One of the treats of my childhood was the arrival each week of LIFE magazine, with its amazing photographs and engaging articles. I learned about dinosaurs, galaxies, Hollywood stars and epic movies, sports events, episodes in U.S. history, and many other topics while leafing through the issues. The large photos and detailed illustrations were often strikingly memorable.

Now I can experience those fascinating photos again. Google is digitizing the entire photographic archives of LIFE magazine—10 million photos in all, most of which were not published and have not been seen previously by the public. The images—negatives, slides, glass plates, etchings, and prints—have been sitting in dusty archives for years. About 20 percent of the collection is already online (http://images.google.com/hosted/life).

The images are organized by decade (when the photos were taken) and by category (people, places, events, sports, and culture). When I searched the photo archive for "mathematics," the results yielded dozens of intriguing and beguiling images.

Many photos display tools of mathematics (abaci, calculators, computers, slide rules, and even protractors) or mathematics in classroom settings. One set by famed photographer Alfred Eisenstaedt, for example, focuses on the U.S. Military Academy at West Point in 1941: West Point cadets marching to mathematics class with slide rules in hand, Cadets reporting to their instructor in a math class.

The archive also includes photos of prominent mathematicians, such as Charles Dodgson, Edouard Cech, Norbert Wiener, Nikolai Bogolyubov, John M. Tukey, and David Blackwell. The archive has many photos of Albert Einstein and several of his cluttered study in Princeton, N.J. Math teachers and students also appear in a variety of settings. One image, for example, shows math senior Judith Gorenstein working at the blackboard at the Massachusetts Institute of Technology in 1956. Another features a student at St. John's College in Annapolis, Md., in 1940 studying a model of geometrical solid.

One is my favorites is a stark display of gadgets for teaching plane and solid geometry.

The LIFE photo archive is available via Google Image Search free for personal and research purposes. Copyright and ownership of all images remain with Time Inc.

## November 26, 2008

## November 11, 2008

### Problems with Slipping Ladders

A ladder leans against a wall. It begins to slide, the top end moving down the wall and the bottom end across the floor away from the wall.

Such a scenario is the basis for a variety of mathematics problems. For example, if the ladder's bottom end moves away from the wall at a constant speed, what is the velocity of the top of the ladder at any given instant? Curiously, the mathematical model indicates that the velocity of the top approaches infinity when the ladder hits the floor.

How can that be? Even with a smooth wall and a frictionless floor, conservation of energy alone dictates that the velocity can't approach infinity. The answer lies not in the mathematics, which is correct, but in the physics of the ladder's motion.

It turns out that the ladder loses contact with the wall when the top has slid one-third of the way down. The original equations describing the situation no longer apply. The ladder then continues to move away from the wall, in a motion determined only by gravity and the reaction force of the floor. A different set of equations describes this part of the motion.

Stelios Kapranidis and Reginald Koo of the University of South Carolina Aiken present this scenario in their article "Variations of the Sliding Ladder Problem," published in the November

"Calculus teachers are familiar with the sliding ladder problem from the study of related rates," Kapranidis and Koo write. "However, it is not well known that some versions of this problem are physically impossible."

What happens when the ladder is somehow linked to the wall and can't leave it as it slides down? In this case, the velocity of the bottom increases to a maximum (when the ladder is one-third of the way down the wall), then decreases to a zero by the time the ladder hits the floor. The velocity of the top increases from zero to some finite value.

What conditions do you need for the top of the ladder to maintain contact with the wall and the bottom to move at a constant speed. Kapranidis and Koo note that such a motion can be realized by introducing extra forces to pull down on the top and push back at the bottom.

"This situation does yield arbitrarily large speeds," the authors say. "Achieving this motion requires the application of constraint forces that approach infinity." In other words, this motion can't happen.

"It is only when we consider the forces that cause the motion that we can resolve the paradox of speeds approaching infinity," Kapranidis and Koo conclude. "They also indicate caution in regarding the sliding ladder as an example of a real world application of mathematics."

Freeman, M., and P. Palffy-Muhoray. 1985. On mathematical and physical ladders.

Kapranidis, S., and R. Koo. 2008. Variations of the sliding ladder problem.

Scholten, P., and A. Simoson. 1996. The falling ladder paradox.

Such a scenario is the basis for a variety of mathematics problems. For example, if the ladder's bottom end moves away from the wall at a constant speed, what is the velocity of the top of the ladder at any given instant? Curiously, the mathematical model indicates that the velocity of the top approaches infinity when the ladder hits the floor.

How can that be? Even with a smooth wall and a frictionless floor, conservation of energy alone dictates that the velocity can't approach infinity. The answer lies not in the mathematics, which is correct, but in the physics of the ladder's motion.

It turns out that the ladder loses contact with the wall when the top has slid one-third of the way down. The original equations describing the situation no longer apply. The ladder then continues to move away from the wall, in a motion determined only by gravity and the reaction force of the floor. A different set of equations describes this part of the motion.

Stelios Kapranidis and Reginald Koo of the University of South Carolina Aiken present this scenario in their article "Variations of the Sliding Ladder Problem," published in the November

*College Mathematics Journal*."Calculus teachers are familiar with the sliding ladder problem from the study of related rates," Kapranidis and Koo write. "However, it is not well known that some versions of this problem are physically impossible."

What happens when the ladder is somehow linked to the wall and can't leave it as it slides down? In this case, the velocity of the bottom increases to a maximum (when the ladder is one-third of the way down the wall), then decreases to a zero by the time the ladder hits the floor. The velocity of the top increases from zero to some finite value.

What conditions do you need for the top of the ladder to maintain contact with the wall and the bottom to move at a constant speed. Kapranidis and Koo note that such a motion can be realized by introducing extra forces to pull down on the top and push back at the bottom.

"This situation does yield arbitrarily large speeds," the authors say. "Achieving this motion requires the application of constraint forces that approach infinity." In other words, this motion can't happen.

"It is only when we consider the forces that cause the motion that we can resolve the paradox of speeds approaching infinity," Kapranidis and Koo conclude. "They also indicate caution in regarding the sliding ladder as an example of a real world application of mathematics."

**References**:Freeman, M., and P. Palffy-Muhoray. 1985. On mathematical and physical ladders.

*American Journal of Physics*53:276-277.Kapranidis, S., and R. Koo. 2008. Variations of the sliding ladder problem.

*College Mathematics Journal*39(November):374-379.Scholten, P., and A. Simoson. 1996. The falling ladder paradox.

*College Mathematics Journal*27(January):49-54.## October 9, 2008

### Improved Pancake Sorting

You have a stack of pancakes, each one a different size. You want to rearrange the pancakes in order of size, with the smallest one on top and the largest on the bottom. But your only option is to insert a spatula at some point in the stack and flip the upper pancakes into reverse order. What's the maximum number of such flips that you'll ever need to sort the pancakes?

This classic mathematical problem, often described as pancake sorting or more formally as prefix reversal, has been around since 1975. Now, a team of computer science students has made the first improvement in nearly 30 years in one theoretical limit on solving the problem.

The maximum number of flips ever needed to rearrange a stack of

Until recently, pancake numbers were known only up to

In the last few years, teams of Japanese computer scientists have used clusters of computers to work out values for

Early on, researchers established theoretical bounds on pancake numbers. Initial work established that P

In 1979, William H. Gates and Christos H. Papadimitriou improved on the upper and lower limits, showing that (5

In 1997, Mohammad H. Heydari and I. Hal Sudborough improved the lower bound to 15

In 2008, students at the University of Texas at Dallas, working with Sudborough, used a lot of computation to establish an improved upper bound, (18/11)

The effort took about two years. "Improving the upper bound for the pancake problem has challenged mathematicians and computer scientists for 30 years," Sudborough noted. "We succeeded through the dedicated efforts of our team, the willingness of all to devote many hours to analysis, verification, and innovation, and our overriding belief that a better bound could be discovered."

This classic mathematical problem, often described as pancake sorting or more formally as prefix reversal, has been around since 1975. Now, a team of computer science students has made the first improvement in nearly 30 years in one theoretical limit on solving the problem.

The maximum number of flips ever needed to rearrange a stack of

*n*pancakes is known as the*n*th pancake number, P*. For five pancakes, for example, no more than five flips are ever needed, so P*_{n}_{5}= 5.Until recently, pancake numbers were known only up to

*n*= 13.In the last few years, teams of Japanese computer scientists have used clusters of computers to work out values for

*n*= 14 to 17: "Computing the diameter of 17-pancake graph using a PC cluster."Early on, researchers established theoretical bounds on pancake numbers. Initial work established that P

*had to be at least*_{n}*n*and no more than 2*n*– 3, for*n*greater than 2.In 1979, William H. Gates and Christos H. Papadimitriou improved on the upper and lower limits, showing that (5

*n*+ 5)/3 flips always suffice and that 17*n*/16 flips may be needed. Bill Gates, co-founder of Microsoft, had worked on the problem when he was a sophomore at Harvard, as described in a recent NPR story, and published the resulting paper, his only technical contribution to the mathematical literature, in collaboration with Papadimitriou, then a young professor at Harvard and now at the University of California, Berkeley. The paper, published in*Discrete Mathematics*, was titled "Bounds for sorting by prefix reversal."In 1997, Mohammad H. Heydari and I. Hal Sudborough improved the lower bound to 15

*n*/14.In 2008, students at the University of Texas at Dallas, working with Sudborough, used a lot of computation to establish an improved upper bound, (18/11)

*n*. A paper describing the results, "An (18/11)*n*upper bound for sorting prefix reversals," was published in*Theoretical Computer Science*(Vol. 410, no. 36, August 31, 2009, pp. 3372-3390). The authors are B. Chitturi, Bill Fahle, Zhaobing Meng, Linda Morales, Charles Shields, Hal Sudborough, and Walter Voit.The effort took about two years. "Improving the upper bound for the pancake problem has challenged mathematicians and computer scientists for 30 years," Sudborough noted. "We succeeded through the dedicated efforts of our team, the willingness of all to devote many hours to analysis, verification, and innovation, and our overriding belief that a better bound could be discovered."

## September 25, 2008

### Waltz of the Cycloids

Mathematical vocabulary isn't something that I normally associate with classical music, so I was caught by surprise recently when an announcer on a local radio station gamely tried to explain the term "cycloid."

Why? The announcer was introducing a piece of music called the

A cycloid is the path traced by a point on the rim of a circle that rolls (without slipping) along a straight line.

Galileo Galilei (1564–1642) studied the cycloid around 1599 and gave the curve its name, using the Greek word for "circle" as its main element.

A segment of a cycloid also represents the shape of the curve along which a bead sliding from rest and accelerated by gravity will slip from one point to another in the least time, a problem originally posed in 1696 by Johann Bernoulli (see the brachistochrone problem). Moreover, a bead sliding on a cycloid will exhibit simple harmonic motion, with a period independent of the starting point.

Galileo suggested that the cycloid would be the strongest possible arch for a bridge, and many concrete viaducts do have cycloidal arches. Cogwheels often have cycloidal sides to reduce friction as gears mesh.

In a chapter on the cycloid in

The version of

Strauss' composition isn't the only one in the musical literature that refers to the cycloid. The Library of Congress collection of American sheet music includes

Why? The announcer was introducing a piece of music called the

*Cycloids Waltz*(*Cycloiden Walzer*), written by Johann Strauss Jr (1825-1899). Strauss had dedicated his composition to the "Gentlemen Technical Students at Vienna University" and conducted its debut at their ball in the Sofienbad-Saal on Feb. 10, 1858. The apt title not only came directly out of mathematical vocabulary that would be familiar to the students but also hinted at the whirling movements of a waltz.A cycloid is the path traced by a point on the rim of a circle that rolls (without slipping) along a straight line.

Galileo Galilei (1564–1642) studied the cycloid around 1599 and gave the curve its name, using the Greek word for "circle" as its main element.

A segment of a cycloid also represents the shape of the curve along which a bead sliding from rest and accelerated by gravity will slip from one point to another in the least time, a problem originally posed in 1696 by Johann Bernoulli (see the brachistochrone problem). Moreover, a bead sliding on a cycloid will exhibit simple harmonic motion, with a period independent of the starting point.

Galileo suggested that the cycloid would be the strongest possible arch for a bridge, and many concrete viaducts do have cycloidal arches. Cogwheels often have cycloidal sides to reduce friction as gears mesh.

In a chapter on the cycloid in

*Martin Gardner's 6th Book of Mathematical Diversions from*Scientific American, Gardner notes that the cycloid has been called the "Helen of geometry," not only because of its beautiful properties but also because it has been the object of so many historic quarrels between eminent mathematicians.The version of

*Cycloiden, Walzer, Op. 207*, that I heard on the radio was recorded for the Marco Polo label by the Slovak State Philharmonic Orchestra, Kosice. The author of the accompanying album notes remarks that the title page design for the first piano edition of the piece featured a circle, encompassing the name of the work and its composer and the dedication to the students. The designer then surrounded this feature with representations of various tools of the technician (set square, compasses, theodolite, and so on) and depictions of several engineering achievements, such as the steamship, steam engine, blast furnace, and plough. But no cycloid.Strauss' composition isn't the only one in the musical literature that refers to the cycloid. The Library of Congress collection of American sheet music includes

*Cycloid Polka*, written by Charles Kinkel and published in 1873. Unfortunately, I can't play the piano or find a recording, so I have no idea what this piece sounds like. But I do know there's usually a lot of lively circling when you dance a polka.## September 4, 2008

### A Fractal in Bach's Cello Suite

Johann Sebastian Bach surely did not have fractals in mind when he composed six suites for solo cello several centuries ago. Nonetheless, at least one movement has the repeating structure on different scales that is characteristic of a fractal.

Harlan J. Brothers of The Country School in Madison, Conn., contends that the first Bourrée in Bach's Cello Suite No. 3 provides a clear example of structural scaling. The recursive form of this musical structure can be visualized as a fractal construction called the Cantor set, he says.

Brothers' findings appear in the paper "Structural Scaling in Bach's Cello Suite No. 3," published in the March 2007 issue of the journal

Examining only the written score, Brothers focused on the phrasing in the first section of the first Bourée. Musical phrasing refers to the way certain sequences of notes are naturally associated with each other, Brothers says.

Brothers detected repeated use of the pattern

For example, the piece starts off with two eighth notes and a quarter note (

Analogously, the first eight measures are repeated, giving two "short" sections that are followed by a 20-measure "long" section.

"Interestingly, although Bach wrote the piece with a repeat symbol at the end of this 20-measure section, anecdotal evidence suggests that some cellists choose to perform it without the second repeat," Brothers noted in his paper. "Performed in this fashion, the Bourrée Part I exhibits a full four levels of structural scaling symmetry."

The structure of Bach's music resembles that of a classic type of fractal known as a Cantor set. Start with a line segment. Remove the middle third. Then remove the middle third from the remaining pieces, and so on. The result is a "Cantor comb."

The hierarchical nesting of the

"The fact that Bach was born almost three centuries before the formal concept of fractals came into existence may well indicate that an intuitive affinity for fractal structure is, at least for some composers, an inherent motivational element in the compositional process," Brothers concluded.

Brothers has set about establishing a mathematical foundation for the classification of fractal music and correcting widespread misconceptions about fractal music. His efforts have revealed that musicians have been composing a form of fractal music for at least six centuries. One example is a type of canon in which different voices repeat the same melody or rhythmic motif simultaneously at different tempos.

Music can exhibit a wide variety of scaling behavior, Brothers says. He has himself written a number of compositions illustrating such properties. And he is keen to have others find further examples of scaling symmetry in what he describes as "the rich and vast body of musical expression."

Harlan J. Brothers of The Country School in Madison, Conn., contends that the first Bourrée in Bach's Cello Suite No. 3 provides a clear example of structural scaling. The recursive form of this musical structure can be visualized as a fractal construction called the Cantor set, he says.

Brothers' findings appear in the paper "Structural Scaling in Bach's Cello Suite No. 3," published in the March 2007 issue of the journal

*Fractals*.Examining only the written score, Brothers focused on the phrasing in the first section of the first Bourée. Musical phrasing refers to the way certain sequences of notes are naturally associated with each other, Brothers says.

Brothers detected repeated use of the pattern

**AAB**on different scales, where each**B**section lasts twice as long as each**A**section.*Analysis of the first 16 measures of the Bourrée from Bach's Suite No. 3*. Courtesy of Harlan Brothers.For example, the piece starts off with two eighth notes and a quarter note (

**m1**), repeats that pattern (**m2**), then continues with a phrase (**m3**) that is twice as long. The same pattern of short, short, long (**s1**) is repeated (**s2**), followed by a longer sequence (**s3**).Analogously, the first eight measures are repeated, giving two "short" sections that are followed by a 20-measure "long" section.

"Interestingly, although Bach wrote the piece with a repeat symbol at the end of this 20-measure section, anecdotal evidence suggests that some cellists choose to perform it without the second repeat," Brothers noted in his paper. "Performed in this fashion, the Bourrée Part I exhibits a full four levels of structural scaling symmetry."

The structure of Bach's music resembles that of a classic type of fractal known as a Cantor set. Start with a line segment. Remove the middle third. Then remove the middle third from the remaining pieces, and so on. The result is a "Cantor comb."

*Four levels in the creation of a Cantor comb*. Courtesy of Harlan Brothers.The hierarchical nesting of the

**AAB**phrasing in the first Bourrée produces a similar pattern."The fact that Bach was born almost three centuries before the formal concept of fractals came into existence may well indicate that an intuitive affinity for fractal structure is, at least for some composers, an inherent motivational element in the compositional process," Brothers concluded.

Brothers has set about establishing a mathematical foundation for the classification of fractal music and correcting widespread misconceptions about fractal music. His efforts have revealed that musicians have been composing a form of fractal music for at least six centuries. One example is a type of canon in which different voices repeat the same melody or rhythmic motif simultaneously at different tempos.

Music can exhibit a wide variety of scaling behavior, Brothers says. He has himself written a number of compositions illustrating such properties. And he is keen to have others find further examples of scaling symmetry in what he describes as "the rich and vast body of musical expression."

## August 8, 2008

### A New Formula for Generating Primes

Some simple expressions can generate a surprisingly large number of primes, whole numbers that are evenly divisible only by themselves and 1. The remarkable formula

Another simple, prime-rich formula,

But there are other possibilities. So people have continued to look for simple prime-generating functions, and Rutgers graduate student Eric Rowland has just found a new one. In a paper published in the

"Blending simplicity and mystery, Eric Rowland's formula is a delightful composition in the music of the primes, one everyone can enjoy," Jeffrey Shallit recently commented on his "Recursivity" blog. A professor at the University of Waterloo, Shallit is editor of the

Here's Rowland's recursive formula for generating primes, as presented by Shallit in his blog.

Define

For

For example, given that

The prime generator is then

Here are the first 23 values of the

7, 8, 9, 10, 15, 18, 19, 20, 21, 22, 33, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 69

Here are the first differences of these values:

1, 1, 1, 5, 3, 1, 1, 1, 1, 11, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 23

Ignoring the 1s, we find that Rowland's formula generates the primes 5, 3, 11, 3 (again), and 23. If we continue the process and remove duplicates, the formula generates the prime sequence 5, 3, 11, 23, 47, 101, 7, 13, 233, 467, 941, 1889, 3779, 7559, 15131, 53, 30323, . . .

Rowland describes his formula in the "A New Kind of Science" blog. He notes that the formula originated in 2003 at the NKS summer school, where participants discover and explore computational systems that exhibit "interesting" behavior.

"This was a surprising discovery, since previously there was no known reliable prime-generating formula that wasn't expressly engineered for this purpose," Rowland said. Rowland went on to prove mathematically that this recurrence produces only 1s and primes. He has created a Mathematica demonstration for exploring the recurrence.

Rowland's formula is unlikely to lead to more efficient ways of generating large primes, a crucial operation in cryptography. His formula produces the prime

Are there other formula's like Rowland's? Recently, French mathematician Benoit Cloitre proved that by setting

Many other questions remain. Is there anything special about the choice of

*x*^{2}+*x*+ 41, for example, yields an unbroken string of 40 primes, starting at*x*= 0.Another simple, prime-rich formula,

*x*^{2}+*x*+ 17, generates prime numbers for all integer values of*x*from 0 through 15. Searching for a polynomial formula that produces all primes, however, would be fruitless. Mathematicians proved years ago that no polynomial expression with integer coefficients has only prime values.But there are other possibilities. So people have continued to look for simple prime-generating functions, and Rutgers graduate student Eric Rowland has just found a new one. In a paper published in the

*Journal of Integer Sequences*, Rowland defines his formula and proves that it generates only 1s and primes."Blending simplicity and mystery, Eric Rowland's formula is a delightful composition in the music of the primes, one everyone can enjoy," Jeffrey Shallit recently commented on his "Recursivity" blog. A professor at the University of Waterloo, Shallit is editor of the

*Journal of Integer Sequences*.Here's Rowland's recursive formula for generating primes, as presented by Shallit in his blog.

Define

*a*(1) = 7.For

*n*greater than or equal to 2, set*a*(*n*) =*a*(*n*– 1) + gcd(*n*,*a*(*n*– 1)). Here "gcd" means the greatest common divisor.For example, given that

*a*(1) = 7,*a*(2) =*a*(1) + gcd(2, 7) = 7 + 1 = 8.The prime generator is then

*a*(*n*) –*a*(*n*– 1). The resulting numbers are the so-called first differences of the original sequence.Here are the first 23 values of the

*a*sequence:7, 8, 9, 10, 15, 18, 19, 20, 21, 22, 33, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 69

Here are the first differences of these values:

1, 1, 1, 5, 3, 1, 1, 1, 1, 11, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 23

Ignoring the 1s, we find that Rowland's formula generates the primes 5, 3, 11, 3 (again), and 23. If we continue the process and remove duplicates, the formula generates the prime sequence 5, 3, 11, 23, 47, 101, 7, 13, 233, 467, 941, 1889, 3779, 7559, 15131, 53, 30323, . . .

Rowland describes his formula in the "A New Kind of Science" blog. He notes that the formula originated in 2003 at the NKS summer school, where participants discover and explore computational systems that exhibit "interesting" behavior.

"This was a surprising discovery, since previously there was no known reliable prime-generating formula that wasn't expressly engineered for this purpose," Rowland said. Rowland went on to prove mathematically that this recurrence produces only 1s and primes. He has created a Mathematica demonstration for exploring the recurrence.

Rowland's formula is unlikely to lead to more efficient ways of generating large primes, a crucial operation in cryptography. His formula produces the prime

*p*only after first generating (*p*– 3)/2 1s. "So it takes a really long time to generate a large prime," Shallit said. Rowland "has a method for skipping over those useless 1s, but doing so essentially requires an independent test for primality."Are there other formula's like Rowland's? Recently, French mathematician Benoit Cloitre proved that by setting

*b*(1) = 1 and*b*(*n*) =*b*(*n*– 1) + lcm(*n*,*b*(*n*– 1)) for*n*greater than or equal to 2,*b*(*n*)/*b*(*n*– 1) – 1 is either 1 or prime.Many other questions remain. Is there anything special about the choice of

*a*(1) = 7? Does Rowland's formula eventually generate all odd primes? Rowland suspects that it does, but there's much more to learn.### A Tetrix at Burning Man

Burning Man is an eccentric cultural gathering, a celebration of community and self expression, held annually in the Black Rock Desert of Nevada. Tens of thousands of participants immerse themselves in a week of art, music, theater, technology, and social activism, while camping out on the playa. The event climaxes with the burning of a giant sculpture, the Man.

In recent years, several installations and camps at Burning Man have had a mathematical flavor. The 2003 edition, for example, featured a contorted Möbius-strip jungle gym. Tom Davis, formerly principal scientist at Silicon Graphics, has web pages describing activities at several Burning Man gatherings, including a rudimentary "math camp" in 2007.

This year's Burning Man will be held Aug. 25 to Sept. 1, and mathematician and beadwork artist Gwen L. Fisher of California Polytechnic State University in San Luis Obispo is working with Paul Brown on an adult-sized jungle gym in the form of a Sierpiński tetrahedron (or tetrix) for the event.

The sculpture, called "Bat Country," will stand 21 feet tall and be built from 384 aluminum baseball bats, which form the structure's edges, and 130 12-inch softballs, with one ball at each vertex. It consists, essentially, of 64 tetrahedra. Steel rods, which thread the bats, stabilize the structure.

"Bat Country" was inspired by Fisher's most popular three-dimensional beaded sculpture, an elegant assemblage of seed beads, bugle beads, and thread. Three views of this beaded fractal framework appeared on the cover of the June 2007

Fisher and her coworkers have now assembled the components into one of the four units that will make up the jungle gym and tested its stability, in preparation for installation at Burning Man 2008.

Fisher notes that the sculpture looks dramatically different from different points of view. From the outside on the ground, "Bat Country" looks like a triangle with a complex lattice of interior edges, she says. From certain angles, however, the bats align, and the structure appears to be a two-dimensional Sierpiński triangle. Looking upward, a viewer can see intriguing arrays of diamonds or triangles.

After Burning Man, Fisher and Brown hope that their sculptural jungle gym can go on public display in the San Francisco area.

In recent years, several installations and camps at Burning Man have had a mathematical flavor. The 2003 edition, for example, featured a contorted Möbius-strip jungle gym. Tom Davis, formerly principal scientist at Silicon Graphics, has web pages describing activities at several Burning Man gatherings, including a rudimentary "math camp" in 2007.

This year's Burning Man will be held Aug. 25 to Sept. 1, and mathematician and beadwork artist Gwen L. Fisher of California Polytechnic State University in San Luis Obispo is working with Paul Brown on an adult-sized jungle gym in the form of a Sierpiński tetrahedron (or tetrix) for the event.

The sculpture, called "Bat Country," will stand 21 feet tall and be built from 384 aluminum baseball bats, which form the structure's edges, and 130 12-inch softballs, with one ball at each vertex. It consists, essentially, of 64 tetrahedra. Steel rods, which thread the bats, stabilize the structure.

*Based on a fractal structure known as a Sierpiński tetrahedron, "Bat Country" will consist of 64 tetrahedra made from aluminum baseball bats connected by softballs*. Courtesy of Gwen Fisher."Bat Country" was inspired by Fisher's most popular three-dimensional beaded sculpture, an elegant assemblage of seed beads, bugle beads, and thread. Three views of this beaded fractal framework appeared on the cover of the June 2007

*Journal of Mathematics and the Arts*, accompanying an article by Fisher and Blake Mellor on the symmetries of beaded beads.Fisher and her coworkers have now assembled the components into one of the four units that will make up the jungle gym and tested its stability, in preparation for installation at Burning Man 2008.

*Fisher stands atop one of four tetrahedral units that will make up the "Bat Country" jungle gym.*Courtesy of Gwen Fisher.Fisher notes that the sculpture looks dramatically different from different points of view. From the outside on the ground, "Bat Country" looks like a triangle with a complex lattice of interior edges, she says. From certain angles, however, the bats align, and the structure appears to be a two-dimensional Sierpiński triangle. Looking upward, a viewer can see intriguing arrays of diamonds or triangles.

*Looking upward from inside "Bat Country," reveals a symmetric array of triangles.*Courtesy of Gwen Fisher.After Burning Man, Fisher and Brown hope that their sculptural jungle gym can go on public display in the San Francisco area.

## May 30, 2008

### Unveiling the Enigma Machine

The Enigma cipher machine figures prominently among the displays at the National Cryptologic Museum, located next to the headquarters of the National Security Agency in Ft. George G. Meade, Md. German military forces used the machine during World War II to encrypt tens of thousands of messages.

In principle, the Enigma's combination of wired rotors and plugs, which changed each letter into a new one as a user typed a message, made the machine highly secure. German officials believed that no enemy could break the coded messages without capturing a keylist, which told the cipher machine operator what rotor and plug settings to use each day.

Nonetheless, the system had vulnerabilities, and mathematicians played a key role in exploiting those flaws and developing methods for breaking Enigma-enciphered messages. Chris Christensen of Northern Kentucky University has recounted the story of three Polish mathematicians who played a crucial role in that effort. His account, titled "Polish Mathematicians Finding Patterns in Enigma Messages," appeared last year in

An illuminating visit to Bletchley Park in England, where the bulk of the codebreaking took place during World War II, recently inspired Danish high school teacher and mathematician Erik Vestergaard to create a Web page devoted to the Enigma. The page includes photos, personal notes, and links to a variety of resources on the topic, including a copy of Christensen's article.

The gradual unveiling in recent decades of the importance of the Bletchley codebreaking effort has also provided the background for a number of fictional accounts. My favorite is the suspenseful novel

"It seems rare that mathematicians are heroes of stories, and it seems even rarer that they are heroes because they are mathematicians," Christensen noted in his article.

"The story of the Polish mathematicians' success against Enigma is well known to cryptologists," he continued. One of the mathematicians, Marian A. Rejewski, "was able to use elementary theorems about permutations to determine the wiring of the Enigma rotors and to determine the Enigma settings."

For a mathematics teacher, the entire episode provides a dramatic real-world example of mathematical theory at work.

In principle, the Enigma's combination of wired rotors and plugs, which changed each letter into a new one as a user typed a message, made the machine highly secure. German officials believed that no enemy could break the coded messages without capturing a keylist, which told the cipher machine operator what rotor and plug settings to use each day.

Nonetheless, the system had vulnerabilities, and mathematicians played a key role in exploiting those flaws and developing methods for breaking Enigma-enciphered messages. Chris Christensen of Northern Kentucky University has recounted the story of three Polish mathematicians who played a crucial role in that effort. His account, titled "Polish Mathematicians Finding Patterns in Enigma Messages," appeared last year in

*Mathematics Magazine*.An illuminating visit to Bletchley Park in England, where the bulk of the codebreaking took place during World War II, recently inspired Danish high school teacher and mathematician Erik Vestergaard to create a Web page devoted to the Enigma. The page includes photos, personal notes, and links to a variety of resources on the topic, including a copy of Christensen's article.

The gradual unveiling in recent decades of the importance of the Bletchley codebreaking effort has also provided the background for a number of fictional accounts. My favorite is the suspenseful novel

*Enigma*by Robert Harris, in which the hero is a mathematician recruited to work at Bletchley Park. The novel later became an entertaining movie directed by Michael Apted. I was less impressed by the Enigma references in Neal Stephenson's sprawling*Cryptonomicon*."It seems rare that mathematicians are heroes of stories, and it seems even rarer that they are heroes because they are mathematicians," Christensen noted in his article.

"The story of the Polish mathematicians' success against Enigma is well known to cryptologists," he continued. One of the mathematicians, Marian A. Rejewski, "was able to use elementary theorems about permutations to determine the wiring of the Enigma rotors and to determine the Enigma settings."

For a mathematics teacher, the entire episode provides a dramatic real-world example of mathematical theory at work.

## May 9, 2008

### The Most Marvelous Theorem

"What makes a theorem great?"

To Dan Kalman of American University, a great theorem is one that surprises. "It is simple to understand, answers a natural question, and involves familiar mathematical concepts," he writes in the April

Kalman's nominee for the "Most Marvelous Theorem in Mathematics" concerns a remarkable relationship between the roots of a polynomial and the roots of its derivative. Kalman calls it Marden's theorem because he first encountered it in the 1966 book

Suppose you have a third-degree polynomial—a cubic function. Marden's theorem specifies exactly where to find the roots of the derivative of this polynomial.

A cubic has three roots in the complex plane. Typically, these roots form the vertices of a triangle. Draw an ellipse inside the triangle so that the ellipse touches the triangle's sides at their midpoints. The two foci of the inscribed ellipse are the roots of the polynomial's derivative.

"How can it be that the connection between a polynomial and its derivative is somehow mirrored perfectly by the connection between an ellipse and its foci?" Kalman asks. "Seeing this result for the first time is like watching a magician pull a rabbit out of a hat."

In a publishing tour de force, Kalman's technical article describing an elementary proof of Marden's theorem appears in the April

What's your candidate for the most marvelous theorem in mathematics?

To Dan Kalman of American University, a great theorem is one that surprises. "It is simple to understand, answers a natural question, and involves familiar mathematical concepts," he writes in the April

*Math Horizons*.Kalman's nominee for the "Most Marvelous Theorem in Mathematics" concerns a remarkable relationship between the roots of a polynomial and the roots of its derivative. Kalman calls it Marden's theorem because he first encountered it in the 1966 book

*Geometry of Polynomials*by Morris Marden (1905–1991). Marden himself traced the theorem to an 1864 paper by Jörg Siebeck.Suppose you have a third-degree polynomial—a cubic function. Marden's theorem specifies exactly where to find the roots of the derivative of this polynomial.

A cubic has three roots in the complex plane. Typically, these roots form the vertices of a triangle. Draw an ellipse inside the triangle so that the ellipse touches the triangle's sides at their midpoints. The two foci of the inscribed ellipse are the roots of the polynomial's derivative.

"How can it be that the connection between a polynomial and its derivative is somehow mirrored perfectly by the connection between an ellipse and its foci?" Kalman asks. "Seeing this result for the first time is like watching a magician pull a rabbit out of a hat."

In a publishing tour de force, Kalman's technical article describing an elementary proof of Marden's theorem appears in the April

*American Mathematical Monthly*and his interactive, online presentation of the theorem is simultaneously available in the MAA's*Journal of Online Mathematics and Its Applications*.What's your candidate for the most marvelous theorem in mathematics?

## March 11, 2008

### Newton's Algebra Book

PROBLEM V. If two Post-Boys A and B, at 59 Miles Distance from one another, set out in the Morning to meet. And A rides 7 Miles in two Hours, and B 8 Miles in three Hours, and B begins his Journey an Hour later than A; to find what Number of Miles A will ride before he meets B.

Math teachers would probably find the substance of this word problem, if not its expression, to be comfortably familiar. They may be surprised to learn, however, that this particular problem comes from the 1769 edition of an English translation of a textbook written in Latin by Isaac Newton (1643–1727).

Celebrated as the inventor of calculus and the author of Philosophiae Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy), Newton also prepared lecture notes during the time that he was at the University of Cambridge as the Lucasian Professor of Mathematics. All undergraduates were required to attend the Lucasian lectures, starting in their third year.

In 1669, Newton had made extensive notes on an algebra textbook written by Gerard Kinkhuysen and published in 1661. Like other 17th-century textbooks on solving equations, Kinkhuysen's book built on traditional material from the previous century, with the addition of some new ideas from René Descartes and Frans van Schooten.

Newton used these notes later when he compiled his Lucasian lectures, organizing the material in much the same way that Kinkhuysen had in his book. It seems unlikely that Newton ever delivered the lectures, but William Whiston (1667–1752), Newton's successor as Lucasian Professor, edited and published the notes in 1707 as the book Arithmetica universalis.

Newton himself was unhappy with the publication of his notes and refused to let his name appear on the book. When the book was printed, he apparently threatened to buy all the copies so that he could destroy them. The book was later translated into English by Joseph Raphson (1648–1715) and published in 1720 as Universal arithmetick.

A second, amended edition of Arithmetica universalis was published in 1722. Newton's name first appears on the 1761 edition of Arithmetica universalis, printed well after his death.

Arithmetica universalis covers a wide range of elementary topics, including algebraic notation, arithmetic, the relationship between geometry and algebra, limits and infinite series, the binomial theorem, and the solution of equations, including the extraction of roots.

Because Newton's book closely follows the structure and content of Kinkhuysen's Algebra, it has "the appearance of being far more elementary than in fact it is," writes Jacqueline A. Stedall of The Queen's College, Oxford. "To the traditional ideas expounded by Kinkhuysen, Newton added some far-reaching insights of his own."

Nonetheless, she adds, Newton often "described rules or procedures without explanation, giving rise to extensive commentary later on."

Today's high school teachers could profit from pondering the way Newton handled fundamental operations and questions of arithmetic and the wide array of examples and problems that he posed, says mathematician Michel Helfgott of East Tennessee State University.

Helfgott and George Baloglou of the State University of New York at Oswego themselves went back to Newton's Universal arithmetick when they recently investigated relations among a triangle's area, perimeter, and angles. They generalized Newton's derivation of the formula x = P/2 – 2A/P, expressing a right triangle's hypotenuse, x, in terms of its area, A, and perimeter, P. The results are in their paper "Angles, Area, and Perimeter Caught in a Cubic," published earlier this year in Forum Geometricorum.

Here's Newton's solution to the problem posed at the beginning of this article, as published in the 1769 edition of Universal arithmetick.

Math teachers would probably find the substance of this word problem, if not its expression, to be comfortably familiar. They may be surprised to learn, however, that this particular problem comes from the 1769 edition of an English translation of a textbook written in Latin by Isaac Newton (1643–1727).

Celebrated as the inventor of calculus and the author of Philosophiae Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy), Newton also prepared lecture notes during the time that he was at the University of Cambridge as the Lucasian Professor of Mathematics. All undergraduates were required to attend the Lucasian lectures, starting in their third year.

In 1669, Newton had made extensive notes on an algebra textbook written by Gerard Kinkhuysen and published in 1661. Like other 17th-century textbooks on solving equations, Kinkhuysen's book built on traditional material from the previous century, with the addition of some new ideas from René Descartes and Frans van Schooten.

Newton used these notes later when he compiled his Lucasian lectures, organizing the material in much the same way that Kinkhuysen had in his book. It seems unlikely that Newton ever delivered the lectures, but William Whiston (1667–1752), Newton's successor as Lucasian Professor, edited and published the notes in 1707 as the book Arithmetica universalis.

Newton himself was unhappy with the publication of his notes and refused to let his name appear on the book. When the book was printed, he apparently threatened to buy all the copies so that he could destroy them. The book was later translated into English by Joseph Raphson (1648–1715) and published in 1720 as Universal arithmetick.

A second, amended edition of Arithmetica universalis was published in 1722. Newton's name first appears on the 1761 edition of Arithmetica universalis, printed well after his death.

Arithmetica universalis covers a wide range of elementary topics, including algebraic notation, arithmetic, the relationship between geometry and algebra, limits and infinite series, the binomial theorem, and the solution of equations, including the extraction of roots.

Because Newton's book closely follows the structure and content of Kinkhuysen's Algebra, it has "the appearance of being far more elementary than in fact it is," writes Jacqueline A. Stedall of The Queen's College, Oxford. "To the traditional ideas expounded by Kinkhuysen, Newton added some far-reaching insights of his own."

Nonetheless, she adds, Newton often "described rules or procedures without explanation, giving rise to extensive commentary later on."

Today's high school teachers could profit from pondering the way Newton handled fundamental operations and questions of arithmetic and the wide array of examples and problems that he posed, says mathematician Michel Helfgott of East Tennessee State University.

Helfgott and George Baloglou of the State University of New York at Oswego themselves went back to Newton's Universal arithmetick when they recently investigated relations among a triangle's area, perimeter, and angles. They generalized Newton's derivation of the formula x = P/2 – 2A/P, expressing a right triangle's hypotenuse, x, in terms of its area, A, and perimeter, P. The results are in their paper "Angles, Area, and Perimeter Caught in a Cubic," published earlier this year in Forum Geometricorum.

Here's Newton's solution to the problem posed at the beginning of this article, as published in the 1769 edition of Universal arithmetick.

## February 14, 2008

### Stooges Statistics

A bop on the head, a punch in the stomach, a poke in the eyes. These were staples of the slapstick comedy of The Three Stooges. Now they've become elements of a novel approach to teaching experimental design, sampling, data gathering, and statistics.

I can remember watching Three Stooges episodes on television when I was a kid, typically on Saturday mornings. In 1963, I was part of a record crowd of more than 35,000 people who jammed Grandstand Stadium in Toronto during the Canadian National Exhibition to see the knockabout trio in person.

The Three Stooges starred in 190 short films, starting in 1934. In the 1950s and 60s, these early black-and-white films were recycled on television, entrancing a new generation of viewers. They remain a popular fixture on cable TV, and you can sample favorite episodes and clips on YouTube. There's even a YouTube-posted 1963 home movie of Moe, Larry, Curly-Joe, and an ape cavorting around an airplane at the airport in Toronto. Last fall, Sony started releasing the complete collection of Stooges shorts, in chronological order.

For mathematicians (and Stooges fans) Robert Davidson and Robert Gardner of East Tennessee State University, the continuing popularity of The Three Stooges and ready availability of the films presented an opportunity. They looked into using these engaging films as data sources for an introductory statistics class.

"People know The Stooges," Gardner says.

Over the 50-year career of The Three Stooges, Moe Howard and Larry Fine were the constants. Curly (Moe's brother) appeared in the first 97 shorts but suffered a stroke in 1947. His brother Shemp took over for 77 episodes between 1947 and 1955. Joe Besser appeared in the last 16 films.

In these shorts, Moe typically doles out the violence, and Larry and especially Curly (and later Shemp) are the unfortunate recipients. When Joe Besser appeared on the scene in 1955, his contract specified that he suffer no slapping or other bodily harm while filming the shorts. Davidson and Gardner wondered whether the level of violence in the films changed as a result.

Data gathering required watching segments of a random sample of shorts starring Curly, Shemp, and Joe. As they watched, the viewers tallied the number of violent acts perpetrated by Moe against his hapless victim.

Davidson and Gardner first tested the null hypothesis that "the average number of violent acts by Moe against Curly per episode is the same as the average number of violent acts by Moe against Shemp." Statistical tests on the tallies suggested that Shemp may have suffered more than Curly did, but the difference wasn't statistically significant enough (only 81% confidence) to warrant rejection of the null hypothesis. The mathematicians needed more data to settle the question.

Comparing Curly and Joe, Davidson and Gardner found that they could safely reject the null hypothesis and affirm that "the average number of violent acts by Moe against Curly per episode is greater than the average number of violent acts by Moe against Joe."

This exercise raises some interesting questions about data collection in the real (or reel) world. In the context of the Stooges films, for example, what constitutes an act of violence? If Curly gets slapped three times in rapid succession, is that one or three instances? What about an accidental poke?

Students "experience the difficulty in designing an experiment, the problems inherent to data collection, and analysis of the collected data," Davidson and Gardner note.

There are plenty of other questions about The Three Stooges that may be worthy of statistical attention. Does the level of violence vary from director to director or from writer to writer? Did Larry suffer less than, as much as, or more than Curly or Shemp? Did that change during the Joe Besser years? You don't even have to focus on violence. What roles did women or minorities play the films? How did that change over time? What sort of language did The Stooges use (or misuse)?

Davidson and Gardner will describe using The Three Stooges as a data source for motivating statistics students at the 15th Georgia Conference on College & University Teaching, Feb. 15, 2008, at Kennesaw State University.

I can remember watching Three Stooges episodes on television when I was a kid, typically on Saturday mornings. In 1963, I was part of a record crowd of more than 35,000 people who jammed Grandstand Stadium in Toronto during the Canadian National Exhibition to see the knockabout trio in person.

The Three Stooges starred in 190 short films, starting in 1934. In the 1950s and 60s, these early black-and-white films were recycled on television, entrancing a new generation of viewers. They remain a popular fixture on cable TV, and you can sample favorite episodes and clips on YouTube. There's even a YouTube-posted 1963 home movie of Moe, Larry, Curly-Joe, and an ape cavorting around an airplane at the airport in Toronto. Last fall, Sony started releasing the complete collection of Stooges shorts, in chronological order.

For mathematicians (and Stooges fans) Robert Davidson and Robert Gardner of East Tennessee State University, the continuing popularity of The Three Stooges and ready availability of the films presented an opportunity. They looked into using these engaging films as data sources for an introductory statistics class.

"People know The Stooges," Gardner says.

Over the 50-year career of The Three Stooges, Moe Howard and Larry Fine were the constants. Curly (Moe's brother) appeared in the first 97 shorts but suffered a stroke in 1947. His brother Shemp took over for 77 episodes between 1947 and 1955. Joe Besser appeared in the last 16 films.

In these shorts, Moe typically doles out the violence, and Larry and especially Curly (and later Shemp) are the unfortunate recipients. When Joe Besser appeared on the scene in 1955, his contract specified that he suffer no slapping or other bodily harm while filming the shorts. Davidson and Gardner wondered whether the level of violence in the films changed as a result.

Data gathering required watching segments of a random sample of shorts starring Curly, Shemp, and Joe. As they watched, the viewers tallied the number of violent acts perpetrated by Moe against his hapless victim.

Davidson and Gardner first tested the null hypothesis that "the average number of violent acts by Moe against Curly per episode is the same as the average number of violent acts by Moe against Shemp." Statistical tests on the tallies suggested that Shemp may have suffered more than Curly did, but the difference wasn't statistically significant enough (only 81% confidence) to warrant rejection of the null hypothesis. The mathematicians needed more data to settle the question.

Comparing Curly and Joe, Davidson and Gardner found that they could safely reject the null hypothesis and affirm that "the average number of violent acts by Moe against Curly per episode is greater than the average number of violent acts by Moe against Joe."

This exercise raises some interesting questions about data collection in the real (or reel) world. In the context of the Stooges films, for example, what constitutes an act of violence? If Curly gets slapped three times in rapid succession, is that one or three instances? What about an accidental poke?

Students "experience the difficulty in designing an experiment, the problems inherent to data collection, and analysis of the collected data," Davidson and Gardner note.

There are plenty of other questions about The Three Stooges that may be worthy of statistical attention. Does the level of violence vary from director to director or from writer to writer? Did Larry suffer less than, as much as, or more than Curly or Shemp? Did that change during the Joe Besser years? You don't even have to focus on violence. What roles did women or minorities play the films? How did that change over time? What sort of language did The Stooges use (or misuse)?

Davidson and Gardner will describe using The Three Stooges as a data source for motivating statistics students at the 15th Georgia Conference on College & University Teaching, Feb. 15, 2008, at Kennesaw State University.

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