The process of turning symbols into a mathematical sculpture has never been easier! This is what you need:
1. a copy of Maple V release 5.
2. a web browser configured with a VRML plugin such as the Cosmo Player.
Maple V furnishes the computing environment allowing you concentrating on the mathematical formulation while silently taking care of the most nonmathematical procedures needed in the visualization. Cosmo Player, on the other hand, creates the kind of visualization environment letting the users to experience the thrill of manipulating virtual geometric objects by rotating, by dragging and by zooming.

Building virtual 3D models creates many fruitful learning activities. We find the following examples particularly exciting.

### Villarceaux Circles

Here is a sculpture which embodies an important mathematical idea: through every point on a torus, there are four circles passing though it!
A complete, rigorous proof of this fact can be found in "Invitation to Geometry" written by Z.A. Melzak. This interesting geometric phenomenon was discovered in 1848 by M. Villarceaux.

It takes seven lines of Maple commands to create such an object:

 > a:=3;b:=7;c:=sqrt(a^2+b^2); > r:=c+a*cos(t); > x:=r*cos(s); > y:=r*sin(s); > z:=a*sin(t); > u:=solve(b*z=a*x,s); > plot3d([x,y,z], s=-u..u, t=0..2*Pi, scaling=constrained,  grid=[30,80]);

Assuming the symbol h stands for the process of drawing the object:

 > h:=plot3d([x,y,z], s=-u..u, t=0..2*Pi, scaling=constrained, grid=[30,80]):

The conversion of the geometric object into the required VRML file is now straightforward.

 >vrml(h,"d:/vrml/villarceaux.wrl", background_color=white);

As a related question, one might ask if a half-torus can have a boundary consisting of two disjoint interlocking circles? The answer: yes, just choose a Villarceaux circle of any torus as a generator of the half-torus and rotate it by 180 degrees about the axis.

Here is the associated code:

 > a:=3;b:=7;c:=sqrt(a^2+b^2); > x1:=b*cos(t); > y1:=a+c*sin(t); > z:=a*cos(t); > x:=x1*cos(u)-y1*sin(u); > y:=x1*sin(u)+y1*cos(u); > plot3d([x,y,z],u=0..Pi,t=0..2*Pi,scaling=constrained);

### Knots

Knot theory involves the embedding of the unit circle into three-dimensional space. Under Maple V, the visual representation of a knot becomes straightforward when the command "tubeplot" is used. For example, the following procedure in Maple:

 > r:=3+cos(t); > x:=r*cos(s); > y:=r*sin(s); > z:=sin(t); > s:=2/5*t; > tubeplot([x,y,z], t=0..10*Pi, radius=0.6,scaling=constrained, grid=[100,10]);

generates a torus knot as in this illustration

By readjusting the ratio between the parameters s and t, we produce this:

### Problems from a Calculus text book

Here is a typical problem that appears in a calculus textbook:

 Find the volume of the region common to the interior of the cylinders  x2 + y2 = a2 and x2 + z2 = a2.

The solid looks like this:

The boundary of the solid consists of two parts: the first part may be created with this code:

 > u:=[cos(t),sin(t),sin(t)]; > v:=[cos(t),sin(t),-sin(t)]; > w:=expand((1-s)*u+s*v); > plot3d(w, s=0..1, t=0..2*Pi, scaling=constrained, color=green);

while the second part is obtained by rotating the first:

 > plot3d([w[1],-w[3],w[2]],s=0..1, t=0..2*Pi, scaling=constrained, color=yellow);

Realizing the parametric description of the solid is the most difficult part in solving the original problem. If we examine how the surfaces are represented symbolically, we find exactly what is need: the cross-sections perpendicular to the x-axis consist precisely of squares with u, v, -u and -v as corners. This serves as an example in which a clear visual understanding of the problem leading to a clear analytical solution.

Here is another similar challenging calculus problem:

 Find the volume of the region common to the interior of the intersection of the cylinder (x - 0.5)2 + y2 = 0.25 and the sphere x2 + y2 + z2 =1.

Again, the vaxed question is: "What does the solid region looks like?" Since a part of the boundary lie on the unit sphere, the attention is thus turned to the spherical coordinates. The Maple V commands used to describe the region is in fact a direct translation of the related mathematical formula:

 > r:=cos(s); > x:=r*cos(t); > y:=r*sin(t); > z:=sin(s); > a1:=plot3d([x,y,z],t=-s..s,s=0..Pi/2,scaling=constrained, color=red,grid=[100,10]): > a2:=plot3d([x,y,-z],t=-s..s,s=0..Pi/2,scaling=constrained, color=red,grid=[100,10]): > t:=s; > b:=plot3d([x,y,(1-2*u)*z],u=0..1,s=0..Pi, scaling=constrained, color=red,grid=[10,100]): > display(a1,a2,b);

Platonic solids and related geometric objects
Subsets of the sphere
Ruled surfaces
Orthogonal grids on a cone