Saturday, January 25, 2014

Potpourri: Links and a French Curve Question

Here's the one I most want to share with everyone, but now I have some reservations...
A lovely documentary on mathematical origami, called Between the Folds, has been posted to youtube. The DVD costs $20, which seems quite reasonable to me. This youtube posting is not from the producers, Green Fuse Films. I've contacted them, in case they want to have it removed. Watch it quickly if you'd like to. The official trailer is also on youtube, so you can get a taste, even after the pirate version is gone.

Many of the links I'm saving these days are ideas I hope to share with my students:

Calculus I
Intuiting the Chain Rule (Girls' Angle)

Calculus II
Volume of a Pendant (SquareCircleZ)

Linear Algebra
What's so special about the 4 fundamental subspaces? (math.stackexchange)

Some of the links aren't for a course, but for me to try out, some day when I have time...

Circles and Equilateral Triangles (Pat's Blog)

A student came to me last week for help. He wants to use fabric (a very loose weave, which will stretch some) to cover a sphere. He needed to know what shape to cut. I had no idea, and first suggested he look at information on world maps. He wants nice seams and no puckering. I thought a shape that wrapped around the equator, with sort of triangular tabs up and down to the poles might work. We knew the sides of those needed to curve, and he wanted an equation. I thought he should experiment with 8 tabs in each direction at first, and those would need to have a 45 degree angle at the pole. I had no idea what kind of equation would fit this.

A former student, who designs and sews clothing, was in the math lab, and I brought him into our conversation. He said he would use a French curve, which is what you see in the image above. We looked it up, and no one seems to know the equations for the curves.

So my question is about the French curves. Can anyone help me figure out their equations?

The coolest thing happened as I began to write this post. I was trying to sketch my pattern idea freehand, and I'm a terrible drawer. So I turned to Geogebra, and started putting in the points and lines. When I got to the curved segments, I was in the right frame of mind. I knew I needed something between the points (0,.5) and (1,3.5), with a vertical tangent at (0,.5) and a slope of 1 (same as 45 degrees) at (1,3.5). I knew that the square root function starts out with a vertical tangent, and so I figured I'd try to modify that. Getting it to go though those two points, I was having trouble getting the slope at the top right. (I think there's a way...) So I figured I could change what root I used. And I solved the problem I had posed! (Always a rush.) The first curve is y = 3 times the cube root of x. The next one in the same direction is y = 3 times the cube root of (x-2), and the one between, that goes in the other direction, is y = 3 times the cube root of (2-x). They may not be just the right shape. My student will have to experiment at this point, I think.

Or is there a way to figure out a better shape mathematically? The problem is that we are really just approximating, because the fabric is a flat surface, and the sphere is everywhere curved (in all directions). So I don't see how we can
 describe perfectly what we want. Ideas?

[Oops! I just tried to explain this to my son, and realized those points have two 45 degree angles in them, for a total of 90 degrees. Back to the drawing board! (More to come...)]

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