All Fields are Required. Web Account created successfully. DOI: Hide All Book Information -. Information Books By Independent Authors. Project Euclid distributor. This will count as one of your downloads. You will have access to both the presentation and article if available. This content is available for download via your institution's subscription. To access this item, please sign in to your personal account. Create an account. Front Pages. David W. Chapter 0. Historical Strands of Geometry.
Chapter 1. What is Straight? Chapter 2. Straightness on Sphere. Chapter 3. What is an Angle? Chapter 4. Straightness on Cylinders and Cones.
Chapter 5. Straightness on Hyperbolic Planes. Chapter 6. Triangles and Congruencies. Chapter 7. Area and Holonomy. Chapter 8. Parallel Transport. Chapter 9. More Triangle Congruencies. Chapter Parallel Postulates. Isometries and Patterns. Dissection Theory. Square Roots, Pythagoras, and Similar Triangles. Projections of a Sphere onto a Plane. Inversions in Circles. Projections Models of Hyperbolic Planes. Geometric 2-Manifolds. Here is another applet , for covering the circle in the Poincare model.
Other good online resources on non-Euclidean geometry can be found on the Visual Mathematics website; the Geometry Junkyard , with its Tilings of Hyperbolic Spaces page; as well as on the pages maintained by independent groups of enthusiasts, like this , or this. The Wikipedia page on non-Euclidean geometry also has merits.
Much weaker in terms of theory but good for some bibliographical references is the entry on non-Euclidean geometry in Wolfram MathWorld. Several websites offer excellent dynamic software. Another one is Geometry and motion , maintained by Daniel Scher. Drexel University maintains a remarkable Math Forum online , with a good page on hyperbolic geometry. Another one is maintained by a group of three mathematicians at various universities.
For millennia, the idea that no-Euclidean geometries might exist was anathema among mathematicians. Karl Friedrich Gauss, arguably the best mathematician ever, delayed publishing his research on non-Euclidean geometry, fearing that he could compromise his reputation. Throughout the last two centuries several intuitive models of non-Euclidean geometries were proposed. In most of them the definitions of basic geometrical notions challenge our commonly held spatial intuitions.
They are, nonetheless, self-consistent within the model to which they belong. One of the many ways of comparing these geometries and the planar Euclidean geometry is to look at the sum of the interior angles of a triangle in each of them. In the spherical geometry the interior angles always add up to more than two right angles degrees , in the planar geometry they add up to exactly two right angles, while in the hyperbolic geometry they add up to less than two right angles.
Here is an example of a triangle on a sphere, with three right angles adding up, therefore, to degrees : [3]. It can be shown that in each type of non-Euclidean geometry the sum of the interior angles of a triangle is directly related to the area of the triangle. Area of a circular surface grows differently in each type of geometry. In Euclidean planar geometry it grows proportional with the square of the radius of the circle.
In hyperbolic geometry it grows exponentially with the growth of the radius. In spherical geometry the area grows with the radius but it cannot exceed the area of the whole spherical surface.
The three geometries also differ is the system of coordinates best implemented in each. This issue is of great importance for the computational treatment of each type of geometry. We are widely acquainted with the rectangular system of coordinates for the Euclidean plane. Yet that one is not without ambiguity, as shown in a public radio interview on the subject of Hurricane Katrina. I insert here an instructional module focused on the two most commonly used coordinate systems, planar and spherical:.
On a hyperbolic plane the most convenient system of coordinates is also rectangular, as shown in the following picture: [7]. Non-Euclidean geometry can also be introduced and studied in a highly technical manner.
For the reader interested in such an approach we offer a brief bibliography. A few I would recommend are the following in alphabetical order of the authors : Bolyai, Janos.
Non-Euclidean Geometry and the Nature of Space. A detailed historical account introduces the reader to the battle of ideas around non-Euclidean geometries. Henderson, David W. Third edition. This is a textbook used in several undergraduate courses in the U. It provides an inviting, detailed, hands-on, inquiry-based approach to learning non-Euclidean geometry. Especially instructive is the comparative view, property by property. Also good some groundbreaking are the illustrations.
Krause, Eugene F. Taxicab geometry: An adventure in non-Euclidean geometry. New York, NY: Dover, This slim booklet is highly entertaining. It contains many exercises in accessible format. Prekopa, Andras, and Emil Molnar Eds. Non-Euclidean geometries. New York, NY: Springer, This is a collective volume published in the memory of Janos Bolyai. It contains contributions of great variety, both in approach and difficulty.
Trudeau, Richerd J. The non-Euclidean revolution. Boston, MA: Birkhauser, Perhaps the most quoted book on non-Euclidean geometry. The approach is more axiomatic than in other books. Weeks, Jeffrey R. The shape of space. Second edition. This is a highly readable introduction to non-Euclidean geometries. The fourth dimension and non-Euclidean geometry in modern art. Most of the book is concerned with multi-dimensional geometry. Plane and spherical trigonometry.
Kulczycki, Stefan. Non-Euclidean geometry. Oxford, UK: Pergamon Press, Manning, Henry Parker. Meschkowski, Herbert. Noneuclidean geometry. Smogorzhevsky, A. Lobachevskian geometry.
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