Mathematica  MINI-PROJECT
Rotation of an Equilateral Triangle

by Sandy Emerick

Introduction

In geometry, we often have difficulty visualizing the geometric figure that is the result of a rotation about a line.  Our purpose here is to pose
questions,  make conjectures, and then utilize the software called Mathematica to verify our
conjectures.  The next step will be to make calculations concerning the figure produced.  For example, we might want to calculate the volume of the figure.

Lesson 1  Rotation about an altitude

The first problem we want to consider is the rotation of an equilateral triangle about one of its altitudes.  Can you visualize the figure you will get when you rotate the equilateral triangle about one of its altitudes?  A good starting place,  is to cut a "triangle" out of paper and rotate it about  one of its altitudes.  Does this give you any good  ideas about the resulting figure of this procedure?
 
Can you name this figure? 

Be brave.  Take a guess. 
Write a paragraph describing your process for finding your result.

If you want to see the figure that this rotation produces,  proceed with this program.
Thanks to Alan and Thea in the Calculus and Mathematica  lab at the University of Illinois, we can now see the result of this rotation through the graphing capabilities of the wonderful software called Mathematica.

Initialization Cells

Needs["Graphics`Colors`"];
Needs["Geometry`Rotations`"];
equil[radius_] =
    {{-radius Sqrt[3]/2,0,0},
     { radius Sqrt[3]/2,0,0},
     {0,0,3/2 radius }};
rotateZ[{p1_,p2_,p3_},t_] :=
          {Rotate3D[p1,t,0,0,{0,0,0}],
           Rotate3D[p2,t,0,0,{0,0,0}],
           p3};
rotateX[{p1_,p2_,p3_},t_]:=
{p1,p2,Rotate3D[p3,0,t,0]}
plotTri[x_] := Graphics3D[Polygon[x]];
triangle1 = equil[1];
triangle1plot = plotTri[triangle1];

Our first frame

Show[triangle1plot,
     ViewPoint->{0.051, -3.100, 1.356},
     PlotRange->{{-2,2},{-2,2},{-2,2}}];

This is simply our original equilateral triangle.  If you would like to see the next step in the process continue to the next frame. To continue:  move the mouse to the arrow at the bottom of
the outside box to the right, click to scroll
downward until you reach the next screen.

Two frames together

triangle2 = rotateZ[triangle1,Pi/6];
triangle2plot = plotTri[triangle2];

Show[triangle1plot,triangle2plot,
     ViewPoint->{0.051, -3.100, 1.356},
     PlotRange->{{-2,2},{-2,2},{-2,2}}];

Does this first stage of the process give you any new clues?
Does your prediction agree with this first stage?
Let's continue with the development of the figure.

Progression showing frames individually

Table[

Show[plotTri[rotateZ[triangle1,theta]],
     ViewPoint->{0.051, -3.100, 1.356},
     PlotRange->{{-2,2},{-2,2},{-2,2}}],

{theta,0,11/12 Pi,Pi/12}];

To see these frames in sequence, hold down on the open apple key and then type y

One frame at a time does not seem to give you too many clues. If the frames are shown
together on one screen, you can get a better idea of the process. To see these frames in an animated sequence, click on the box below the next frame, and then ,simultaneously hold downthe open apple key and the y key.

Progression showing frames together

Clear[list,theta]
list = {};
Do[AppendTo[list,plotTri
[rotateZ[triangle1,theta]]];
Show[list,
   ViewPoint->{0.051, -3.100, 1.356},
   PlotRange->{{-2,2},{-2,2},{-2,2}}],

   {theta,0,11/12 Pi,Pi/12}];

Can you now "see" the figure formed?
Click anywhere to stop the animation.
Mathematica  now shows the surface.

Surface formed

Clear[h,r,t,radius]
radius = 1;
ParametricPlot3D[{(1 - h) radius Sqrt[3]/2 Cos[t],
				(1 - h) radius Sqrt[3]/2 Sin[t],
				 h 3 radius/2},
				{h,0,1},{t,0,2 Pi},
				ViewPoint->{0.051, -3.100, 1.356},
				PlotRange->{{-2,2},{-2,2},{-2,2}},
				Ticks->None];

Were you correct in your original prediction?  
Can you name the figure formed when you rotate an equilateral triangle about one of its altituded?
You got it.  It is a cone!!

Lesson 2   Volume and Surface Area
Can you find the volume of this figure?  What kind of
information is necessary for this calculation?
Find the volume of this geometric figure if the side of the
original equilateral triangle is 10 cm