Mathematical Experiment 7


Construct the torus using the command "tubeplot".


Construct two tori as thus:


Construct the torus using the command "plot3d".


Construct the torus by pasting two patches as thus:


Construct one half of a torus so its cross-section consists of two concentric circles.


Construct one half of a torus so its cross-section consists of two disjoint circles.


Construct one half of a torus so its cross-section consists of two intersecting circles.
These circles are called Villarceaux circles. Reference: Z.A. Melzak, Invitation to Geometry, pp. 63-68.

Construct one half of a torus so its cross-section consists of two interlocking circles in space.
> restart;
> a:=3:b:=4:c:=sqrt(a^2+b^2);
> x0:=c*cos(t);
> y0:=a+c*sin(t);
> z0:=0;
> s:=arccos(b/c);
> x1:=x0*cos(s)-z0*sin(s);
> y1:=y0;
> z1:=x0*sin(s)+z0*cos(s);
> x:=x1*cos(u)-y1*sin(u);
> y:=x1*sin(u)+y1*cos(u);
> z:=z1;
> plot3d([x,y,z],t=0..2*Pi,u=0..Pi,scaling=constrained);


Construct a knot as thus:


Construct knots as thus:


Construct a twisted torus whose sections are formed by regular triangles:


Construct the VRML files associated with all models in this exercise.