Reblog: Patty Paper Trisection

[Feature photo above by Michael Cory via Flickr (CC BY 2.0).]

I hear so many people say they hated geometry because of the proofs, but I’ve always loved a challenging puzzle. I found the following puzzle at a blog carnival during my first year of blogging. Don’t worry about the arbitrary two-column format you learned in high school — just think about what is true and how you know it must be so.

I hope you enjoy this “Throw-back Thursday” blast from the Let’s Play Math! blog archives:

One of the great unsolved problems of antiquity was to trisect any angle using only the basic tools of Euclidean geometry: an unmarked straight-edge and a compass. Like the alchemist’s dream of turning lead into gold, this proved to be an impossible task. If you want to trisect an angle, you have to “cheat.” A straight-edge and compass can’t do it. You have to use some sort of crutch, just as an alchemist would have to use a particle accelerator or something.

One “cheat” that works is to fold your paper. I will show you how it works, and your job is to show why …

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More Than One Way To Find the Center of a Circle

[Feature photo above by hom26 via Flickr.]

My free time lately has gone to local events and to book editing. I hope to put up a series of blog posts sometime soon, based on the Homeschool Math FAQs chapter I’m adding to the paperback version of Let’s Play Math. [And of course, I’ll update the ebook whenever I finally publish the paperback, so those of you who already bought a copy should be able to get the new version without paying extra.]

But in the meantime, as I was browsing my blog archives for an interesting “Throw-Back Thursday” post, I stumbled across this old geometry puzzle from Dave Marain over at MathNotations blog:

Is it possible that AB is a chord but NOT a diameter? That is, could circle ABC have a center that is NOT point O?

Jake shows Jack a piece of wood he cut out in the machine shop: a circular arc bounded by a chord. Jake claimed that the arc was not a semicircle. In fact, he claimed it was shorter than a semicircle, i.e., segment AB was not a diameter and arc ACB was less than 180 degrees.

Jack knew this was impossible and argued: “Don’t you see, Jake, that O must be the center of the circle and that OA, OB and OC are radii.”

Jake wasn’t buying this, since he had measured everything precisely. He argued that just because they could be radii didn’t prove they had to be.

Which boy do you agree with?

• Pick one side of the debate, and try to find at least three different ways to prove your point.

If you have a student in geometry or higher math, print out the original post (but not the comments — it’s no fun when someone gives you the answer!) and see what he or she can do with it.

Dave offers many other puzzles to challenge your math students. While you are at his blog, do take some time to browse past articles.

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Logic: The Centauri Challenge

Another fun discovery from the #MTBoS Challenge: Brian Miller (@TheMillerMath) posted this interstellar puzzle on his blog today.

More Logic Puzzles

If you liked the Centauri Challenge, you may also enjoy the following blog posts:

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Math That Is Fun: Infinite Primes

Oh, my! Ben Orlin over at Math with Bad Drawings just published my new favorite math proof ever:

I had a fight with Euclid on the nature of the primes.
It got a little heated – you know how the tension climbs.

It started out most civil, with a honeyed cup of tea;
we traded tales of scholars, like Descartes and Ptolemy.
But as the tea began to cool, our chatter did as well.
We’d had our fill of gossip. We sat silent for a spell.
That’s when Euclid turned to me, and said, “Hear this, my friend:
did you know the primes go on forever, with no end?” …

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What Is a Proof?

I’ve been enjoying the Introduction to Mathematical Thinking course by Keith Devlin. For the first few weeks, we mostly talked about language, especially the language of logical thinking. This week, we started working on proofs.

For a bit of fun, the professor emailed a link to this video. My daughter Kitten enjoyed it, and I hope you do, too.

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The World of Mathematical Reality

I wanted to include this video last week when I mentioned Paul Lockhart’s new book, but I couldn’t figure out how to copy it from Amazon. So today I read Shecky’s review of Measurement, which included the YouTube video. Thanks, Shecky!

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Lockhart’s Measurement

After watching the video on the Amazon.com page, this book has jumped to the top of my wish list.

You may have read Paul Lockhart’s earlier piece, A Mathematician’s Lament, which explored the ways that traditional schooling distorts mathematics. In this book, he attempts to share the wonder and beauty of math in a way that anyone can understand.

According to the publisher: “Measurement offers a permanent solution to math phobia by introducing us to mathematics as an artful way of thinking and living. Favoring plain English and pictures over jargon and formulas, Lockhart succeeds in making complex ideas about the mathematics of shape and motion intuitive and graspable.”

If you take any 4-sided shape at all — make it as awkward and as ridiculous as you want — if you take the middles of the sides and connect them, it always makes a parallelogram. Always! No matter what crazy, kooky thing you started with.

That’s scary to me. That’s a conspiracy.

That’s amazing!

That’s completely unexpected. I would have expected: You make some crazy blob and connect the middles, it’s gonna be another crazy blob. But it isn’t — it’s always a slanted box, beautifully parallel.

WHY is it that?!

The mathematical question is “Why?” It’s always why. And the only way we know how to answer such questions is to come up, from scratch, with these narrative arguments that explain it.

So what I want to do with this book is open up this world of mathematical reality, the creatures that we build there, the questions that we ask there, the ways in which we poke and prod (known as problems), and how we can possibly craft these elegant reason-poems.

— Paul Lockhart
author of Measurement

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Sample from the Introduction to Mathematical Thinking Class

I’m really looking forward to Keith Devlin’s free Introduction to Mathematical Thinking class, which starts in mid-September. There are more than 30,000 nearly 40,000 students signed up already. Will you join us?

These days, mathematics books tend to be awash with symbols, but mathematical notation no more is mathematics than musical notation is music.

A page of sheet music represents a piece of music: the music itself is what you get when the notes on the page are sung or performed on a musical instrument. It is in its performance that the music comes alive and becomes part of our experience. The music exists not on the printed page but in our minds.

The same is true for mathematics. The symbols on a page are just a representation of the mathematics. When read by a competent performer (in this case, someone trained in mathematics), the symbols on the printed page come alive — the mathematics lives and breathes in the mind of the reader like some abstract symphony.

— Keith Devlin
Introduction to Mathematical Thinking

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Thinking (and Teaching) like a Mathematician

photos by fdecomite via flickr

Most people think that mathematics means working with numbers and that being “good at math” means being able to do (only slower) what any \$10 calculator can do. But then, most people think all sorts of silly things, right? That’s what makes “man on the street” interviews so funny.

Numbers are definitely part of math — but only part, and not even the biggest part. And being “good at math” means much more than being able to work with numbers. It means making connections, thinking creatively, seeing familiar things in new ways, asking “Why?” and “What if?” and “Are you sure?”

It means trying something and being willing to fail, then going back and trying something else. Even if your first try succeeded — or maybe, especially if your first try succeeded. Just knowing one way to do something is not, for a mathematician, the same as understanding that something. But the more different ways you know to figure it out, the closer you are to understanding it.

Mathematics is not just memorizing and following rules. If we want to teach real mathematics, we teachers need to learn to think like mathematicians. We need to see math as a mental game, playing with ideas. James Tanton explains:

Homeschooling is much more than just doing school at home — it’s a lifelong lifestyle of learning. And thanks to the modern miracle of inter-library loan, even those of us who live in the middle of nowhere can get just about any book sent directly to our tiny home-town libraries.

As I mentioned in Math Teachers at Play 46, I’m trying to add more living books about math to our homeschool schedule, including my own self-education reading. So, a copy of Fermat’s Enigma: The Epic Quest to Solve the World’s Greatest Mathematical Problem finally showed up at my library, and I am thoroughly enjoying it.

A Mathematician for President

[Image courtesy of the Images of American Political History.]

In 1876, a politician made mathematical history. James Abram Garfield, the honorable Congressman from Ohio, published a brand new proof of the Pythagorean Theorem in The New England Journal of Education. He concluded, “We think it something on which the members of both houses can unite without distinction of party.”

Remember the Math Adventurer’s Rule: Figure it out for yourself! Whenever I give a problem in an Alexandria Jones story, I will try to post the answer soon afterward. But don’t peek! If I tell you the answer, you miss out on the fun of solving the puzzle. So if you haven’t worked these problems yet, go back to the original post. Figure them out for yourself — and then check the answers just to prove that you got them right.

Euclid’s Geometric Algebra

Euclid’s Geometric Algebra

Picture from MacTutor Archives.

After the Pythagorean crisis with the square root of two, Greek mathematicians tried to avoid working with numbers. Instead, the Greeks used geometry to demonstrate mathematical concepts. A line can be drawn any length, so straight lines became a sort of non-algebraic variable.

You can see an example of this in The Pythagorean Proof, where Alexandria Jones represented the sides of her triangle by the letters a and b. These sides may be any length. The sizes of the squares will change with the triangle sides, but the relationship $a^2 + b^2 = c^2$ is always true for every right triangle.

The Pythagorean Proof

[In the last episode, Alexandria Jones received a letter from archaeologist Sofia Theano, asking for help with a Pythagorean puzzle.]

Answers to Leon’s Figurate Number Puzzles

Remember the Math Adventurer’s Rule: Figure it out for yourself! Whenever I give a problem in an Alexandria Jones story, I will try to post the answer soon afterward. But don’t peek! If I tell you the answer, you miss out on the fun of solving the puzzle. So if you haven’t worked these problems yet, go back to the original post. Figure them out for yourself — and then check the answers just to prove that you got them right.

Backwards Math

Photo by Complicated.

Princess Kitten is recovering from her cold and getting some energy back. She came to me and said wistfully, “I wish I could do backwards math.”

I looked up from my keyboard. “Backwards math? What do you mean?”

“Umm. It’s kinda hard to explain, but I can show you.”

April Fool’s Day: Fun with Math Fallacies

Photo by RBerteig.

Take a break from “serious” math and have a little fun today with some classics of recreational mathematics. Do you have a favorite math or logic fallacy? Please share it in the Comments below.

Rewriting the History of Math

Here are a couple of quick links to math in the news:

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Hints and Solutions: Patty Paper Trisection

No peeking! This post is for those of you who have given the trisection proof a good workout on your own. If you have a question about the proof or a solution you would like to share, please post a comment here.

But if you haven’t yet worked at the puzzle, go back and give it a try. When someone just tells you the answer, you miss out on the fun. Figure it out for yourself — and then check the answer just to prove that you got it right.

Puzzle: Patty Paper Trisection

[Feature photo above by Michael Cory via Flickr (CC BY 2.0).]

One of the great unsolved problems of antiquity was to trisect any angle using only the basic tools of Euclidean geometry: an unmarked straight-edge and a compass. Like the alchemist’s dream of turning lead into gold, this proved to be an impossible task. If you want to trisect an angle, you have to “cheat.” A straight-edge and compass can’t do it. You have to use some sort of crutch, just as an alchemist would have to use a particle accelerator or something.

One “cheat” that works is to fold your paper. I will show you how it works, and your job is to show why.

Historical Tidbits: Alexandria Jones

[Read the story of the pharaoh’s treasure: Part 1, Part 2, and Part 3.]

Here are a few more tidbits from math history, along with links to relevant Internet sites or books, and three more math puzzles for you to try. I hope you find them interesting.

Next time, a new adventure (sort of)…

The Secret of the Pharaoh’s Treasure, Part 3

[In the last episode, Alexandria Jones discovered a mysterious treasure: three wooden sticks, like tent pegs, and a long loop of rope with 12 evenly spaced knots. Her father explained that it was an ancient Egyptian surveyor’s tool, used to mark right angles.]

Back at the camp, Fibonacci Jones stacked multi-layer sandwiches while Alexandria poured milk and set the table for supper.

“Geometry,” Fibonacci said.

“What?”

Geo means earth, and metry means to measure. So geometry means to measure the earth. That is what the Egyptian rope stretches did.”

Alex thought for a moment. “So in the beginning, math was just surveying?”

“And taxes…”

Geometry: Can You Find the Center of a Circle?

For the last couple of days, I have been playing around with this geometry puzzle. If you have a student in geometry or higher math, I recommend you print out the original post (but not the comments — it’s no fun when someone gives you the answer!) and see what he or she can do with it.

[MathNotations offers many other puzzles for 7-12th grade math students. While you are at his blog, take some time to browse past articles.]

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The Secret of the Pharaoh’s Treasure, Part 2

[In the last episode, Alexandria Jones, daughter of the world-famous archaeologist, caught her father’s arch-enemy trying to uncover the Pharaoh’s Treasure.]

…”I can’t believe it!” Simon Skulk threw down the last stone in disgust and walked away. At the mouth of the cave, he turned back and shook his fist. “You haven’t seen the last of me, Alexandria Jones.”

Her muscles aching, Alex sank to the ground and hugged her dog. The she gave him a little push toward the front of the cave. “Rammy, go get Dad.”

Ramus barked once and took off running.

Alex turned back to look at the Pharaoh’s Treasure. Where the last stone had stood was a hole. In the hole lay three wooden sticks, like tent pegs, and a long loop of rope with 12 evenly-spaced knots.

What could it be?

Carnival of Mathematics, ordinal 5

I missed getting an entry into the latest Carnival of Mathematics, which went up a day early at Science and Reason. (Serves me right for procrastinating!) As usual, most of the articles are well over my head.

The carnival begins with a tribute to Field’s Medalist Paul Cohen (April 2, 1934 – March 23, 2007), the man who settled the first of the famous Hilbert Problems, the Continuum Hypothesis. Then come the math articles. Here are my favorites:

Formalisation and de-formalisation [That page has disappeared, but I think this one will work. If that link goes AWOL, too, just do a search for “Scooping the Loop Snooper” by Geoffrey K. Pullum.]
In which we find a delightful informal proof that the Halting Problem is undecidable. Wouldn’t it be fun if all math proofs could be written in Dr. Seuss-style verse?

Introducing the Surreal Numbers (Edited rerun)
Which explains a method to create and represent the real numbers that I could almost convince myself I understood.

The old new math
In which JD teaches his algebra class a bit of twentieth-century history. If you aren’t familiar with Jonathan’s blog, be sure to spend some time browsing his “puzzle” posts.

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Math Quotes VII: Problems Worthy of Attack

Time to catch up on our blackboard quotes.