Geometry [Millman]

[mklabgr]

Book:Geometry: A Metric Approach with Models
Authors: Richard S. Millman & George D. Parker
Publication date: 1991, 2nd edition (hardcover published December 17, 1990)

Summary
Geometry: A Metric Approach with Models is a rigorous undergraduate introduction to Euclidean and non-Euclidean geometry, distinguished by its emphasis on axioms and mathematical models. Rather than treating Euclidean geometry as the only natural setting, Millman and Parker develop geometry axiomatically and repeatedly test definitions and theorems in different models. Examples include the ordinary Cartesian plane, the Poincaré upper half-plane, the taxicab plane, and the Moulton plane. This approach helps the reader distinguish what follows logically from a particular axiom from what merely appears obvious in a familiar geometric diagram. 

The book begins with basic ideas about axioms, models, sets, functions, incidence and metric geometry, before introducing betweenness, segments, rays, angles, plane separation and angle measurement. It then develops neutral geometry—the geometry obtained without assuming Euclid's parallel postulate—and uses this framework to show precisely where Euclidean and hyperbolic geometry diverge. The chapters on the theory of parallels lead naturally into hyperbolic geometry, including asymptotic rays, the angle defect of triangles and properties of parallel lines. Euclidean geometry is then recovered by adding an appropriate version of the Euclidean parallel postulate. 

The later chapters treat area and transformations, culminating in an extensive study of isometries. Topics include reflections, collineations, the Klein and Poincaré disk models, invariant sets and the classification and groups of isometries. A notable pedagogical feature is that models which fail to satisfy particular axioms are used deliberately: such countermodels demonstrate why hypotheses are necessary and give students a much stronger understanding of the logical structure of geometry. The second edition also adds expository exercises and originally had accompanying computational material. 

Key takeaways

• Main area: Geometry — especially axiomatic, Euclidean and hyperbolic geometry.
  
• The central idea is to understand geometry through axioms and models, rather than relying purely on diagrams.
  
• It clearly demonstrates the relationship between neutral, Euclidean and hyperbolic geometry.
  
• Models such as the Poincaré plane, taxicab plane and Moulton plane show which geometric statements depend on particular axioms.
  
• The final treatment of isometries and transformation groups gives the book a bridge toward more advanced geometry.
  
• Best suited to undergraduate mathematics students, teachers, or readers who already have some experience with mathematical proofs. 
  

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