e Tecnologia – Ministério da Ciência e Ensino Superior Co-ordination: José Francisco Rodrigues, Suzana Nápoles and Jorge Nuno Silva

5 Video-based Hypermedia for Communicating Mathematics Visual representations provide an essential aid to communicate mathematical concepts increase efficacy of the mathematics message improve learning processes Hypervideo provides mechanisms to structure, navigate and interact with information spaces open and flexible integration of multiple media a new communication tool adaptable to learning styles and situations

6 Video-based Hypermedia for Communicating Mathematics The combi Different Video Maps Text centered and Video centered pages

7 Video-based Hypermedia for Communicating Mathematics The combi Hypervideo, a more powerful model: Has spatial and temporal dimensions Truly integrate different media New concepts of navigation Hypervideo allows: Select what is to be seen Control the pace of the material Easy go back and forth, stop, annotate Compare and relate to other material

8 Video-based Hypermedia for Communicating Mathematics The combi In 2000 the CMAF/University of Lisbon, in collaboration with the town of Óbidos, the University of Madeira and the Portuguese Mathematical Society, recreated a virtual exhibition of high visual quality together with its authors, T. Banchoff and D. Cervone. This itinerant exhibition aims to stimulate the imagination and rouse the curiosity of young people and the general public, by showing the potential of new information technologies and computational geometry through a fascinating experience that relates mathematics, art and computer graphics.

9 Video-based Hypermedia for Communicating Mathematics This exhibition can be visited in This virtual exhibition of mathematical objects evokes the notion of dimension

10 Video-based Hypermedia for Communicating Mathematics The combi In and Outside the Torus This image of the exhibition depicts the three-dimensional projection of a torus in four-space. It is a projection of a torus lying originally in four-space, representing the cross product of two circles. Bands on the torus have been removed to make it possible to "see through" it. The exhibition includes several MPEG movies of the torus rotating in four dimensions.

11 Video-based Hypermedia for Communicating Mathematics The combi Movie of In and Outside the Torus This torus is the three-dimensional projection of a torus in four-space. It is projected stereographically from a point on the torus itself, so the projected surface seems to stretch out to infinity. This movie shows the surface rotating in four-space. As the torus passes through the projection point, its image in three- space appears to extend to infinity, and it turns inside out: the region that was outside is now inside, and vice versa.

12 Video-based Hypermedia for Communicating Mathematics Triple Point Twist This surface is a ruled surface given by the equations (x,y,z) = (u, v 3 + cv, uv + v 5 + cv 3 ). The rulings for the surfaces are the straight lines produced when v is held fixed and u is allowed to vary. This movie shows the surface being swept out by these ruling lines. The line first sweeps out a portion of the surface, then seems to backtrack, causing the surface to intersect itself. Finally, it curves back on itself a third time, and passes through the curve of self-intersection again, forming the triple point.

13 Video-based Hypermedia for Communicating Mathematics The parametrization of the surface has a constant c that varies from -1 to 1 in this movie. At c = 0, the singularity disappears and the surface becomes embedded. For values of c greater than 0, the surface has no self-intersection, but for values of c less than 0, a triple point and two pinch points appear. The movie shows a series of different values of c as it varies from -1 to (x,y,z) = (u, v 3 + v, u v + v 5 + v 3 )

14 Video-based Hypermedia for Communicating Mathematics Pithagoras theorem The history of Pi The Tunnel of Samos Polinomials Similarity Sinus, Cosinus I Sinus, Cosinus IISinus, Cosinus III Early History of Mathematics The 8 +1 films of the “Project Mathematics” by Tom M. Apostol, have been translated in Portuguese by CMAF/UL and have been distributed in more than 600 high schools by the Ministry of Education in Portugal.

15 Video-based Hypermedia for Communicating Mathematics The combi The History of PI  A videotape of 30 min + Workbook  To introduce basic concepts, grab attention and motivate to learn more  Live action, music, special effects, computer animations  Historical documents and applications to real world  Additional mathematical contents and exercises  Links to web pages and to integrated applets

16 Video-based Hypermedia for Communicating Mathematics The combi The History of Pi in Hypervideo is a prototype of a multimedia tool produced by CMAF/Projecto Matemática em Acção at the University of Lisbon Based in the homonimous film by T. Apostol

17 Video-based Hypermedia for Communicating Mathematics Access to the Text... Follow spacio-temporal link Follow common text index link

18 Video-based Hypermedia for Communicating Mathematics Spacio-temporal links: shortcuts to specific information with just 1 click alternative path: 3 clicks 1st click: index 2nd click: next page 3rd click: link on the text

19 Video-based Hypermedia for Communicating Mathematics shows adicional link information1 - On mouse over: 2 - On mouse click:opens text page3 - Link to Cinderella application