Sunday, 29 May 2011

Legos and Recreating Lost Science

Not sure how I missed this the first time around, but there is a really cool video about the reconstruction of the ancient Greek eclipse calculator, the Antikythera ....made of LEGOS!!!! produced by Nature Magazine. Gotta love it...

Now that you are excited, you can read the Nature Article here at the project web site.

Neat Geometry

Just came across a nice geometric idea from Antonio Gutierrez. Not surprising since he has been posting nice geometry on the internet for quite a while.   If you teach geometry, or have a kid who wants to learn geometry, this ought to be on your reader list. 
So for this one, take any scalene triangle and pick a point on the same plane... any point.. go on, I'll wait. 

OK, Ready?  Now draw  a segment from the point to any vertex.  I chose A above in Red.

Now draw a perpendicular to that segment through your point. Also draw the extension of the side of the triangle opposite the vertex you chose to connect (in my case, segment BC).   Finally mark the point where the perpendicular you created passes the opposite side. 

Now do that with the other two vertices.  What happens?  Turns out the three intersections all lie on the same line. 

Friday, 27 May 2011

A Geome-Treat with a Calculus Twist

So you start with a circle, let’s use x^2 + y^2 = 25 as a specimen.   We find a point on the curve, say (3,4), and decide we want to find the line tangent to the circle at that point. 

For me it is(was) always a rush showing kids after a year of calculus that it can be done by just thinking of the equation as x(x) + y(y) = 25 and then replacing the x and y in parentheses with the coordinates 3 and 4.  The line 3x +4y = 25 not only passes through the point (3,4), but it is also tangent to the circle. 

 Ok, so suppose we do that again.. maybe we pick (4,-3) and write the line 4x-3y = 25. 
Now we have two lines. 
Perhaps we take the time to figure out that the two lines must intersect at (7,1) and in passing we wonder what happens if we plug that point into the x(x) + y(y) = 25 the same way? We look at the result, 7x+y=25, and decide that this time it not only doesn’t go through the point (7,1), it  certainly will not be tangent. 
But what the heck. We have come this far so we can graph that line, too…

Ok, so the treat here is that you can do this with ANY conic; ellipses, hyperbolas, circles or parabolas.   If you have a point on the curve, you can find the tangent line with a simple substitution.  And if you have a point not on the conic and want to draw the tangents to the curve through that point, you just plug in that point and it gives you the line through the two tangents.  

Parabolas can be a little trickier so here is an example of finding the two tangents through a point not on the curve. 
I’ll use the simple y=x2 for the parabola, and pick a point (1,-1) for the exterior point.  We can replace one of the x variables in y= xx with the 1, but what about the y value.  Since there is only the one y, we can replace it with the average of y and -1, or (y-1)/2.  That gives us the line y-1 =2x , which is just y= 2x+1.  When we graph the whole thing we see that, indeed, the line cuts through points from which tangents to the curve would pass through our given point. 

But, of course, the calculus student must now show that it always works.  
Shall we make that due Wednesday?

Thursday, 26 May 2011

A Pretty Construction of a Parabola from Archimedes

I was recently re- reading through some old (1920) notes from the Philosophical Magazine that Dave Renfro sent me (THANKS, Dave) and came across a nice problem based on an old principal of parabolas known to Archimedes.  The method, I learned in the article, was used by Archimedes in his On Floating Bodies, book two.. in the course of investigating the equilibrium of a floating parabaloid of revolution.. In the article the author gives this theorem and quotes it as if it is well known... Yet it seems not to appear in texts much then (1920) or now.  I did find the question in An elementary treatise on pure geometry: with numerous example by John Wellesley Russell

Here is the problem: Given a point on the curve  A,  and the slope of the tangent at that point, (AC) and a second point on the curve B,  construct (in the classical sense) additional points on the parabola...  C was place above B by chance, and can be anywhere along the tangent.  I have placed the problem on a coordinate grid to present it as a function of x, but the actual coordinates of the points have no influence on the construction, although it is assumed that you know the direction of the axis of symmetry (in this case, vertical).

The calculus student in you might want to attack this analytically, but time for that later.  Let me show you the Geometric method of Archimedes. 
We begin by constructing a vertical line through B, and selecting a point D, somewhere along this line .  Through this point draw another line parallel to the tangent and a second through point A.  Finally draw a  secant AB of the parabola. 
And then the final act.  Mark the point where the parallel to the tangent intersects the secant AB.  From this point, extend a vertical line to find the point where it intersects AD.  This final point P is on the parabola AB with a tangent of AC at A, and it will be for whatever point D you picked originally. 
With straight edge and compass, you would have to pick a new point D, then recreate another parallel to the tangent, find another intersection at E, and then vertically transfer that up to the line AD for each new point.  But with Geogebra, you can construct, and then just move D and watch P trace out the parabola. 
So NOW let’s do a little calculus. 
If the original points are at (0,0) [why not]  and (p,q) and the slope of the tangent is m, then we  need to find A, and B (C=0 by a clever choice of coordinates) for the parabola y= Ax2 + Bx.   We also know that at x= 0, dy/dx = m  so 2Ax +B = m so B must be the slope m.  Now we just need to fill in y= Ax2 + mx  and passing through (p,q).  This gives us  q = Ap2+mp  and we can solve for A = (q-mp)/p2. 
With my selected  easy values of m=1 and (p,q) = (4,1) we see that y= -3x2/16 + x . 
Two more nice problems for Calculus students that point out things that are easy not to notice in the rush to memorize rules and such..
PROVE each:
1)  If you draw to tangents to a parabolic function, the x-coordinates of their intersection is the arithmetic average of the x-coordinates of the two points of tangency. 

2)  If you draw the tangents to any parabola at the endpoints of the latus-rectum, they will always be perpendicular. 

Wednesday, 25 May 2011

A Letter to the Governor of Michigan

I got this in an email this morning from my beautiful Jeannie.  It has come third hand and I have not verified that the content is absolutely true, but the message is certainly right on the spot.. Enjoy
Dear Governor Snyder, (Gov. of Michigan)

In these tough economic times, schools are hurting. And yes, everyone  
in Michigan is hurting right now financially, but why aren't we  
protecting schools? Schools are the one place on Earth that people  
look to "fix" what is wrong with society by educating our youth and  
preparing them to take on the issues that society has created.

One solution I believe we must do is take a look at our corrections  
system in Michigan. We rank nationally at the top in the number of  
people we incarcerate. We also spend the most money per prisoner  
annually than any other state in the union. Now, I like to be at the  
top of lists, but this is one ranking that I don't believe Michigan 
wants to be on top of.

Consider the life of a Michigan prisoner. They get three square meals  
a day. Access to free health care. Internet. Cable television. Access  
to a library. A weight room. Computer lab. They can earn a degree. A  
roof over their heads. Clothing. Everything we just listed we DO NOT  
provide to our school children.

This is why I'm proposing to make my school a prison. The State of  
Michigan spends annually somewhere between $30,000 and $40,000 per  
prisoner, yet we are struggling to provide schools with $7,000 per  
student. I guess we need to treat our students like they are  
prisoners, with equal funding. Please give my students three meals a  
day. Please give my children access to free health care. Please  
provide my school district Internet access and computers. Please put  
books in my library. Please give my students a weight room so we can 
be big and strong. We provide all of these things to prisoners because 
they have constitutional rights. What about the rights of youth, our 

Please provide for my students in my school district the same way we 
provide for a prisoner. It's the least we can do to prepare our 
students for the giving our schools the resources 
necessary to keep our students OUT of prison.
     Respectfully submitted,
     Nathan Bootz
     Ithaca Public Schools

Thursday, 19 May 2011

The Mathematics of Big Game Hunting

 What follows appeared in American Mathematical Monthly in 1938.  The actual author is NOT H. Petard..but Petard is an interesting term...Google translate replaces it with the English "Petard" which is not much help..but if you study the can be rewarding.  The footnotes and references are a critical part of the document.

A contribution to the mathematical theory of big game hunting

– H. Petard
Princeton, New Jersey
This little known mathematical discipline has not, of recent years, received in the literature the attention which, in our opinion, it deserves. In the present paper we present some algorithms which, it is hoped, may be of interest to other workers in the field. Neglecting the more obviously trivial methods, we shall confine our attention to those which involve significant applications of ideas familiar to mathematicians and physicists.
The present time is particularly fitting for the preparation of an account of the subject, since recent advanaces both in pure mathematics and theorectical physics have made available powerful tools whose very existence was unsuspected by earlier investigators. At the same time, some of the more elegant classical methods acquire new significance in the light of modern discoveries. Like many other branches of knowledge to which mathematical techniques have been applied in recent years, the Mathematical Theory of Big Game Hunting has a singularly unifying effect on the most diverse branches of the exact sciences.
For the sake of simplicity of statement, we shall confine our attention to Lions (Felis leo) whose habitat is the Sahara Desert. The methods which we shall enumerate will easily be seen to be applicable, with obvious formal modifications, to other carnivores and to other portions of the globe. The paper is divided into three parts, which draw their material respectively from mathematics, theoretical physics, and experimental physics.
The author desires to acknowledge his indebtedness to the Trivial Club of St. John’s College, Cambridge, England; to the MIT chapter of the Society for Useless Research; to the F o P, of Princeton University; and to numerous individual contributors, known and unknown, conscious and unconscious.

1. Mathematical methods

1. The Hilbert, or axiomatic, method. We place a locked cage onto a given point in the desert. After that we introduce the following logical system:
  • Axiom I. The set of lions in the Sahara is not empty.
  • Axiom II. If there exists a lion in the Sahara, then there exists a lion in the cage.
  • Rule of procedure. If P is a theorem, and if the following is holds: “P implies Q”, then Q is a theorem.
  • Theorem 1. There exists a lion in the cage.
2. The method of inversive geometry. We place a spherical cage in the desert, enter it and lock it from inside. We then perform an inversion with respect to the cage. Then the lion is inside the cage, and we are outside.
3. The method of projective geometry. Without loss of generality, we can view the desert as a plane. We project the surface onto a line, and then project the line onto an interior point of the cage. Thereby the lion is projected onto that same point.
4. The Bolzano-Weierstrass method. Divide the desert by a line running from N-S. The lion is then either in the E portion or in the W portion; let us assume him to be in the W portion. Bisect this portion by a line running from E-W. The lion is either in the N portion or in the S portion; let us assume him to be in the N portion. We continue this process indefinitely, constructing a sufficiently strong fence about the chosen portion at each step. The diameter of the chosen portions approaches zero, so that the lion ultimately surrounded by a fence of arbitrarily small perimeter.
5. The “Mengentheoretisch” method. We observe that the desert is a separable space. It therefore contains an enumerable dense set of points, from which can be extracted a sequence having the lion as limit. We then approach the lion stealthily along this sequence, bearing with us suitable equipment.
6. The Peano method. Construct, by standard methods, a continuous curve passing through every point of the desert. It has been remarked [1]that it is possible to traverse such a curve in an arbitrarily short time. Armed with a spear, we traverse the curve in a time shorter than that in which a lion to move a distance equal to its own length.
7. A topological method. We observe that a lion has at least the connectivity of a torus. We transport the desert into four-space. Then it is possible [2] to carry out such a deformation that the lion can be returned to three-space in a knotted condition. He is then completely helpless.
8. The Cauchy, for function theoretical, method. We examine a lion-valued function f(z). Let ζ be the cage. Consider the integral

where C represents the boundary of the desert. Its value is f(ζ), i.e. there is a lion in the cage [3].
9. The Wiener-Tauberian method. We obtain a tame lion, L0, from the class L(-¥, ¥), whose Fourier transform vanishes nowhere, and release it in the desert. L0 then converges toward our cage. By Wiener’s General Tauberian Theorem [4], any other lion, L (say), will converge to the same cage. Alternatively we can approximate arbitrarily closely to L by translating L0 through the desert [5].)
10. The Eratosthenian method. Enumerate all the objects in the desert. Examine them one by one, and discard all those that are not lions. A refinement will capture only prime lions.

2. Methods from theoretical physics

11. The Dirac method. We observe that wild lions are, ipso facto, not be observable in the Sahara desert. Consequently, if there are any lions at all in the Sahara, they are tame. We leave catching a tame lion as an exercise to the reader.
12. The Schroedinger method. At any given moment there is a positive probability that there is a lion in the cage. Sit down and wait.
13. The nuclear physics method. Place a tame lion into the cage, and apply a Majorana exchange operator [6] on it and a wild lion.
As a variant, let us suppose, to fix ideas, that we require a male lion. We place a tame lioness into the cage, and apply the Heisenberg exchange operator [7] which exchanges spins.
14. A relativistic method. We distribute about the deser lion bait containing large portions of the Companion of Sirius. When enough bait has been taken, we project a beam of light across the desert. This will bend right around the lion, who will hen become so dizzy that he can be approahced with impunity.

3. Experimental physics methods

15. The thermodynamics method. We construct a semi-permeable membrane, permeable to everything except lions, and sweep it across the desert.
16. The atom-splitting method. We irradiate the desert with slow neutrons. The lion becomes radioactive, and a process of disintegration set in. When the decay has proceeded sufficiently far, he will become incapable of showing fight.
17. The magneto-optical method. We plant a large lenticular bed of catnip (Nepeta cataria), whose axis lies along the direction of the horizontal component of the earth’s magnetic field, and place a cage at one of its foci. We distribute over the desert large quantities of magnetized spinach (Spinacia oleracea), which, as is well known, has a high ferric content. The spinach is eaten by herbivorous denizens of the desert, which in turn are eaten by lions. The lions are then oriented parallel to the earth’s magnetic field, and the resulting beam of lions is focus by the catnip upon the cage.


[1] After Hilbert, cf. E. W. Hobson, “The Theory of Functions of a Real Variable and the Theory of Fourier’s Series” (1927), vol. 1, pp 456-457
[2] H. Seifert and W. Threlfall, “Lehrbuch der Topologie” (1934), pp 2-3
[3] According to the Picard theorem (W. F. Osgood, Lehrbuch der Funktionentheorie, vol 1 (1928), p 178) it is possible to catch every lion except for at most one.
[4] N. Wiener, “The Fourier Integral and Certain of its Applications” (1933), pp 73-74
[5] N. Wiener, ibid, p 89
[6] cf e.g. H. A. Bethe and R. F. Bacher, “Reviews of Modern Physics”, 8 (1936), pp 82-229, esp. pp 106-107
[7] ibid
This first apperared in Amer. Math. Monthly 45 (1938) p446-447. In fact, the Method 10 included here did not actually appear in the original Monthly version, but in a slightly expanded version of that appeared in Eureka.
* It is perhaps generally not known that Petard’s** full initials are H. W. O., standing for “Hoist With Own.”
** Actually, H. Petard is the pen-name for the mathematician E. S. Pondiczery***, who preferred to publish the paper pseudonymously.
*** In fact, Pondiczery himself was a fictious mathematician invented by Ralph P. Boas and Frank Smithies.****
**** This was published in The American Mathematical Monthly, 1938, with one editorial alternation: a “footnote to a footnote was ruthlessly removed.” Consider this last footnote to a footnote to a footnote of a footnote as our nod to the masters.

Wednesday, 18 May 2011

Science "Humor"

Someone started telling jokes last night on twitter... some are old, some are new,  and a few of them are actually funny...  so, here are

The BEST of Joke-night on  Twitter
for Science-nerds
 Some jokes require a minimal level of science education,
 and others are just bad)
1)      I worked out the momentum so accurately that I cant find it anymore!

2)      What do you get when you cross an elephant with a banana?
Elephant banana sine theta.

3)      And the bartender says, "We don't serve tachyons here." A tachyon walks into a bar.

4)      2 monkeys in a bath - one says " ooo ooo aah aah aah ooo ooo" The other says "Well, put some cold in".

5)      What's red and stands in the corner? A naughty London bus

6)    Heisenberg gets stopped on the motorway by the police.
Cop: Do you know how fast you were going sir?
Heisenberg: No, but I know exactly where I am.

7)      A photon checks into a hotel. The bell hop asks him " Can I help you with your luggage?" To which the photon replies, "I don't have any. I'm traveling light."

8)    A neutron goes into a bar and orders a beer. As the neutron is reaching for its wallet, the bartender looks at it and says, "Oh, for you--no charge."

9)    everyone knows that math puns are the first SINE OF MADNESS!

10) What do you say to a guy who walks into a bar three times?   Move the bar stupid!
Alternate Version: What did the man say who walked into a bar? Ouch!

11) Decartes walks into a bar. The bartender says: Rene! Great to see you! How about a beer? Decartes says: I think not. And promptly disappears

12)  Only two things are infinite, the universe and human stupidity, and I'm not sure about the former. Albert Einstein

13)  How do farmers do trigonometry? Using swine functions..

14)  What do Kermit the Frog and  John the Baptist have in common? Same middle name.

15)  Wanted, dead AND alive, Schrödinger's Cat

16)  I dream of a better future. where chickens can cross the road without being asked about their motives.

17) Actual article title... Frank Harary and Ronald Read wrote a 1974 paper entitled "Is the null graph a pointless concept?"

18) From Zain, the Amazing.. I'm reading a great book on anti-gravity. Can't put it down!

19)  Lee Sent (see comments)  
Two atoms accidently bumps into one another.

"Ouch, I think I lost an electron."

"Are you sure?"

"Yes! I'm positive!"

20 )Q. What's woolly and equivalent to the axiom of choice? A. Zorn's Llama.  Richard Elwes

21)  Q. What's an anagrams for Banach-Tarski?  A.  Banach-Tarski Banach-Tarski. from Nat Stahl

22) (and one of the oldest ever) Q. What's purple and commutes?   A.  Abelian Grape

23)   Q: What is normed, complete, and yellow?  A: Bananach space From Derek Orr

24) And a single question with three nice answers:
   Q.  How many mathematicians does it take to screw in a lightbulb?
A1.  0.99999999999...  Murray Bourne
A2.  1.0000000000... 1    Edward Shore
A3.  Infinitely many ..  The first screws it in half way, the next 1/4 of the way, ........  David Marain
A4.(and my favorite) One, he gives it to three physicists, thus reducing it to a problem that has already been solved.

25)Q: What does the B in Benoit  B. Mandelbrot stand for?
A:   Benoit B. Mandelbrot

Can I have a rim shot Ringo!
***YOUR BEST JOKE GOES HERE ... As soon as you send it in that comment box below..

Sunday, 15 May 2011

Art in Motion

Just found this on the  blog of Richard Wiseman, a magician/mathematician/author from here in England.. Anyone know a name for this art, and how they are done? 

Thanks to Nate, (see comments) it seems that something very similar has ben done in childrens books and walls all over America.

Who Needs Libraries Anyway?

Alcohol tax? No.... Cigarette Tax ?  No... 

Fire the Librarians... Can Art Teachers, Music teachers, ..... Math teachers be far behind...

 From the LA Times 

If state education cuts are drastic, the librarians' only chance of keeping a paycheck is to prove they're qualified to be switched to classroom teaching. So LAUSD attorneys grill them.

In a basement downtown, the librarians are being interrogated.

On most days, they work in middle schools and high schools operated by the Los Angeles Unified School District, fielding student queries about American history and Greek mythology, and retrieving copies of vampire novels.
But this week, you'll find them in a makeshift LAUSD courtroom set up on the bare concrete floor of a building on East 9th Street. Several sit in plastic chairs, watching from an improvised gallery as their fellow librarians are questioned.

A court reporter takes down testimony. A judge grants or denies objections from attorneys. Armed police officers hover nearby. On the witness stand, one librarian at a time is summoned to explain why she — the vast majority are women — should be allowed to keep her job.

Thursday, 12 May 2011

Texas, Teaching, and Too Crazy to Believe

Got an e-mail today from a teacher in my school whose home is in Texas.  Makes you go "Hmmmmm."  Fact is, I grew up there... believe me these kids know how to drive fast... what they don't know is how to ...pick one or several (read, write, do math)


Texas Funds Formula One Race, May Fire Teachers

By Darrell Preston and Aaron Kuriloff - May 11, 2011 5:43 PM GMT+0100
·         Texas, which may balance its budget by firing thousands of teachers, plans to commit $25 million in state funds to Formula One auto racing each year for a decade.
Four years after motorsports’ most popular series left the U.S., Texas investors including Clear Channel Communications Inc. co-founder B.J. “Red” McCombs are building a 3.4-mile (5.5-kilometer) track to bring the event to Austin. Comptroller Susan Combs has agreed to pay $25 million for races through 2022, a subsidy questioned by critics and lawmakers as the state cuts costs to close an estimated $15 billion two-year deficit.
“I don’t understand why 25 people in Austin could not put up $1 million each if they thought this was a good opportunity instead of the state making a $25 million commitment,” said Senator Dan Patrick, a Houston Republican. “The developers should find the money through private sources.”
As many as 100,000 teachers in Texas may be fired because of spending cuts to cope with the state’s budget crisis, according to Moak Casey & Associates, an Austin-based education consultant. For $25 million a year, the state could pay more than 500 teachers an average salary of $48,000.
“I have to wonder why the state of Texas is all over funding for this racetrack and not the school-funding crisis,” said Ewa Siwak, 44, who teaches German in the Austin Independent School District and whose job at Bowie High School is being cut. “Tax dollars for education should be a higher priority.”

(it goes on to say how much money this will generate---billions—and it will be good for the economy…  couldn’t stand to read much more….)

Wednesday, 11 May 2011

Quaternions and Quilts

As I write this my beautiful Jeannie is in Paducah working on our retirement home.  I mention this because Paducah happens to be the home of the National Quilt Museum (which just managed to stay above water level in recent weeks).  And I mention that  because a tweet from @MathBits just came across my desk with the image below.
Now if that looks like just any old pretty quilt, look again.  The quilt is designed to represent the Quaternions (sometimes called the H group, and sometimes Q8) multiplication table ... In case you've forgotten (shame) the multiplication rules for Quaternions here they are :
Group equations

As best I can tell, the quilt is the handy work of Professor Gwen L. Fisher of the Department of Mathematics at California State Polytechnic Univ., San Louis Obispo.
And if you need to see the whole multiplication relation explained, see the page here, with that and much more about Quaternions and quilts. 

Tuesday, 10 May 2011

The Man of Numbers

Just got a tweet from Keith Devlin reminding all that his soon to be released biography of  Fibonacci is on sale at 35% off for preorders.  This is one of those characters in math history who should be on the reading list of every middle and high school teacher.  I wasn't asked to review it (shame on you Mr. Devlin) but given the subject, his level of knowledge, and the number of really great books he has already written, I'm betting it will be a great one.  I put mine on order today...  Just in time for summer reading on the beach...
The Man of Numbers: Fibonacci's Arithmetic Revolution

Sunday, 8 May 2011

The Four Color Problem

My pre-calc class were full of questions the other day after a reading prompt in their English classes.  I tried to give answers and tell them the gist of the story, but was not satisfied with my own information on the early history of the problem.  So today I was researching in the hope I could put together a much better description of the story in this blog...but while researching, found a really great covering of the story with discrete math details and images that would take hours to concoct... so instead, All my students should read this really nice version of the events at the NRich math site here in the UK... A really good story...

now can someone explain what  that has to do with this event, occurring in the same room?

Saturday, 7 May 2011

Swinging Science, Harmony of the Spheres

I found this at John D Cook's The Endeavour web page and wanted to make sure my students get to see it...

It is described as "Fifteen uncoupled simple pendulums of monotonically increasing lengths dance together to produce visual traveling waves, standing waves, beating, and (seemingly) random motion." but I think the label should just be "SPOOKY". 


This is from the folks at Harvard Natural Science, and they give a little detail for the curious (that's you, right, you are curious)...

How it works: The period of one complete cycle of the dance is 60 seconds. The length of the longest pendulum has been adjusted so that it executes 51 oscillations in this 60 second period. The length of each successive shorter pendulum is carefully adjusted so that it executes one additional oscillation in this period. Thus, the 15th pendulum (shortest) undergoes 65 oscillations. When all 15 pendulums are started together, they quickly fall out of sync--their relative phases continuously change because of their different periods of oscillation. However, after 60 seconds they will all have executed an integral number of oscillations and be back in sync again at that instant, ready to repeat the dance.

Wednesday, 4 May 2011


Every kid learns it in science class.. the colors of the rainbow (visible light spectrum) are given by the mnemonic ROY G BIV for Red, Orange, Yellow, Green, Blue, Indigo, and Violet. But really, before you knew the mnemonic if someone asked, "What colors do you see?" what would you have said. I don't know if anyone has ever tested young kids before they are exposed to science classes, but it seems it used to be common to describe four colors, Red, Yellow, Green, and Blue... So why do we see (or why are we told to see) seven?

Turns out, it was old Isaac Newton, and his dedication to the ancient Greek ideas. Here is a clip from a blog by Patricia Fara, a senior lecturer at Cambridge, and author of "Science: A Four Thousand Year History" at the Nature web site.
Consider Isaac Newton. He believed so firmly in the Greek idea of a harmonic universe that he divided the rainbow into seven colours to correspond with the musical scale. Before then, although opinions varied, artists mostly showed rainbows with four colours. It is, of course, impossible to make any objective decision about the correct number, because the spectrum of visible light varies continuously: there is no sharp cut-off between bands of different colours, so how you think about a rainbow affects how you see it. Be honest - can you tell the difference between blue, indigo and violet?
Dr. Len Fisher has written that, “The mediaeval rainbow had just five colours: red, yellow, green, blue and violet. But Newton added two more – orange and indigo.... The background, though, is that Newton believed that the rainbow should have seven colours, because his view of the harmony of nature required that the colours should be “divided after the manner of a Musical Chord”

I found this on Wikipedia:  "n Classical Antiquity, Aristotle had claimed there was a fundamental scale of seven basic colors. In the Renaissance, several artists tried to establish a new sequence of up to seven primary colors from which all other colors could be mixed. In line with this artistic tradition, Newton divided his color circle, which he constructed to explain additive color mixing, into seven colors. His color sequence with the unusual color indigo is still kept alive today by the Roy G. Biv mnemonic. Originally he used only five colors, but later he added orange and indigo, in order to match the number of musical notes in the major scale."

I have found a number of illustrations related to light in which the letters ROYGBIV were spelled out showing the spectrum, such as this clip from The Sunday School Teacher(1867)

One earliest use I have found for the use as a mnemonic, or "aid memorie" as it is called in the article, is from the May 2, 1890 issue of the magazine, Photographic Times:

An even earlier use appears in School managment, by Joseph Landon, in 1883, in a section of the book called Memory in Education :

So I can't find out who first came up with the mnemonic, but I did find a song about it...lyrics are here.  And  Aaron Wagner sent me a link to a video of a different ROY G BIV song by "They Might Be Giants" that his four-year old highly endorses.  Thanks to Aaron and his child for the tip.

Say it with Numbers, but NOT Prime Numbers

Just read an article summary from the Journal of Marketing Research on a twitter tip by Alex Bellos about the preference for composite numbers in product names... Her is most of the good stuff. 

A series of experiments documents the influence of numbers on the liking of brands. For example, an imaginary brand name for anti-dandruff shampoo (Zinc) is more liked when it includes a common product number (e.g., Zinc 24) than when with includes a prime number (e.g., Zinc 31). The research also shows that the presence of the operands responsible for the sum or product further enhance the liking of a brand name. For example, not only is a Volvo S12 more liked than a Volvo S29, but liking is further enhanced when an advertisement for a Volvo S12 includes a license plate with the numbers 2 and 6. The operands 2 and 6 make 12 more familiar because they encourage the subconscious generation of the number 12.
The influence of available operands on liking for the number brand extends to advertising claims. The authors conducted a study in which consumers were asked to make a choice between V8 and Campbell’s tomato juice. Some consumers saw a V8 advertisement that stated, “Get a full day’s supply of 4 essential vitamins and 2 minerals with a bottle of V8” whereas other saw an advertisement that stated “Get a full day’s supply of essential vitamins and minerals with a bottle of V8.” More consumers chose the bottle of V8 when the number 4 and 2 were explicitly mentioned in the claim. Creating similar advertisements for Campbell’s tomato juice did not influence preferences for Campbell’s.

Tuesday, 3 May 2011

More on Pi and the 47 Ronin

A few years ago I wrote this post... Since then I have learned enough to add an addendum at the bottom...
If you take the Asakusa Line from Shinagawa, just one stop away you will come to one of the most famous shrines in all of Tokyo, the Sengakuji Temple. It isn't the biggest, prettiest, or most ornate, but it is rich with the kind of history the Japanese love. This is the resting place of the 47 Ronin, one of Japan's most popular samurai stories.

"The story has all the elements for a Hollywood production: a good, noble guy who dies unfairly; a corrupt court official and cunning villain who is disliked by everyone but seems to be always ahead of the game; the good guy’s loyal subordinates who are totally determined to avenge their master’s death at whatever price, even with their own lives... In the end, the story has sparked the imagination and inspired the utmost respect from an entire nation for over 300 years."[Luis Estrada's Travel Blog]

I came across a mathematical reference to the story in a March 1908 article in the American Mathematical Monthly I received recently from Dave Renfro.

"In Tokyo, at the Buddhist temple of Sengakuji lie buried the forty-seven Ronin, the national heroes of feudal Japan. Just within the gate, in a two-storied building, swords, armor and other relics of these heroes are shown on payment of a fee. By the side of the path leading to the tombs is a well with the inscription, 'Here they washed it.' No one in Japan needs to be told that it was the bloody head they were bringing to the grave of their lord, that dead master for whom they considered it the highest privilege thus to forfeit all their lives. The popular reverence for these heroes is still attested not only by the incense perpetually kept burning before their tombs but in stranger fashion by the fresh visiting cards constantly left upon their graves. [To someone who is still there, do the Japanese still leave these visiting cards?]"

"All the world knows their exploit, but who knows that one of them, Shigekiyo Matsumura, was the greatest Asiatic mathematician of his age, who in his work Sanso, published in 1663, calculated the length of one side of a regular inscribed polygon of 32768 or 215 sides, obtaining 0.000095873798655313483 and thence for the value of pi 3.141592648, which is accurate to seven places of decimals, to eight significant figures..."

I would be thrilled if any of the folks who still read this in the Tokyo area would send a digital picture of the tomb of Matsumura so that I can add it to this note.
ADDENDUM -----------------------------------
The very kind Arjen Dijksman connected me with Japanese Physicist Tasuo Tabata who gave me some detail.  It seems that  Matsumura might be more appropriately called Shigekiyo Muramatsu.  He did write the sanso, and all the math things described seem to be a modest description of his contributions.  However, Professor Tabata tells me that he was NOT one of the 47 Ronin.  He is, however, connected to the story. The professor says, "Shigekiyo had only a daughter. Her husband Hidenao and their son Takanao joined the 47 ronin. --"... So now, I guess I'm down to wanting a picture of the 47 Ronin from Sengakuji Temple. And if anyone knows where the grave of Shigekiyo Muramatsu is located, and/or has a picture I would love to have one.
Professor Tabata has written the details he found in slightly more detail here.

Thanks again to professor Tabata and Arjen for their help.

Sunday, 1 May 2011

Gauss' Missing Blackboard

I just came across a document on the web that I can't find the source of but it has a really nice story about Gauss and the constructable 17-gon story. It seems to be related to someone named Alex Anderson...(if you know that person, he has a picture I want a copy of very much)..
"Gauss was visiting Braunschweig (Brunswick) and still lying in bed on 29 Mar 1796 when he realised the connection of the cyclotomic polynomial to the construction of the n-gon and how to construct the 17 gon. Ahrens [p. 11] says Gauss sent the slate on which he had done the calculation to Wolfgang Bolyai, who preserved it." Does anyone know where that slate is today?

Tangent Sequences of a Cubic

Old mathematicians never die;
they just go off on a tangent.

Recently re-introduced to a pretty property of the tangents to cubics from Ross Honsberger's book, More Mathematical Morsels (Dolciani Mathematical Expositions).  I will illustrate a simple part of the property, then provide a source for a really nice extension that seems to be pretty new.

As the magician says, take a cubic, take any cubic... OK, so graph any y=f(x) where f(x) is a third degree polynomial. (you really don't have to graph it, but it might help you see a second property).  

1)  Pick any point other than the inflection point and record the x-value as A

2) Write the equation of the tangent line at that point
3)  Find where the tangent line intersects f(x) , call this B
4)  Repeat the process starting at Point B to get a point C,
5)  Do it again starting at C to get point D...
Look at the sequence A, B, C, D... What do you observe?

Ok, so let's practice a little integration. Find the area between the tangents and the curve.  Now think of the ratio of the two areas big/little...
Focus deeply, I'm reading your mind...
AHA, the ratio was 16.

"How does he do it?"

There is an additional nice extension about this idea explained by Alexander Bogomolny at his Cut-the-Knot web site.  Enjoy.