• Adam Kanigowski Awarded European Mathematical Society Prize

    He is the first member of UMD’s Department of Mathematics to receive this prestigious award for young mathematicians. The European Mathematical Society (EMS) awarded a 2024 EMS Prize to Adam Kanigowski, a Polish-born associate professor in the University of Maryland’s Department of Mathematics. Established in 1992, the prize is presented every four years to Read More
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  • Alfio Quarteroni receives the Blaise Pascal Medal in Mathematics

    Congratulations to Alfio Quarteroni for winning the 2024 Blaise Pascal Medal in Mathematics The message from the European Academy of Sciences reads: We are excited to announce that Professor Alfio Quarteroni has been awarded the esteemed 2024 Blaise Pascal Medal in Mathematics for his outstanding contributions to the field, particularly in Read More
  • Archana Receives the Donna B. Hamilton Award

    Archana Khurana has been selected to receive the Donna B. Hamilton Award for Excellence in Undergraduate Teaching in a General Education Course.  Awards are based solely on student nominations and are solicited from across campus.  From the many nominations received, the selection committee was very impressed by the student experience Read More
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Description

Hilbert's axioms for Euclidean Geometry. Neutral Geometry: The consistency of the hyperbolic parallel postulate and the inconsistency of the elliptic parallel postulate with neutral geometry. Models of hyperbolic geometry. Existence and properties of isometries.

Prerequisites

1 course with a minimum grade of C- from (MATH240, MATH341, MATH461).


Level of Rigor

Standard


Sample Textbooks

The Four Pillars of Geometry, by J. Stillwell

Foundations of Geometry, by G.A. Venema.


Applications


If you like this course, you might also consider the following courses


Additional Notes

Topics

Logical deficiencies in Euclidean Geometry

Flawed proofs

Correction of some flawed proofs by clearly stating certain postulates and giving rigorous deductions

Overview of the structure of Euclidean Geometry

Importance of Euclid's Parallel Postulate as opposed to the other postulates

Equivalence of certain postulates (Play fair, etc) to Euclid's Parallel Postulate

Absolute or Neutral geometry. Work of Legendre and Saccheri

Negation of Euclid's Parallel Postulate; non-Euclidean geometry.

Discussion of models for non-Euclidean geometry

Axiom systems

Incidence axiom and ruler postulate

Betweenness

Segments, Rays and Convex sets

Angles and Triangles

Pasch's Postulate and Plane Separation Postulate

Perpendiculars and inequalities

SAS postulate

Parallel postulates

Models for non-Euclidean geometry

Proof of the consistency of non-Euclidean geometry by means of models

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