Donal O’Shea’s Poincare Conjecture: In Search for the Shape of the Universe is an excellent rendition on the classical theories of shape for the planet earth and the universe itself utilizing Pythagoras, Euclidean Geometry, Einstein and modern mathematicians. Fractal geometry is another possible application because the earth’s spherical presentation is non-linear in many places.
A basic assumption is that we do not know the shape of earth for sure. In fact, the shape of the planet is not constant. Instead, there are finite and not so finite changes in topography due to volcano activity, earthquakes, and other significant disturbances that literally change the face of the planet on a continuing basis.
Continuous space has infinite dimensions. The essence of the Poincare Conjecture is that there is no boundary for earth in the classic sense of a beginning and an end. Every loop on a sphere shrinks to a point. In addition, there are no two parallel lines on a sphere because any two lines intersect at some point.
The Poincare Conjecture also asserts that three compact manifolds on which a closed path shrinks to a point is the exact topology as its 3-sphere. An equivalent form of the conjecture involves a homotopy equivalence.
In mathematical topology, two continuous functions from one topological space to another are called homotopic when one can be “continuously deformed” into the other. If a 3-manifold is homotopy equivalent to the 3-sphere, then it is homeomorphic to it. A homeomorphism or topological isomorphism or bi-continuous function is a continuous function between topological spaces that have a continuous inverse function.
Grigori Perelman proved the full geometry conjecture in 2003 by employing the Ricci flow. In differential geometry, the Ricci flow is an intrinsic geometric flow. It is a process that deforms the metric of a Riemannian manifold by smoothing out irregularities . A Riemannian metric opens the possibility to define various geometric notions on a Riemannian manifold such as areas, angles, and curve lengths.
Poincare Conjecture: In Search for the Shape of the Universe is a perfect acquisition for physicists, mathematicians, logicians, and an audience of professionals in the allied areas of mathematics and computer science. O’Shea’s presentation is strong in many spots and difficult to understand in others.