This was posted recently on FAIL blog. I try to be a bit nicer when I’m grading. It’s still funny though.
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Pythagorean Theorem
Just when I said the other day that we care less about the ancient Greek mathematicians than we did 200 years ago, I turn the attention of my readership to the Pythagorean Theorem. It’s a simple repost: I liked the following video that Denise @ Let’s Play Math posted yesterday and wanted to share it. It also connects nicely to the post my dad made the other day about beautiful proofs.
Fractals
Just read two nice posts about fractals today. One from wired.com about electricsheep.com, which produces these amazing fractal videos like the one below by using crowd computing power
And also, I just saw this post from Walking Randomly that talks about the extensive abilities of wolframalpha to generate fractals on command.
Give the Babylonian’s some credit
So says this CNN article from Friday.
Over 1,000 years before Pythagoras was calculating the length of a hypotenuse, sophisticated scribes in Mesopotamia were working with the same theory to calculate the area of their farmland.
Working on clay tablets, students would “write” out their math problems in cuneiform script, a method that involved making wedge-shaped impressions in the clay with a blunt reed.
These tablets bear evidence of practical as well as more advanced theoretical math and show just how sophisticated the ancient Babylonians were with numbers — more than a millennium before Pythagoras and Euclid were doing the same in ancient Greece.
“They are the most sophisticated mathematics from anywhere in the world at that time,” said Alexander Jones, a Professor of the History of the Exact Sciences in Antiquity at New York University.
He is co-curator of “Before Pythagoras: The Culture of Old Babylonian Mathematics,” an exhibition at the Institute for the Study of the Ancient World in New York.
[Hat tip: Mr. Gherman]
Google Ngram
This is super fun. Google has just released this tool for playing with word frequency data from a huge amount of scanned literature (5 million books dating as far back as 500 years). You can read more about it here, including some nice research that’s already being done with the full data set that’s also been released. (also here)
For example, here’s a graph of the appearance of the word “homeschool” in the collective Google corpus.
You can also compare the appearance of words. For example, here’s informal evidence that we care less about ancient Greek mathematicians (BC) and more about European mathematicians (17th and 18th century) than we did 100 years ago.
Not very rigorous, I’ll admit. But it’s an example of what kind of interesting trends can be instantly teased out. As this article quotes Erez Lieberman-Aiden of Harvard University, “It’s not just an answer machine. It’s a question machine.” I think that’s a nice way to put it.
Mathematical modeler fails to learn from history
Posted by guest blogger Dr. Gene Chase.
For all of you who love mathematical modeling and love (unintentional) humor, here’s a link for you.
Apparently researcher M. M. Tai invented a method for finding the area under “glucose tolerance and other metabolic curves,” a method which has now come to be called — in American Diabetes Association circles at least — “Tai’s model.”
We of course call it the Trapezoidal Rule. ::sigh:: As historian George Santayana once said, “Those who cannot remember the past are condemned to repeat it.”
[Hat off to Slashdot for breaking the news.]
The Important Theorems Are the Beautiful Ones
Dr. Gene Chase guest blog author here again.
What makes a math theorem important?
The usual answer is that it is either beautiful or useful. If like me you think that being useful is a beautiful thing, then important theorems are the beautiful ones.
But what makes a theorem beautiful? For example, why is the Theorem of Pythagoras widely regarded as beautiful: and a, b, and c are not 0 if and only if a, b, and c are the sides of a right triangle? (OK, break into small groups and discuss this among yourselves! An answer appears at the bottom of this post.)
But the theorem 1223334444 = 1223334443 + 1 is not beautiful, won’t you agree?
If the theorem is geometric, we can appeal to visual beauty. For example, three circles pairwise tangent have a beautiful property that is animated here.
But beautiful theorems do not have to be geometric. Numbers are beautiful. For example, Euclid’s theorem that there are an infinity of primes is beautiful. No one has been able to draw a beautiful picture about that, although people have tried from astronomer and mathematician Eratosthenes in 200 BC to science fiction writer and mathematician Stanislaw Ulam in 1963.
For $15 you can have a mathematical theorem named after you. But I can guarantee that it won’t be beautiful. So if you want a theorem named after you, give Mr. Chase the $15 instead and he’ll find one for you. Don’t use 1223334444 = 1223334443 + 1. I claim that as “Dr. Gene Chase’s theorem.”
Answer to discussion question above: Most folks say that a beautiful theorem has to be “deep,” which is just a metaphor for “having many connections to many other things.” For example, the Theorem of Pythagoras has to do with areas, not squares specifically. The semicircle on the hypotenuse of a right triangle has an area equal to the sum of the areas of the semicircles on the adjacent sides. And so for any three similar figures.
Do you remember the joy that you feel when you first learned that two of your friends are also friends of each other? That’s the joy that a mathematician feels when she discovers that the Theorem of Pythagoras and the Theorem of Euclid are intimate with each other. But I’ll leave that connection to another post.
Math is about surprising connections. Which is to say, it’s about beauty.
Math Mondays
If I haven’t said so before, you should check out the Museum of Mathematics’ Math Monday, written by George Hart, whose work I love (I think every fan of math loves his work). Last week’s Math Monday was on cross sections of bagels. This week Hart is making sculptures from egg cartons. Check it out!
The Lecture
The Lecture. Is it so bad? This recent post by Dr. John Fea at his history blog addresses this question (be sure to check out the links at the bottom too). And I’d like to weigh in on the issue too.
The trendy education gurus would tell tell you ‘lecturing is bad.’ As my students know, I use a mixture of lecture, guided practice, group work, games, etc. Even though I’d love to say I do mostly creative out-of-the-box activities, the truth is that I mostly lecture. I’m not sure that’s entirely bad, though.
When I was in college, I really enjoyed a good lecture–key word good. Dr. Fea was one of my history professors and he was a fantastic lecturer. He would get into it, he moved around, he was well-spoken, and he knew his stuff. I enjoyed many of my other professors for the same reasons (one of my philosophy professors, in particular). And I still enjoy listening to good lectures when I get the chance. A good lecture captures the audience like any other good performance would. Why do you think the lectures on TED.com are so popular?
I’m going to keep lecturing. In fact, I have a great lecture planned for my Precalculus class tomorrow and I’m excited to give it. I’ll be presenting a beautiful proof, and I’ve got a well-planned powerpoint to go with it. In fact, for this particular lecture, I tell the students they don’t even have to take notes…just soak it in!
Three cheers for the lecture! 🙂
Cryptography
Sorry for not posting much recently. The reason is simply life’s busyness, partially due to work I’ve been doing for my cryptography class (I’m getting my masters). We just took the final exam last night, so perhaps I’ll be blogging more in the coming month. That being said, I enjoyed the class and learned lots about cryptography.
Much of my time recently was spent working on our final paper. I, with a colleague of mine (Mr. Davis!), chose to investigate secure variants on the Hill Cipher, which is a matrix multiplication encryption scheme first purposed in 1929 by Lester Hill. We implemented the modern versions of the cipher with computer programs and learned quite a bit in the process. Here’s a link to our paper and our presentation.
Also related to cryptography, I’ve been following a bit of the news on Kryptos, a sculpture at the CIA created by Jim Sanborn. On it there are four ciphers, three of which have been cracked. As of the writing of this post, the fourth has yet to be cracked. But just a few weeks ago (Nov 20), “Berlin” was revealed by Sanborn as a clue to the last unsolved section of the cipher. This information was released by the New York Times, but here’s the wired.com article where I first read about it. The clue decrypts 6 ciphertext letters of the last section. This is what cryptanalysts call a ‘crib.’ And most classic ciphers like substitution, shift, and the hill cipher easily succumb to a known-plaintext attack: If you give me some plaintext and the corresponding ciphertext, I can recover the key. It doesn’t sound like this last section is quite as easy as the classic ciphers, though.
One of the other project presentations in our cryptography class was on Kryptos. I also found a presentation on the recently cracked chaocipher very engaging.
Random Walks 