Boy's Surface by tleathrum

Last crawled date: 3 years ago

A 3-D rendering of Boy's surface. The design starts in Sage, export to X3D (resulting file included), then import X3D into Blender. In Blender, edit to remove artifacts Sage leaves behind (like a cube of lines around the figure, which can also be removed in Sage using option "frame=False" as seen in Sage code below), then "thicken" the surface so that Cura can recognize it as a solid object:
select Properties -> Modifiers (wrench icon) -> Add Modifier -> Solidify
set Thickness to 0.1 and click Apply
Now export to STL, then import STL into Cura. In Cura, scale (because graph units in Sage become millimeters in Cura) and situate figure on print platform, then export STL again to form the STL thing file here.
Here is the Sage code to generate the figure:
p,t=var('p,t')
u=cos(t)*sin(p)
v=sin(t)*sin(p)
w=cos(p)
x=(1/2)*((3*u^2-1)+2*v*w*(v^2-w^2)+u*w*(u^2-w^2)+u*v*(v^2-u^2))
y=(sqrt(3)/2)*((v^2-w^2)+u*w*(w^2-u^2)+u*v*(v^2-u^2))
z=(u+v+w)*((u+v+w)^3-4*(u-v)*(v-w)*(w-u))
parametric_plot3d([x,y,z], (t,0,2*pi), (p,0,pi), frame=False)
Since Sage is Python-based, be sure to check line breaks.
The parameterization in the Sage code is based on the variables t and p representing spherical coordinate angles, giving u, v, and w as coordinates of points on the unit sphere in space, then x, y, and z on the actual surface. The surface is an immersion of the projective plane into 3-space. The immersion is self-intersecting but does not have any singularities ("pinch points").
This surface presents a minor issue for slicer software because of its topology. Even setting infill in the slicer, this figure remains hollow -- topologically, it doesn't have an "inside," similar to a Klein bottle. This is why it is necessary to "solidify" the surface in Blender, even though by all appearances the surface seems closed.
There is a statue of Boy's surface at the Mathematics Research Institute at Oberwolfach:http://www.mfo.de/general/boy/boy_small.jpg
The statue was a gift to the Institute from the Mercedes-Benz corporation, designed on a computer and manufactured using equipment at Mercedes-Benz.