## Archive for February, 2016

### Angle of Opportunity

My wife and I noticed that one of our sons has been getting his pants wet while urinating. He’s 8; these things happen. But when it occurred twice on consecutive days, we had reason for concern. When we inquired, he explained, “Sometimes when I start to pee, I hit the back of the seat. So I push my penis down, but then I hit the front of the toilet, and the pee ricochets and gets my pants wet.”

My wife began to pursue a line of investigative questioning, but I stopped her. “This is just simple geometry,” I explained.

I could have predicted my wife’s reaction. She said:

Not

everythinghas to be a math problem.Especiallythis.

Even if that were true (it’s not), this situation still begs for some trigonometric analysis.

I’m just over 6 feet tall, so my fire hose is approximately 20″ above the toilet when I urinate. As shown in Figure A, when I stand a reasonable horizontal distance from the commode, my *angle of opportunity* is approximately 30°.

My son, on the other hand, barely clears 4 feet. His water gun is less than 6″ above the toilet when he urinates, so his angle of opportunity is a mere 20°, as illustrated in Figure B.

The images clearly indicate why mothers tell their sons (and husbands), “Stand closer to the toilet when you go!” Doing so increases the angle of opportunity and thus decreases the likelihood of a “clean-up in Aisle 3.”

But more importantly, the above images and some quick trig calculations show that an adult male — who probably has greater control than a young boy, anyway — also has a 50% greater range through which to aim when making a deposit.

Upon completing my explanation, I turned to my son. “Though it may be harder for you to hit the mark, that doesn’t excuse peeing on your pants. I think you need to be more careful.”

I then addressed my wife. “I also think we need to cut him a little slack on this one.”

“And *I* think,” she said, “that *you* are absolutely unbelievable.”

With that, she excused herself.

I’m not sure where she went, but I suspect it was to text one of her friends about how lucky she was: not only is her husband good at math, but he can apply it in extremely esoteric situations.

—

Rather remarkably, there has actually been serious scientific investigation into this phenomenon:

More importantly, there are a number of jokes at the intersection of math and urination:

Why do statisticians choose the last urinal?

Because there’s only a 50% change of being splashed by someone else.What’s in the toilet of the math department restroom?

A natural log.What does a mathematician call a toilet seat?

An ass-toroid.

### Mathematically Unconscious

Both of my sons sleepwalk. At least once a week, one of them will wake up an hour after bedtime, walk down the stairs, and start speaking gibberish. They have no idea what they’re saying, because they aren’t awake — even though their eyes are open. (Freaky!)

During a recent somnambulation, Alex stood at the top of the stairs. He appeared frustrated. Finally, he said:

I just need to find the numbers. It shouldn’t take long.

As you might well imagine, it’s a little scary to have your son walking and talking while asleep. The only solace is that his subconscious thoughts are about math.

I don’t sleepwalk. But I recently had a dream in which I attended a cocktail party and asked the other attendees a most unusual question:

I suspect that my 7 years as an editor and 4 years as a question writer for MathCounts are to blame, but that doesn’t make it any easier to accept.

I vividly remember a dream I had in college, on the night prior to my Linear Algebra midterm. Feeling unprepared for the exam, my nightmare consisted of two brackets pinching my head like a vice, while numbers floated past.

I awoke in a cold sweat at 5 a.m., and proceeded to a study carrel for more test prep.

I was happy to learn that other folks dream about math, too. While subscribed to a listserve for former instructors of the Center for Talented Youth, I received a message from Mark Jason Dominus that read, “I dreamt of the following problem while I was sleeping last night. When I woke up, I convinced myself that it was a good problem, so I’ve decided to share it.”

The volume of a 3 × 3 × 3 cube is 27 cubic units, and the volume of a 2 × 2 × 1 rectangular prism is 4 cubic units. Theoretically, six prisms should be able to fit inside the cube, with three cubic units empty. But can you arrange six 2 × 2 × 1 prisms so they fit inside a 3 × 3 × 3 cube?

Good luck, and sweet dreams!

### 4 Folds, 40 Theorems, and Chinese New Year

If you asked a mathematician how to make an origami coin purse for Chinese New Year, you might receive the following instructions:

- Rotate a square piece of paper 45°.
- Reflect an isosceles right triangle about a diagonal of the square so that congruent isosceles right triangles coincide.
- Reflect the lower left vertex of the isosceles right triangle so it coincides with the opposite leg, such that the base of the isosceles right triangle and one side of the newly formed isosceles acute triangle are parallel.
- Reflect the other vertex of the isosceles right triangle to coincide with a vertex of the isosceles triangle formed in the previous step.
- Fold down a smaller isosceles right triangle from the top.
- Fill the coin purse with various denominations of currency, such as a $2 bill, a $1 gold coin, a quarter, or a penny.

Don’t like those directions? Perhaps these pictures will help.

To be truly authentic for Chinese New Year, you’ll want to use red paper instead of white.

The Common Core State Standards for Mathematics (CCSSM) have added rigor to geometry by requiring students to prove no fewer than a dozen different theorems. Further, the geometry progressions document formulated by Dr. Hung-Hsi Wu suggests that a high school geometry course should use a transformational approach, basing a majority of the foundational axioms on rotations, translations, and reflections.

What struck me about the origami coin purse was the myriad geometry concepts that could be investigated during this simple paper-folding activity. Certainly, the idea to use origami in geometry class in nothing new, but this one seems to have a particularly high return on investment. Creating the coin purse requires only **four folds**, and though perhaps I lied by claiming **forty theorems** in the title, the number of axioms, definitions, and theorems that can be explored is fairly high. (Claiming “forty theorems” wasn’t so much untruthful as much as it was archaic. The word *forty* has been used to represent a large but unknown quantity, especially in religious stories or folk tales, as in *Ali Baba and the 40 Thieves*, it rained “40 days and 40 nights,” or Moses and the Israelites wandered the desert for 40 years.)

For instance, the fold in Step 3 requires parallel lines, and the resulting figures can be used to demonstrate and prove that parallel lines cut by a transversal result in alternate interior angles that are congruent.

No fewer than four similar isosceles right triangles are formed, and many congruent isosceles right triangles result.

The folded triangle in the image above is an isosceles acute triangle; prove that it’s isosceles. And if you unfold it from its current location, the triangle’s previous location and the unfolded triangle together form a parallelogram; prove it.

The final configuration contains a pair of congruent right triangles; you can use side-angle-side to prove that they’re congruent.

But maybe you’re not into proving things. That’s all right. The following are all of the geometric terms that can be used when describing some facet of this magnificent creation.

- Parallel
- Congruent
- Similar
- Alternate
- Angle
- Square
- Reflection
- Rotation
- Translation
- Transversal
- Bisect
- Coincide
- Right Angle
- Perpendicular
- Isosceles
- Vertex
- Triangle
- Trapezoid
- Parallelogram
- Pentagon

And the following theorems and axioms can be proven or exemplified with this activity.

- Sum of angles of a triangle is 180°.
- Alternate interior angles formed by a transversal are congruent.
- Any quadrilateral with four congruent sides is a parallelogram.
- Congruent angles are formed by an angle bisector.
- Triangles with two sides and the included angle congruent are congruent.

There is certainly more to be found. What do you see?