Chapter 3
... asymptotes are the lines z = ± x . Now rotate the hyperbola and its asymptotes about the zaxis. The asymptotes generate the cone L, and the hyperbola generates a two-sheeted hyperboloid lying inside L; denote the upper hyperboloid by B. Then every line through the origin in 3-space intersects B exac ...
... asymptotes are the lines z = ± x . Now rotate the hyperbola and its asymptotes about the zaxis. The asymptotes generate the cone L, and the hyperbola generates a two-sheeted hyperboloid lying inside L; denote the upper hyperboloid by B. Then every line through the origin in 3-space intersects B exac ...
3379 NonE hw
... If you send a pdf file, please send it as ONE file not as individual pages. My mailbox: 651 PGH for turn in by hand If you turn it in by hand, please have it date and time stamped. ...
... If you send a pdf file, please send it as ONE file not as individual pages. My mailbox: 651 PGH for turn in by hand If you turn it in by hand, please have it date and time stamped. ...
Printout
... Exercise 2.1. Identify the error or errors in the proof that all triangles are isosceles. Exercise 2.2. Identify the error or errors in the proof of the Rusty Compass Theorem. Exercise 2.3. For each model (Euclidean, Taxicab, Max-distance, Missing-Strip, and Poincaré Halfplane), find the distance be ...
... Exercise 2.1. Identify the error or errors in the proof that all triangles are isosceles. Exercise 2.2. Identify the error or errors in the proof of the Rusty Compass Theorem. Exercise 2.3. For each model (Euclidean, Taxicab, Max-distance, Missing-Strip, and Poincaré Halfplane), find the distance be ...
The discovery of non-Euclidean geometries
... By the middle of the 19th century (in the world of pure mathematics research at least), it was starting to be accepted that Euclid's geometry is not the only one possible Namely, there is another form of plane geometry, now called hyperbolic geometry, in which Euclid's Postulate 5 does not hold an ...
... By the middle of the 19th century (in the world of pure mathematics research at least), it was starting to be accepted that Euclid's geometry is not the only one possible Namely, there is another form of plane geometry, now called hyperbolic geometry, in which Euclid's Postulate 5 does not hold an ...
MATH 498E—Geometry for High School Teachers
... points) and two exams – a take-home midterm exam (100 points) and an in-class, closed-book final exam (150 points). You may work with others on your homework but what you turn in should be in your own words and mathematics (not copied from anyone else or any other source). Homework is due at the beg ...
... points) and two exams – a take-home midterm exam (100 points) and an in-class, closed-book final exam (150 points). You may work with others on your homework but what you turn in should be in your own words and mathematics (not copied from anyone else or any other source). Homework is due at the beg ...
Hypershot: Fun with Hyperbolic Geometry
... 1. A straight line segment can be drawn joining any two points. 2. Any straight line segment can be extended indefinitely in a straight line. 3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center. 4. All right angles are congruent. 5. If tw ...
... 1. A straight line segment can be drawn joining any two points. 2. Any straight line segment can be extended indefinitely in a straight line. 3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center. 4. All right angles are congruent. 5. If tw ...
The Rise of Projective Geometry
... The 5 Postulate Proclus (410-485) wrote a commentary on The Elements where he comments on attempted proofs to deduce the fifth postulate from the other four, in particular he notes that Ptolemy had produced a false 'proof'. Proclus then goes on to give a false proof of his own. However he did give t ...
... The 5 Postulate Proclus (410-485) wrote a commentary on The Elements where he comments on attempted proofs to deduce the fifth postulate from the other four, in particular he notes that Ptolemy had produced a false 'proof'. Proclus then goes on to give a false proof of his own. However he did give t ...
Poincaré Conjecture
... Poincaré and Geometrization Conjectures Application of the Hamilton-Perelman Theory of the Ricci Flow" in the Asian Journal of Mathematics The paper contains 328 pages ...
... Poincaré and Geometrization Conjectures Application of the Hamilton-Perelman Theory of the Ricci Flow" in the Asian Journal of Mathematics The paper contains 328 pages ...
What is the Poincaré Conjecture?
... Poincaré and Geometrization Conjectures Application of the Hamilton-Perelman Theory of the Ricci Flow" in the Asian Journal of Mathematics The paper contains 328 pages ...
... Poincaré and Geometrization Conjectures Application of the Hamilton-Perelman Theory of the Ricci Flow" in the Asian Journal of Mathematics The paper contains 328 pages ...
Henri Poincaré
Jules Henri Poincaré (French: [ʒyl ɑ̃ʁi pwɛ̃kaʁe]; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and a philosopher of science. He is often described as a polymath, and in mathematics as The Last Universalist by Eric Temple Bell, since he excelled in all fields of the discipline as it existed during his lifetime.As a mathematician and physicist, he made many original fundamental contributions to pure and applied mathematics, mathematical physics, and celestial mechanics. He was responsible for formulating the Poincaré conjecture, which was one of the most famous unsolved problems in mathematics until it was solved in 2002–2003. In his research on the three-body problem, Poincaré became the first person to discover a chaotic deterministic system which laid the foundations of modern chaos theory. He is also considered to be one of the founders of the field of topology.Poincaré made clear the importance of paying attention to the invariance of laws of physics under different transformations, and was the first to present the Lorentz transformations in their modern symmetrical form. Poincaré discovered the remaining relativistic velocity transformations and recorded them in a letter to Dutch physicist Hendrik Lorentz (1853–1928) in 1905. Thus he obtained perfect invariance of all of Maxwell's equations, an important step in the formulation of the theory of special relativity.The Poincaré group used in physics and mathematics was named after him.