A Cross-Section of the Calabi-Yau Quintic

According to string theory, space-time isn't 4-dimensional as you might expect, but 10-dimensional.  Where are the extra six dimensions?  One answer is that they're "compactified": roughly speaking, rolled up into such a small space as to be unobservable at human scales.

Calabi-Yau spaces may be that microscopic shape, deep inside the hidden dimensions of string theory.  There are many such spaces, but since they are six-dimensional, they're not easy to draw!  This model shows a cross-section through a likely space. The cross-section is a surface that can be projected into 3-dimensional space, and that makes it possible to draw it.

Physically it's a 3 1/8" cube, and the surface within is a wildly self-intersecting ride through space:

Warning: Math Ahead

This surface was chosen and projected into 3-space by Andrew Hanson at Indiana University.  (He does fantastic work in visualization – scroll down his page for many interesting images.)  Dr. Hanson explains how this model was chosen from among the many spaces in the family:

“This particular space is one of the most appealing candidates, because there's a series of Calabi-Yau spaces embedded in CPN (N-dimensional complex projective space) described by homogeneous polynomials of degree (N+1).  These spaces have real dimension 2(N-1), so the hypothesis that there are six hidden dimensions in string theory means that there is a unique choice within this series of Calabi-Yau spaces, namely N=4, and the polynomial must be this quintic (degree N+1=5):
z1⁵ + z2⁵ + z3⁵ + z4⁵ + z5⁵ = 0.
The 2D surface is computed by dividing by z5 and setting z3/z5 and z4/z5 to be constant.  This defines a 2-manifold slice of the 6-manifold; we then normalize the resulting inhomogeneous equations to simplify them, yielding the complex equation that is actually solved for the surface,
z1⁵ + z2⁵ = 1.
The resulting surface is embedded in 4D and projected to ordinary 3D space for display.”