Calabi-Yau Manifold
Calabi-Yau Manifolds play an important role in string theory, which suggests that there are at least 10 dimensions. Other than the 4 spacetime dimensions which we can perceive, physicists anticipate that the additional 6 dimensions are so compact and small that they cannot be observed with contemporary methods. The extra dimensions of spacetime are sometimes conjectured to take the form of a 6-dimensional Calabi–Yau manifold.
The compactification in Calabi–Yau manifolds is important because it makes the string theory of 10 dimensions consistent with the perceptible 4-dimensional theory.
String Theory
String theory is a model of fundamental physics whose building blocks are one-dimensional extended objects (strings) rather than the zero-dimensional points (particles) that are the basis of the Standard Model of Particle physics. String theorists are attempting to adjust the Standard Model by removing the assumption in quantum mechanics that particles are point-like. By removing this assumption and replacing the point-like particles with strings, it is hoped that string theory will develop into a sensible quantum theory of gravity. Moreover, string theory appears to be able to “unify” the known natural forces (gravitational, electromagnetic, weak and strong) by describing them with the same set of equations.
No experimental verification or falsification of the theory has yet been possible, though the Large Hadron Collider near Geneva, Switzerland may produce relevant data.
Studies of string theory have revealed that it predicts not just strings, but also higher-dimensional objects (branes). String theory strongly suggests the existence of ten or eleven (in M-Theory) spacetime dimensions, as opposed to the relativistic four (three spatial and one time).
source: www.newuniverse.co.uk
In string theory, the extra dimensions of spacetime are curled up into calabi-yau shapes like these.
Rotation around the vertical axis of a tridimensional representation of a quadridimensional Calabi-Yau manifold
(c) 2001-2009 Jean-Francois Colonna and France Telecom R&D and CMAP (Centre de Mathematiques APpliquees) / Ecole Polytechnique
If it is seen in a three dimensional space, the Calabi-Yau manifold will look like this. Image source: Simple English Wikipedia
A two-dimensional hypersurface of the quintic Calabi-Yau three-fold. Image source: Wikipedia
Wow this forms look awesome wich software was used to generate the surfaces, Wolfram Mathematica ?
I would like to see the equations used in the exercises I bet they are really complex ?
Hi Rodrigo, thanks for your comment.
The pictures are wikipedia commons. I guess the blue one was made with Mathematica, and for the top image, there´s more info on this site: http://www.cs.indiana.edu/~hanson/