Chapter 15: Rubber Sheet Geometry
“The hidden ingredient that makes all of Euclid’s geometry work is length, a metric quantity, one which is unchanged by rigid motions and defines Euclid’s equivalent concept to motion, congruence.” (page 265)
Topology: rubber-sheet geometry; is the geometry of shapes that can be deformed or distorted in extremely convoluted ways. All that matters here is continuity. Holes. Knots. It studies the shapes of things in their own right, not as part of something else.
Important Figures: Euler (Polyhedra, Königsberg Bridges), Descartes, Leibniz, Gauss (linking number, winding number, Fundamental Theorem of Algebra), Johann Listing, Augustus Möbius (The Möbius Band), Riemann (Riemann sphere, singularities), Grigori Perelman (2002, manifold, solved the Poincaré Conjecture).
“In retrospect, the main difficulties in developing topology were internal ones, best solved by abstract means; connections with the real world had to wait until the techniques were sorted out properly.” (page 286)
Topology: rubber-sheet geometry; is the geometry of shapes that can be deformed or distorted in extremely convoluted ways. All that matters here is continuity. Holes. Knots. It studies the shapes of things in their own right, not as part of something else.
Important Figures: Euler (Polyhedra, Königsberg Bridges), Descartes, Leibniz, Gauss (linking number, winding number, Fundamental Theorem of Algebra), Johann Listing, Augustus Möbius (The Möbius Band), Riemann (Riemann sphere, singularities), Grigori Perelman (2002, manifold, solved the Poincaré Conjecture).
“In retrospect, the main difficulties in developing topology were internal ones, best solved by abstract means; connections with the real world had to wait until the techniques were sorted out properly.” (page 286)
Chapter 16: The Fourth Dimension: Geometry out of this world
“In his science fiction novel The Time Machine... “There are really four dimensions, three which we call the three planes of Space, and a fourth, Time””. (page 288)
Important Figures: William Rowan Hamilton (1837, quaternions), Hermann Günther Grassmann (1862, hypernumbers), Josiah Williard and Oliver Heaviside (represent vectors algrebraically), Edwin Wilson, James Joseph Sylvester (linear algebra), Einstein, Hermann Minkowski (space-time, events), Lagrange, Edwin Abbott (Flatland).
“The advantage of high-dimensional geometry is that it brings human visual abilities to bear on problems that are not initially visual at all. Because our brains are adept at visual thinking, this formulation can often lead to unexpected insights, not easily obtainable by other methods. Mathematical concepts that have no direct connection with the real world often have deeper, indirect connections. It is those hidden links that make mathematics so useful.” (page 307)
Important Figures: William Rowan Hamilton (1837, quaternions), Hermann Günther Grassmann (1862, hypernumbers), Josiah Williard and Oliver Heaviside (represent vectors algrebraically), Edwin Wilson, James Joseph Sylvester (linear algebra), Einstein, Hermann Minkowski (space-time, events), Lagrange, Edwin Abbott (Flatland).
“The advantage of high-dimensional geometry is that it brings human visual abilities to bear on problems that are not initially visual at all. Because our brains are adept at visual thinking, this formulation can often lead to unexpected insights, not easily obtainable by other methods. Mathematical concepts that have no direct connection with the real world often have deeper, indirect connections. It is those hidden links that make mathematics so useful.” (page 307)