Math Art
Klein Bottle

Klein Bottle

Klein Bottle is a topological surface, given by a parametrisation, immersed, with a point group.

Open Klein Bottle in the interactive viewer →

aperiodic closed closed-nonorientable def-parametric immersed implemented non-orientable topological tradition-classical tradition-physical-model

About

A bottle whose inside is its outside: a surface with no boundary and no two sides at all. It cannot be built in three dimensions without passing through itself, and the familiar glass models cheat exactly there. In four dimensions the self-intersection can be undone.

Formula

x(u, v)
2⁢sin⁡(u)2⁢cos⁡(u)+(12−130⁢(2⁢u−π)⁢2⁢u⁢(2⁢π−2⁢u))⁢cos⁡(v)⁢5⁢cos⁡(u)25⁢cos⁡(u)2+4⁢sin⁡(u)2⁢(3⁢cos⁡(u)2−1)22 \sin\left(u\right)^{2} \cos\left(u\right) + \frac{\left(\frac{1}{2} - \frac{1}{30} \left(2 u - \pi\right) \sqrt{2 u \left(2 \pi - 2 u\right)}\right) \cos\left(v\right) \cdot 5 \cos\left(u\right)}{\sqrt{25 \cos\left(u\right)^{2} + 4 \sin\left(u\right)^{2} \left(3 \cos\left(u\right)^{2} - 1\right)^{2}}}
y(u, v)
(12−130⁢(2⁢u−π)⁢2⁢u⁢(2⁢π−2⁢u))⁢sin⁡(v)\left(\frac{1}{2} - \frac{1}{30} \left(2 u - \pi\right) \sqrt{2 u \left(2 \pi - 2 u\right)}\right) \sin\left(v\right)
z(u, v)
5⁢sin⁡(u)−(12−130⁢(2⁢u−π)⁢2⁢u⁢(2⁢π−2⁢u))⁢cos⁡(v)⁢2⁢sin⁡(u)⁢(3⁢cos⁡(u)2−1)25⁢cos⁡(u)2+4⁢sin⁡(u)2⁢(3⁢cos⁡(u)2−1)25 \sin\left(u\right) - \frac{\left(\frac{1}{2} - \frac{1}{30} \left(2 u - \pi\right) \sqrt{2 u \left(2 \pi - 2 u\right)}\right) \cos\left(v\right) \cdot 2 \sin\left(u\right) \left(3 \cos\left(u\right)^{2} - 1\right)}{\sqrt{25 \cos\left(u\right)^{2} + 4 \sin\left(u\right)^{2} \left(3 \cos\left(u\right)^{2} - 1\right)^{2}}}

Properties

Family
topological
Given by
a parametrisation
Curvature
no condition imposed
Periodicity
not periodic
Ends
0
Embedding
immersed
Orientable
no
Symmetry kind
point group
Fidelity
exact
Exactness
elementary

Definition

Franzoni's classical bottle shape, the operator's default rendition: a tube of radius r(u) about the stretched dumbbell directrix, upright with the long axis on Oz. The u = pi tube end meets the u = 0 end in the same circle under v -> pi - v, and the mesh is glued closed there. The paper's shape numbers (a, b, c, d) = (20, 8, 11/2, 2/5) are baked in at their dumbbell scaling (5, 2, 1/2, 1/30); the older polynomial immersion stays reachable as the operator's POLYNOMIAL rendition

Sources

Chart reproduced numerically against the shipped implementation: fwd max 0.48 of tol, 1.0% of oracle beyond tol (tol floor 2.5 x 0.04376).