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Endraß surfaces are a pair of octic surfaces which have 168 ordinary double points. This
is the maximum number known to exist for an octic surface,
although the rigorous upper bound is 174. The equations of the surfaces 
are
![64(x^2-w^2)(y^2-w^2)[(x+y)^2-2w^2]
[(x-y)^2-2w^2]-{-4(1+/-sqrt(2))(x^2+y^2)^2+[8(2+/-sqrt(2))z^2+2(2+/-7sqrt(2))w^2](x^2+y^2)-16z^4+8(1∓2sqrt(2))z^2w^2-(1+12sqrt(2))w^4}^2=0,](/images/equations/EndrassOctic/NumberedEquation1.svg) |
where 
is a parameter. All ordinary double points
of 
are real, while 24 of those in 
are complex. The surfaces were discovered
in a five-dimensional family of octics with 112 nodes, and are invariant under the
group 
.
The surfaces illustrated above take 
. The first of these has 144 real ordinary
double points, and the second of which has 144 complex ordinary
double points, 128 of which are real.
