While on the beach one day, I found a triangular rock. So I started on some beach relief.
Then I iterated.
And I iterated again.
And I continued to iterate, until the whole beach was one big Sierpinski Triangle!
Challenge Geometry
Math and Art: An Impossible Construction
A favorite pastime of mine is offering impossible problems to students as extra credit, like asking them to find the smallest perfect square that has a remainder of 3 when divided by 4. I don’t tell them the problems are impossible, of course, as that would ruin the fun. Usually it engages and confuses them, and it makes them suspicious of me. That’s a win-win-win in my book.
So while discussing some three-dimensional geometry, I offered extra credit to anyone who could build a model of a Klein bottle. The Klein bottle is a hard-to-imagine surface that has neither an inside nor an outside; it’s like a bag that is sealed up, but somehow the bag is inverted in on itself. If you are familiar with the Mobius strip, the Klein bottle is basically a Mobius strip, one dimension up.
One reason that the Klein bottle is hard to visualize is that it can’t exist in three dimensions. It needs a fourth dimension in order to twist around on itself, kind of like the way the Mobius strip (which itself is two-dimensional) needs that third dimension to twist through before you tape it back together. So, I was pretty impressed with the student who made this:
Not bad at all, for someone who is dimensionally challenged. Here’s a nice representation for comparison, although it’s still a cheat: the Klein bottle really doesn’t intersect itself.
A nice example of student work!
Appreciation Art Geometry
Math and Art: Smashing Cones
During an exploration into solid geometry, we discovered we could make these lovely flowers
by smashing the paper cones we had made.
Check out more math and art with paper on my Fun With Folding page.
Application Geometry Photography
Math and Art: Knitting the Plane
As opposed to tiling the plane, here’s knitting the plane. Or, the fence.
As you can see, they’ve still got a ways to go!
And this isn’t the first example of mathematical knitting I’ve seen: check out this knitted Klein bottle!
Application Geometry
Math and Art: Math and Computer Animation
This is a clear, concise, and fascinating overview of how some very advanced mathematical ideas are making their way into 3-D animation.
http://www1.ams.org/samplings/feature-column/fcarc-harmonic
Here’s the basic setup. In order to efficiently model a character, you approximate it with a frame that is built around a few important points. To move the character, you focus on moving just those points that define the frame. Thus, moving the character from point A to point B boils down to understanding where those handful of crucial points go.
The tricky part is figuring out a way to smoothly bring all those in-between points along for the ride, and that’s where the math comes in. The secret is to think of those in-between points as averages of the points that define the frame. The article explains how barycentric coordinates, harmonic functions. and a surprising amount of calculus are being used to pull off this movie magic!