Geometry

Challenge Geometry

The Art Gallery Problem

I recently attended a function at the Museum of Modern Art hosted by Math for America. In typical MfA fashion, the invitation was designed around a relevant mathematical idea, in this case, the Art Gallery problem.

The Art Gallery problem (aka the museum problem) basically asks “What is the minimum number of guards required to keep an entire region under watch?”.

We make the usual idealizing assumptions for the purposes of elegant mathematics. For instance, we assume that the guards can see infinitely far, that there are no obstructions (other than the walls), that the guards can see everything in front of them, and the like.

So looking at the map at the right, how many guards would be needed to watch over the fifth floor at the MoMA?

The Art Gallery problem is a classic question from computational geometry, and its solution involves a lot of great ideas from graph theory and graph coloring.

There are a number of fun extensions to this problems, too, including the watchmen route problem (one watchman guarding the entire museum) and the fortress problem (guarding the exterior, rather than the interior).

The Art Gallery problem is the best kind of math problem: easy to state and understand, surprisingly rich and complex, and lots of fun to play around with!

By patrick honner, 14 years9 years ago

Appreciation Challenge Geometry

Geometry Photography Competition

The New Scientist magazine has a monthly photo contest, and the theme this month is Geometry!

I’m thinking of submitting my picture of Parabolas in Nature, but probably not my picture of my imperfect division of a squash.

The contest ends November 30th, at midnight. The winner and select runners-up will be posted on the New Scientist website. All the details can be found here:

http://www.newscientist.com/article/dn19684-photo-competition-geometry.html

So keep your eyes open and your camera at hand–there’s a lot of Geometry out there!

By patrick honner, 14 years9 years ago

Appreciation Geometry

Twirling Tori

This is a mind-blowing animation of arms circulating around a torus, created by Emilio Gomariz. I definitely became entranced for a bit, trying to follow a single hand all the way around the donut. (Click the image or this link if you can’t see the animation).

Apart from being visually amaazing, this puts me in mind of a result about fluid flows on surfaces. This animation demonstrates that a liquid, for example, can flow over the surface of a torus in such a way that every point moves: nothing appears stationary here, and everything is moving in a smooth (i.e., continuous) fashion.

The remarkable result is that this same can not be done on the surface of a sphere! There will always (at least) one point on the surface of the sphere that doesn’t move. A popular interpretation of this result is that however windy it might be outside, there is always at least one point on the Earth that is perfectly calm.

By patrick honner, 14 years9 years ago

Appreciation Challenge Geometry

Proofs Without Words

Here are two of my favorite Proofs Without Words. I’ve been thinking about infinite geometric series a lot lately, and these are two lovely, well-known, visualizations of two amazing infinite sums:

In a square of side length 1 (and therefore, area 1), cut the square in half; then cut one half in half (that’s a quarter); now cut one of the quarters in half (that’s an eighth); and so on and so on and so on (this puts the infinite in infinite sum). Eventually you’ll fill up the whole square! So this is a demonstration of the following amazing, and somewhat counterintuitive, fact that

Similarly, this diagram

is a visual representation of the following sum:

As any good, lazy mathematician would say, the details are left to the reader.

Appreciation Art Geometry

Wireframe Torus

This is a cool example of wire-sculpture: a single piece of wire woven into the shape of the torus.

http://makezine.com/2010/10/24/math-monday-wire-torus-challenge/

(I doubt you can play pool on this one, though.)

The author of the blog post, George Hart, is the proprietor of the soon-to-be Math Museum, and it seems he is something of a sculptor himself. According to the article, this piece was on display at a conference held by the European Society for Mathematics and the Arts.

Poking around their website and admiring the the multitudinous mathart is a pleasant way to pass a little time.

By patrick honner, 14 years9 years ago

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