June 7th, 2011 | 6 Comments

A Klein bottle is a non-orientable surface, which has no defined left and right, as stated on Wikipedia. There we can also find a Gnuplot plot of the bottle, which we want to fine-tune a little bit in order to reach the result in Fig. 1.

In order to reach Fig. 1 we start with the definition of the parametric functions given in Wikipedia and do a simple `pm3d`

plot with them.

set parametric x(u,v)= v<2*pi ? (2.5-1.5*cos(v))*cos(u) : \ v<3*pi ? -2+(2+cos(u))*cos(v) : \ -2+2*cos(v)-cos(u) y(u,v)= v<2*pi ? (2.5-1.5*cos(v))*sin(u) : \ sin(u) z(u,v)= v<pi ? -2.5*sin(v) : \ v<2*pi ? 3*v-3*pi : \ v<3*pi ? (2+cos(u))*sin(v)+3*pi : \ -3*v+12*pi splot x(u,v),y(u,v),-z(u,v) w pm3d

The result is shown in Fig. 2.

Now we add some lines to the surface and hide parts, which are not visible in 3d.

set pm3d depthorder hidden3d 1 set hidden3d

Here the `depthorder`

option takes care of the right positioning of the bottleneck going back into the bottle, which is not correct in Fig. 2. The `hidden3d 1`

option draws lines in the right order for a correctly looking 3d plot using line style 1. The additional `set hidden3d`

command takes care of showing only those lines that are visible in 3d. These options will result in Fig. 3.

In order to reach at Fig. 1 we just have to set the surface to be transparent, which can be done by the `set style fill`

command.

set style fill transparent solid 0.65

[…] following color settings are from: http://www.gnuplotting.org/klein-bottle/ set pm3d depthorder hidden3d 1 set hidden3d set style fill transparent solid 0.65 set palette rgb […]

[…] Image by Gnuplotting under CC-BY-NC […]

[…] http://www.gnuplotting.org/klein-bottle/ […]

[…] Klein bottle […]

[…] before, so I don't know if it can help. The only places I found online that seemed related were Klein bottle pgfplots in combination with gnuplot requires additional semicolon Edit: Thanks to hpesoj626's […]

I notice in fig. 3 that the shading for surfaces is piecewise constant. Is it possible to have the shading be continuously varying in height? (First time poster but I’ve admired your posts for many many years. Thank you!)