11 Describe how 8 identical pipes are joined together symmetrically.

> restart:

> with(plots):with(plottools):

Warning, the name changecoords has been redefined

> u:=x*2^(1/2):z:=2*(1-s)+s*u:

> a1:=plot3d([x,y,z],s=0..1,t=-Pi/3..Pi/3,scaling=constrained,grid=[30,8],color=yellow):

> 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):

> b:=display(a,b1,b2,b3):

> c:=reflect(b,[0,0,0]):

> display(b,c);

12 Describe how 6 identical pipes are joined together symmetrically.

> restart:

> k:=1:

> with(plots):with(plottools):

Warning, the name changecoords has been redefined

> u:=x*k:z:=2*(1-s)+s*u:

> a1:=plot3d([x,y,z],s=0..1,t=-Pi/4..Pi/4,scaling=constrained,grid=[25,8]):

> a2:=rotate(a1,0,0,2*Pi/4):

> a3:=rotate(a1,0,0,4*Pi/4):

> a4:=rotate(a1,0,0,6*Pi/4):

> a:=display(a1,a2,a3,a4):

> b1:=reflect(a,[[0,0,0],[1,0,k],[1,1,k]]):

> b2:=rotate(b1,0,0,2*Pi/4):

> b:=display(a,b1,b2):

> c:=reflect(b,[0,0,0]):

> display(b,c);

13 Describe how 12 identical pipes are joined together symmetrically.

> restart:

> k:=5^(1/2)/2+1/2:

> with(plots):with(plottools):

Warning, the name changecoords has been redefined

> u:=x*k:z:=2*(1-s)+s*u:

> a1:=plot3d([x,y,z],s=0..1,t=-Pi/5..Pi/5,scaling=constrained,grid=[15,6]):

> 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,k],[1,1,k]]):

> 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):

> b:=display(a,b1,b2,b3,b4,b5):

> c:=reflect(b,[0,0,0]):

> display(b,c);

14 Describe how 4 identical pipes are joined together symmetrically.

> restart:

> k:=1/sqrt(2):

> with(plots):with(plottools):

Warning, the name changecoords has been redefined

> u:=x*k:z:=2*(1-s)+s*u:

> a1:=plot3d([x,y,z],s=0..1,t=-Pi/3..Pi/3,scaling=constrained,grid=[25,7]):

> 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,k],[1,1,k]]):

> b2:=rotate(b1,0,0,2*Pi/3):

> b3:=rotate(b1,0,0,4*Pi/3):

> display(a,b1,b2,b3);

>