Isometries WIMS Marina Cazzola Dipartimento di Matematica e - PDF document

Isometries • Teaching geometry • Why isometries? • Geometry or Geometries? • Isometries and teacher training Symmetry Isometries, symmetry, teacher training and WIMS Tools Isometries WIMS Marina Cazzola Dipartimento di Matematica e Applicazioni Universit` a di Milano-Bicocca 12 June 2014 University of Milano-Bicocca – Dip. di Matematica e Applicazioni Symmetry and WIMS – pagina 1 University of Milano-Bicocca – Dip. di Matematica e Applicazioni Symmetry and WIMS – pagina 2 Teaching geometry Why isometries? Isometries Cycles in Italy Isometries • Teaching geometry • Teaching geometry The scope of geometry was spectacularly • Why isometries? • Why isometries? • Geometry or • Geometry or • Primaria: grade 1 (6 years) to grade 5. broadened by Klein in his Erlanger Programm Geometries? Geometries? • Isometries and teacher • Isometries and teacher training • Secondaria di primo grado: grades 6, 7 and 8. training (Erlangen program) of 1872, which stressed Symmetry Symmetry the fact that, besides plane and solid • Secondaria di secondo grado: grades 9 to 13. Tools Tools Euclidean geometry, there are many other WIMS WIMS Teacher training geometries equally worthy of attention. • Primary: University teacher training degree (H. S. M. Coxeter, Introduction to geometry , “Scienze della formazione primaria” (5 years) John Wiley & Sons Inc., second edition edition, 1969, p. ix) • Secondary: University degree (3 year + 2 year) and “Tirocinio formativo attivo” (1 year) University of Milano-Bicocca – Dip. di Matematica e Applicazioni Symmetry and WIMS – pagina 3 University of Milano-Bicocca – Dip. di Matematica e Applicazioni Symmetry and WIMS – pagina 4 Geometry or Geometries? Isometries and teacher training Isometries Prospective teachers • Teaching geometry [. . . ] Euclidean geometry is by no means the only possible • Why isometries? • Geometry or • I hate mathematics, I never understood Geometries? geometry: other kinds are just as logical, almost as useful, • Isometries and teacher training mathematics, I do not want to have anything to do and in some respect simpler. According to the famous Symmetry with mathematics Enlargen program (Klein’s inaugural address at the Tools • I already know everything I need to know WIMS University of Erlangen in 1872), the criterion that distinguishes one geometry from another is the group of In both cases we need to show them “something new” transformations under which the proposition remain true. (possibly something likable). ( ibid. , p. 67) fet University of Milano-Bicocca – Dip. di Matematica e Applicazioni Symmetry and WIMS – pagina 5 University of Milano-Bicocca – Dip. di Matematica e Applicazioni Symmetry and WIMS – pagina 6 Sym´ etrie Isometries Isometries Symmetry Symmetry • Sym´ etrie • Sym´ etrie • Beautiful images • Beautiful images • Deep mathematical • Deep mathematical concepts concepts • Groups through images • Groups through images • Rosettes with flowers • Rosettes with flowers • Breaking symmetry • Breaking symmetry Symmetry Tools Tools WIMS WIMS University of Milano-Bicocca – Dip. di Matematica e Applicazioni Symmetry and WIMS – pagina 7 University of Milano-Bicocca – Dip. di Matematica e Applicazioni Symmetry and WIMS – pagina 8

Beautiful images Beautiful images Isometries Symmetry • Sym´ etrie • Beautiful images • Deep mathematical concepts • Groups through images • Rosettes with flowers • Breaking symmetry Tools WIMS University of Milano-Bicocca – Dip. di Matematica e Applicazioni Symmetry and WIMS – pagina 9 University of Milano-Bicocca – Dip. di Matematica e Applicazioni Symmetry and WIMS – pagina 10 Beautiful images Beautiful images University of Milano-Bicocca – Dip. di Matematica e Applicazioni Symmetry and WIMS – pagina 11 University of Milano-Bicocca – Dip. di Matematica e Applicazioni Symmetry and WIMS – pagina 12 Beautiful images Wallpaper patterns University of Milano-Bicocca – Dip. di Matematica e Applicazioni Symmetry and WIMS – pagina 13 University of Milano-Bicocca – Dip. di Matematica e Applicazioni Symmetry and WIMS – pagina 14 Analogy Difference University of Milano-Bicocca – Dip. di Matematica e Applicazioni Symmetry and WIMS – pagina 15 University of Milano-Bicocca – Dip. di Matematica e Applicazioni Symmetry and WIMS – pagina 16

Deep mathematical concepts Groups through images Isometries • Groups Isometries Given a figure, you can find its symmetry group Symmetry Symmetry • Sym´ etrie • Sym´ etrie ◦ the tool to describe “symmetry” of a figure is its • Beautiful images • Beautiful images • Deep mathematical • Deep mathematical symmetry group i.e. the set of all isometries of concepts concepts r q t • Groups through images • Groups through images the plane that leave the figure unchanged four reflections • Rosettes with flowers • Rosettes with flowers • Breaking symmetry • Breaking symmetry ( σ r , σ t , σ s e σ q ) Tools Tools with respect to WIMS WIMS s the dashed lines (Something is missing) University of Milano-Bicocca – Dip. di Matematica e Applicazioni Symmetry and WIMS – pagina 17 University of Milano-Bicocca – Dip. di Matematica e Applicazioni Symmetry and WIMS – pagina 18 Groups and images Groups and images Isometries Given a symmetry group you can build images with that Isometries Given a symmetry group you can build images with that Symmetry Symmetry symmetry symmetry • Sym´ etrie • Sym´ etrie • Beautiful images • Beautiful images • Deep mathematical • Deep mathematical concepts concepts • Groups through images • Groups through images • Rosettes with flowers • Rosettes with flowers • Breaking symmetry • Breaking symmetry Tools Tools WIMS WIMS The composition of two reflection with intersecting axes D4 is a rotation University of Milano-Bicocca – Dip. di Matematica e Applicazioni Symmetry and WIMS – pagina 19 University of Milano-Bicocca – Dip. di Matematica e Applicazioni Symmetry and WIMS – pagina 20 Shaping an idea Rosettes with flowers Milano Falso gelsomino Trachelospermum jasminoides 5 . ( C 5 ) Trento Verbena Verbena officinalis ∗ 5 . ( D 5 ) C5 Seoul University of Milano-Bicocca – Dip. di Matematica e Applicazioni Symmetry and WIMS – pagina 21 University of Milano-Bicocca – Dip. di Matematica e Applicazioni Symmetry and WIMS – pagina 22 Breaking symmetry Breaking symmetry University of Milano-Bicocca – Dip. di Matematica e Applicazioni Symmetry and WIMS – pagina 23 University of Milano-Bicocca – Dip. di Matematica e Applicazioni Symmetry and WIMS – pagina 24