As nice as the various graphing programs are, sometimes there is no substitute for a real model. Something you can hold and talk to late into the night. Everyone knows that it is pretty easy to make regular polyhedra, but not everyone knows how easy it can be to make a beautiful surface. The key is selection.
The first rule of elementary surface building is to feign indifference to any surface that is not neatly ruled. So if someone asks you about building a hyperbolic paraboloid with equation z = x^2 - y^2, JUST SAY NO. Quickly volunteer to make a model of z = x * y, because it is the same thing, just rotated. And since it is a model, you can just turn it on its side!!
Right now we are not interested in something so boring as a hyperbolic paraboloid [note to Fred: no, you are not boring, it was just a joke]. We are going to make a model of.......drum roll please....
x = r cos[t]
y = r sin[t]
z = z
If we substitute these values into the original equation (and wave the magic trig identity wand) we end up with z = .5 sin[2t]. Oh my goodness! The value of z is now dependent only on t, not on the radius. This means that it is a RULED SURFACE! Time to begin construction.
Since this is in cylindrical coordinates, it is easiest to construct this surface within a cylinder. I recommend a clear plastic soda bottle, either one or two liter. The one liter size makes a convenient sized model, easily transportable and such. Remove the colored base and cut the bottle in half. You will use the dome shaped half. If you have a really long needle, you can also leave the bottle intact. That is what Prof. Klotz did when he made his version of this surface.
Basically, the problem is to draw the function z = .5 sin[2t] around the circumference of the bottle. I will describe a simple, non-technological method for doing this. Some of you may be able to do it with a computer application. For us model building purists, this is blasphemy.
(1) Cut a piece of graph paper the length of the circumference of the bottle. In the case of a one liter bottle, it will be about 27 cm long, but you should not measure it. Just base it off of the bottle itself. Cut it to any width you want, keeping in mind that it would be ideal if the grid boxes lent themselves to a convenient scale for drawing a sine curve with extrema of ± .5. The surface turns out looking the best if the extrema are about 5 cm apart.
(2) You need to draw the function on the paper. It will be two complete sine waves. One easy way to do this is to just fold the paper in half over and over. This way you can have evenly spaced x-values. I folded mine so that there were 32 divisions in the strip, making each interval equal to pi/16 for a complete length of 2pi. Then it will be easy to plot values for z every pi/16. [note: you should only have to compute five values!]
(3) After you have the function drawn, carefully tape the paper to your bottle. Then take a pin or needle and poke a hole through each point that you have marked. Make sure that the pin goes all the way through the plastic.
(4) Remove the paper. You will have 32 perfectly spaced pin holes tracing a sine curve around your bottle. Notice that the holes opposite each other are of the same height. That means that any diameter of the the bottle is of constant height.
(5) This is the only dangerous part, so the standard disclaimers about parental supervision apply. Take your mental age into consideration. Some adults really need parental supervision. Heat a pin or needle in a candle (make sure that you hold the pin with some tape or cloth or something). Then use the hot needle to widen each of the holes. You will have to reheat the pin at least once per hole, so use a candle with a longer life span than a birthday candle. The holes need to be 1 - 2 mm in diameter, depending on how thick the string is that you want to use.
(6) Now is the rewarding part. Thread string or yarn back and forth through the holes, connecting the holes opposite each other. The lines should all be parallel to the xy plane and go through the z axis. When you have finished, tie off the ends. You should see a sine curve traced in yarn around the bottle and a beautiful surface inside the bottle.
(7) All surfaces need names. My model of this particular surface is named Leslie [in honor of Leslie Jean Schramer, for anyone who is reading this in Columbia, MO]. In general, you should not name your surface after yourself. It is poor form. Only theorems should be named for yourself. Name your surface after your best friend, one of your parents, the teacher you are trying to kiss up to, or the guy that sits next to you in your algebra class. It is a great way to flirt. What better compliment to give a person than to name a surface after him or her (or it)?
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