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Cayley's cubic surface is the unique cubic surface having four ordinary double points (Hunt), the maximum possible for cubic surface (Endraß). The Cayley cubic is invariant under the tetrahedral group and contains exactly nine lines, six of which connect the four nodes pairwise and the other three of which are coplanar (Endraß).
If the ordinary double points in projective three-space are taken as (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1), then the equation of the surface in projective coordinates is
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(1)
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(Hunt). Defining "affine" coordinates with plane at infinity 
and
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(2)
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(3)
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(4)
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then gives the equation
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(5)
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plotted in the left figure above (Hunt). The slightly different form
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(6)
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is given by Endraß (2003) which, when rewritten in tetrahedral coordinates, becomes
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(7)
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plotted in the right figure above.
The Hessian of the Cayley cubic is given by
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(8)
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in homogeneous coordinates 
, 
,
, and 
. Taking the plane at infinity as 
and setting 
, 
,
and 
as above gives the equation
![25[x^3(y+z)+y^3(x+z)+z^3(x+y)]+50(x^2y^2+x^2z^2+y^2z^2)
-125(x^2yz+y^2xz+z^2xy)+60xyz-4(xy+xz+yz)=0,](/images/equations/CayleyCubic/NumberedEquation6.svg) |
(9)
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plotted above (Hunt). The Hessian of the Cayley cubic has 14 ordinary double points, four more than the general Hessian of a smooth cubic surface (Hunt).
