A Togliatti surface is a quintic surface having 31 ordinary double points. Togliatti (1940, 1949) showed that such surfaces exist, although he did not explicitly derive equations for them. Beauville (1978) subsequently proved that 31 ordinary double points are the maximum possible. van Straten (1993) subsequently constructed a three-dimensional family of solutions and in 1994, Barth derived the example
![64(x-w)[x^4-4x^3w-10x^2y^2-4x^2w^2+16xw^3-20xy^2w+5y^4+16w^4-20y^2w^2]
-5sqrt(5-sqrt(5))(2z-sqrt(5-sqrt(5))w)[4(x^2+y^2-z^2)+(1+3sqrt(5))w^2]^2=0,](/images/equations/TogliattiSurface/NumberedEquation1.svg) |
where 
is a parameter (Endraß 2003), illustrated above for 
.
This surface is invariant under the dihedral group 
of group
order 10 and contains exactly 15 lines. Five of these are the intersection of
the surface with a 
-invariant
cone containing 16 nodes, five are the intersection of the surface with a 
-invariant plane containing 10 nodes, and the last five are
the intersection of the surface with a second 
-invariant plane containing no nodes (Endraß 2003).
ordine con soli punti doppi isolati." Vierteljschr.
Naturforsch. Ges. Zürich 85, 127-132, 1940.van Straten,
D. "A Quintic Hypersurface in 
with 130 Nodes." Topology 32, 857-864,
1993.