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 Seminar Course on the mathematical model collection

of Leipzig University

  • Details of the second online meeting, 

          Zoom link will be shared upon request.

      Time: 28/05/2020 - 16:00

      The tentative program is as follows: 

-- Mahsa Sayyary Namin  ~ 16:00

          Kummer Surfaces

--  Miruna-Ştefana Sorea ~ 16:20

           Boy's Surface 

--  coffee break in distance    ~  16:40

--  Doreen Wetzel        ~ 17:00

           Conic Sections and Quadric Surfaces 

 

  • Details of the first online meeting, 

    • Time: 07/05/2020 - 17:00

    • Description of the course, especially detailed information will be provided about the structure of the course which is highly influenced by the circumstances due to the pandemic.

    • A presentation on the collection by Julia Struwe.

    • A presentation on the subcollection of cubic surfaces by Türkü Özlüm Çelik.

Instructors: Türkü Özlüm Çelik, Julia Struwe

 

Contact: turkuozlum (at) gmail (dot) com 

               struwe (at) math (dot) uni-leipzig (dot) de (German speaking)

                  

Textbook: Geometry and Imagination 

                  by David Hilbert and Stephan-Cohn Vossen

                  Original title: Anschauliche Geometrie.

 

Information

        

  • The students are invited to give an approx. 50 minutes seminar talk on a part of the textbook (see below) which seems to be influenced by the models in the collection. The talks are expected preferably in English, however, German talks are also welcome. At the end of each talk, there will be a discussion phase over the presentation. The talks are planned to be promoted by software illustrations under the direction of the instructors. 

      (see SageMathPolymakeJuliaMathematica, Surfer etc.)

 

  • Grading will be based on the quality of the presentation. Broadly, criteria are outlined as follows

    • The topics were addressed clearly in the first few minutes of the presentation. 

    • The results were presented in an organized manner.

    • Visual aids were appropriately used.

    • The presentation was appropriately paced. 

    • The speaker responded to questions in a forthright manner. 

 

  • Most part of the book is "quite elementary" as David Hilbert says. Please see the preface of the book. Guidance by the instructors in office hours will be available for students throughout the preparations of their talks to discuss over the topic together.

 

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Photos from the collection.

Modell 16.jpg

 Dodecahedron with 5 Hexahedra inside

Helicoid.png

Helicoid

MaS127_2.JPG

Boy's Surface

These are the (tentative) topics for the seminar talks which may change based on the number of participants.  

 

Topics (listed as in the textbook)

  • Plane Curve. The Cylinder, the Cone, the Conic Sections and Their Surfaces of Revolution. The Second-Order Surfaces. The Thread Construction of the Ellipsoid, and Confocal Quadrics. The Pedal-Point Construction of the Conics. The Directrices of the Conics. The Movable Rod Model of the Hyperboloid. (31 Pages)

 

  • Plane Lattices. Plane Lattices in Theory of Numbers. Lattices in Three and More than Three Dimensions. Crystals as Regular Systems of Points. (24 Pages)

 

  • Regular Systems of Points and Discontinuous Groups of Motions. Plane Motions and Their Composition; Classification of the Discontinuous Groups of Motions in the Plane. The Discontinuous Groups of Plane Motions with Infinite Unit Cells. The Crystallographic Groups of Motions in the Plane. Regular Systems of Points and Pointers. Division of the Plane into Congruent Cells. Crystallographic Classes and Groups of Motions in Space. Groups and Systems of Points with Bilateral Symmetry. (30 Pages)

 

  • Preliminary Remarks about Plane Configurations. The Configurations 7and 83 . The configurations 93. Perspective, Ideal Elements, and the Principle of Duality in the Plane. (24 Pages)

 

  • Ideal Elements and the Principle of Duality Space. Desargues' Theorem and the Desargues Configuration (103). Comparison of Pascal's a and Desargues Theorems. Preliminary Remarks on Configurations in Space. Reye's Configuration. (24 Pages)

 

  • The Regular Polyhedra. Regular Polyhedra in Three and Four Dimensions, and Their Projections. Polyhedra. (25 Pages)

 

  • Plane Curves. Space Curves. Curvature of Surfaces. Elliptic, Hyperbolic, and Parabolic Points. Lines of Curvature and Asymptotic Lines. Umbilical Points, Minimal Surfaces, Monkey Saddles. (21 Pages)

 

  • The Spherical Image and Gaussian Curvature. Developable Surfaces, Ruled Surfaces. The Twisting of Space Curves. (22 Pages)

 

  • Eleven Properties of the Sphere. Bendings Leaving a Surface Invariant. Elliptic Geometry. Hyperbolic Geometry, and its Relations to Euclidean and to Elliptic Geometry. (33 Pages)

 

  • Stereographic Projection and Circle-Preserving Transformations. Poincaré's Model of the Hyperbolic Plane. Methods of Mapping, Isometric, Area-Preserving, Geodesic, Continuous and Conformal Mappings. Geometrical Function Theory. Riemann's Mapping Theorem. Conformal Mapping in Space. Conformal Mappings of Curved Surfaces. Minimal Surfaces. Plateau's Problem. (24 Pages)

 

  • Linkages. Continuous Rigid Motions of Plane Figures. An Instrument for Constructing the Ellipse and its Roulettes. Continuous Motions in Space. (18 Pages)

 

  • Surfaces. One-Sided Surfaces. The Projective Plane as Closed Surface. (29 Pages)

 

  • Topological Mappings of a Surface onto Itself. Fixed Points. Classes of Mappings. The Universal Covering Surface of the Torus. The Problem of Contiguous Regions, The Thread Problem, and the Color Problem. The Projective Plane in Four-Dimensional Space. The Euclidean Plane in Four-Dimensional Space.  (21 Pages)

 

 

 

 

 

 

 

 

 

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