• Four Science Terps Awarded 2025 Goldwater Scholarships

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  • Announcing the Winners of the Frontiers of Science Awards

    Congratulations to our colleagues who won the 2025 Frontiers of Science Award: - Dan Cristofaro-Gardiner, for his join paper with Humbler and Seyfaddini: “Proof of the simplicity conjecture”, Annals of Mathematics 2024. - Dima Dolgopyat & Adam Kanigowski, for their joint paper with Federico Rodriguez Hertz: “Exponential mixing implies Bernoulli”, Annals of Mathematics… Read More

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    Maryland Global, published a great interview with our visiting professor (and diplomat), Qendrim Gashi. The interview is available at https://marylandglobal.umd.edu/about/news/math-olympiads-diplomacy-meet-visiting-math-professor-qendrim-gashi Read More

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    Eugenia Brin, a Russian immigrant and retired NASA scientist who, with her family of accomplished Terps, became an important benefactor of the University of Maryland, died on Dec. 3, 2024. She was 76 years old. The rest of the article can be read here: https://cmns.umd.edu/news-events/news/eugenia-brin-1948-2024 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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