Here's an idea for illustrating all the different shapes you can get out of conic sections: get some Play-Doh, roll it out into a cone shape (the "conic" part) -- and also a reasonably sharp knife (for the "sections" part).

First, here's our starting cone:

Note that if you cut off just the tippy-top part then you get a single point:

On the other hand, if you carefully take a shaving down the very edge then it produces a line:

But if you make a slice perpendicular to the base, then you get a perfect circle (of any size you want, depending on how far down the cone you take it):

Make a similar slice at a slight angle and the cross-section you get is now an ellipse:

Take the slice at a steeper angle and you'll produce our old quadratic friend, the parabola:

And increase the angle a bit more (greater than the edge of the cone itself), and you'll produce the parabola's angry cousin, the hyperbola (or really a half-branch of such):

Kind of neat. Full disclosure: the cone gets pretty "smooshed" on each cut (kind of like a loaf of bread with a dull knife), and I had to gently re-shape back into the proper section before each photo. Therefore, this demonstration probably works best in static photography, and would be somewhat less elegant live or in a video. But the nice thing about the Play-Doh is that you can sticky it back together pretty well after each sectional cut, and it's the only material I could think of that would work well in that way. Can you think of anything else?

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Go for a dual cone with touchy tips and chop both simultaneously to get both halves of the hyperbola. Also chop thru the where the tips touch at various interesting angles to generate all manner of intersecting lines.

ReplyDeleteTheres a way to display the solution of a system of equations using intersecting conics but I don't think its terribly amenable to play dough.

If you want to really mess with people you can use play dough as an analog computer to square the circle, leading to all kinds of interesting and valuable discussion about rounding and irrationals, although this is getting kinda off topic.

Yeah, I wish I could make the double cone happen for the hyperbola (et. al.), but of course standing a mass of Play-Doh up on a point wasn't going to be feasible (and also be possibly hard to see). Great other ideas once people get the basics in their head, though!

DeleteFollow-up from Isabelle: "And to answer your question at the bottom, there are oil-based clays that are very soft to shape and probably hold a better shape than play-doh, especially after a few cuts. One brand I believe is called Plastelina (found in art stores)."

ReplyDeletePerhaps you'd get less smooshing if you used a taut length of thin wire or fishing line instead of a knife.

ReplyDeleteOoh, nice idea!

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