How much is enough?

If you notice a pattern, how many times do you have to check that it works before being certain. Six? Twenty? Two thousand?

Two favorite examples of mine that demonstrate that patterns can continue for a long time before going awry:

The first example is the polynomial f(n)=n²+41n+41. If you plug in any whole number for n from 1 to 40, then f(n) is a prime number: f(1)=83, f(2)=127, all the way to f(40)=3281. But f(41)=3403=41·83 is not prime. So something that works 40 times in a row might fail.

The second example are the cyclotomic polynomials. Look at polynomials of the form xⁿ-1 that have been factored:

x-1=(x-1)
x²-1=(x-1)(x+1)
x³-1=(x-1)(x²+x+1)
x⁴-1=(x-1)(x+1)(x²+1)
x⁵-1=(x-1)(x⁴+x³+x²+x+1)
x⁶-1=(x-1)(x+1)(x²+x+1)(x²-x+1)

The last polynomial in each case is the cyclotomic polynomial of order n. [It has a much more technical definition using imaginary numbers and the product of primitive roots of unity]. And at first glance it looks like the coefficients are 0, 1, or -1. Even at second glance or sixth glance, since it’s true for the first 104 cyclotomic polynomials. But not the 105^th.

x¹⁰⁵-1=(x-1)•(x²+x+1)•(x⁴+x³+x²+x+1)•
(x⁶+x⁵+x⁴+x³+x²+x+1)•(x⁸-x⁷+x⁵-x⁴+x³-x+1)•
(x¹²-x¹¹+x⁹-x⁸+x⁶-x⁴+x³-x+1)•
(x²⁴-x²³+x¹⁹-x¹⁸+x¹⁷-x¹⁶+x¹⁴-x¹³+x¹²-x¹¹
+x¹⁰-x⁸+x⁷-x⁶+x⁵-x+1)•
(x⁴⁸+x⁴⁷+x⁴⁶-x⁴³-x⁴²-2x⁴¹-x⁴⁰-x³⁹+x³⁶+x³⁵
+x³⁴+x³³+x³²+x³¹-x²⁸-x²⁶-x²⁴-x²²-x²⁰+x¹⁷+x¹⁶
+x¹⁵+x¹⁴+x¹³+x¹²-x⁹-x⁸-2x⁷-x⁶-x⁵+x²+x+1)

See those two coefficients of 2 in that last polynomial? So the pattern of coefficients being only 0, 1, or -1 fails. Interestingly, the reason for this initial failure occurring so late in the game is that 105 is the smallest number that has three distinct odd prime factors (105=3·5·7). The integer 385 is also a product of three distinct odd primes (385=3·5·11) and the 385^th cyclotomic polynomial is the first one to have a 3 as a coefficient (see Wolfram Mathworld).

Patterns. You just can’t trust them.

Decagon Mug

TwoPi and I went out for lunch to the Cheesecake Factory the other day, and noticed partway through that the mugs were regular decagons! So he snuck a photo while pretending to check text messages.

Is it hard to see? Here’s the outline on top:

Cool, isn’t it? I’m not sure what the most common shape would be, especially since our own glasses are all circles (though I noticed our department chair has octagon glasses). If anyone else has any neat polygonal glasses, send me a photo* and I’ll post them here. I love finding real life polygons!

* hlewis5 following by the @ sign and then after that @ sign put naz.edu

Toilet Paper Math

Really, is there a more appropriate follow-up to yesterday’s featured theorem?

Last night young Quentin, age 4½, went to get some toilet paper to clean toothpaste out of the sink after brushing his teeth (because — get this — he likes to clean up after himself. I can hardly believe it.). As he pulled off a strip of TP, he suddenly held it against himself and got all excited: “This is as big as my belly!” I pointed out that his belly was three squares big, and asked how long his arm was. He measured, and exclaimed, “My arm is three squares long!” When he tried to measure his leg, it fell short so I suggested he might need one more square. He immediately went to the roll, counted off a strip four squares long, and held it against his leg. Yup, four squares worked.

The sink stayed dirty for a while after that while he want around measuring his hand (one square), our arms, etc. The nice thing about toilet paper is that he could take strips of various sizes and just pick the one that seemed best. His measurements weren’t exact (I’m not going to hire him to build a bookcase, for example) but he did seem to have the basic idea of measurement and that’s a topic that several K-6 teachers I’ve talked to say is the one that students need the most help with after number sense. (Speaking of which, Denise on Let’s Play Math had a great post Tuesday about helping kids learn number sense.) And I think non-standard measurement is one of the NYS math standards. [Quick check -- yup, it's 1.M.2, 1.M.11, 2.M.1, 2.M.10, and 3.M.10. I spent a while last year putting all the NYS math standards into Excel worksheets for easy searching and posted them here if anyone would find that useful.]

Thinking about blogging this, I googled “Toilet Paper Math” and found some other interesting ways to use toilet paper to do math. You can determine the least expensive choice of TP at the grocery store. You can fold it in half twelve times. You can find the thickness of a sheet of TP (although it seems like density — aka fluffiness — might make that inexact). You can calculate how much text you can print on a roll of toilet paper.

And finally, you can read about how Sir Roger Penrose sued the Kimberly Clark Corporation back in 1997 because one of the designs that was printed on Kleenex quilted toilet paper looked like Penrose tiles (see Wolfram’s Mathworld or this more detailed summary from Professor Richard H. Stern’s Computer Law Class at George Washington University.)

Update 5/10: I think these are the kinds of pictures that mbork was referring to below. In the first two examples the triangle has a right angle, but in the third the angle a bit larger.

The Geometry Van

I taught Geometry this spring, and we spent about half the semester working through Euclid (Book I and a smattering of some others). We proved SAS (Side Angle Side), ASA, AAS, but not ASS (Angle Side Side). Because there is no ASS in Geometry.

Here <A=<A’, AB=A’B’, and BC=B’C’ but the two triangles are not congruent. All we get are bad jokes.

Except that’s not quite true! If <A=<A’, AB=A’B’, BC=B’C’, and BC≥AB, then the two triangles are congruent! In class, I referred to this theorem as ASS.

(The Hypotenuse-Leg Theorem is a special case of this, since the hypotenuses, as the sides across from the corresponding right angles, are certainly longer than the corresponding legs of the triangle).

We didn’t refer to this theorem very often, but it is, well, memorable. And so Adele, one of our majors, was thrilled when she noticed ASS written on a van this past weekend! She snuck a photo of it:

According to Adele and everyone who was with her, this is exactly how I would write the shorthand. We have no idea why it’s on this van. I like to think that this is a Secret Geometry Van, coming out to help students everywhere by providing them with extra theorems.

Pi Day Sudoku Solution

Remember Pi Day Sudoku from Brainfreeze Puzzles?

If you were wanting the answer, it’s been posted!  You can find the solution here.

The Spring Newsletter is here!

We’ve just posted the Spring 2008 issue of Our Newsletter (the seasonal newsletter of the Nazareth College Math Department).   This issue is named The Rudin in honor of Mary Ellen Rudin, according to our tradition of naming each issue after a mathematician (which would have been a great idea to come up with from the start, but in fact only evolved three issues in when we STILL didn’t have any ideas for a Newsletter name).  The lead articles on Extracting Square Roots and Cube Roots come straight from TwoPi’s posts here; other articles are about conferences and competitions our students were in, Alumni News, Toilet Seat Gymnastic (which was in fact our second post ever here, although the article is a little longer than the blog entry), Math Club news, and Problems.  And if any of you submit solutions to problems, we promise to post your name for laud and admiration in the next issue!

Cinco de Mayo Math

Today is the day many people (mostly in the US) celebrate Mexican Independence Day (nope that’s in September) the day in 1862 that the Mexican Army beat the French at Puebla, about 70-80 miles east of Mexico City. As part of this celebration, you might do some Cinco de Mayo Math by Paula J. Maida, which includes ideas like finding the proportion of colors on a Mexican Flag or writing the eleven letters CINCODEMAYO on wooden cacti and letting kids choose a letter randomly as a way to explore both probability and fractions.

Or, expanding beyond Mexico (as the celebration has in many places), you might read “Spanish Colonial Mathematics: A Window on the Past” by Ed Sandifer, which was published in the College Math Journal in September 2002 and is available as a .pdf file here from Ed’s homepage. As Ed explains in this interview:

[S]ome Latino students feel disconnected to math. But we can help make a connection by teaching about Spanish-colonial math and pointing to facts such as there were 11 math books published in Spanish in the New World before there were any in English, with the first being published in 1556, only 100 years after the printing press was invented.

The article above examines 7 of those math books. There’s the 1556 Sumario Compendioso de las quentas de plata y oro que in los reynos del Piru son necessarias a los mercadores: y todo genero de tratantes. Los algunas reglas tocantes al Arithmetica. Fecho por Juan Diez freyle. (Isn’t that a GREAT title!?! It translates as Compendious summary of the counting of silver and gold that are necessary in the kingdoms of Peru to merchants and all kinds of traders. The other rules touching on Arithmetic. Made by Juan Diez, friar.) which was full of all sorts of tables to help you out of you were a tax collector (Hey! More Tax Math!) and includes among other things the following example of how to multiply 875 by 978:

Isn’t this wild? Some of it is explained in the article: the initial 8×9 of 800×900 becomes the 72 of the upper left. Then 8×7 of 800×70 is 56, but this is written with the 5 below the 2 in 72 and the 6 next to the 72, making the 72 look like 726. Then 8×8 of 800×8 is 64, and the 6 of 64 is put next to the 5 while the 4 is next to the 726.

It continues in this fashion, with 7×9=63 of 70×900 put underneath the 56 and with all the numbers crammed together like Galley Division of the same time period. David Smith write in his book History of Mathematics (p. 119) that this method is essentially The Method of the Cup (per copa) because it looks like a goblet.

Several books followed the Sumario Compendioso…, including King Philip of Spain’s Pragmática sobre los diez días del año in 1584, which wasn’t so much about math as about how to deal with the fact that switching from the Julian to the Gregorian calendar involved skipping ten days, and several books that look at military mathematics and formations. In the latter category is the Breve aritmética por el mas sucinto modo, que hasta oy se ha visto. Trata en las quentas que se pueden ofrecer para formar campos y esquadrones by Benito Fernandez de Belo— another fabulous title which means Brief arithmetic for the most succinct method which has been seen up to today. Treating calculations that one can do for the formation of camps and squadrons — which shows how to align 278 men in a squadron into a regular pentagon and contains a woodcut doodle in the back. The final book in the article is the 1696 Cubus, et sphaera geometrice duplicata by Juan Ramón Coninkius about straightedge and compass constructons. Granted, the constructions (doubling the volume of a cube or sphere) were impossible, but that was still unknown at the time.

Happy Cinco de Mayo!

Music to Study By

Or to work by.  Or to procrastinate by.  Here’s a video by starkravenmadd of Doctor Steel’s Fibonacci Sequence:

Kepler’s First Attempt

When it comes to orbits, Johannes Kepler knew his stuff. He’s the one who in 1602 realized that planets orbit in ellipses rather than circles, which became the first of his Three Planetary Laws. But no one is perfect, and these were not his first attempts at describing the motions of the Heavens. In 1596 he published Mysterium Cosmographicum (The Mysteries of the Cosmos), in which he proposed the following model for the solar system:

In this model, the six known planets were envisioned as traveling in circles, along the equators of six giant spheres. The six giant spheres were separated by the five platonic solids. Saturn and Jupiter were separated by a giant cube, and Jupiter and Mars by a giant tetrahedron. It’s harder to see the interior planets in the drawing above, so here’s a close up:

Mars and Earth were separated by a giant dodecahedron, Earth and Venus by a giant icosahedron, and, finally, Venus and Mercury by a giant octahedron. And then, in center of all the orbits, was the Sun.

Let’s see how accurate this model is. If you start with a giant platonic solid, like a cube, you can circumscribe a sphere on the outside and inscribe a sphere on the inside, and then compare the ratio of the radii of the two spheres. It turns out to be √3≈1.73. And lo, if you look at the average radius of Saturn’s orbit (9.021 Astronomical Units) and divide it by the average radius of Jupiter’s orbit (5.20336 AU), it rounds to 1.73. Let’s see how the other ratios match up:

	Giant Polyhedron	Ratio of spheres in model	Ratio of Actual Planet Orbits
Saturn to Jupiter	cube	1.73	1.73
Jupiter to Mars	tetrahedron	3.00	3.42
Mars to Earth	dodecahedron	1.26	1.52
Earth to Venus	icosahedron	1.26	1.38
Venus to Mercury	octahedron	1.73	1.87

Not too shabby! Plus, as a bonus, you can see that the cube and the octahedron, which are dual polyhedra, have the same ratios of the radii of the circumscribed and inscribed spheres (√3≈1.73); likewise, the dodecahedron and the icosahedron (which are also duals of each other) have the same ratio of the radii of circumscribed and inscribed sphere (≈1.26). And unlike the Titius-Bode law, the big gap between Jupiter and Mars didn’t really cause any problems since the tetrahedron fit nicely in there. But a few years later Kepler realized it was wrong, and Uranus’s discovery later would have sealed the deal in any case. Poor Kepler. But it’s still an impressive idea, and was deemed important enough even recently to put on a 2002 commemorative 10-Euro coin in Austria (designed by Thomas Pesendorfer).

Yeah Kepler!

The planet data came from NASA; the data on the radii of circumscribed and inscribed spheres came from Wolfram MathWorld. It’s not clear if the coin is copyrighted or even copyrightable or not; it seems to fall under fair-use guidelines, however. You can find the coin at the Austrian Mint.

Carnival of Mathematics #32!!!

The 2⁵ Carnival of Mathematics is in town at Teaching College Math Technology Blog, run by Maria Anderson.   Most of her entries are about technology — using it, improving it, etc, although there are some non-tech ones like this Mathmaticious video.  Some of my recent favorite entries are a simulation of Leibniz’s calculating machine and a recent discussion about open source textbooks.

The Carnival itself has an assortment of articles from Tic Tac Toe to Linear Equations.  Enjoy!

Finessing Grades

Put these numbers in order: , , and , assuming that .

Here’s the context: Suppose (hypothetically, you know) you’re grading exams on a percentage scale (93% and above is an A, 90-92% is an A-, etc) and after adding scores you realize that maybe because of an unusually difficult problem, the points don’t quite match your overall impression in the sense that the exams that indicate a deep understanding should be the ones that get an A. Now suppose that by adjusting the scores by 5 points they do seem to match up and this is simpler than going back and revising the partial credit schemes on the whole exam.

One way (which only works with foresight) is to add a question, like “What’s your name?” or “What was your favorite topic in the course?” that, in theory, everyone should get right (a freebie question, as it were). This adds 5 points to each person’s score (x) and also to the total (N), raising the overall percentage from to .

A second way is to just drop 5 points from the “Points Possible” column, resulting in a score of . Finally, a third way is to just give everyone 5 points extra credit, giving . What’s a little surprising is that these have different effects, depending on the original score. Take a look:

x=0	0.0%	4.8%	0.0%	5.0%
x=25	25.0%	28.6%	26.3%	30.0%
x=50	50.0%	52.4%	52.6%	55%
x=75	75.0%	76.2%	78.9%	80.0%
x=100	100.0%	100.0%	105.2%	105.0%

For everything but the very highest scores (x=95 and above, it turns out), adding 5 points is the most generous option. But for the lowest scores (x=47.5 and below, it turns out) the freebie question is actually a better option than just dropping 5 points from the total. Indeed, the freebie question is almost as good as extra credit for people who otherwise missed all the points, and does nothing for those at the higher end, while dropping 5 points from the total does the reverse.   (Which leads me to think that analyzing which scheme to use is no simpler than going back and adjusting the partial credit after all.)

Disclaimer: this year I think my exams are on target, and I won’t be adjusting any scores. I have, however, used each of these variations at one time or another in the past although I hadn’t looked closely at the distinctions before today.

1, 2, 4, 8, what comes next?

Suppose you take a circle, put some dots along the outside, and then connect them, as in the picture to the left (which has 5 dots on the outside). If only two lines cross at any point, how many regions will the circle be divided into?

Let’s find out! If you place 1 dot along the outside you can’t connect it to anything so you get one region (the entire circle). If you place 2 dots along the outside and connect them you divide the circle into 2 regions. By placing 3 dots along the outside you divide the circle into 4 regions.

Looking at the bottom row, if you place 4 dots on the circle and connect them you get 8 regions. Hey, notice that neat 1, 2, 4, 8 pattern of the title? What about five dots?

At this point it’s getting a little hard to keep track of all the regions, so I colored them in the pictures below.

Looking at the middle of the bottom row, putting 5 dots along the edge divides the circle into 16 pieces, as might be expected.

What if there are 6 dots along the outside? And that’s where everything goes terribly wrong. It only divides the circle into 31 pieces, not 32. Putting 7 dots along the outside is even worse: you get 57 pieces.

It turns out that even though the numbers 1, 2, 4, 8, 16, 31 start off as powers of 2, they’re really following the formula , where n is the number of dots on the outside. (Where the heck does that come from, you might wonder? It actually comes from the simpler-looking (see Wolfram’s Mathworld)).

So much for our nice pattern.

Fractured Theorems

The Bovino-Weierstrass Theorem:  In any bounded pasture containing an infinite number of cow pies, you can stand in one location where no matter how close you look, there are an infinite number of cow pies near your feet.

 Surely there must be more such fractured theorems lurking out there in our collective imagination… If you come up with any new ones, share them in the comments.

Almost Perfect: The Titius-Bode Law

It’s not perfect, and it only works for 7 of the 8 planets, but it’s still great for getting an approximation for how far apart to hang the glow-in-the-dark planets from your ceiling.

The distance from the Sun to Mercury is approximately 4 tenths of an AU (Astronomical Unit — the average distance from the Sun to the Earth), and the average distance from the Sun to Venus, Earth, Mars, the asteroid belt, Jupiter, Saturn, and Uranus is approximately [4+(3·2^N)] tenths of an AU, where N=0,1,2,3,4,5, and 6 respectively. Isn’t that a neat equation? It was first observed by David Gregory in 1702 (in Latin; 1715 in English) in his book The Elements of Astronomy and is therefore named after Johann Titius (who published a German translation of a 1724 book by Christian Wolff that contained the same description) and Johann Bode (who read Titius’s translation and put it as a footnote in his own textbook). And here’s where I’m wondering if I can get my name added to the Law by virtue of just mentioning it here.

Outrageous Interest Rates

It’s the end of the semester here, which means lots and lots of grading to do (I know, I know—if I didn’t assign it, I wouldn’t have to grade it, right?). So I’m grading papers last night, and I have MTV Hits on for background noise (they’re showing Yo! MTV Raps reruns from 10-15 years ago—how cool is that?), and a commercial comes on as I happen to take a break from eyestrain reading. It’s for a place called Cash Call, which offers fast loans (as in 1 day) over the phone or online, perfect for those times when life throws you a curve. Right?

Here’s the catch: If you take out, say, a $2600 loan from Cash Call (the example they give in the commercial’s fine print), you’ll pay—are you sitting down?—99.25% interest for 42 months! You end up repaying nearly $9100, or 3.5x the original loan! How about a $10,000 loan, with payments spread out over 10 years? Sure, if you’re willing to pay 59.46%, or almost $60,000. (”Exceptionally qualified applicants” may qualify for a 29.26% loan. Woo-hoo!)

I believe it would actually be more financially responsible to buy a $15,000 car at 12% for 6 years ($293/month) and sell the car for $10,000 than to take one of these loans. There are plenty of loan calculators available online. All I can say is… do the math.