### The famed Klein Bottle

This is an unfamiliar view of the Klein bottle lifted from Ian Stewart. For a more familiar
look, use the parametric version below
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)+

16*x*z*(x^2+y^2+z^2-2*y-1)=0

Parametric form:

x = cos(u)*(cos(u/2)*(sqrt_2+cos(v))+(sin(u/2)*sin(v)*cos(v)))

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)

Figure 8 parametric form:
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

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