Modern GeometriesThis comprehensive, best-selling text focuses on the study of many different geometries -- rather than a single geometry -- and is thoroughly modern in its approach. Each chapter is essentially a short course on one aspect of modern geometry, including finite geometries, the geometry of transformations, convexity, advanced Euclidian geometry, inversion, projective geometry, geometric aspects of topology, and non-Euclidean geometries. This edition reflects the recommendations of the COMAP proceedings on Geometry’s Future, the NCTM standards, and the Professional Standards for Teaching Mathematics.References to a new companion text, Active Geometry by David A. Thomas encourage students to explore the geometry of motion through the use of computer software. Using Active Geometry at the beginning of various sections allows professors to give students a somewhat more intuitive introduction using current technology before moving on to more abstract concepts and theorems. |
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Page 365
Chapters 2 through 5 extended the study of Euclidean geometry by introducing
many modern ideas. Chapter 6 through 8 incorporated modern transformations
that resulted in very general geometries or a geometry that differed from
Euclidean. Various sets of axioms for Euclidean geometry appeared in Chapter 1
. Before introducing, in the next section, an axiomatic treatment of a particular non
-Euclidean geometry, we need to summarize some properties that apply to all
sets of axioms ...
Chapters 2 through 5 extended the study of Euclidean geometry by introducing
many modern ideas. Chapter 6 through 8 incorporated modern transformations
that resulted in very general geometries or a geometry that differed from
Euclidean. Various sets of axioms for Euclidean geometry appeared in Chapter 1
. Before introducing, in the next section, an axiomatic treatment of a particular non
-Euclidean geometry, we need to summarize some properties that apply to all
sets of axioms ...
Page 368
pology, is significant in that it is something other than a generalization of
Euclidean geometry. Non-Euclidean geometry does not include ordinary
geometry as a special case. From the time Euclid stated his postulates, about 300
B.C., controversies arose concerning the fifth postulate. Many substitutes (
including Playfair's axiom) were suggested for it, but, more significantly,
mathematicians felt that it was not a postulate at all. For almost 2000 years, many
people attempted to show that ...
pology, is significant in that it is something other than a generalization of
Euclidean geometry. Non-Euclidean geometry does not include ordinary
geometry as a special case. From the time Euclid stated his postulates, about 300
B.C., controversies arose concerning the fifth postulate. Many substitutes (
including Playfair's axiom) were suggested for it, but, more significantly,
mathematicians felt that it was not a postulate at all. For almost 2000 years, many
people attempted to show that ...
Page 398
sure that non-Euclidean geometry is as consistent as Euclidean geometry or as
the algebra of real numbers. E. Beltrami is given the credit for first proving the
relative consistency of non- Euclidean geometry, in 1 868. The proof of the
relative consistency of a non-Euclidean geometry consists of finding a model
within Euclidean geometry that, with suitable interpretations, has the same
postulational structure as the non-Euclidean geometry. Then any inconsistency in
the non-Euclidean ...
sure that non-Euclidean geometry is as consistent as Euclidean geometry or as
the algebra of real numbers. E. Beltrami is given the credit for first proving the
relative consistency of non- Euclidean geometry, in 1 868. The proof of the
relative consistency of a non-Euclidean geometry consists of finding a model
within Euclidean geometry that, with suitable interpretations, has the same
postulational structure as the non-Euclidean geometry. Then any inconsistency in
the non-Euclidean ...
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Contents
Sets of Axioms and Finite Geometries | 1 |
Geometric Transformations | 37 |
Convexity | 109 |
Copyright | |
11 other sections not shown
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application bisector boundary point center of inversion Chapter circle of inversion circular region collinear complete quadrangle concept congruent conic contains convex body convex hull convex set corresponding points definition determined distance distinct points elliptic geometry endpoints equations Euclidean geometry exactly example Exercises fifth postulate Find the image finite geometry fractal given line given point glide reflection group of transformations harmonic set homogeneous coordinates hyperbolic geometry ideal point interior point invariant inverse points isogonal conjugates length mapping mathematics measure midpoint motions nine-point circle non-Euclidean geometry omega triangle perpendicular perspective plane dual point of intersection problem Problem-Solving Idea projective geometry proof of Theorem properties Prove radius ratio real numbers Reuleaux triangle rotation Saccheri quadrilateral Section segment set of axioms set of points shown in Figure shows similar simple closed curve statement straight line straightedge supporting line symmedian tangent three points three-space topological translation vector vertex vertices