May 2012

Pendulum Waves

This week’s entry in the “Wow, the Internet is Awesome!” file:

http://www.youtube.com/watch?v=yVkdfJ9PkRQ

These pendula of staggered lengths are set swinging, and the result is fascinating. At first, the weights seem to trace out sine waves, but they quickly start to cycle through a wide range of behaviors.

At times the pendula appear highly organized, and at times almost chaotic, as they cycle through various patterns. Amazingly, the weights eventually return to their initial state!

A really beautiful mathematical demonstration.

By patrick honner, 12 years4 years ago

Application Geometry

CD Packing Problems

I consider myself an expert arranger of things. I enjoy rearranging storage space, packing things away, and helping people fill up moving trucks. It’s a way to apply geometry and optimization techniques, two of my favorite things.

In general, the packing problem entails trying to find the most efficient way to pack a certain kind (or kinds) of object into a certain fixed space. Packing problems are, generally speaking, very challenging because every packing problem is unique. There isn’t a good, efficient procedure that solves them all.

Here is yet another example of problems with packing problems. After shedding a bunch of CD cases, I thought I’d try to pack them up in a box. Here was my first attempt.

I got 49 CDs in the box, but there was a bit of unused space left over. I couldn’t fit a CD into that unused space, but I thought maybe I could rearrange everything to make some of that space usable.

So I tried again.

The number of CDs in this new arrangement differed by one. While I can compare which of these packings is more efficient, the problem is comparing all possible packings! There are a lot of options to consider.

As useless as they are, I ended up having a lot of fun with these CD cases. I made some parallelepipeds with them and used them to demonstrate Cavalieri’s Principle!

By patrick honner, 12 years8 years ago

Geometry Resources

Folding Website

This is great website from Joseph O’Rourke, author of “How to Fold It: The Mathematics of Linkages, Origami,and Polyhedra” .

www.howtofoldit.org

The website has several videosand cool animations that demonstrate some of the basic ideas in mathematical folding, like the one-cut problem, the map puzzle, and folding polygons into convex polyhedra.

There are also a few folding patterns available for download, just in case you’d like to produce a turtle with one cut!

And for more resources on math and origami, check out my fun with folding page!

By patrick honner, 12 years6 years ago

Appreciation

Uneven Haircuts

Barbers strike me as a practical lot. Which makes this pricing structure seem especially strange to me.

I mean, isn’t a kid going to have less overall hair than an adult? I understand that an adult head may have less coverage percentage-wise, but the larger head-radius more than makes up for the lower proportion of hair. I’ve got to believe that the price-per-hair is considerably higher for a child than an adult under this pricing system.

Of course, I would not be so naive as to expect understandable pricing structures from hair stylists.

Related Posts

Sandals, Similarity, and Scaling

By patrick honner, 12 years8 years ago

History Resources

BBC Podcasts: A Brief History of Mathematics

This is a set of ten podcasts from the BBC titled “A Brief History of Mathematics“:

http://www.bbc.co.uk/podcasts/series/maths

Narrated by mathematician Marcus du Sautoy, these podcasts begin with Newton, Leibniz, and Calculus, and cover other great personalities in mathematics like Euler, Fourier, Ramanujan, Poincare, and, of course, Gauss.

Each podcast is about 15 minutes in length, and all 10 are freely available for download.