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):

z1^{5} + z2^{5} + z3^{5} + z4^{5} + z5^{5} = 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,

z1^{5} + z2^{5} = 1.

The resulting surface is embedded in 4D and projected to ordinary
3D space for display.”