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	4	2+2+0	speciality	compulsory	
Didactic Mathematics - in Hungarian	4	2+2+0	speciality	compulsory	

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&#8217; 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&#8217;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)
	
Bibliography	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%