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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.