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, 16 yearsOctober 26, 2010 ago

Geometry Technology

The Perfect Parallelepiped

In general, it’s unusual for a rectangle to have sides and diagonals whose lengths are all integers (i.e., whole numbers). Consider the following three rectangles, all of width 3:

Looking at the different lengths, we see one place where the diagonal length is an integer, but in the other cases, the diagonal length happens to be a non-terminating, non-repeating decimal (i.e., irrational). Indeed, the diagonal length will be an integer exactly when the length and width are part of a Pythagorean triple, but compared to the alternative, this is uncommon. (While there are infinitely many occurrences of this, we can still meaningfully consider it uncommon).

Now, imagine the situation in three dimensions. A rectangular prism (think of a cardboard box) has 12 sides, 12 face diagonals, and four space diagonals. It would be extremely unusual for all of those 28 lengths to be integers. Even if we didn’t limit ourselves to rectangular prisms, but we allowed for the box to be slanted in all directions (that is, a parallelepiped), it would still be a numerical miracle for all those lengths to be integers.

Well, meet the perfect parallelepiped!

This was discovered by a couple of mathematicians at Lafayette College in Pennsylvania, using brute-force computer trials. It looks like they found some others, too. So thank you, Clifford Reiter and Jorge Sawyer, for giving me an extra credit problem for my next exam!

Related Posts

CDs, Prisms, and Parallelepipeds

By patrick honner, 16 yearsOctober 23, 2010 ago

Geometry

The Final Word in Triangle Appreciation

As has been previously noted, October has been a nice month for triangle appreciation. Today, 10/19/10, offers us one more polygon to ponder.

Consider the 10-19-10 triangle.

It’s a fairly ordinary triangle, as triangles go. It’s a little short compared to the other triangles we’ve looked at recently, but there’s nothing wrong with that.

What’s special about the 10-19-10 triangle is that it’s our last chance this month to enjoy triangularity. Tomorrow, the Triangle Inequality steps in. You can’t have a 10-20-10 triangle, because once the third side hits 20, you need all of the 10s to get from point A to point B. There’s no wiggle room for the triangle’s interior.

Although I must admit, I do find much to admire in this degenerate triangle.

Geometry History

Benoit Mandelbrot

Benoit Mandelbrot died this week at the age of 85, leaving a giant mark on the worlds of mathematics and science. Mandelbrot coined the term fractal, writing the seminal book on the topic in 1982–The Fractal Geometry of Nature. By rejecting the generally accepted, if never actually articulated, notion that things were smooth, Mandelbrot challenged everyone’s notion of shape, distance, and dimension.

As is often the case, Mandelbrot was considered crazy at first, but he dies with almost legendary status. We will likely be talking about fractals and Mandelbrot sets hundreds if not thousands of years from now, the way we talk about the Pythagorean Theorem and Euler’s Number today.

And as is also often the case, Mandelbrot’s brilliant and revolutionary ideas can be traced to a simple question that he chose to think of differently: how long is the coast of Britain?

The answer, he was surprised to discover, depends on how closely one looks. On a map an island may appear smooth, but zooming in will reveal jagged edges that add up to a longer coast. Zooming in further will reveal even more coastline.

“Here is a question, a staple of grade-school geometry that, if you think about it, is impossible,” Dr. Mandelbrot told The New York Times earlier this year in an interview. “The length of the coastline, in a sense, is infinite.”

From Mandelbrot’s NYT’s obituary.

By patrick honner, 16 yearsOctober 17, 2010 ago

Proofs Without Words

Wireframe Torus

The Perfect Parallelepiped

The Final Word in Triangle Appreciation

Benoit Mandelbrot

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