Mathematical Experiment 10

Construct the portion of the cylinder x2 + y2 = 1 lying between the planes z = 0 and z = 1 + x.

Describe how two pipes of the same size are joined perpendicularly.

Construct two pipes joined as thus:

Describe how three pipes are symmetrically joined.

Wrap the graph of y = cos 2x around a cylinder.

Wrap the graph of y = cos 3x around a cylinder.

Construct the graph of z = x2 - y2 above the unit circle.

Construct the monkey's saddle given by z = x3 - 3xy2 above the unit circle.

Construct this model:

Construct this model:

Describe how 8 identical pipes are joined together symmetrically.
> restart;
> with(plots):with(plottools):
> x:=cos(t);
> y:=sin(t);
> u:=x*sqrt(2);
> z:=(1-s)*2+s*u;
> a1:=plot3d([x,y,z],t=-Pi/3..Pi/3,s=0..1,scaling=constrained,grid=[30,8]):
> a2:=rotate(a1,0,0,2*Pi/3):
> a3:=rotate(a1,0,0,4*Pi/3):
> a:=display(a1,a2,a3):
> b1:=reflect(a,[[0,0,0],[1,0,sqrt(2)],[1,1,sqrt(2)]]):
> b2:=rotate(b1,0,0,2*Pi/3):
> b3:=rotate(b1,0,0,4*Pi/3):
> c:=display(a,b1,b2,b3):
> cc:=reflect(c,[0,0,0]):
> display(c,cc);

Describe how six identical pipes are joined together symmetrically.

Describe how 12 identical pipes are joined together symmetrically.

Describe how 4 identical pipes are joined together symmetrically.