Monday, December 13, 2010

Pi Song Lyrics

It's never too soon to start looking ahead to Pi Day 2011. I'm trying to reword Elton John's "Your Song" to make it more pi-centric, but haven't gotten very far. Suggest a line or several, and your name may end up in the credits in the next music video!

It's a little bit funny this feeling inside
It's a little bit funny, this number called pi

I'm not one of those who can easily hide

I don't have much money but boy if I did

I'd buy a big house where we both could live

If I was a sculptor, but then again, no

Or a man who makes potions in a traveling show

I know it's not much but it's the best I can do

My gift is my song and this one's for you

And you can tell everybody this is your song

It may be quite simple but now that it's done

I hope you don't mind

I hope you don't mind that I put down in words

How wonderful life is while you're in the world

I sat on the roof and kicked off the moss

Well a few of the verses well they've got me quite cross

But the sun's been quite kind while I wrote this song

It's for people like you that keep it turned on

So excuse me forgetting but these things I do

You see I've forgotten if they're green or they're blue

Anyway the thing is what I really mean

Yours are the sweetest eyes I've ever seen

Friday, December 3, 2010

"Warrior mathematicians"?

Sure, there's the story of Archimedes using the power of the parabolic mirror to burn up Roman warships.  But for the most part, mathematicians have been non-warriors, interested more in uncovering beauty and truth for its own sake, rather than in crushing enemies or hearing the lamentation of women.  Today NPR reported on the way network analysis is being used to locate key players in terrorist networks in order to reduce the number of IEDs:




Wednesday, November 24, 2010

Error: Divide by Zero

I recently came across this car on the road.

This struck me as (1) inaccurate and (2) ineffective. But then I didn't know it would be better if it said "undefined mpg", "infinite mpg," or something else entirely?

Thursday, November 11, 2010

Pulsar music

Speaking of globular clusters, some work has been done to convert the electromagnetic pulses of pulsars near the core of globular cluster 47 Tucanae into audible sounds for our listening pleasure. 

More info here.

A lovely video of this cluster:

Tuesday, November 2, 2010

Autumn leaves as globular clusters?

As I looked at the red leaves falling around a maple tree in my yard, I was reminded of an appendix in Pickover's Chaos in Wonderland called "Build Your Own Globular Cluster".  I dug it out and found the equation
p = 1/(1+r²)^n
where p is the density of stars in the cluster, r is the radius, and n is a so-called polytrophic index, "typically 2.5<n<3.2".  The index also provided a program for plotting globular clusters, but I found it difficult to understand, so I went about creating my own from the equation.  I used the inverse function r(p), and let p be a random number between 0 and 1.  The result looked very similar to the pattern of leaves I saw on the ground, but I still wonder if this was the right equation to us, or if a normal distribution would be more appropriate.

Sunday, October 24, 2010

R.I.P. Mandelbrot

As I was choosing a suitable region of Mandelbrot's famous fractal set for a shirt design, I was unaware that he was passing away at the same time.

Saturday, September 25, 2010

Paint can cardioids

As I looked into a nearly empty paint can, I noticed a familiar heart-shaped graph:


I'm used to seeing the cardioid-shaped region of the Mandelbrot set (what my students have sometimes called "the butt crack"), but also noticed it in my Spirograph program, where a family of graphs can be generated that seem similar to the paint can cardioids (or caustics):



You can get the same cardioid graph by drawing a cycloid around a circle of the same diameter (an epicycloid):

Monday, September 13, 2010

Updated Teacher Soundboard

Here's a classroom soundboard that I have found useful.  It's 2mb, so be patient.

Thursday, September 9, 2010

Fibonacci right outside

After all the fun I've had with simulated sunflowers, I thought it was time to grow some real ones.  They're probably 8 feet tall now.



I wondered if the Fibonacci sequence (1, 2, 3, 5, 8, 13, 21, 34, 55...) really showed up in the seeds, or whether it was just a close approximation.


Opening this image in Paint and putting a dot on each of the spirals, I found Fibonacci numbers in full bloom: 55 clockwise spirals and 34 counterclockwise!

Ian Stewart's Nature's Numbers gives a brief but sufficient explanation for this phenomenon in plants, and doesn't go overboard with finding the Golden Ratio everywhere, as I have been annoyed to see some do.

Friday, August 27, 2010

How much tar paper?

My dad got a partially used role of tar paper and wanted to know how many square feet it has left.


The roll is 36.0 in. wide, with a 6.00 in. diameter.  The hollow core of the roll has a 2.88 in. diameter, and the tar paper is 0.02 in. thick.

Is there enough to cover 4 faces of an 8-ft. cube?

I like this problem because it can be solved in at least two ways.  What is the answer?

Saturday, August 21, 2010

Tapering off my drug dependence

I had back surgery recently and have been taking the narcotic oxycodone, which has a half-life of 3.8 hours approximately.  I have been tapering my down my dose, taking

10mg / 4 hours

10 mg / 5 hours

10 mg / 6 hours

And now I want to drop to 5 mg, but wondered with what frequency I should take it.  I'll look at the levels remaining in my system after the doses given above, so I need to solve the half-life equation 0.5 = e^(3.8k), then evaluate

10e^(4k)

10e^(5k)

10e^(6k)

I get 4.83mg, 4.03mg, 3.36mg, respectively.  So when I drop down to the 5mg tablet, I'd like to take another when I have about 3.5mg left.  Let's see when that occurs:

3.5 = 5e^(kt)

Using k from above (have you found it yet?), I get t =~2 hours.  So I'll start with that period and increase from there.

Thursday, August 19, 2010

Blank numbers

A few years ago I was teaching complex numbers in an algebra 2 class.  It took students awhile to grasp the idea of i.  I told them it was defined as the square root of negative one.

"But you can't square root a negative."

"That's why we have to create a new definition - we want to see what results from this definition," I said.

One student was particularly enamored by this idea, and I suggested he look at the other forbidden operation (besides square rooting a negative): dividing by zero.  He defined 1/0 as u, perhaps for "undefined," but called numbers involving u "blank numbers," since a calculator would return a blank screen after the error.

Over several days I prodded this student to create theorems about blank numbers, and to avoid contradictions by adding "special rules."  I have since lost his work, and don't remember the stumbling block that kept him from continuing.

Perhaps you can find it. Let 1/0 = u.

Thm. 1: u² = u

Where does it go from here?

Friday, August 6, 2010

Sketchy recursion in Inception

In the movie Inception, most of which takes place inside people's dreaming minds, we are told that 5 minutes of dreaming equates to 1 hour inside that dream (a 1:12 ratio) because "your mind functions more quickly."  If, once you are dreaming, you go to sleep and have a dream (within a dream), the same ratio holds, so that 5 minutes in the "real world" equates to 12 hours in the 2nd-layer dream.

We are told that the more layers of dreams-within-dreams you have, the more unstable your situation becomes, so the pharmacist administers some powerful sedatives to help.  But this changes the ratio from 1:12 to 1:20 somehow, and now 5 real minutes should give you 100 minutes in the first level of dreaming, 2000 minutes (33) hours in the 2nd level, and so on.

The numbers given in the movie were a bit sketchy, though - instead of "100 minutes, 33 hours, 28 days," they said something like "2 hours, 1 week, 6 months," or something like that.

The other fishy math was the amount of dreaming that got done during the time the van was falling from the bridge to the water.  If we're generous and say the bridge was 100 ft above the water, it should have been about a 4.5-second fall.  The should have given them 90 seconds in the hotel and 30 minutes in the snowy wonderland.  Max.

Do I have this right?

Thursday, July 1, 2010

Apollonian artichoke juice

I had just finished eating an artichoke when I noticed that the oil and vinegar in the bowl had formed a circle fractal like the Apollonian gasket!


click image below for wikipedia info.

Tuesday, May 4, 2010

dy/dx

Here's a new song to help you remember your derivative rules.  It was based on the classic Aicha video by GellieMan (bottom of post below).



Lyrics
(Am F C G)
cosine, negative sine
f is tangent, secant squared is f prime
dy/dx is rise over run
x to the n, n x to the n minus 1
Square root, one over 2 root x
natural log is just 1 over x
log base b, 1 over x ln b
b to the x gains an ln b Oooooh
dy/dx, isn't it strange
dy/dx, rate of change
dy/dx, slope at a point
tangent line, tangent line
in my life.

Sunday, April 11, 2010

Pythagorean triples

Do you ever need to come up with an unusual Pythagorean triple (a set of 3 natural numbers such that a² + b² = c²) quickly?  If you're not a math teacher, maybe you don't.

Here's what I use:

3,4,5 and multiples (6,8,10; 9,16,20; ...)

5,12,13 and multiples

7, 24, 25 and multiples

8,15,17 and multiples

But then my memory gives out, so I have to revert to a formula. For any natural number p, you get

2p, p² - 1, p² + 1.

So for p = 8  you get 16,63,65.

I wondered what other algebraic tricks I could find, and came upon

4p, p² - 4, p² + 4.

So for p = 5 I got 20, 21, 29. Since 29 is prime I know the triple isn't just a multiple of an earlier one.  Odd p values work well with this one:

p = 7: 28, 45, 53

p = 9: 36, 77, 85

Where can we go from here?

Saturday, April 3, 2010

Open Letter to Demetri Martin

Dear Demetri,

As a fan of your work I humbly offer the following for your consideration.  In your "Large Pad" routine, you refer to a "simple chart" relating three variables: how short someone is, how drunk he is, and how funny it is.  You provide a 2-dimensional graph when a 3D graph is required to show the amount of funniness for each combination of shortness and drunkness:




The graph you provided appeared to depict shortness as [exponentially] dependent on drunkness, which is not what you intended.

I hope this clarifies the relationship you seek to communicate.

Sincerely,

Nathan Shields

Friday, March 12, 2010

My dog Phi Phi

I wanted a nice region bounded by two curves (who wouldn't?), so I graphed a parabola and a reciprocal function passing through (0,1):

y = -x² + 1
y = 1/(x + 1)



The second intersection looked vaguely familiar... 0.61803398875... where had I seen that before?

Oh yeah, the golden ratio!  Phi = 1.61803398875...

An interesting fact is that Phi - 1 = 1/Phi ... are there any other numbers that have this quality?

Sunday, February 14, 2010

Triangle problem...again

Hey, you haven't answered my first triangle problem, but here's a related video to inspire you:

Scientific notation in Avatar

Despite complete disregard for their work by leadership, scientists are the heroes of Avatar.  And who better to teach scientific notation?
GRACE: Alright, look -- I don’t have the answers yet, I’m just now starting to even frame the questions. What we think we know -- is that there’s some kind of electrochemical communication between the roots of the trees. Like the synapses between neurons. Each tree has ten to the fourth connections to the trees around it, and there are ten to the twelfth trees on Pandora --

SELFRIDGE: That’s a lot I’m guessing.

GRACE: That’s more connections than the human brain. You get it? It’s a network -- a global network. And the Na’vi can access it -- they can upload and download data -- memories -- at sites like the one you destroyed.

So we multiply 104 and 1012 (perhaps dividing by 2 so as not to count connections twice), and get 1016 (within 1 order of magnitude).  A 1988 article in Annual Review of Neuroscience suggests a human brain has  100 billion (1011) neurons and 100 trillion (1014) synapses, so the Avatar claim seems reasonable.

Can we quantify human memory like this?  Yeah, probably.

Thursday, February 4, 2010

Pascal in Excel

As an enrichment activity I had a student fill a large sheet of graph paper with Pascal's triangle - just the ones digits, anyway.  Then he colored the digits in different ways to discover patterns.  After doing this by hand, the student was interested in saving some time with Excel, so I devised this method:

  1. Make a grid of 0's.  Really you just need a border of 0's, but filling an area is faster.  It should be twice as wide as tall.

  2. Select cell B2, and give it the formula =MOD(A1+c1,10).
    pascalexcel1

  3. Copy B2 and paste it into all the cells in the grid except for the top row and the left and right columns.

  4. Now they are all adding the values of the two cells diagonally above them.  Just put a "1" in the middle of the top row and see what happens.
    pascalexcel2

  5. To color the different digits, click Format, Conditional Formatting, and make some stuff up.
    pascalexcel3

  6. I changed the font color to white and clicked File > Web Page Preview:
    pascalexcel4


You could do some fun experiments with this, leading kids toward the idea of cellular automata!

Wednesday, January 20, 2010

Two sweet math books of 2009

A couple sweet math books for a broad audience (and, coincidentally, the last two additions to my collection) include:
logicomix
Logicomix, a graphic novel about Bertrand Russell's quest for certainty in the foundations of math, and the madness that mysteriously infuses the study of logic.
the-math-book
The Math Book, a survey of important ideas in mathematics throughout history, distilled into 1-page summaries with full-color graphics.