Thanks Dave!
Here is the Magma code I used for producing the cubic surface in Nanoblocks:
R<u,v,w>:=PolynomialRing(RealField(8),3);
U:=2^(1/2)*u;
V:=6^(1/2)*v;
W:=(4*3^(1/2))*w/5;
x:=(3*U+V+2*W)/6;
y:=(-3*U+V+2*W)/6;
z:=(W-V)/3;
f:=R!(81*(x^3+y^3+z^3)-189*(x^2*y+x*y^2+x^2*z
+x*z^2+y^2*z+y*z^2)+54*x*y*z+126*(x*y+x*z+y*z)
-9*(x^2+y^2+z^2)-9*(x+y+z)+1);
g:=R!(3/5-u^2-v^2);
[[[(1+Sign(Evaluate(f,<i/20+1/40,j/20+1/40,k/20+1/40>)))
*(1+Sign(Evaluate(g,<i/20+1/40,j/20+1/40,k/20+1/40>)))/4
: i in [-16..15]] : j in [-16..15]] : k in [-13..26]];
The last line turns the equations into a set of matrices of zeros and
ones describing the layers, much like 3D printing. In each layer, the
ones are where both f and g are positive; the role of g is only to
intersect with a cylinder. The line defining f gives the equation of the
cubic in (x,y,z) coordinates. But this equation has a symmetry of order
three permuting x, y and z, so the cubic would point diagonally away
from the origin. I really wanted it to point upwards, so u, v, and w are
rotated from x, y, z by a transformation given in the first few lines.
The final piece of information you need is that nanoblocks are only 4/5
as high as they are long and wide, which is why the definition of W in
terms of w has a spurious looking factor of 4/5 in it. Hope that helps.
The particular equation of the cubic that I chose is the "Clebsch
diagonal cubic." In general, a nonsingular cubic surface has 27 lines on
it, over the complex numbers. This is one where all 27 of the lines are
real.
