Mathematical Experiment 6

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 6 identical pipes are joined together symmetrically. 
Describe how 12 identical pipes are joined together symmetrically. 
> restart;tau:=(sqrt(5)+1)/2:
> with(plots):with(plottools):
> x:=cos(t):
> y:=sin(t):
> u:=x*tau:
> z:=(1-s)*2+s*u:
> a1:=plot3d([x,y,z],t=-Pi/5..Pi/5,s=0..1,scaling=constrained,grid=[20,8]):
> a2:=rotate(a1,0,0,2*Pi/5):
> a3:=rotate(a1,0,0,4*Pi/5):
> a4:=rotate(a1,0,0,6*Pi/5):
> a5:=rotate(a1,0,0,8*Pi/5):
> a:=display(a1,a2,a3,a4,a5):
> b1:=reflect(a,[[0,0,0],[1,0,tau],[1,1,tau]]):
> b2:=rotate(b1,0,0,2*Pi/5):
> b3:=rotate(b1,0,0,4*Pi/5):
> b4:=rotate(b1,0,0,6*Pi/5):
> b5:=rotate(b1,0,0,8*Pi/5):
> c:=display(a,b1,b2,b3,b4,b5):
> cc:=reflect(c,[0,0,0]):
> display(c,cc);
Describe how 4 identical pipes are joined together symmetrically.