This is a challenging model to build. It is definitely not an undertaking for the faint of heart. It took me about four hours to build it, not including the planning. You, however, don't even have to do the planning, since I have done it for you. Gosh, I feel like I am allowing a legion of lazy math modellers to develop. You do need to plan on spending a little time with a 3-D graphing program on a computer. If you are a really talented conic section sketcher, you won't have to, but some of us are conically-challenged.

First of all, the theory behind the madness: We start by looking at the level curves (surprise, surprise). Set z equal to a constant that we will call t and assume that t is not equal to zero. Then the equation can be rearranged to look like:

y ^ 2 - ( ( x ^ 2 ) / t ) * y + ( x ^ 4 ) = 0

Now you can see that it is a quadratic equation in y, with x in the coefficients. Use the quadratic formula to solve for y. You will find that for a given value of t, there are two equations for y, both of them parabolas through the origin. Also note that the absolute value of t must be less than or equal to .5, since the discriminant must not be less than zero. To make things really easy for you, here is a chart of the values you find if you do this. For maximum educational benefit, you should still do it yourself.

Cheat Sheet for Drawing Parabolas
Z = Y =Y =
.5x ^2x ^ 2
.42 * x ^ 2.5 * x ^ 2
.33 * x ^ 2.33 * x ^ 2
.24.79 * x ^ 2.209 * x ^ 2
.19.90 * x ^ 2.101 * x ^ 2
-.1-9.90 * x ^ 2-.101 * x ^ 2
-.2-4.79 * x ^ 2-.209 * x ^ 2
-.3-3 * x ^ 2-.33 * x ^ 2
-.4-2 * x ^ 2-.5 * x ^ 2
-.5-x ^ 2-x ^ 2

Materials: foam board (foam core, the stuff you used for your science project backdrop), a ruler, pencil, a sharp X-Acto knife (preferably with some replacement blades; the #11 blade works very nicely), glue or rubber glue, and a set of parabolas

First off, we need to make the parabolas. If you look closely, you will note that you need only make nine parabolas, since you can flip a piece of cardboard very easily to change the sign. Use your favorite computer graphing program to make graphs of the nine different parabolas, making sure that the scale is the same on each. I graphed mine with a range of +/- 10 and made the widest one about 14 cm. Print a copy of each, then rubber glue them to thin cardboard. Cut out the parabolas, and, voila, you have a set of templates. While you are at the computer, take a look at a graph of the surface just to get an idea of what is going on. There is a great picture courtesy of Jeremy Dilatush in the Limits section of the Forum. You should note that for negative z values, there is a canyon with a small mountain in the middle of it, while for positive z values, there is a large mountain with a small gourge in it. Also note that y values and z values always have the same sign.

Now cut six squares of foamboard, with sides equal in length to the widest parabola. Do this as carefully as possible, since accuracy in measurement really helps the appearence of the final model. Then carefully draw lines dividing each square into four equal quadrants. Do this on both faces of the squares.

Start with the very bottom layer. The bottom surface of this square will be at z = -.5, so carefully trace a the parabola y = x ^ 2 there. The upper surface will be at z = -.4. Trace just y = .5 * x ^ 2 for now, since we want to make the canyon part of the surface first. Now what you want to do is connect the two curves in a relatively smooth manner. So cut out around the narrower parabola (y = x ^ 2 , in this case) with your knife and remove it. If you cut neatly, you can use this piece later. Then carefully shave the foam at an angle to connect this edge to the parabola on the other side of the foam.

You will repeat this process to build up the negative z values. When you get to z = 0, just smooth off the foam from the curve at -.1 to the edges of the square (but not past the x axis). Now you can use a similar process to build the 'mountain' part of the other half. You should have half of your cutting already done, as you can use pieces that you just removed from the squares. After you do this, the second set of parabola equations is used to make the little mountain in the canyon and the gourge in the mountain.

When you are all done, carefully align all of the pieces and glue them together. If you are lucky, it will look even better than the one shown below.

Inevitably, there will be singularities at the vertices of the parabolas, thus proving that singularities are conserved, if not preserved, in the construction of surfaces.

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Joan Hoffmann
8 November 1995