English: This image shows a local 2D cross-section of the real 6D manifold
known in string theory as the Calabi-Yau quintic. This is an
Einstein manifold and a popular candidate for the wrapped-up 6 hidden
dimensions of 10-dimensional string theory at the scale of the Planck
length. The 5 rings that form the outer boundaries shrink to points
at infinity, so that a proper global embedding would be seen to have
genus 6 (6 handles on a sphere, Euler characteristic -10).
The underlying real 6D manifold (3D complex) has Euler characteristic
-200, is embedded in CP4, and is described by this homogeneous
equation in five complex variables:
-
z05 + z15 + z25 + z35 + z45 = 0
The displayed surface is computed by assuming that some pair of complex inhomogenous
variables, say z3/z0 and z4/z0, are constant (thus defining a 2-manifold slice of the 6-manifold), renormalizing the resulting inhomogeneous equations, and plotting the local Euclidean space solutions to the inhomogenous complex equation
-
z15 + z25 = 1
This surface can be described as a family of 5x5 phase transformations
on a fundamental domain, 1/25th of the surface, shown (slightly
hidden) in blue. Each of the first set of phases mixes in a brighter
red color to its patch, and the second set mixes in green. Thus the
color alone shows the geometric parentage of each of the 25 patches.
The resulting surface, which is embedded in 4D, is projected to 3D
according to one's taste to produce the final rendering. Further
details are given in
Andrew J. Hanson, "A construction for computer
visualization of certain complex curves," Notices of the
Amer. Math. Soc. 41 (9): 1156-1163, (November/December 1994).