The Kummer Quartic is one of the most classical mathematical shapes; it was studied by E.E. Kummer around 1870. This smoothed version is highly symmetric; actually, it has the symmetry of one of the Platonic Solids, namely the regular tetrahedron. In modern mathematical classification terms, this surface is called a smooth K3 surface of degree 4.
More info on
MO-Labs.com.