Math Teachers at Play #4

It’s been two weeks since the last Mathy carnival (oh Carnival of Mathematics, where aaaaaaarrrrrrrrrreeeeeeee you?), and on this snowy (seriously) April day nothing could be better than a nice long list of great blog posts.  And that’s exactly what there is at  Math Teachers at Play #4!    It’s being hosted over at Homeschool Bytes, a blog that shares a whole bunch of resources for homeschooling.  (Although they’re not limited to homeschooling, of course.  Our older kiddo is learning multiplication, and Timez Attack looks like a great way to  help him get the number facts at his fingertips.)

But back to the Carnival.  The carnival divided into different steps/levels, from counting money to the SATs [both the modern version and some questions that could have occurred if there were SATs in 1557], so there’s something for everyone.  Time to play!

The Abel Prize in rhyme

There once was a math guy from Russia
Which is just to the east of East Prussia.
He proved many a theorem.
Geometers revere him.
The Abel Committee did gusha.

They said to Mikhail Gromov,
(Whose sweater in this picture looks mauve)
“You are so creative!
A mathematical native!
You’re analysis’s own Brett Favre!”*

His work (expanding thresholds
Of n-dimensional manifolds
And related measure)
Gave people such pleasure
That this year’s Abel Prize he holds.

* No doubt they were referring to Brett’s heyday in the 90s, and not to last season.

For a more interesting but less poetic summary of his life and work, check out the official summary.

The Creative Commons picture of Mikhail Gromov was taken in 2007 by Gert-Martin Greuel.

In Good Company

We were thrilled to learn that we were mentioned in an article about math blogs that Jenni and Jon Ingram wrote for the March 2009 issue of Mathematics Teaching, a journal of The Association of Teachers of Mathematics in the UK.  Most of the articles (like the evocatively named “It would be 19 if 10 was an odd number” or “Puppets Count”) are only available to members, but a few, including theirs, are open to all.

They also wrote about dy/dan, Let’s Play Math!, and MathNotations,  with descriptions both of the blogs and of some samples posts for each.

Thanks for putting us in such good company!

The Beach Photo was taken by Richard Marris during the Kioloa Flickrmeet, Kiola, New South Wales, Australia [creative commons license].

Definitions

What is a planet?

The current definition, from Resolution B5 of the International Astronomical Union, is that

A planet is a celestial body that
(a) is in orbit around the Sun,
(b) has sufficient mass for its self-gravity to overcome rigid body forces so that it assumes a hydrostatic equilibrium (nearly round) shape, and
(c) has cleared the neighbourhood around its orbit.

Pluto hasn’t cleared its orbit, so it’s not a planet.  There’s actually some controversy about this (and as usual, we turn to Wikipedia to learn all about it), but essentially that’s how things stand.  Pluto can at least rest assured that it’s not the first planet to be demoted.

But a lot of people don’t like that Pluto isn’t a planet, and last month the Illinois State Senate adopted a resolution that declared March 13, 2009 as “Pluto Day” and also included the line:

that as Pluto passes overhead through Illinois’ night skies, that it be reestablished with full planetary status

You can read the formal resultion here; it’s a short read, thought it packs no fewer than 10 “Whereas”s into just 1½ pages.   Here’s what is bothering me:  the bill’s sponsor, Senator Gary Dahl, apparently said:

I don’t think we are changing the status of the planet. We’re simply asking that March 13 be declared Pluto Day and that, for the day, Pluto is a planet.

But, if you add Pluto, aren’t you actually changing the definition of Planet?  If not in broad terms, at least by adding “and Pluto”? I get the distinct impression that he doesn’t understand what a definition is.  It reminds me of how our math majors often struggle with definitions when they’re first learning how to write proofs:  starting from the definition of an even integer (twice an integer), it’s initially hard for them to prove that if n is even, then so is n+2.

And yes, I might be making too much of this.  I don’t want to begrudge Pluto its special day.  But still.

One Two Three

Time for some math problems!  I was initially going to share two interesting problems featuring the number 2, but then it turned out I’d done one of them wrong and the answer wasn’t 2 at all.  But then I turned to the digits 1 and 3, and they saved the day.  Hooray!

Problem #1 is about amicable pairs.  These are numbers each of whose proper divisors add to the other number.  For example, 220 and 284 are amicable, because the proper divisors of 220 (1,2,4,5,10,11,20,22,44,55,110) add to 284, and the proper divisors of 284 (1,2,4,71,142) add to 220.  And that’s the example you’re going to find if you do any sort of search for amicable pairs, because the next pair is 1184 and 1210.  Not that they are so uncommon (there are about 12 million known pairs), but they do tend to be pretty big.

So anyway, take any amicable pair.  Start with the first number, add the reciprocals of all of its divisors, and then take the reciprocal of the sum [i.e. (1/1+1/2+1/4+...+1/220)^-1].  Do the same thing to the other number [i.e. (1/1+1/2+1/4+...+1/284)^-1].  Then add those two numbers together, and you’ll get 1.

Pretty neat, huh?  And the proof isn’t too bad (it involves putting each sum over a common denominator and falls into place from that, where “falls into place” can mean anything from you see it right away to six hours of staring).

Problem #2:  I learned of this problem via Ted’s comment here.  (Thanks Ted!)   He linked to Ron Knott’s site, which has all sorts of stuff about generating Pythagorean Triples, including this algorithm:

• Start with two fractions that multiply to 2:  for example, 3/4 and 16/6.
• Add 2 to each number:  11/4 and 28/6
• Put the numbers over a common denominator:  33/12 and 56/12
• The numerators will be the first two numbers of a Pythagorean Triple:  33²+56²=65²

Isn’t that cool??  Furthermore, you’ll get primitive triples exactly when you start with reduced fractions (and use the least common denominator in the third step).

Problem #3 is about base 3.  Find all positive integers n such that neither n nor n² have any 1s when written in Base 3.  And since I told you the answers to the first two problems and just left off the proofs, I think I’ll leave off the answer to this and let you see if you can find them all.    Happy Problem Solving!

Two great sites

Here’s what I saw when I checked out xkcd this morning:

(I’ll admit, it took me a moment to get, after which I decided it was one of my favorite strips.)

So that was one great way to start the day.  Then I found that Denise had posted Math Teachers at Play #2.  And it has, like, a million fantastic posts, plus the picture are really cute.  So that was another great way to start the day.

Plus it’s Friday.  It doesn’t get much better than that.

Square Root Day!

I was driving into work today, and when I turned on my radio I heard that it was Square Root Day.  Except I missed the part where they were explaining what it is, and all I heard was that it won’t happen again for a while, and I thought to myself, “Huh?  But isn’t EVERY day a square root day, since 3 is the square root of 9, and tomorrow, the 4^th, will be the square root of 16?”  And I was completely confused.

Then just a few minutes ago, my friend Cheryl popped into my office and said, “HAPPY SQUARE ROOT DAY!”  And she was able to explain this holiday:  it’s because it’s 3-3-09, and 3 is the square root of 09.  Then the light shown shone down and angles sang* and it all made sense.  And the radio was right about the rarity — the next one isn’t until 4-4-16.

And as soon as Cheryl left, my student Adele popped into my office and apologized for overhearing our conversion (nice of her, but unnecessary — my office is right off of the Math Center and I listen to student conversations all the time).  Anyway, Adele passed along that she also had heard about Square Root Day!  It turns out that it’s all over the internet.  Apparently you’re supposed to eat root vegetables cut into squares.  This might explain why it’s not as celebrated a holiday as Pi(e) Day.  So without further ado,

Happy Square Root Day!

Just so I’m not behind on another holiday, Thursday and Saturday are Odd Days, since 3-5-9 and 3-7-9 are odd numbers in increasing order.  Celebrate away!

*I started to write “angels” and misspelled it, but decided that angles singing was more appropriate.

There’s a new Carnival in Town!

Posted today is the inaugural edition of a new Carnival: Math Teacher at Play. This carnival is the brainchild of Denise at Let’s Play Math, who wanted a gathering that focused on posts that were appropriate to teaching elementary school through high school mathematics.  Because this is complementary to the Carnival of Mathematics, it will appear every other Friday, when the Carnival of Mathematics is not in session.  This means that we’ll have Math Carnivals every Friday — hooray!

The Carnival will initially be hosted at Let’s Play Math, but Denise welcomes other hosts. This first edition has 20 entries (wow!) gathered by Denise, and I look forward to reading them all!

The photo above is from Diliff (posted under GNU-FDL) and it actually shows a crowd at a carnival in Nottingham, but I’m sure that they would have wanted even more to be at this carnival.

Unzipping the Klein Bottle

The Klein Bottle.  I remember first learning about it,  and it was, well, hard to visualize.  (“Self intersecting?  The outside and inside all the same?  Hanging out in four-dimensional space?  Ummm, okay.”)  Then I discovered ACME Klein bottles, and they made my life happy because I could sort of understand what a Klein bottle was, and because their web page is fun to read.  Plus, they sell Klein Steins:

(Godzilla finished his beverage before I got a chance to photograph it.)

One feature of a Klein bottle is that it can be made from two Möbius strips.   Another thing that was hard for me to picture (though again Clifford Stoll at ACME comes to the rescue), but I just ran across this video of a Klein bottle that had been made by zipping together two Möbius strips, and I thought it was so neat that I wanted to share it.  I might just have to figure out how to work the sewing machine just so I can make one.

(And really, I’d just intended this post to end here, but then I found the following clip and couldn’t resist adding it!)

ACME Klein Bottle photo by Lethe, published under GNU Free Documentation License.

Carnival #49 this Friday!

We’ll be hosting the Carnival of Mathematics #49 on Friday the 13^th, so send in your submissions by midnight Thursday (which really means early in the morning Friday)!  You can post them in the comments here, or use the official form (which seems to be working fine — I’ve been getting submissions that way), or send them by email to hlewis5 followed by the @ sign followed by naz.edu.

Ode to Eday

Today is e-day — the day that millions come together worldwide in praise and glory of the number e, which is approximately 2.718…

But this is not the only e-day.  Indeed, there are many others that appear to have nothing at all to do with the number.

• In Paducah, Kentucky E-day falls on February 21 and refers to Engineers Day.  They have an egg drop contest, tape people to walls, and create edible cars.
• January 1, 2002 was Euro Day, when a whole bunch of countries simultaneous adopted the Euro.  Right now 1 Euro is worth 1.2944 US dollars according to Google.   Speaking of Google, did you know that their Zurich office has twirly slides, fireman poles, and meeting rooms in the shape of  igloos?
• eDay in New Zealand is the day that people can get rid of their e-waste (which to me sounds like spam, but means old computers).  Old electronics get sent out to recycling places instead of the landfill.
• And then there’s my favorite:  the island of Eday in northern Scotland, one of the Orkney islands.
  131 people live there (as of last summer), and the main exports have been peat and limestone (and pirates!  Or at least one pirate:  John Gow).  They have a new Heritage and Visitor Centre, and enough B&Bs for a local tourists.  There is however, no evidence of a “Spend e-day on Eday!” marketing  campaign — they might want to try that.*

*Using the 2 July version of e-Day, presumably. Though today it’s a balmy 34°F up on the isle, so perhaps it could be a biannual tradition.

The fancy e was created by Acf and is available under the GNU Free Documentation License.

Dimension Confusion

So how about that Super Bowl?   Having analyzed the game here yesterday (OK, not the actual game, but potential scores) it’s time to reflect on the commercials.  The sense seems to be that the commercials were…OK.  Some pretty funny ones, but not one of the absolute best years ever for commercials.

And from a math perspective, one commercial stood out.

Although come to think of it, the 1D version of Chuck might be really entertaining!  The ultimate stick figures.

What’s the pattern?

1, 11, 21, 1211, 111221, 312211, 13112221, …

This sequence was the favorite pattern of my former department chair, Nelson Rich.  I thought he invented it, but a quick search on the internet reveals that it’s pretty easy to find if you search for the first few terms.

But don’t do that! It’s a sneaky one, but fun to try and figure out.

Polygons in the Smithsonian

While we were in DC, we managed to sneak in a visit to the Museum of Natural History.  And right next to the Hope Diamond (which, umm, looked surprisingly small.  I’d envisioned something fist-sized, which goes to show how little I know about diamonds) there were some really cool rocks.

The first thing I saw were balalt columns.

I got really excited because I wrote about last April and how they often form hexagons.  So I took a closer look:

Yup, a perfect hexagon pentagon.  But there were a few hexagons around it, and it was pretty neat looking.

Then I moved to the minerals.  There was some neat symmetry in these twin crystals:

(Here’s the info on them.)

Then I saw cubes.  Lots and lots of cubes, because a bunch of crystals grow that way.  There was this Fluorite from Spur Mountain mine, Cave in Rock in Illinois.

And these cubes of fluorapophyllite from Poona, Maharashtra in India.They were clearer in real life.

Then there’s this fluorite in gypsum, which is neat both because the cube is embedded in a see-through rock and because it’s from Penfield, which is just down the street from Rochester.  Seriously, you could pretty much walk there, and it’s an easy bike ride.

Does anyone know if these rocks are related to the stuff that’s in toothpaste?

If shiny is more to your liking, here’s a whole bunch of Galite cubes from Missouri.

Here’s the sign.  I like taking pictures of signs.  Otherwise, how would you know that there’s sphalerite mixed in?

And then check out this Pyrite.  It grows in two ways:  cubes and dodecahedrons, and this photo shows examples of both.  I love pyrite. 

I can’t wait to go back to the Smithsonian again.

Hot off the Press! A History of Math resource

We’re hanging out in Washington, DC right now at the Joint Mathematics Meetings, which is lots of fun because the weather is beautiful (no snow on the ground and Actual Sun peeking through!  Except for the sleet predicted for tomorrow, but I’m going to ignore that) and there are a bunch of good talks.

One I heard of today (by Sarah J. Greenwald, with Gregory Rhoads listed as a co-presentor) was of a brand-spanking-new web site, online as of yesterday, put together for people who want to incorporate some history of math into Multivariable Calculus or Differential Geometry.  The intent was to be a general resource of information, but also to have classroom activities.

The link is http://mathhistory.mathsci.appstate.edu

So hey, be one of the first to visit!  And check back if you don’t see what you need, since the site is still in its early stages.

Note 1: I had wondered at first if this site would be like Convergence, but it’s geared towards a more specific topic so the two are complementary.

Note 2: When I saw Sarah J. Greenwald’s name, it was so familiar to me that I figured that I knew her from some math thing or other.  But then when I saw her I didn’t recognize her, and then I was too embarrassed to start a conversation with her even though she was really friendly because clearly I’d forgotten where I knew her from.  And then when I was looking up her name for this post to make sure I had spelled it right I discovered that the reason her name was so  familiar was that she’s the Simpson’s Math person (along with Andrew Nestler).  And she talked to David X. Cohen about Futurama Math in an interview that has been referenced on this blog.    So the mere fact that I knew her name so well that I thought we must be best buds is a testament to how delusional I am much I enjoy her work.