String theory is a prime candidate for unifying quantum mechanics with Einstein's general theory of relativity (a "Theory of Everything"). In string theory, the fundamental physical objects are vibrating strings that sweep out world sheets in spacetime, with elementary particles represented not as points but as different modes of vibration of these strings on the scale of the Planck length, of the order of m.
In one version of a consistent string theory, the strings must live in a 10-dimensional spacetime. Since human physical experience appears to be that of four-dimensional spacetime (three space dimensions plus time), it is presumed that if 10-dimensional string theory is correct, there must be six additional dimensions that are curled up into complicated undetectably small shapes at the Planck scale known as Calabi-Yau spaces. Every point in spacetime would therefore possess six additional dimensions whose topology is described by a Calabi-Yau space, of which there are a large number of possibilities.
This Demonstration depicts what many believe is the simplest and most elegant of these possibilities, a quintic (fifth degree) polynomial in four-dimensional complex projective space. When two of the four complex variables are fixed, the surface that remains can be displayed using ordinary graphics projected from 4D into 3D; in this Demonstration you can interactively alter the 4D viewpoint to see a variety of different aspects of this fundamental shape. The fact that the surface is derived from a quintic polynomial can be perceived directly by noting the five-fold pie-slice arrangements that occur in the color-coded variant of the image of this Calabi-Yau shape.
For details on the surface, the color coding, and the mathematics involved, see A. J. Hanson's page.
A. J. Hanson, "A Construction for Computer Visualization of Certain Complex Curves," in "Computers and Mathematics" column, K. Devlin, ed., Notices of the American Mathematical Society, 41(9), 1994 pp. 1156–1163.