Archive for the ‘Books’ Category

Time and Tisza = living on the river

Saturday, January 2nd, 2010
Mayflys on the Tisza River, Silas 2009

GIANT Mayflys on the Tisza River, Silas 2009

I just listened to a wonderful interview of Susan Silas, by Will Corwin of, on the occasion of the opening of her exhibit, Helmbrecht’s Walk. It is at the Hebrew Union College Museum on One West 4th Street, NYC, and will be up for the academic year ending in June 2010.

She talks about the walk, of course, but also covers decaying birds and giant mayflys on the Tisza River in Hungary, all apropos of time passing. My dad often swam in the Tisza when he lived in Szolnok and worked at the Cukorgyar (beet sugar factory) before WWII. He always had plenty of sweets to treat the girls on the beach.

Makes me anticipate even more Sean Carroll’s book, From Eternity to Here, due out this week. Then I’ll have the art and science of time covered.

Wishing everyone a sweet New Year!

!eggstrondinary = BA on APOD (again)

Sunday, September 21st, 2008

Okay, its Autumnal not Vernal, but can’t an Equinox go by without egg balancing, and the obligatory reference to the Bad Astronomer’s explanations.  According to the Astronomy Picture of the Day (available as a Windows background), edited by Robert Nemiroff, who has a free Astronomy course online, and Jerry Bonnell, apparently no.   APOD again published the picture of BA Plait (whose new book is out October 20th) balancing eggs in his kitchen.  The kids above show Phil up by balancing eggs on their (Blefuscudian) small end.

Update: eggPOD

September 19 = TLAPD

Friday, September 19th, 2008

Again.  So Shiva ma Timbas, and buysteal the book.

Steve Sigur = Teacher, Music & Nature Lover, Triangle Geometer

Sunday, August 17th, 2008

In 2004 he was the first high school teacher to be an invited speaker at the MAA’s MathFest.  Steve Sigur also co-wrote a book with arguably the most famous mathematician of our time, John H. Conway.  Aside from being a gifted mathematician - as the many testimonies of his students attest [1][2][3][4], Steve Sigur was, for nearly 30 years, a gifted teacher.  When stricken with the same deadly brain cancer that recently afflicted Senator Edward Kennedy, he underwent experimental procedures at Duke University that enabled him to continue teaching for over a year; and incidentally, finish his triangle-shaped-full-color-with-software-included-book, The Triangle Book (not available yet).

As Philip Davis’ 1995 paper (pdf) points out, triangles have a long and rich history, from Euler’s Elements to the work of Emile Lemoine.  To understand beyond 1995 The Modern Geometry of the Triangle you must read Sigur’s paper (pdf).  His web pages are chock full of triangle math and other delights.  But the crown jewel is a set of interviews the Paideia School did with him.  Here he discusses his love for life.

Steve Sigur died last month.

Car Car = Fuelish Hyperbolae

Thursday, October 4th, 2007

My blogoverse buddy (BvBTM) Jonathan asked me to contribute to CoM-18, which he is hosting and I am happy to do so. But first a belated shoutout to my BvB, and best reviewer, Dave Marain, who interviews Professor Lynn Steen, a principal architect of the original NCTM Standards and a highly respected voice in reform mathematics.

car car

Don’t expect any research topics here. This is about solving a practical problem in automotive gas economy which involves a pricing anomoly, a Greek mathmatician who may have tutored Alexander the Great, and an 18th century Scottish math professor who almost loses his job by taking an unauthorized 2-year sabbatical to tour the Continent. Can you imagine not losing your job today?

So what is the problem? I have 2 cars, an older model which uses regular gas, and a slightly newer model which only uses the more expensive high-test gas, but also gets more miles per gallon (mpg). The question is which is more economical to  drive? The pricing anomaly I have observed is that no matter the price per gallon (ppg) of gasoline, and the price has fluctuated up and down quite a bit, the difference in price between regular and high-test is almost always a constant, viz., 25 cents. So which car  is more economical to drive actually depends on the price of gasoline! Observe.

The price per mile, ppm = price per gallon / miles per gallon.  The percent difference in ppm between regular and high-test,

%diff = (ppmregular / ppmhigh-test) – 1 = (p / m) – 1,

where p = ppgregular / ppghigh-test and

m = mpgregular / mpghigh-test.

If we take

d = ppghigh-testppgregular = $0.25 and

m = 25 / 27 and

then plot ppghigh-test against %diff we get:


So if the price of high-test is below about $3.35 the gas guzzler is more economical fuel-wise, and versa vice. But what is this  curve? A cubic polynomial trend line fits it almost perfectly. So is it a polynomial? Zooming out gives us a clue.


It is a rectangular hyperbola (first studied by Menaechmus, a student of Plato, about 350 BC) flipped on the X axis and asymptotic to X = 0 and Y = 0.08 = (1 / m) -1. You can easily see this by rewriting the equation for %diff as

%diff = A / ppghigh-test + B where

A = -d / m and

B = (1 / m) – 1

So this leaves the question of why a cubic polynomial fits the data so well. And here is where our Scottish math professor, Colin MacLaurin, comes in. In 1742 he wrote A Treatise on Fluxions (pdf), the first systematic application  of Newton’s calculus, in which he shows, among other trigonometric marvels, how the equation for a hyperbola could be closely approximated by truncating an infinite polynomial series.

NY X Colbert = Re – Viewing Claudette

Friday, September 21st, 2007

friedman hale hale dowd colbert 

Content published for the NYTimes TimeSelect program is now free! This not only includes Op-Ed columnists Thomas Friedman and Maureen Dowd, but the venerable paper’s archives. Last year, when Time magazine opened its archives we looked at the history of Star Trek through Time‘s articles.  This times(sic) I started to investigate the ever-popular Stephen Colbert.  But the best I could find was a Correction and a funny ellipsis.  I couldn’t even find that he doesn’t even read his own books, even though it is all over the blogoverse.

Correction     ellipsis

However, I did find reviews of Claudette Colbert’s movies – starting with her first film, the silent 1927 Frank Capra rowing film, For the Love of Mike, and culminating in the 1934 five Oscar winner, Capra’s It Happened One Night (trailer), the prototype for the Hollywood screwball comedy.

For Love of Mike is a presumed lost movie. The following complete review by Mordaunt Hall (great name) is most of what is known about it’s content.  Any rowers have a print? When this film was completed there was no money left to pay the director, Frank Capra (who considered it the worst film of his career), so he had to hitchhike back to Hollywood from New York City!

August 24, 1927 by Mordaunt Hall. “For the Love of Mike” makes no pretensions of being anything but a movie. It seemed to satisfy the audience at the Hippodrome yesterday afternoon, for there was laughter and, at the end, applause. It is not a dignified piece of work and neither is it ever convincing. Ben Lyon looks no more like a Yale athlete than Richard “Skeets” Gallagher looks like a coxswain of a racing shell. But during the boat race scenes, supposedly between Yale and Harvard, there is quite a good deal of interest, even though you have a hazy idea that Ben Lyon’s arms are not stroking the Yale crew. Here, Yale wins. It would be interesting to see this production in Cambridge, Mass., if the exhibitor had forgotten to change a few titles and still have Yale win. Mike, the hero of this extravaganza, is first perceived as an active and attractive baby, not quite a year old. He is left on the doorstep and brought up by Abraham Katz, Herman Schultz and Patrick O’Malley, who, as the child’s adopted fathers, show their ignorance of infants. It is a case of necessity being the mother of invention when it comes to dressing the baby and arranging it so that the child can drink its milk. Sometimes this baby looks at George Sidney, Ford Sterling and Hugh Cameron, who officiate in the rôles of the fathers, as if wondering why these grown-up, plump specimens of humanity were making so much fuss and being so inordinately ridiculous. What this infant prodigy might have muttered if he had been able to read the comic subtitles would perhaps have been even more interesting than his expression of contempt at some of the pantomime. Rudolph Cameron figures as the man with a past who is eager to collect a few sheckels wherever possible. He contributes some really thoughtful acting, which is more than Ben Lyon does. In fact, Mr. Lyon suffers by comparison in those scenes in which he appears with Rudolph Cameron (there are two Camerons in this film). Claudette Colbert, who was seen in Kenyon Nicholson’s play “The Barker,” lends her charm to this obstreperous piece of work. She seems quite at home before the camera.

September 7, 1929 by Mordaunt Hall. The closing scene of the talking picture, “The Lady Lies,” evoked a hearty round of applause from an audience that filled the Paramount Theatre yesterday afternoon. It is a good, popular entertainment, a film that sustains the interest and has the advantage of intelligent acting by the principals. … Miss Colbert is charming.

April 15, 1929 by Mordaunt Hall. “The Hole in the Wall,” an audible screen adaptation of a play by Fred Jackson, is a queer combination of senseless drama and some excellent pictorial direction. The plot of this mystic melodrama fails to be in the least disconcerting, but the idea of the imaginative swindlers having a mirror they use as a giant periscope to see who is at the door is interesting. … Another unfortunate feature of this production is that the able Claudette Colbert was called upon to act in it.

May 19, 1930 by Mordaunt Hall. Above the tumult, the shouting, the slang, the wisecracking and crudities of the audible pictorial version of the play “The Big Pond” looms the figure of Maurice Chevalier, whose whole-hearted acting compensates in no small degree for the farcical extravagances in this unimaginative narrative. … Claudette Colbert plays Barbara pleasingly, but through poor photography she does not look nearly as attractive as she is.

September 2, 1930 by Mordaunt Hall. It is, of course, rather amusing to perceive a Frenchman impersonating a nouvenu riche from Chicago and to observe the gestures of his wife. … But as good as Miss Colbert and the others are in their rôles, it is Mr. Menjou who shines in this production.

 I highly recommend The Smiling Liutenant,an early (1931) Lubitsch classic, along with A Merry Widow, Ninotchka, Heaven Can Wait … . 

May 23, 1931 by Mordaunt Hall. Wit and melody swing through Maurice Chevalier’s latest picture, “The Smiling Lieutenant,” which is now adorning the Criterion screen. That cinematic artist, Ernst Lubitsch, supplies the rapier-like comedy and none other than Oscar Straus is responsible for the charming musical compositions.
Besides M. Chevalier’s clever acting and singing, there are splendid performances by Claudette Colbert as Franzi, Miriam Hopkins as the Princess and George Barbier as Adolf.

February 28, 1931 by Mordaunt Hall. Granted that “Honor Among Lovers,” the current film attraction at the Times Square Paramount and the Brooklyn Paramount, strikes a popular note, its incidents are frequently far from credible, particularly when the law enters into the story. It has, however, the virtue of being exceptionally well cast, with Claudette Colbert and Fredric March officiating as the principal characters. … Miss Colbert is excellent.

December 5, 1931 by Mordaunt Hall. “His Woman,” which was known as “Sal of Singapore” during most of its journey through the studied mills in Astoria, blew feverishly across the Paramount’s screen yesterday. It opens in one of those cinematic hell-holes in the Caribbean, where the sailors order gin straight and the dancing girls draw red roses across their lips to show the devil in them. Into this maelstrom of dank passions stalks honest Sam Whalan, skipper of a tramp freighter. He pushes a Spaniard. The Spaniard pushes back. Sam is a little tight, so he picks up a chair and dashes madly about in all directions. Maybe it is the camera, but Sam seems to be giving an imitation of an old woman stirring pudding as he swings his chair. That is only the beginning of Sam’s adventures. There are many more laughs later on; some of them intentional. … Gary Cooper and Claudette Colbert give their usual competent performances in the leading rôles, but they are beaten down by the general burlesque qualities of the story.

March 12, 1932 by Mordaunt Hall. There is nothing in “The Wiser Sex,” the present picture at the Paramount, that recalls in the remotest way the clever writing of Clyde Fitch, and yet this film story is said to be an adaptation of his play, “Her Confessions.” The introductory scenes rather lead one to expect a narrative inspired by Samuel Seabury’s recent activities in connection with the Hofstadter legislative committee, but soon it developes into a somewhat clumsy melodrama, with dialogue that is beyond even the ken of the competent cast to make effective. … Miss Colbert is charming, but often she might have been photographed to better advantage.

October 1, 1932 by Mordaunt Hall. Notwithstanding the acrimony that burst forth intermittently between George M. Cohan and Paramount Studio officials during the filming of “The Phantom President,” this feature, in which Broadway’s little giant makes his talking-picture début, is a crackerjack show. It is an adaptation of a novel by George F. Worts and was directed by Norman Taurog, who has to his credit among other productions that splendid shadow venture, “Skippy.” This film is an excellent example of technical skill and many of the episodes are highly imaginative. It was not surprising that it kept the audience at its first exhibition yesterday in a constant state of glee. In an introductory interlude there are the paintings of Washington, Jefferson, Lincoln and Roosevelt, which come to life, sing and deliver opinions concerning the country’s need for a man. Then one is introduced to Theodore K. Blair, a Presidential prospect, who, however, in the opinion of several of his friends lacks political charm and other characteristics. It is granted, however, that if he were to become engaged to the charming Felicia Hammond, they might nominate him with a certain assurance. But Miss Hammond, impersonated by Claudette Colbert, does not fancy being Mrs. Blair. … Miss Colbert is more attractive than ever as the bright-eyed, winsome Felicia.

April 9, 1932 by Mordaunt Hall. A wild affair, brimming over with clever nonsense, is now occupying the Paramount screen. It bears the title of “The Misleading Lady” and is an adaptation of a play of the same name by Charles W. Goddard and Paul Dickey which was presented in this city as far back as 1913. Its humor affords rich opportunities for Claudette Colbert, Edmund Lowe and Stuart Erwin. … Miss Colbert gives a capital portrayal as Helen.

June 25, 1932 by Mordaunt Hall. There is much to admire in the acting and in the composition of the scenes in the Paramount’s film, “The Man From Yesterday,” a story of the war which hails from a play known as “Wound Stripes.” The dialogue, however, is frequently open to adverse criticism, for the lines are often unconsciously humorous. … Claudette Colbert is gracious and competent as Sylvia.


The un-reviewed 1932 movie, Sign of the Cross, is the pious DeMille spectacular where Colbert takes her famous milk bath.

January 21, 1933 by Mordaunt Hall. While George Arliss is to be seen at the Radio City Music Hall as a monarch who is only too glad to get rid of his throne, Claudette Colbert is appearing on the screen of the Paramount as a Queen who pleases her people by marrying a commoner—the man she has been in love with since she left her country a year before. It is interesting to note that Mr. March and Miss Colbert are also lending their talents to Cecil B. De Mille’s film of “The Sign of the Cross,” which is now at the Criterion … Miss Colbert, with all due respect for royalty, is almost too good looking for a queen.

May 18, 1933 by Mordaunt Hall. Culled from Max Miller’s book, “I Cover the Waterfront,” there has reached the Rivoli a stolid and often grim picture, the principal asset of which is the clever acting of the principals, particularly that of the late Ernest Torrence. Its drama is not nearly as successful as one might expect in a film directed by James Cruze. The sullen happenings are more shocking than suspenseful, and for moments of levity there are the mumblings and doings of a bibulous newspaper man and the Hollywood conception of the manner in which another reporter berates his city editor. … Miss Colbert does well as Julie, but she is scarcely convincing as a fisherman’s daughter, chiefly because she does not look the type.

October 7, 1933 by A.D.S. The unwed mother continues her losing fight with the cinema in “Torch Singer,” which, with appropriate wails and shrieks, was unfurled at the Paramount yesterday. This time it is the story of a handsome young woman (Claudette Colbert) who signs her baby away and becomes a grim and bitter dispenser of torch songs in midnight rendezvous. Her favorite is one called “Give me liberty or give me love.” The idea is that men made her suffer and she is going to make them pay. Her favorite epigram: “I’m like glass. Nothing will cut me but diamonds.” … Miss Colbert adds a fillip to her generally fine performance by singing her own songs in an attractive contralto.

February 23, 1934 by Mordaunt Hall. There are few serious moments in “It Happened One Night,” a screen feast which awaits visitors to the Radio City, and if there is a welter of improbable incidents these hectic doings serve to generate plenty of laughter. The pseudo suspense is kept on the wing until a few seconds before the picture ends, but it is a foregone conclusion that the producers would never dare to have the characters acted by Clark Gable and Claudette Colbert separated when the curtain falls. … Miss Colbert gives an engaging and lively performance.

This film feature is followed by a new Walt Disney “Silly Symphony” called “The China Shop,” which is as delightful as any of these prismatic cartoons. Like his other “Symphonies,” “The China Shop” constantly brings to mind aspects of “Alice in Wonderland.” A conception of Oscar Wilde’s “The Birthday of the Infanta” is the principal stage attraction.

radio city

Unknown couple outside Radio City 

Radio City used to be such a bargain.  Incidently, this movie also featured Alan Hale Sr., father of Alan Hale Jr., aka “the Skipper” on Gilligan’s Island.

The Secret Theorem of M. E. Midy = Casting In Nines

Thursday, September 6th, 2007

“Your small minds are musclebound with suspicion … because the only exercise you ever get is jumping to conclusions.”

That’s Danny Kaye as Walter Mitty in a 1947 movie to be remade casting Mike Myers. [Kaye, author James Thurber and many others were accused of being Communists in the propaganda pamphlet Red Stars #3 published in 1960 (!) by the Cinema Educational Guild.]

In the 1830s a Professor of Mathematics to the College of Nantes France published the following works:

  • The Calculation of simple and compound interest rates, simplified with a mental addition of two numbers,
  • Arithmetic, commercial, industrial and administrative, simplifed to addition by the Reformed system of calculation,
  • A Theorem of Mr. Sturm, and its numerical applications,
  • A novel Shorthand, improvement to what is written in the margins (re: Fermat’s Last Theorem?), and
  • a 21-page book, De Quelques Propriétés des Nombres et des Fractions Décimales Périodiques proving a theorem eventually named after him.

Midy’s Theorem is a very pretty result.  It was neglected by mathematicians until 2003 when Yale undergraduate Brian Ginsberg published an extension in a student paper.  Since then there have been numerous papers with generalizations. So what is this theorem?

Rational numbers, which are defined as fractions of two integers p and q, viz., p/q, when represented as a decimal expansion come in two forms: finite decimals, e.g., 1/5 = 0.2; and repeating decimals, e.g., 1/6 = 0.1666…, which is usually shown using the vinculum, viz., 1/6 = 0.16.  Since 1/6 is the product of a finite decimal 0.5 and a repeating decimal 0.3, as David Wells shows in his wonderful book, 1/6 has a nonperiodic part whose length is the same as that of 1/2, and a periodic part with period equal to the period of 1/3.  By multiplying by a power of 10 we can convert any p/q to  p’/q, where p’/q is a pure repeating decimal.

Near Calamity The nice thing is that the period only ever depends on q.  Anticipating by 120 years mathematicians’ keen interest in modular forms, these periods were first related to the multiplicative order of 10 (mod q), in other words, the smallest exponent, e, such that 10e (mod q) = 1, by J. W. L. Glaisher in 1878! [Papa Glaisher’s balloon exploits] Thus:

    • 10 mod 7 = 3
    • 100 mod 7 = 2
    • 1000 mod 7 = 6
    • 10000 mod 7 = 4
    • 100000 mod 7 = 5
    • 1000000 mod 7 = 1

So the multiplicative order of 10 (mod 7) is 6, and 1/7 as a repeated decimal has period 6.

Now consider rationals that have representations as pure repeating decimals where the period is even, such as:  

  • 1/7 = 0.142857
  • 1/77 = 0.012987
  • 1/121 = 0.0082644628099173553719
  • 1/803 = 0.00124533

Breaking these periodic strings in half and adding yields 999, 999, 99999999999, and 4545, respectively.  Those strings of 9‘s can’t be a coincidence, and Midy’s Theorem tells us exactly when to expect them, namely:

If p/q is a fraction written in lowest terms and a pure repeating decimal of period 2k, and q is not divisible by 10, then the ‘nines property’ holds iff:

  1. q is a prime or
  2. q is a prime  power; or
  3. the gcd(q, 10k – 1) = 1, where gcd is the greatest common denominator.

Thus, 7 is a prime, 121 is a prime power (112), and gcd(77, 999) = 1, whereas gcd(803, 9999) = 11.  In 2003 Ginsberg extended Midy’s result and things progressed quickly.

Ginsberg chopped in three repeating decimals where the period is of length 3k, and showed that the nines property holds if p is 1 and q is prime.  In 2005 Gupta and Sury (pdf) solve the problem of 1/q where q is prime in complete generality.  Also in 2005 Abdul-Baki (pdf) has some imaginative, true things to say about Midy.

In early 2006 Gil and Weiner (pdf) extend Midy’s Theorem, for 1/q where q is prime, to other bases. Later in 2006 Lewittes (pdf) extends Midy in general to other bases, and extends Ginsberg’s extension, e.g., it holds for 1/21 = 0.047619, 04 + 76 + 19 = 99.  Also in 2006 Martin (Integers vol. 7) generalizes Midy’s result for chopping pure repeating decimals of the form 1/q into arbitrary fixed size commensurable pieces. 

Chopping into arbitrary fixed size commensurable pieces in arbitrary bases does not seem to obey any obvious pattern.  There is more work ahead on extending this no-longer-so-secret theorem.

If only one person knows the truth, it is still the truth. – Mahatma Gandhi

Thanks, Bassam!

Crises += PhDs – Math SATs

Wednesday, August 29th, 2007

Sputnik stamp     911 stamp

Khodayar Akhavi points out that education funding is nowhere near as generous after the 911 crisis as it was after the Sputnik crisis (which may have ended with Gene Cernan’s last footprint on the Moon). Nevertheless, PhD production in the U.S. seems to have picked up after both crises.

PhD Production

The College Board just released figures for the 2007 SATs.  I looked at the scores for Math in States where at least 40% of the student population took the exams, and compared the rankings for 1997 (left) to 2007 (right).  The rankings are color coded in quintiles, with legend Red, Yellow, Green, Blue, and Violet (Red the worst).  Making double jumps up in the rankings are NC, VA, and VT; double jumps down are MD and ME.  Maine should get a bye, since this year it forced 100% of its eligible students to take the exam.  IN, TX, and NV should get special mention as the only non-coastal States to have at least 40% SAT participation. 

Math SAT Rankings by State 1997 & 2007

Update: While SC as a State improved, Miss SC Teen has issues with geography & numericity.