3. Double Torus and MultiTorus in R3

The ordinary torus is essentially just a product of two circles and hence has a familiar picture in R3 that is easily drawn parametrically, that is, as the image of a mapping into R3. But the double torus--the torus with two holes--has no simple parametrization. So to draw one, we use Mathematica's ability to plot the level surfaces of a function f on R3.

For the double torus, a suitable function f has the formula (h(x)+y2)2+z2, where h is given to Mathematica as

    h[x_]:= (x-1)*x^2*(x+1),

with graph as shown on the right.

Suppose the function f above does have a double torus as level surface. To write the commands that will plot it, we would like to know, ahead of time, a coordinate box that will contain it. To find such limiting coordinates, we draw the intercepts of the torus with the coordinate planes of R3.


Part 1: Intercepts

To get the intercept with the xy plane, set z=0 in the formula for f. Then the command

 ContourPlot[(h[x]+y^2)^2,

  {x,-1.1,1.1},{y,-.63,.63},Contours->{.02},

  AspectRatio->Automatic,PlotPoints->40]

will produce the intercept.

The parameters used here were found as much by trial-and-error as by computation. However, for an indication of how the function f can produce a double torus,. consider the line x=0.7 in the figures. If y=0, then since h(.7) ~ -0.22, the value (h(.7)+0)2 ~ 0.04 already exceeds 0.02, so there can be no point (.7,0,z) on the level surface f=0.02. By constrast, if y=±0.5, then clearly (h(.7)+.25)2 is less than 0.02, so there are points (.7,±.5,z) on f=0.02.

The intercept in the xz plane is given by

  ContourPlot[h[x]^2+z^2, {x,-1.1,1.1},{z,-.2,.2},

   Contours->{.02},AspectRatio->Automatic, PlotPoints->40]

This plot will show that the z values of torus lie in the interval -0.2 < z < 0.2

The yz intercept is a single oval, giving no new information.


Part 2: DoubleTorus

We construct the double torus as a union of three parts, an approach that will make it easy to produce plots of multitori, those with any number of holes. To plot level surfaces of a function on 3-space, we install a command from Mathematica's Graphics package:

    <<Graphics`Contour3D`

(Note that backquotesappear here, not ordinary single quotes.)

Mathematica does not find it easy to produce a good double torus from a single command. But asking for less gives something much more useful.

    bigX = ContourPlot3D[(h[x]+y^2)^2+z^2,

    {x,-.7,.7},{y,-.63,.63},{z,-.2,.2},

    Contours->{.02},AspectRatio->Automatic,PlotPoints->{4,7}]

This will become a double torus if we attach the following ends.

The ends are plotted by

    lftend = ContourPlot3D[(h[x]+y^2)^2+z^2,

    {x,-1.5,-.7},{y,-.63,.63},{z,-.2,.2},

    Contours->{.02},AspectRatio->Automatic,PlotPoints->{4,7}]

and

    rtend = ContourPlot3D[(h[x]+y^2)^2+z^2,

   {x,.7,1.5},{y,-.63,.63},{z,-.2,.2},

   Contours->{.02},AspectRatio->Automatic,PlotPoints->{4,7}]

Then the ends are attached by the following command, which also removes the enclosing boxes.

   Show[{lftend,bigX,rtend}, Boxed->False]


Part 3: MultiTori

To construct a multitorus with any number of holes, just glue another bigX onto the first one, and then another, and so on--finishing by attachinging left and right ends, as above.

We illustrate this in the case of the three-hole torus. The second bigX--the one to be attached to the first--is just a copy of the first that has been moved to the right by distance 1.4 (the width of a bigX). Thus in the command building bigX, we increase the limits on x by 1.4, and in the formula for the function f, subtract 1.4 from x.

    bigX2=ContourPlot3D[ (h[x-1.4]+y^2)^2+z^2,

    {x,.7,2.1},{y,-.63,.63},{z,-.2,.2},

    Contours->{.02},AspectRatio->Automatic,PlotPoints->{4,7}]

Similarly, the right end must be translated by 1.4 in the x direction

    rtend2=ContourPlot3D[ (h[x-1.4]^2+z^2)^2+z^2,

    {x,2.1,2.9},{y,-.63,.63},{z,-.2,.2},

    Contours->{.02},AspectRatio->Automatic,PlotPoints->{4,7}]

If all has gone well, these parts fit together to give a triple torus.

   Show[{lftend,bigX,bigX2,rtend2}, Boxed->False]


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