Infinite Integer Triangles

Here’s an interesting question.

Given the side of a triangle with integer length, what is the set of all points in the plane for which the other two sides will also have integer lengths?

And by interesting, I mean that the answer wasn’t immediately obvious to me.

So I drew a segment 5 units long in Geometer’s SketchPad, created a bunch of concentric circles with integer radii and centers at the endpoints of the segment, identified the intersection points of those circles, and finally hid the circles. The result was the following beautiful image:

The points in the plane that yield triangles with integer side lengths.
(The given segment is 5 units long.)

And by beautiful, I mean that the result is, well, beautiful. At least to a math dork. If this had been painted by Van Gogh, it would have been called Triangle in a Starry Night. (Okay, maybe not.)

The triangle indicated by the dashed lines is the famous 3-4-5 right triangle. The points in the upper right and upper left corners yield the less well known but similarly intoxicating 5-5-8 triangle. If the limitations of the web allowed this image to extend infinitely in all directions, the result would be infinite beauty. Alas, reality confines us.

I have an infinity of jokes that deal with triangles and circles, but I’ll only share a subset of them here.

What did the triangle say to the circle?
Your life is pointless.

Why don’t circles hang out with ellipses?
Too eccentric.

What did the hypotenuse say to the other two sides?
Nice legs!

Where do circles and ellipses spend their vacations?
Coney Island.

What’s a circle?
A round, straight line with a hole in the middle.

What did the circle say to the tangent line?
Stop touching me!

Impossible Tasks for Rabbits

Too cold to run to the convenient store for a six-pack? Too many Allen wrenches necessary for you to put together that Vrådal loft bed that you bought from Ikea? Want someone to walk Fido while you finish the last 1,126 pages of War and Peace?

If you need something done but you’re too lazy or unqualified to do it yourself — and if you live in certain metropolitan areas — then you can have one of the “runners” at TaskRabbit do it for you. It’s a service that turns grunt work into a game. Simply post the task that needs to be done, and runners bid for the opportunity to do it.

There is no limit to the types of tasks that are listed. A recent posting asked, “Do you know a Girl Scout from whom I can place a cookie order?” One of the top runners completed the task “Prank Call Our Customer Service Rep” in which he posed as an animal-welfare activist who sought donations to buy Viagra for soon-to-be-extinct Siberian tigers [“Call of Duty,” Wired, Aug 2011].

That’s funny stuff.

I received no compensation from TaskRabbit to write this post, nor am I an angel investor; I just think it’s a great idea. There ought to be a similar service for intellectual tasks…

• Statistician needed to massage data from a research study to get the results we’d like.
• Geometer needed to square a circle and/or trisect an angle.
• Looking for contractor to build real-life version of Waterfall by M. C. Escher.
• Puzzler needed to solve the following slider puzzle:
• Logician needed to figure out who shaves the Barber of Seville.
• Number theorist needed to rationalize the denominator of a fraction with sqrt(pi) in the denominator.
• Spatial geometrician needed to get Dirk Gently’s couch unstuck, or to walk through a revolving door with a pair of skis on her shoulder.
• Engineer and hopeless romantic needed to kiss a girl by walking halfway to her, then halfway again, then halfway again, …

Twin Pi Day

Today (June 23) around 7:41 a.m. and 8:05 a.m., respectively, my twin boys passed 3.1415926 years! My wife didn’t think it was a milestone worthy of celebration, but 154 days ago, I missed the chance to celebrate their e birthday, and I wasn’t going to let another opportunity slip away. This morning, we celebrated with circular pancakes for breakfast, and we had pizza for lunch. Which reminds me of a joke…

What is the volume of a disk with radius z and height a? pi · z · z · a

At lunch, I planned to explain pi to the boys. I wrote the letters p and i on a piece of paper and set the paper on the table, and Eli asked, “Daddy, why did you put pee on the table?” That was the end of the lesson. (It was funny to me that he’d pronounce it this way, since the i at the end of his name makes a long i sound. But then I thought of some other words he knows — such as ski, mi (Spanish for my), and his friend Ani (pronounced ah–nee) from school — and realized his mistake was completely understandable.

In regards to mispronunciation, here’s a thought to ponder.

Ya ever wonder how Euler pronounced Euclid?