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Lecturer(s)
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Jukl Marek, doc. RNDr. Ph.D.
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Vítková Lenka, Mgr. Ph.D.
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Course content
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1. Euclid's Stoicheia. Historic remarks. Axiomatic principles in geometry. 2. Abstract geometries, incidence geometry. Models of the cartesian plane, the Poincaré plane, the Riemann sphere. Parallel lines. 3. Hilbert's point of view (incidence, order, continuity, congruence), absolute geometry. 4. Theorems equivalent to the Euclide parallelism axiom, its negation, hyperbolic planes. 5. Metric approach of G. H. Birkhoff: Distance function, postulate on a coordinate system on a line, coordinates of a point on a line, metric geometry, examples, the "taxicab" metric, the Moulton plane. Ordering, line segments and rays, angles, triangles. The Pasch geometries.
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Learning activities and teaching methods
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Lecture, Activating (Simulations, Games, Dramatization)
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Learning outcomes
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Examine classical geometrical topics from the view-point of axiomatic method, add geometric axioms step by step, reach also other geometries distinct from the Euclidean one. Make acquainted with metric approach of Birkhoff.
1. Knowledge Recall metric approach of Birkhoff for building Pasch geometries.
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Prerequisites
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unspecified
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Assessment methods and criteria
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Student performance
Credit: the student should be present at least at 70% of classes, and participate actively. Colloquium: the student has to understand the subject.
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Recommended literature
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Hilbert D. (1903). Grundlanden der Geometrie. Leipzig: B. G. Treubner.
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J.Gómez. (2018). Neeukleidovské geometrie.
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kolektiv autorů. (1985). Konstrukčná geometria. SPN Bratislava.
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Kutuzov B. V. (1952). Lobačevského geometrie a elementy základů geometrie. ČSAV Praha.
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Millman R. S., Parker G. D. (1991). Geometry. A Metric Approach with Models. Springer.
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Sekanina M. (1988). Geometrie II. SNTL Praha.
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