So the Futility Closet, a constant source of neat results, puzzles, quotes and more, posted this pretty result:
As with many visual theorems, my first impulse was to make a dynamic visualization. Off to GeoGebra!
But I quickly hit a snag... I didn't know how to construct the tangent circles. I made the basic set up and then starting monkeying around. Eventually I thought about how to just make a circle tangent to one of the interior circles and the line. I made a tracing point with the distance to the interior circle and the dividing line and traced along where it would be equidistant - a requirement for a circle tangent to both.
Of course - a parabola. To be tangent to a circle and a line is like the definition of a parabola as all the points equidistant from the directrix and focus. The point D had to be on the parabola, which helped me to find the directrix. The focus had to be the center of the circle.
So then the other parabolas weren't too hard to find. The center of one of the twins had to be both on a parabola of tangents to an interior circle and the line, and a parabola for the enclosing circle and the line.
And now we can see that Archimedes result was true in general.
This is a special case of the Apollonian Circle problem (finding a circle tangent to three non-concentric circles &/or lines), and I feel like it was helpful in deepening my understanding of that. To be specific, a special case of the Circle-Circle-Line special case. But it was fun.
Now that we can construct them, how would you prove the twin-ness of these circles?
The sketch is, as usual, up on GeoGebraTube for you to play with yourself.
Postscript: the always educational Alexander Bogomolny (proprietor of Cut-the-Knot) had this to add on Google+:
They are no longer twins. E.g., Circle Triplets.
But there are more, Arbelos, and even more: C.W. Dodge, T. Schoch, P.Y. Woo, and P. Yiu, Those ubiquitous Archimedean circles, Mathematics Magazine, Vol. 72 (1999), 202-213. (JSTOR) and Some More Archimedean Circles in the Arbelos, Frank Power, Forum Geometricorum, 2005. (postscript).
Tuesday, March 25, 2014
Gaussian Skirmish
I have a particular fondness for Gauss, as he was my entry into math history.
John Hocking, whom I usually refer to as my undergraduate math guru, was, by great fortune, my math teacher for the first two years at Michigan State. A good researcher himself, he brought math alive with deep connections, intriguing applications, lively stories of discovery and some history of the great mathematicians. It had never really occurred to me to wonder who invented the stuff we were studying. Unthinkable in most other subjects.
But when Karl Friedrich Gauss came up, the usually passionate Gibb kicked it into another gear. "This is a man whose name should be as famous as Beethoven's!" And yet none of us had heard of him! "Like the magnetic unit?" one person asked.
In that pre-Google world of 1982, it took some effort to find out more.
One day in our capstone class is devoted to Gauss, then. Given an hour to talk about him, what should you cover?
Excuse the Google docs, but here's what I share:
One interesting thing in a math history class is that the closer we get to being relevant to today, the harder the math is to understand. And class is diverse in the sense that despite all being math majors, there are future elementary teachers, secondary teachers, grad students and applied mathematicians. For Gauss, he loved number theory so, and modular arithmetic was so crucial to his number theory, I came up with a game to increase experience with modular addition. It seems to me that in a half hour students go from hesitant to fluent, and the discussions at the start of playing are powerful.
These directions are in particularly good shape, as Sue Van Hattum helped edit them for her upcoming book.
Hope you give it a try and enjoy it as much as we do!
P.S. Looking for more Gaussian fun? Here's today's longer than usual list of To Do choices.
To do:
Daily: lots to choose from, of course...
John Hocking, whom I usually refer to as my undergraduate math guru, was, by great fortune, my math teacher for the first two years at Michigan State. A good researcher himself, he brought math alive with deep connections, intriguing applications, lively stories of discovery and some history of the great mathematicians. It had never really occurred to me to wonder who invented the stuff we were studying. Unthinkable in most other subjects.
But when Karl Friedrich Gauss came up, the usually passionate Gibb kicked it into another gear. "This is a man whose name should be as famous as Beethoven's!" And yet none of us had heard of him! "Like the magnetic unit?" one person asked.
In that pre-Google world of 1982, it took some effort to find out more.
One day in our capstone class is devoted to Gauss, then. Given an hour to talk about him, what should you cover?
Excuse the Google docs, but here's what I share:
One interesting thing in a math history class is that the closer we get to being relevant to today, the harder the math is to understand. And class is diverse in the sense that despite all being math majors, there are future elementary teachers, secondary teachers, grad students and applied mathematicians. For Gauss, he loved number theory so, and modular arithmetic was so crucial to his number theory, I came up with a game to increase experience with modular addition. It seems to me that in a half hour students go from hesitant to fluent, and the discussions at the start of playing are powerful.
These directions are in particularly good shape, as Sue Van Hattum helped edit them for her upcoming book.
Hope you give it a try and enjoy it as much as we do!
P.S. Looking for more Gaussian fun? Here's today's longer than usual list of To Do choices.
To do:
Daily: lots to choose from, of course...
- Make sure you understand the Fundamental Theorem of Algebra.
- One neat result of Gauss was proving Fermat’s conjecture that every counting number can be expressed as the sum of three triangular numbers. Good thing to play with, and fun that it connects to Gauss’ 7 year old result.
- work on a classic Gauss problem: how many ways are there to write a whole number as a sum of perfect squares? For example, 5 has 8 ways, not using zeroes but yes using negatives. So (-1)^2+2^2=5 is one way.
- Follow Mike’s process through explaining that problem, Keep notes and make sense yourself of what he and his kids share.
- Gauss Facts (Particularly fun; be sure to check these out.)
- The book Disquisitiones Arithmeticae
- Gauss’ proofs of the Fundamental Theorem (pdf; tough read)
- Gauss was connected with the first great female mathematician of the modern era, Sophie Germain, who had to overcome terrible sexism.
- Gauss was an advisor to Bernhard Riemann, who made groundbreaking advances in geometry by describing geometric surfaces and more with complex-coefficient equations. Also responsible for the mysterious zeta function, and the as-yet-unsolved Riemann hypothesis.
- Jim Loy has a crazy collection of math resources, including a neat page on Gaussian primes (prime complex numbers): http://www.jimloy.com/algebra/gprimes.htm
- Why the bell curve is called a Gaussian distribution, and its odd connection with pi. http://www.cut-the-knot.org/proofs/ClassicalIntegral.shtml
- Gauss’ modular arithmetic is the foundation of Public Key Cryptography. Here’s a clear exposition.
- A pretty clear explanation of the connection between AGM and ellipse perimeter.
Decimal Games: Burger Time
My last three games for Mr. Schiller's 5th graders have all been about
decimals. They worked pretty well, and I definitely tried new things for
myself as an educational game designer.
The first game came when they were first digging into decimal multiplication, just doing whole number times the decimals. I went through a large number of gyrations about a good context, but I kept coming back to measurement. I thought about a race game, where students measured out multiple decimal portions of a meter or centimeter - and I still think that has some potential. If it was warm, especially, I'd love to see them out with meter sticks on a course they made around the school. Of course, we were in the midst of the harshest winter in 70 years. No one even remembered what the sidewalks looked like. I also thought about physically stacking things, but I wasn't sure about what materials we had to stack. I'd like to see more of what game designers call dexterity games in math.
I finally went with burgers - I suppose with Robert Kaplinsky's In-n-Out burger activity rattling around my skull. The game itself is almost more of an activity. Some students built their burgers and didn't care about the game layer on top of that. Choosing ingredients, making the picture - that was very engaging for most of them.
Build a Burger
Who makes the best burger in town?
Materials: 5 dice (or 5 dice per team/player)
Idea: Roll 5 dice to get ingredients for your burger. The numbers correspond to how many mm tall each part is. Three 5s means 3 all beef patties. Two 6s means two layers of bun. You get to reroll one time. Pick the dice you want to reroll.
1 – sauce, .1 cm. Choose from ketchup, mustard, mayo, hot sauce, , barbecue sauce, secret sauce.
2 – cheese, .2 cm.
3 – bacon OR onions (Mix if you have 2 or more)
4 – tomato OR lettuce, .4 cm. (Mix if you have 2 or more)
5 – patty, .5 cm. Maximum: 3 patties, UNLESS you have 3 buns, then you can have 4 patties.
6 – bun layer, .6 cm. Maximum: 3 layers. Every burger needs at least 1 layer of bun!
The choice on the different rolls seemed crucial - their customization increased variety and student ownership. The rerolling mechanic is the child of Yahtzee, of course, but a great way to add choice and chance for redemption.
We launched the game with stories of great burgers with excessive description. I rolled up a burger with help from the class and demonstrated the multiplication. As the students played, they ignored some of the rules - which I see as making the game their own. In the end of class discussion, that gave them plenty of fodder for suggestions for the game. Add another ingredient, let people go meatless or bunless, and - certainly - we should actually make the burgers.
The game was an undeniable hit, and seemed to provide some good experience for multiplication of decimals as groups of decimal quantities. As you can see in the student work below, there was lots of material generated for discussion between student and teacher and students with each other.
And finally, a student who clearly has a future in fast food marketing:
"See, it has meat instead of a bun, but still the regular meat, with the bacon and cheese in the middle."
Here's the form as a pdf if you're interested. Email me for the Word doc if you want to modify.
The first game came when they were first digging into decimal multiplication, just doing whole number times the decimals. I went through a large number of gyrations about a good context, but I kept coming back to measurement. I thought about a race game, where students measured out multiple decimal portions of a meter or centimeter - and I still think that has some potential. If it was warm, especially, I'd love to see them out with meter sticks on a course they made around the school. Of course, we were in the midst of the harshest winter in 70 years. No one even remembered what the sidewalks looked like. I also thought about physically stacking things, but I wasn't sure about what materials we had to stack. I'd like to see more of what game designers call dexterity games in math.
I finally went with burgers - I suppose with Robert Kaplinsky's In-n-Out burger activity rattling around my skull. The game itself is almost more of an activity. Some students built their burgers and didn't care about the game layer on top of that. Choosing ingredients, making the picture - that was very engaging for most of them.
Build a Burger
Who makes the best burger in town?
Materials: 5 dice (or 5 dice per team/player)
Idea: Roll 5 dice to get ingredients for your burger. The numbers correspond to how many mm tall each part is. Three 5s means 3 all beef patties. Two 6s means two layers of bun. You get to reroll one time. Pick the dice you want to reroll.
1 – sauce, .1 cm. Choose from ketchup, mustard, mayo, hot sauce, , barbecue sauce, secret sauce.
2 – cheese, .2 cm.
3 – bacon OR onions (Mix if you have 2 or more)
4 – tomato OR lettuce, .4 cm. (Mix if you have 2 or more)
5 – patty, .5 cm. Maximum: 3 patties, UNLESS you have 3 buns, then you can have 4 patties.
6 – bun layer, .6 cm. Maximum: 3 layers. Every burger needs at least 1 layer of bun!
The choice on the different rolls seemed crucial - their customization increased variety and student ownership. The rerolling mechanic is the child of Yahtzee, of course, but a great way to add choice and chance for redemption.
We launched the game with stories of great burgers with excessive description. I rolled up a burger with help from the class and demonstrated the multiplication. As the students played, they ignored some of the rules - which I see as making the game their own. In the end of class discussion, that gave them plenty of fodder for suggestions for the game. Add another ingredient, let people go meatless or bunless, and - certainly - we should actually make the burgers.
The game was an undeniable hit, and seemed to provide some good experience for multiplication of decimals as groups of decimal quantities. As you can see in the student work below, there was lots of material generated for discussion between student and teacher and students with each other.
And finally, a student who clearly has a future in fast food marketing:
"See, it has meat instead of a bun, but still the regular meat, with the bacon and cheese in the middle."
Here's the form as a pdf if you're interested. Email me for the Word doc if you want to modify.
Sunday, March 9, 2014
Carnival of Mathematics 108
Welcome to this months Carnival of Mathematics!
I thought it was auspicious that this is 108, a number closely connected with the Golden Ratio, because of its appearance in the regular pentagon. (I am also Golden, if you can't see the auspices.)
108 is pretty interesting in other ways, too. I got wondering if many other numbers are multiples of the same number with zeroes removed. (What about other digits?) New to me was the idea of a refactorable number: a number divisible by the count of its divisors. They are rarer than I would have first expected; also called the tau numbers. 108 is the 18th tau number. Can you find all 17 prior?
Let's get to the submissions! Where I could find them, I linked the author's twitter account as well.
There are many states with a Highway 108, but Kansas gets the picture, since they use a sunflower (their state flower), which is also - of course - associated with the Golden Ratio.
Patrick Honner asks who has done a billion dollars worth of work? Wages, worth and gender bias all figure in.
Cav has a post trying to make sense of maths testing. Maybe separating it out into two different subjects... On a more personal note, he looks at amusing, infuriating and worrying answers on his own students' exams.
Jennifer Silverman shares her visual approach to the quadratic formula at her tumblr. College math majors often are missing this connection between completing the square and the QF.
Malke Rosenfeld writes about Beautiful Objects at Math in Your Feet. The objects in consideration are lovely group symmetries from Christopher Danielson.
Mike Lawler got great reasoning out of Fawn Nguyen's bridge problem. Inspired me to try it with freshmen and grad students to great effect. And now on to the grad students classrooms, too!
Sue Van Hattum makes an argument that optimization is the best application topic in standard math classes, and uses her recent calc lesson as an example.
This graphic designer, Jack Hagley, made a neat logo based on his research at Wolfram-Alpha. He thought 108=1x2x2x3x3x3 was beautiful in its own right. (I agree!) These are hyperfactorial numbers.
Edmund Harriss has several threads converging in his post at Maxwell's Demons about Rational Parameterisation of the Circle. Very novel idea, connections with stackexchange, and beautiful images from Lissajous variations. He also has some very mathy fun with generating "huge worlds of potential logos" for Twitter Math Camp 14. (Full, but there is a wait list.)
John D. Cook accesses Category Theory to determine whether the stars go up or down at his blog, the Endeavor. Hint: yes.
Dan McQuillan writes about induction in the Unreasonable Efficiency of Mathematical Writing. I love his advice: "focus on the beautiful idea."
At the By Way of Contradiction blog, there is a probability post on how to pick odds in the most favorable way. Includes this sentence: "Satisfied with your math, you share your probability, he puts 13.28 on the table, and you put 2.72 on the table." This is, I think, the pub for me.
In some Hindu and Buddhist practice, a mantra is recited 108 times. This leads to malas (prayer beads) often having 18, 27, 54 or 108 beads. This may be connected to the Sri Yantra, which has "9 interlocking triangles forming 43 smaller triangles."
Cool geometry, regardless.
108 figured prominently on LOST. (Here's some of the connections from the show.) Most significantly, it was the sum of the numbers 4 8 15 16 23 42. Never sufficiently explained, I think, and I'm still mad about the ending. So there.
Evelyn Lamb digs into the history of the Parallel Postulate at her Scientific American blog Roots of Unity. Triangles with angle sums of 0 degrees, rectangles held hostage - exciting stuff. Evelyn also has a post at the AMS (her Blog on Math Blogs) looking at Michael Pershan's Math Mistakes site and contemplating SBG. (Go for it, Evelyn!)
Fiona Keates, who blogs in The Repository at the Royal Society, has a piece on a mathematician in the movies. Yes, Ramanujan is coming to a theatre near you!
Shecky Riemann interviews the fascinating Cathy O'Neil, author of the Mathbabe blog, at Math Tango. (Cathy's on Twitter, too.)
The oddest 108 connection I found was 108 Rock Guitars. Since Math Rocks!, they get a link. (Maybe I meant Math Rock?)
Antonio ChinchÃ³n has a post on the sound of the Mandelbrot Set. A neat combination of pretty, computation and sonification of data. He also has a post about Warholing data (after the pop artist), which he does to Grace Kelly.
Mike Croucher at Walking Randomly has several different mathematizations of the heart that go far beyond the standard cardioid.
Sam Shah derived the curve found in string art as he did Doodling in Math Class.
Hopefully you got a chance to see Carnival 107 at White Group Mathematics. Next month's 109 is at Tony's Maths Blog. Be sure to check out Sue Van Hattum's hosting of the Math Teachers at Play carnival 71 (with 71 links). You can submit a post at the Carnival's homepage at the Aperiodical. Katie Steckles makes these happen, and submitted several interesting items above; Thony Christie also made some nice picks. Gracias!
There were some posts I wasn't sure what to make of... John Gabriel arguing that no valid construction of the reals exists. Katie submitted a Windows 8 math game app called Equations. I haven't been able to test it out, though. The reviewer dislikes it for pretty valid sounding reasons.
I thought it was auspicious that this is 108, a number closely connected with the Golden Ratio, because of its appearance in the regular pentagon. (I am also Golden, if you can't see the auspices.)
108 is pretty interesting in other ways, too. I got wondering if many other numbers are multiples of the same number with zeroes removed. (What about other digits?) New to me was the idea of a refactorable number: a number divisible by the count of its divisors. They are rarer than I would have first expected; also called the tau numbers. 108 is the 18th tau number. Can you find all 17 prior?
There are many states with a Highway 108, but Kansas gets the picture, since they use a sunflower (their state flower), which is also - of course - associated with the Golden Ratio.
Patrick Honner asks who has done a billion dollars worth of work? Wages, worth and gender bias all figure in.
Cav has a post trying to make sense of maths testing. Maybe separating it out into two different subjects... On a more personal note, he looks at amusing, infuriating and worrying answers on his own students' exams.
Jennifer Silverman shares her visual approach to the quadratic formula at her tumblr. College math majors often are missing this connection between completing the square and the QF.
Malke Rosenfeld writes about Beautiful Objects at Math in Your Feet. The objects in consideration are lovely group symmetries from Christopher Danielson.
Mike Lawler got great reasoning out of Fawn Nguyen's bridge problem. Inspired me to try it with freshmen and grad students to great effect. And now on to the grad students classrooms, too!
Sue Van Hattum makes an argument that optimization is the best application topic in standard math classes, and uses her recent calc lesson as an example.
This graphic designer, Jack Hagley, made a neat logo based on his research at Wolfram-Alpha. He thought 108=1x2x2x3x3x3 was beautiful in its own right. (I agree!) These are hyperfactorial numbers.
Edmund Harriss has several threads converging in his post at Maxwell's Demons about Rational Parameterisation of the Circle. Very novel idea, connections with stackexchange, and beautiful images from Lissajous variations. He also has some very mathy fun with generating "huge worlds of potential logos" for Twitter Math Camp 14. (Full, but there is a wait list.)
John D. Cook accesses Category Theory to determine whether the stars go up or down at his blog, the Endeavor. Hint: yes.
Dan McQuillan writes about induction in the Unreasonable Efficiency of Mathematical Writing. I love his advice: "focus on the beautiful idea."
At the By Way of Contradiction blog, there is a probability post on how to pick odds in the most favorable way. Includes this sentence: "Satisfied with your math, you share your probability, he puts 13.28 on the table, and you put 2.72 on the table." This is, I think, the pub for me.
In some Hindu and Buddhist practice, a mantra is recited 108 times. This leads to malas (prayer beads) often having 18, 27, 54 or 108 beads. This may be connected to the Sri Yantra, which has "9 interlocking triangles forming 43 smaller triangles."
Cool geometry, regardless.
Edward Frenkel, author of the recent hit Love and Math, had a much talked about editorial in the LA Times: How our 1000 year old math curriculum cheats America's kids. (That's what we get for buying an 800 year old curriculum when we started, I say.)
Askhat Rathi at the Conversation has my favorite recent math in the news story, the discovery of a new class of polyhedra. Yeah!
108 figured prominently on LOST. (Here's some of the connections from the show.) Most significantly, it was the sum of the numbers 4 8 15 16 23 42. Never sufficiently explained, I think, and I'm still mad about the ending. So there.
Evelyn Lamb digs into the history of the Parallel Postulate at her Scientific American blog Roots of Unity. Triangles with angle sums of 0 degrees, rectangles held hostage - exciting stuff. Evelyn also has a post at the AMS (her Blog on Math Blogs) looking at Michael Pershan's Math Mistakes site and contemplating SBG. (Go for it, Evelyn!)
Fiona Keates, who blogs in The Repository at the Royal Society, has a piece on a mathematician in the movies. Yes, Ramanujan is coming to a theatre near you!
Shecky Riemann interviews the fascinating Cathy O'Neil, author of the Mathbabe blog, at Math Tango. (Cathy's on Twitter, too.)
The oddest 108 connection I found was 108 Rock Guitars. Since Math Rocks!, they get a link. (Maybe I meant Math Rock?)
Antonio ChinchÃ³n has a post on the sound of the Mandelbrot Set. A neat combination of pretty, computation and sonification of data. He also has a post about Warholing data (after the pop artist), which he does to Grace Kelly.
Mike Croucher at Walking Randomly has several different mathematizations of the heart that go far beyond the standard cardioid.
Sam Shah derived the curve found in string art as he did Doodling in Math Class.
Hopefully you got a chance to see Carnival 107 at White Group Mathematics. Next month's 109 is at Tony's Maths Blog. Be sure to check out Sue Van Hattum's hosting of the Math Teachers at Play carnival 71 (with 71 links). You can submit a post at the Carnival's homepage at the Aperiodical. Katie Steckles makes these happen, and submitted several interesting items above; Thony Christie also made some nice picks. Gracias!
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