That’s right, I did say kids. This cool app was designed and programmed completely by two high school students in California. ABC News has a nice article about them.

The 87th Carnival of Mathematics has arrived!! Here’s a simple computation for you:

What is the sum of the squares of the first four prime numbers?

That’s right, it’s

Good job. Now, onto the carnival. This is my first carnival, so hopefully I’ll do all these posts justice. We had lots of great submissions, so I encourage you to read through this with a fine-toothed comb. Enjoy!

Rants

Here’s a post (rant) from Andrew Taylor regarding the coverage from the BBC and the Guardian on the Supermoon that occurred in March 2011. NASA reports the moon as being 14% larger and 30% brighter, but Andrew disagrees. Go check out the post, and join the conversation.

Have you ever heard someone abuse the phrase “exponentially better”? I know I have. One incorrect usage occurs when someone makes the claim that something is “exponentially better” based on only two data points. Rebecka Peterson has some words for you here, if you’re the kind of person who says this!

John D. Cook highlights a question you’ve probably heard before: Should you walk or run in the rain? An active discussion is going on in the comments section. It’s been discussed in many other places too, including twice on Mythbusters. (I feel like I read an article in an MAA or NCTM magazine on this topic once, as well. Anyone remember that?)

Murray Bourne submitted this awesome post about modeling fish stocks. Murray says his post is an “attempt to make mathematical modeling a bit less scary than in most textbooks.” I think he achieves his goal in this thorough development of a mathematical model for sustainable fisheries (see the graph above for one of his later examples of a stable solution under lots of interesting constraints). If I taught differential equations, I would absolutely use his examples.

Last week I highlighted this new physics blog, but I wanted to point you there again: Go check out Five Minute Physics! A few more videos have been posted, and also a link to this great video about the physics of a dropping Slinky (see above).

Statistics, Probability, & Combinatorics

Mr. Gregg analyzes European football using the Poisson distribution in his post, The Table Never Lies. I liked how much real world data he brought to the discussion. And I also liked that he admitted when his model worked and when it didn’t–he lets you in on his own mathematical thought process. As you read this post, you too will find yourself thinking out loud with Mr. Gregg.

Card Colm has written this excellent post that will help you wrap your mind around the number of arrangements of cards in a deck. It’s a simple high school-level topic, but he really puts it into perspective:

the number of possible ways to order or permute just the hearts is 13!=6,227,020,800. That’s about what the world population was in 2002. So back then if somebody could have made a list of all possible ways to arrange those 13 cards in a row, there would have been enough people on the planet for everyone to get one such permutation.

I think it’s good to remind ourselves that whenever we shuffle the deck, we can be almost certain that our arrangement has never been created before (since arrangements are possible). Wow!

Alex is looking for “random” numbers by simply asking people. Go contribute your own “random” number here. Can’t wait to see the results!

Quick! Think of an example of a real-world bimodal distribution! Maybe you have a ready example if you teach stat, but here’s a really nice example from Michael Lugo: Book prices. Before you read his post, you should make a guess as to why the book prices he looked at are bimodal (see histogram above).

Philosophy and History of Math

Mike Thayer just attended the NCTM conference in Philadelphia and brings us a thoughtful reaction in his post, The Learning of Mathematics in the 21st Century. Mike wrote this post because he had been left with “an ambivalent feeling” after the conference. He wants to “engage others in mathematics education in discussions about ways to improve what we do outside of the frameworks that are being imposed on us by those outside of our field.” As a secondary educator, I agree with Mike completely and really enjoyed his post. Mike isn’t satisfied with where education is going. In his post, he writes, “We are leaping ahead into the unknown with new educational models, and we never took the time to get the old ones right.”

Edmund Harriss asks Have we ever lost mathematics? He gives a nice recap of foundational crises throughout the history of mathematics, and wonders, ultimately, if we’ve actually lost any mathematics. There’s also a short discussion in the comments section which I recommend to you.

Peter Woit reflects on 25 Years of Topological Quantum Field Theory. Maybe if you have degree in math and physics you might appreciate this post. It went over my head a bit, I’m afraid!

Book Reviews

In this post, Matt reviews a 2012 book release, Who’s #1, by Amy N. Langville and Carl D. Meyer. The book discusses the ranking systems used by popular websites like Amazon or Netflix. His review is thorough and balanced–Matt has good things to say about the book, but also delivers a bit of criticism for their treatment of Arrow’s Impossibility Theorem. Thanks for this contribution, Matt! [edit: Thanks MATT!]

Shecky R reviews of David Berlinski’s 2011 book, One, Two Three…Absolutely Elementary mathematics in his Brief Berlinski Book Blurb. I’m not sure his review is an *endorsement*. It sounds like a book that only a small eclectic crowd will enjoy.

Peter Rowlett also weighs in on the recent news about a German high school boy who has (reportedly) solved an open problem. Many news sources have picked up on this, and I’ve only followed the news from a distance. So I was grateful for Peter’s comments–he questions the validity of the news in his recent post “Has schoolboy genius solved problems that baffled mathematicians for centuries?” His comments in another recent post are perhaps even more important though–Peter encourages us to think of ways we can remind our students that lots of open problems still exist, and “Mathematics is an evolving, alive subject to which you could contribute.”

Jess Hawke IS *Heptagrin Girl*

Here’s a fun-loving post about Heptagrins, and all the crazy craft projects you can do with them. Don’t know what a Heptagrin is? Neither did I. But go check out Jess Hawke’s post and she’ll tell you all about them!

Any Lewis Carroll lovers out there? Julia Collins submitted a post entitled “A Night in Wonderland” about a Lewis Carroll-themed night at the National Museum of Scotland. She writes, “Other people might be interested in the ideas we had and also hearing about what a snark is and why it’s still important.” When you check out this post, you’ll not only learn about snarks but also about creating projective planes with your sewing machine. Cool!

Mike Croucher over at Walking Randomly gives a shout out to the free software Octave, which is a MATLAB replacement. Check out his post, here. MATLAB is ridiculously expensive, and so the world needs an alternative like Octave. He provides links to the Kickstarter campaign–and Mike has backed the project himself. I too believe in Octave. I’ve used it a few times for my grad work and I’ve been very grateful for a free alternative to MATLAB.

The End

Okay, that’s it for the 87th Carnival of Mathematics. Hope you enjoyed all the posts! Sorry it took me a couple days to post it–there was a lot to digest :-).

If you missed the previous carnival (#86), you can find it here. The next carnival (#88) will be hosted by Christian at checkmyworking.com. For a complete listing of all the carnivals, and more information & FAQ about the carnivals, follow this link.

Shout out to Chase Martin, who has just started a great physics blog, Five Minute Physics. My friend Chase and I have a lot in common:

Our names

Our love for juggling

Our love for math & physics

Our love for teaching

Our blogs

Chase is awesome, and you’ll love his fun-loving lecture style in these videos. His goal is ambitious: to put the entire lecture content of his high school physics course on youtube. Wow! File this under ‘flipping the classroom.’

Here are a few of his first videos for your enjoyment.

By the way, the perfect place to collect random photos and other things you love is on Pinterest. I’ve been collecting pins for the last year or so, and you may want to check out these two boards of mine, at least:

By now perhaps you’ve seen this floating around the internet. It was reported here and here and here and here, at least.

Physicist Dmitri Krioukov got a $400 ticket for not making a full stop at at stop sign. He wrote a paper explaining why the police officer could have been wrong, went to court, and got the fine lifted.

If you haven’t read the paper, I encourage you to do it. It’s fairly short and only requires knowledge of a Calculus. Here is a direct link to the pdf. Here’s the abstract:

We show that if a car stops at a stop sign, an observer, e.g., a police officer, located at a certain distance perpendicular to the car trajectory, must have an illusion that the car does not stop, if the following three conditions are satisfied: (1) the observer measures not the linear but angular speed of the car; (2) the car decelerates and subsequently accelerates relatively fast; and (3) there is a short-time obstruction of the observer’s view of the car by an external object, e.g., another car, at the moment when both cars are near the stop sign.

What do you think? Is Professor Krioukov just trying to buffalo the court, or does he have a legitimate case? I guess if there’s any doubt at all about his guilt, then he should be forgiven the fine. And that’s what the court did rule.

However, there is one particular assumption that he makes which is absolutely way-off (look at the paper and notice the key on Figure 3, labeling the blue curve). But the court certainly didn’t catch that. Like I said, he may have been trying to pull the wool over the eyes of the court with lots of math and physics.

Despite the fact that he published his paper on April 1st, I do think this story is true. Like I said, though, the paper does contain an error. So despite all of his good effort, I think he should have been given the ticket. The court didn’t notice it, but he pulled a fast one on them! (pun intended!)

Suppose two equally weighted cars collide in a head-on collision, each traveling at 50 miles per hour. Do you think that the impact for one car will be more severe on the car and driver than the impact of that car’s hitting a brick wall?

To be fair, we have to assume that neither the cars nor the wall compress at all. If the wall is as soft as a pillow, I’ll take the wall every time.

Marilyn vos Savant’s recent column in Parade Magazine says that hitting an oncoming car in that way is no more severe than hitting a solid wall. They both stop dead, whether the wall or the other car causes it.

Each experiences a momentum change that is the same as if they hit a wall, not twice as much. That’s clear when I think of it now, using the law that momentum = impulse (that is, mass * velocity = force * time) but I’ve been mistaken when I’ve only thought about it casually, thinking it must be a 100 mph impact..

If a bike hits a car head-on, the situation is different, because the “bike-car” combination will continue to move in the direction of the car, so my intuition is correct in that case: The bike driver fares worse than the car driver. Comments at Marilyn vos Savant’s blog say as much.

I used to think that car bumpers that collapse at the slightest impact were poorly made. In fact, if momentum is constant, extending the time of impact will decrease the force, to keep force * time constant.

Give me “cheap” bumpers and a wall made of pillows every time.

Hat tip to Hackaday for this one. It turns out that a nanosecond is the time it takes light to travel 11.8 inches. A very handy rule of thumb to have around as you explain large distances and small times.

Even for those of us who finished high school algebra on a wing and a prayer, there’s something compelling about equations. The world’s complexities and uncertainties are distilled and set in orderly figures, with a handful of characters sufficing to capture the universe itself.

For your enjoyment, the Wired Science team has gathered nine of our favorite equations. Some represent the universe; others, the nature of life. One represents the limit of equations.

We do advise, however, against getting any of these equations tattooed on your body, much less branded. An equation t-shirt would do just fine.

The Beautiful Equation: Euler’s Identity

Also called Euler’s relation, or the Euler equation of complex analysis, this bit of mathematics enjoys accolades across geeky disciplines.

Swiss mathematician Leonhard Euler first wrote the equality, which links together geometry, algebra, and five of the most essential symbols in math — 0, 1, i, pi and e — that are essential tools in scientific work.

Theoretical physicist Richard Feynman was a huge fan and called it a “jewel” and a “remarkable” formula. Fans today refer to it as “the most beautiful equation.”

I was glad to see that the first “must have” equation was Euler’s Identity (note that “Euler’s Identity” is the accepted name for this, not to be confused with Euler’s Formula or Euler’s Polyhedron Formula or any of the other amazing facts named for Euler). I think there’s large consensus in the math community that this is, indeed, a breathtaking equation. It may not be the most fundamentally important, but it definitely showcases why mathematicians delight in math.

I’m ashamed to say it, but I hardly knew any of the other equations. I knew Boltzman’s equation; Maxwell’s equations and Schrödinger’s equation have come up in some of my graduate coursework, but the others I hadn’t ever seen. One might argue that the other equations are not so important. (If you like arguing about such things, join those commenting on the article). You should still look through the list yourself; how many of these equations do you know?

Granted, this was a general article that encompased all “true geeks” not just math geeks. But still, don’t we all want to be a true geek?

(Oh, and happy birthday to Johan (III) Bernoulli, who had no notable equations named for him :-))

Today we had a “Tower of Terror” competition in our Calculus classes (just for Halloween fun :-)). The rules are: You get 5 pieces of 8.5″x11″ paper and 8″ of masking tape. The goal is to build the tallest free-standing tower.

Here’s a photo of the best tower from all the classes, standing at an amazing 57″. Great job, ladies!

Designing a good roller-coaster loop is a balancing act. The coaster will naturally slow down as it rises, so it has to enter the loop fast enough to make it up and over the top. The curving track creates a centripetal force, causing the cars to accelerate toward the center of the loop, while momentum sweeps them forward. Loose objects like riders are pinned safely to their seats. The acceleration gives the ride its visceral thrill, but it also puts stress on the fragile human body—and the greater the velocity, the greater the centripetal acceleration.

Coney Island’s Flip-Flap Railway, built in 1895, reached a neck-snapping 12 times the force of gravity at the bottom of its loop—more than enough to induce what pilots call G-LOC, or gravity-induced loss of consciousness. In other words, riders often passed out. In fact, any vehicle trying to get around a perfectly circular loop has to hit at least 6 g’s—enough to render most people unconscious.

To solve the problem, modern designers adopted an upside-down teardrop shape called a clothoid, in which the track curves more sharply up top than at the bottom. Then most of the turn happens at the peak, when the coaster is moving the slowest and the acceleration is least. Result: no G-LOC, just screams. The formula that helped them do it? a_{c} = v^{2}⁄ r.