Course guide of Geometry and Topology Workshop (27011C3)

It is recommended that the student has passed the subjects Geometry I-II-III, Algebra II, Curves and Surfaces and Topology II.

When using AI tools for the development of the subject, the student must adopt an ethical and responsible use of them. The recommendations contained in the following url must be follwed: Recomendaciones para el uso de la inteligencia artificial en la UGR.

En el caso de utilizar herramientas de IA para el desarrollo de la asignatura, el estudiante debe usarlas de forma ética y responsable. Se deben seguir las recomendaciones contenidas en el documento Recomendaciones para el uso de la inteligencia artificial en la UGR.

Euclidean geometry: Constructions with ruler and compass. Polyhedra. Universes of dimension two. Non -euclidean geometries. Computer aided geometry.

• Recognize symmetries in planar and spatial figures.
• Determine the group of symmetries of a rosette, frieze and mosaic.
• Know the different types of non-Euclidean geometries, as well as the aspects in which they differ.
• Model geometric problems, especially using geometry oriented software.

• Lesson 1. Classical geometry: Euclid's Elements and other axiomatic approaches. Constructions.

• Lesson 2. Non-Euclidean Geometries: The postulate of the parallels. The birth of non-euclidean geometries. The sphere and the hyperbolic plane. Constructions in the hyperbolic plane: parallel and ultrapararallel lines, angles and triangles, regular polygons.

• Lesson 3. Symmetries: Movements in the plane and symmetries of objects.

• Lesson 4. Introduction to polyhedra: Constructing polyhedra. Platonic and Archimedian polyhedra. Descartes' formula. Euler's formula: different proofs.

• Lesson 5. Surfaces and 3-manifolds: Exploring the shape of a universe. Connected sums. Orientability vs. two faces. Geometry of a Universe. flat manifolds.

1. C. ALSINA, R. PEREZ y C. RUIZ, Simetría Dinámica, Síntesis 1989.

2. W. BARKER, H. ROGER. Continuous Symmetry. American Mathematical Society, Providence, RI. 2007.

3. M. BERGER. Geometry, vol. 1 y 2, Springer-Verlag 1987.

4. R. BONOLA, Non-Euclidean Geometry, A critical and Historical Study and its Development, Dover Publications, Inc., New York (USA) 1955.

5. F. BORCEUX, An axiomatic approach to geometry (Geometric Trilogy I), Springer 2014.

6. O. BYRNE. The First Six Books of the Elements of Euclid in Which Coloured Diagrams and Symbols Are Used Instead of Letters for the Greater Ease of Learners. Taschen GmbH, Cologne 2010.

7. J. N. CEDELBERG, A Course in Modern Geometries, Undergraduate Texts in Mathematics, Springer, New York (USA) 2001.

8. J.H. CONWAY, H. BURGIEL y C. GOODMAN-STRUSS, The Symmetries of things, A K Peters Ltd. 2008.

9. H.S.M. COXETER, Introduction to Geometry, John Wiley 1969.

10. P.R. CROMWELL, Polyhedra, Cambridge University Press 1999.

11. EUCLID. The Thirteen Books of Euclid’s Elements Translated from the Text of Heiberg. Introduction and Books I, II.Translated by Thomas L. Heath. Vol. I. 3 vols. Dover Publications, Inc., New York. 1956.

12. R. L. FABER, Foundations of Euclidean and non-Euclidean Geometry, Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York (USA) 1983.

13. M. J. GREENBERG. Euclidean and Non-Euclidean Geometries: Development and History. 3rd ed. New York: W.H. Freeman 1993.

14. D. HILBERT y S. COHN-VOSSEN, Geometry and the imagination, Nueva York: Chelsea Publishing Co. 1952 (AMS Chelsea Publishing 1999)

15. M. KLEIN, Mathematical Thought From Ancient to Modern Times, Volume 3. Oxford University Press 1972.

16. J. M. LEE, Axiomatic Geometry, Pure and applied undergraduate texts 21, American Mathematical Society 2013.

17. W. P. THURSTON. “The Geometry and Topology of Three-Manifolds.” Princeton, N.J. 1980.

18. J.R. WEEKS, The Shape of Space, Marcel Dekker, New York 2002.

• Geogebra (gratis and free software for dynamical geometry): https://www.geogebra.org/classic
• Euclid's Elements, José Luis Bueso Montero, https://www.ugr.es/~jlbueso/euclides/1/index.html
• Byrne’s Euclid: A reproduction of Oliver Byrne’s celebrated work from 1847 plus interactive diagrams, cross references, and posters designed by Nicholas Rougeux, Nicholas Rougeux, https://www.c82.net/euclid/
• Byrne’s Euclid, Jimmy Button, https://github.com/jemmybutton/byrne-euclid/releases/download/Current/byrne_context.pdf

• MD01. Lección magistral/expositiva 
• MD02. Sesiones de discusión y debate 
• MD03. Resolución de problemas y estudio de casos prácticos 
• MD04. Prácticas en sala de informática 
• MD05. Seminarios 
• MD06. Análisis de fuentes y documentos 
• MD07. Realización de trabajos en grupo 
• MD08. Realización de trabajos individuales 

In accordance with the "Evaluation and Qualification Regulations of the Students of the University of Granada", in order to evaluate the acquisition of the contents and competencies to be developed in the field, continuous evaluation systems and unique evaluation systems will be used alternately.

Continuous assessment:

It will be the preferential evaluation method. To evaluate the acquisition of the contents and competencies, a diversified evaluation system will be performed, selecting the most appropriate evaluation techniques for the subject at all times, which allows to highlight the different knowledge and capacities acquired by students, according to the following criteria:

• A test will take place at the end of each lesson, including written assignments, individual activities in PRADO, and/or presentations. These eliminatory tests will be held during class time. In the ordinary exam, the last lesson will be graded in a similar manner, and students will also be allowed to increase their grades for each lesson.

As established by the regulations, students who have failed the ordinary call will have an extraordinary call. All students have the right to attend it, regardless of whether or not they have followed the continuous evaluation process. The qualification of the students in the extraordinary call will be adjusted to the rules established in this course guide. This way, those students not following the continuous evaluation system will be granted the possibility of obtaining up to 100% of the qualification by carrying out an exam of the same characteristics as the single final assessment.

This will be an exceptional evaluation method. A student who cannot follow the continuous evaluation system may be accepted after proper justification. This evaluation will be carried out in a single academic act that may include as many tests as necessary to prove that the student has acquired all the skills described in this guide. Essentially, it will consist of a theoretical-practical written or oral examination of the previously detailed syllabus. The final mark will be expressed by numerical qualification in accordance with the provisions of article 5 of the R.D. 1125/2003, of September 5, which establishes the European Credit System and the system of qualifications in the university degrees of an official nature and validity in the national territory. The global qualification will correspond to the weighted score of the different aspects and activities that make up the corresponding evaluation system.

Assessment by incidents

In this case, the evaluation regulations approved on November 6, 2016 by Governing Council of the University of Granada (BOUGG No. 112, of November 9, 2016) will be taken into account. In this way, students who cannot attend evaluation tests at the assigned date by the Faculty of Sciences, may request the Director of the Department for assessment by incidents in the cases indicated in the aforementioned regulations. In addition, the evaluation by court and the evaluation of students with disabilities or other specific educational support needs will follow the provisions of the aforementioned regulations.

Información de interés para estudiantado con discapacidad y/o Necesidades Específicas de Apoyo Educativo (NEAE): Gestión de servicios y apoyos (https://ve.ugr.es/servicios/atencion-social/estudiantes-con-discapacidad).

• Geogebra (gratis and free software for dynamical geometry): https://www.geogebra.org/classic