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Course Description form
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Course Title
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English Code /No
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ARABIC code/no.
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credits
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Th.
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Pr.
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Tr.
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TCH
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Euclidean and non-Euclidean Geometries
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Math 462
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3
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3
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Pre-requisites
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Math 343
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Brief contents, to be posted in university site and documents(4-5 lines):
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Fundamentals of Euclidean and non-Euclidean geometries. Axioms of Incidence, betweenness, congruence, Hilbert-Euclidean, Dedekind. Elliptic, Euclid’s and hyperbolic parallel properties. Affine and projective geometries. Ordered field. Defect of a triangle. Saccheri’s quadrilateral. Laboachevskian geometry. Independence of Parallel Postulate. Beltrami-Klein Model
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Faculties and departments requiring this course (if any): Faculty of
Science/Department of Mathematics.
Objectives: Prefer In points
The objectives of this course are as follows:
1. To introduce to the students the basic concepts of Euclidean and non-Euclidean geometries.
2. To improve the students logical thinking.
3. To understand how the ideas in Euclidean geometry can be extended to several non-Euclidean cases.
4. Using elliptic property, the student will learn geometry without parallel lines.
5. Using hyperbolic parallel property, the student will learn geometry with several parallel lines through a point.
6. They will learn the use of ordered fields in various geometries.
7. They will learn the extension of affine plane as a projective plane.
8. They will learn several models of abstract geometries such as Laboachevskian, Beltrami-Klein Models, etc.
Contents: Prefer In points
1- Fundamental of Euclidean geometry
2- Logical axioms and Incidence geometry
3- Axioms of betweenness
4- Axioms of congruence
5- Affine geometry, ordered field
6- Saccheri and Lambert Quadrilaterals
7- Lobachevski, Gauss, Bolyai
8- Beltrami-Klein Model
Course Outcomes:
A- Knowledge:
(Specific facts and knowledge of concepts, theories, formula etc.)
From this course the students will get a general background of non-Euclidean geometries. Knowledge of incidence, affine, Sahheri, Lambert, Lobachavsky, Gauss, Bolyai, and Beltrami-Klein Models of non-Euclidean geometries is useful to understand modern concepts of Mathematics.
B-Cognitive Skills:
(Thinking, problem solving )
This is a very theoretical as well as practical course in nature. The students will develop their thinking and their cognitive skills will be improved by studying several abstract axioms, hypothesis, theorems and their logical proofs involved in this course.
C- Interpersonal skills and responsibilities:
The students are encouraged to participate in class and discuss mathematics among themselves. They are also expected to submit home works to continuous assessment.
D- Analysis and communication:
(communication, mathematical and IT skills)
The students learn from this course how the practical ideas in nature can be given a mathematical formulation and how to solve them by using tools in this subject.
Assessment methods for the above elements:
Exam I (25 %)
Exam II (25 %)
Assignments and quizzes (10 %)
Final Exam (40 %)
Text book:
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Euclidean and non-Euclidean Geometries, Development and History
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Author: Marvin Jay Greenberg
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Publisher: W.H. Freeman and Company, New York, 2007
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Supplementary references
Anderson, James W. Hyperbolic Geometry, second edition, Springer, 2005.
Jeremy Gray (1989) Ideas of Space: Euclidean, Non-Euclidean, and Relativistic, 2nd edition, Clarendon Press.
Other Information Resources:
MacTutor Archive article on non-Euclidean geometry
Non-Euclidean geometry on Planet Math
Synthetic Space-time, a digest of the axioms used, and theorems proved, by Wilson and Lewis. Archived by WebCite.
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Time table for distributing Theoretical course contents
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Remarks
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Experiment
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weak
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Fundamental of Euclidean geometry, historical background
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1
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Euclidean axioms and their several applications
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2
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Axioms of Incidence
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3
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isomorphisms and duals ,sModel
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4
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Axioms of betweenness
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5
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Axioms of congruence
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6
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Elliptic, Euclid’s and hyperbolic parallel properties
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7
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Hilbert-Euclid and Dedekind axioms
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8
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Fields and ordered fields
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9
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Affine planes and projective planes
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10
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Saccheri and Lambert Quadrilaterals
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11
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The defect and similar triangles
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12
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Lobachevski, Gauss, Bolyai
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13
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Independence of Parallel Postulate,
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14
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Beltrami-Klein Model
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15
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Final exam.
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