#
Mathematical Experiment 10

Construct the portion of the cylinder x^{2}
+ y^{2} = 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 = x^{2} - y^{2} above the unit circle.

Construct
the monkey's saddle given by z = x^{3} - 3xy^{2} 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.