1. SylabUZ
2. Wydział Matematyki, Informatyki i Ekonometrii
3. semestr zimowy 2022/2023
4. Mathematics - pierwszego stopnia z tyt. licencjata
5. Elements of Modern Geometry

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Elements of Modern Geometry - opis przedmiotu

Cel przedmiotu

To familiarize students with the basic concepts of mosaic theory.

Wymagania wstępne

Zapisz zmiany

Elementary knowledge of geometry, linear algebra and general topology.

Zakres tematyczny

Lecture 

Two dimensional tilings

Definition of a tilng and related concepts (edge, vertex, etc.) (2 h)

Existence theorem (1 hour)

Tilings using selected heptiamonds (2 h)

Symmetry group of a tiling (2 h)

Lattice tilings (2 h)

Regular tilings  (3 h)

Examples of k-uniform tilings (2 h)

Dissection tilings (1 hour)

Normal tilings (1 h)

Euler's theorem for normal tilings (2 hours)

Mosaics by M. C. Escher (2 hours) 

Tilings of n-dimensional space.  (1 hour)

Fedorov polyhedra (1 hour)

Minkowski's theorem on parallelepipeds (2 h)

Minkowski and Keller hypotheses for tilings of n-dimensional space using unit cubes (2 h)

Aperiodic tilings Regular sets and Voronoia cells (1 h)

Shechtman's discovery - quasicrystals (1 h)

Aperiodic tilings - Wang, Berger, Robinson (1 hour) Penrose tilings (3 h)

Class

Two dimensional tilings (4 h)

Determining edges and vertices of a tiling (1 h)

Determining symmetry groups of selected tilings (2 h)

Lattice tilings(3 h)

Determining dissection tilings (1 h)

Mosaics in architecture, design, art and nature - outdoor classes (4 h)

Tilings of  n-dimensional space Construction of paper models of selected mosaics in 3-dimensional space (2 h)

Keller's conjecture in 3-dimensional space (proof) (2 h)

2-period divisions of space into cubes. Counterexample to Keller's conjecture (2 h)

Aperiodic tilings. Tilings constructions from two Penrose rhombuses (3 h)

Wang tiling in 3-dimensional space (2 h)

Penrose tilings in pop culture (1 h)

Talks (2 h)

Metody kształcenia

Efekty uczenia się i metody weryfikacji osiągania efektów uczenia się

Opis efektu	Symbole efektów	Metody weryfikacji	Forma zajęć

Warunki zaliczenia

1. The condition for passing the exercises is a positive grade in the test. It is allowed to present a paper on topology. The topic is to be chosen independently by the student. Papers can be prepared by a group of two or three students. The topic of the paper must be approved by all students and the instructor.
2. Completion of the lecture is in written form with the possibility of discussion of solutions between the examiner and the examined student.

The grade for the course consists of the grade for the exercises (40%) and the grade for the lecture (60%). The condition for passing the lecture is a positive grade from the exercises. The condition for passing the course is a positive grade from the lecture.

Literatura podstawowa

1. B. Grunbaum and G. C. Shephard, Tilings and Paterns, W. H. Freeman, New York 1987.
2. M. Senechal, Quasicrystals and geometry, Cambridge University Press, 1995.

Literatura uzupełniająca

Magia M. C.  Eschera, Wydawnictwo Solis, 2009 (praca zbiorowa).

Uwagi

Zmodyfikowane przez dr Ewa Sylwestrzak-Maślanka (ostatnia modyfikacja: 27-03-2024 16:25)