Babes-Bolyai University of Cluj-Napoca
Faculty of Mathematics and Computer Science
Study Cycle: Master

SUBJECT

Code
Subject
MMG1006 Geometrical Constructions
Section
Semester
Hours: C+S+L
Category
Type
Computational Mathematics - in Hungarian
2
2+1+0
speciality
compulsory
Didactic Mathematics - in Hungarian
2
2+1+0
speciality
compulsory
Didactic Mathematics - in Hungarian
4
2+1+0
speciality
compulsory
Teaching Staff in Charge
Prof. VARGA Csaba Gyorgy, Ph.D.,  csvargacs.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