A question struggled me for a long time: If a $3D$ Klein Bottle represented by Robert Israel function. How to determine the exact boundary. I mean if you put it in a cuboid, what will the exact values of width, length, and height of the cuboid be?
$$\begin{align} x(u,v) =& -\frac{2}{15}\cos u(3\cos v - 30\sin u + 90 \cos^4 u\sin u\\ & - 60\cos^6 u \sin u + 5\cos u \cos v\sin u)\\ y(u,v) =& -\frac{1}{15}\sin u(3\cos v - 3\cos^2 u\cos v - 48\cos^4 u\cos v + 48\cos^6 u \cos v\\ &-60\sin u + 5\cos u\cos v\sin u - 5\cos^3 u\cos v\sin u \\ &-80\cos^5 u\cos v\sin u + 80\cos^7u \cos v \sin u )\\ z(u,v) =& \frac{2}{15}(3 + 5\cos u\sin u)\sin v \end{align} $$ for $0 \le u < \pi$ and $0 \le v < 2\pi$.
original image: https://i.stack.imgur.com/BM00U.png
