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.”