Mathematical Experiment 2

Construct a torus. > a:=2;c:=5;r:=c+a*cos(t);
> x:=r*cos(s);
> y:=r*sin(s);
> z:=a*sin(t);
> plot3d([x,y,z], s=0..2*Pi,t=0..2*Pi, scaling=constrained);

Construct one-half of a torus as thus:

Construct one-half of a torus as thus:

Construct one-half of a torus as thus:

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Design a pattern on the torus appearing as thus:

Construct one-half of a torus consisting of two intersecting circles: These circles are known as the Villarceaux circles.

> a:=2;b:=5;c:=sqrt(b^2-a^2);
> r:=b+a*cos(t);
> x:=r*cos(s);
> y:=r*sin(s);
> z:=a*sin(t);
> w:=solve(c*z=a*x,s);
> h:=plot3d([x,y,z],s=-w..w,t=0..2*Pi,scaling=constrained,lightmodel=light3,grid=[100,100]):
> with(plottools):
> vrml(h,"z:/usr1/data/disk3/exp02/2/4.wrl",background_color=white);
Construct one-half of a torus consisting of two linking circles by rotating the circle

[b cos(t), a + c sin(t), a cos(t)]
(a2 + b2 = c2)

about the z-axis by p.

Construct this twisted prismatic ring to be found on the stairs between two floors of the Math Library: