Mathematical Experiment 10

Construct a knot like this:

Construct a knot based on the parametric representation

[cos 3t, sin 4t, sin 5t]

Construct a knot based on the parametric representation

[cos 13t, sin 4t, sin 17t]

Construct a chain of five rings each encloses the other four.

Construct a chain of five rings of the same size each encloses exactly two others.

Construct the image of the Cartesian grid lines under the stereographic projection

> with(plots):with(plottools):
> a:=coordplot(cartesian,[-10..10,-10..10],scaling=constrained):
> f:=transform((x,y) -> [2*x/(x^2+y^2+1),2*y/(x^2+y^2+1),(x^2+y^2-1)/(x^2+y^2+1)]):
> b:=display(f(a),scaling=constrained):
> c:=sphere([0,0,0],0.99,style=patchnogrid,color=yellow):
> h:=display(b,c):
　

Construct the cyclide by inverting a torus.

Construct any four mutually tangent spheres of different sizes.

Apply the command "tubeplot" to construct the seashell.

Construct the astroidal ellipsoid given by

x = (cos u cos v)³

y = (sin u cos v)³

z = sin³v

Reference: A. Gray, Modern Differential Geometry of Curves and Surfaces, p. 304.

Construct the cupola with vertices at

where