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!
Appreciation Geometry Photography
Math and Art: Shadows and Triangles
The sun sometimes turns rectangles into triangles. A nice Sunday morning image.
Appreciation Art Geometry
Math and Art: The Art of the Ellipse
This is a cool article about how important the ellipse is to the artist.
http://opinionator.blogs.nytimes.com/2010/09/23/the-frisbee-of-art/
The author gives a nice, if long, explanation about the significance of the ellipse, but it basically boils down to this: circles are everywhere. And often, when we are looking at circles, we’re looking at them atilt. We see projections of the circle, and projections of circles are ellipses.
Think of it this way: suppose you have a hula hoop and you hold it parallel to the ground. The shadow you see is circular, but if you tilt the hula hoop, the shadow will change–into an ellipse. I don’t have a hula hoop, so I made do with a spare key ring:
As the circular key ring is rotated, it becomes less parallel to the ground; the shadow becomes less circular and more elliptical. And at the end, the ellipse vanishes–an ellipse eclipse!