Think of it as a rectangle where one pair of opposite sides are joined directly, and the other pair are joined with a half twist. One sided surface first described by Felix Klein.
Polynomial form; Looks odd, from Ian Stewart.
(x^2+y^2+z^2+2*y-1)*((x^2+y^2+z^2-2*y-1)^2-8*z^2)+Parametric form:
16*x*z*(x^2+y^2+z^2-2*y-1)=0
x = cos(u)*(cos(u/2)*(sqrt_2+cos(v))+(sin(u/2)*sin(v)*cos(v)))Figure 8 parametric form:
y = sin(u)*(cos(u/2)*(sqrt_2+cos(v))+(sin(u/2)*sin(v)*cos(v)))
z = -1*sin(u/2)*(sqrt_2+cos(v))+cos(u/2)*sin(v)*cos(v)
x = (a+cos(uu/2)*sin(vv)-sin(uu/2)*sin(2*vv))*cos(uu) y = (a+cos(uu/2)*sin(vv)-sin(uu/2)*sin(2*vv))*sin(uu) z = sin(uu/2)*sin(vv)+cos(uu/2)*sin(2*vv)
Here is the PoV3-source I used
And here is the cat'ed figure 8 PoV3-source