Babes-Bolyai University of Cluj-Napoca
Faculty of Mathematics and Computer Science
Study Cycle: Master
SUBJECT
| MMG1006 | Geometrical Constructions |
| Teaching Staff in Charge |
Prof. VARGA Csaba Gyorgy, Ph.D., csvarga cs.ubbcluj.ro |
| Aims |
|
The purpose of the course is to familiarize the students with the notions and methods of Algebra and Geometry, which are used in the Theory of Geometric Constructions , such as geometric transformations, algebraic tools, noneuclidean methods. Using these knowledges the students will be able to decide whether a construction can be made using the ruler and compasses.
The students can use these methods in teaching. |
| Content |
|
Course 1. Geometric construction problems.
-The axioms of geometric constructions. - Methods for solving Geometric construction problems. The method of intersection, the method of geometric transformations. Algebraic method. Course 2. Isometries. -Isometries of the plane. Symmetries. Rotations Course 3. Homotheties and inversions. Course 4. Elements of Projective Geometry. -Harmonic division. -Theorems of Desargues, Pappus, Brianchon. Course 5. Algebraic bases of Euclidean geometric constructions. - Euclidean geometric constructions. - Euclidean geometric constructions using coordinates Course 6. Some Geometric construction problems. -Constructing the roots of polynoms with at most 4th grade. -Delos problem. -trisection of the angles Course 7. Geometric construction problems leading to polynoms with higher grade -Ireductible Polynoms. - Sufficient condition for constructing the roots of a polynom -Constructing regulate polygons. Course 8. Extensions of fields. Polynom ring. Extensions of transcendent fields. Course 9. Notions from the Theory of Polynoms. - Minimal Polynoms. - Symmetric polynoms. - Sufficient condition for constructions. Course 10. Galois groups Course 11. Parametric constructions. -Schonemann-Eisenstein Theorem. Course 12. Kronecker’ Method. Applications. Course 13. Construction problems using only the compasses. - Mohr-Mascheroni theorem. - Construction problems using only the ruler. Course 14. Construction problems using methods of Noneuclidean Geometry. - Constructions solved using square ruler, parabola, ellipse, resp. hyperbola in plane. Seminars Seminar 1: Classical Geometric construction problems. Seminar 2: Geometric construction problems solved using Symmetries and Rotations Seminar 3: Geometric construction problems solved using Homotheties Seminar 4: Solving Geometric construction problems using Inversions. Seminar 5: Solving Geometric construction problems using the Theorems of Pappus and Desargues Seminar 6: angle’s trisection Seminar 7: constructing regulate polygons using the ruler and compasses Seminar 8: Galois groups Seminar 9: Solving Geometric construction problems using only the compasses Seminar 10: Solving Geometric construction problems using only the ruler Seminar 11: parametric constructions Seminar 12: Solving Geometric construction problems using non-Euclidean tools Seminar 13: presentation of individual projects (I) Seminar 14: presentation of individual projects (II) |
| References |
|
1. V.T. Baziljev, K.I. Dunyicsev, Geometria, Tankönyvkiadó, Vol. I., II, Budapest, 1985.
2. Tóth, A., Noţiuni de teoria construcţiilor geometrice, E.D.P. Bucureşti, 1963. 3. Szökefalvi Nagy-Gyula, A geometriai szerkesztések elmélete, Kolozsvár, 1943. 4. Buicliu, Gh., Probleme de construcţii geometrice cu rigla şi cu compasul, Ed. Tehnică, 1957. 5. Czédli, G., Szendrei, Á. , Geometriai szerkeszthetőség, Polygon, Szeged, 1997. 6. D. Andrica, Cs. Varga, Văcăreţu, D. Teme alese de geometrie, Ed. Plus, 2002 |
| Assessment |
|
Activity on Seminars 30%
Presentation of an essay 30% Viva voce final exam 40% |
| Links: | Syllabus for all subjects Romanian version for this subject Rtf format for this subject |