August 31, 2020

Hankies, Snarks, and Triangles

Charles Lutwidge Dodgson (1832-1898) was a competent mathematician who taught at Christ Church, Oxford, during the nineteenth century. His creativity manifested itself not in proving important theorems but in the realm of recreational mathematics. His enduring fame rests on two books he wrote under the name Lewis Carroll.

Both Alice's Adventures in Wonderland and Through the Looking-Glass and What Alice Found There contain many examples of Dodgson's passion for mathematical games, puzzles, logic paradoxes, riddles, and all sorts of word play. Indeed, his fascination with card games and chess provided the background for his two Alice books.

Under the name of Lewis Carroll, Dodgson also published a variety of articles and leaflets devoted to puzzles, games, magic tricks, riddles, puns, anagrams, and ingeniously constructed verse. In the book The Universe in a Handkerchief: Lewis Carroll's Mathematical Recreations, Games, Puzzles, and Word Plays (Copernicus, 1996), Martin Gardner offers a varied selection of these little treasures, gleaned from Carroll's publications, diaries, and letters.

Gardner's title comes from a brief reference in a Carroll book called Sylvie and Bruno Concluded, in which a German professor explains how a Möbius band has only one side and one edge. He then goes on to demonstrate how to sew together two handkerchiefs to make a three-dimensional one-sided surface known to topologists nowadays as a projective plane. Because this closed surface has neither an outside nor an inside, you can say it contains the entire universe.

In Carroll's lengthy nonsense ballad, "The Hunting of the Snark," the Butcher tries to convince the Beaver that 2 + 1 = 3. He adopts the following procedure:

"Taking Three as the subject to reason about—
A convenient number to state—
We add Seven, and Ten, and then multiply out
By One Thousand diminished by Eight.

"The result we proceed to divide, as you see,
By Nine Hundred and Ninety Two:
Then subtract Seventeen, and the answer must be
Exactly and perfectly true."

Writing out and simplifying the algebraic expression for the operations, [(x + 7 + 10)(1,000 − 8)/992] − 17, you find that the procedure always yields the same number that you start with!

Dodgson's fascination with puzzles is evident in many diary entries. On one occasion, he wrote: "Sat up last night till 4 a.m., over a tempting problem, sent me from New York, 'to find three equal rational-sided right-angled triangles.' I found two, whose sides are 20, 21, 29; 12, 35, 37; but could not find three."

Had Dodgson doubled the sides of the two triangles he had found, he would have obtained the first two triangles of the triple he was looking for. The smallest solution consists of triangles of sides 40, 42, 58; 24, 70, 74; and 15, 112, 113, all of which have the same area, 840.

There are actually infinitely many such right-angled triangle triples. Beyond the smallest triple, however, the integral sides of other triples are each at least six digits long.

Lewis Carroll certainly had a lot up his sleeve, and The Universe in a Handkerchief provides a welcome survey of Carroll's mathematical magic tricks.

Originally posted January 13, 1997

August 30, 2020

Points of View

A rectangular slab of polished granite gives an impression of solemn immutability. With its straight lines and smooth surface, it is an elegant, humanmade artifact meant to stand as a monument or serve as a skin for a sleek skyscraper.

Breaking a granite slab produces jagged fragments. The fractal geometry of the granite's fractured edges bears witness to the raw stone's natural history and contrasts sharply with the classical geometry of its manufactured facade.

In working with stone, a sculptor inevitably confronts breakage. When mathematician and sculptor Nat Friedman first looked at the shards of a broken granite slab, he saw something more than an unfortunate accident.

Friedman had become aware of how twentieth-century sculptors such as Henry Moore (see "Composition in Form and Space") had opened up the solid form to create space so they could revel in the interplay of form and light created by the opening.

Friedman envisioned how he could create space by leaving gaps between the fragments when he reassembled a broken slab. He then developed a technique for making prints from the granite assemblage.

A fractal stone print by Nat Friedman.

The resulting images, made with deep-blue or black ink on sheets of thin, porous paper, leave a vivid impression of jagged lightning bolts slashing across an otherwise dark sky. Some resemble drawings of crumpled coastlines edging unknown landmasses on an explorer's map. The stark contrasts in form and color tease the eye.

"Visual thinking leads to seeing that mathematical forms can also generate art forms," Friedman maintained.

Trefoil Knot Minimal Surface by Nat Friedman. Carved from limestone, this form is based on the shape of the soap-film minimal surface on a wire bent into a trefoil knot.

An artist can look at a mathematical shape and envision unlimited possibilities, even from a shape as seemingly simple as a tetrahedron, a trefoil knot, a Möbius strip, or a fractal surface. An artist can transform a mathematical idea into an evocative artwork.

Spiral Möbius by Nat Friedman.

To sculptor Arthur Silverman, for example, tetrahedrons were very special. He spent more than three decades investigating variations of tetrahedral forms, creating sculptures displayed in public spaces in New Orleans and other cities from Florida to California.

One of the more intriguing of Silverman's tetrahedral creations was an ensemble of sculptures he called Attitudes. The six pieces were spread across a grassy area at the Elysian Fields Sculpture Park in New Orleans.

Attitudes by Arthur Silverman.

All the pieces had the same geometry. Each one was made up of two identical tetrahedrons, having faces in the form of tall isosceles triangles that were welded together to form a single object. In the park, each piece had a strikingly different orientation.

To Nat Friedman, Silverman's creation was an example of a hypersculpture. Its ensemble arrangement represented a way of seeing a three-dimensional form from many different viewpoints at once.

To see every part of a two-dimensional painting, you have to step away from it in the third dimension. To see a three-dimensional sculpture in its totality, you need a way to slip into the fourth dimension. Friedman called this hypothetical process "hyperseeing."

A hypersculpture consisting of a set of several related sculptures provides one way to approximate that experience. Silverman's Attitudes, for instance, presents multiple views of an object from a single viewpoint, because copies of the same object lie in different orientations.

Another set of related sculptures is Rashomon by Charles Ginnever. The basic piece is an angular steel framework that can stand stably in fifteen different orientations. In each position, it looks startlingly different and tells a unique story.

A metal model demonstrates one of the stable positions available for Charles Ginnever's sculpture Rashomon.

By having several copies of the same sculpture in the same setting, each positioned differently, the artist can tell a remarkably complex, multidimensional tale.

Hyperseeing is easiest when a sculpture is highly symmetrical. For example, if the front and back views of a sculpture are identical, you can readily reconstruct from one view what the entire sculpture looks like. Simplest of all, a featureless sphere can be understood with just one glance.

Another strategy to encourage hyperseeing is to make the form at least partly transparent or to create a ribbed structure, as seen in many sculptures by Ginnever and Charles Perry.

Such approaches are somewhat reminiscent of the X-ray and time-lapse presentations of such artists as Pablo Picasso and Marcel Duchamp, who used multiple, fractured images of the same object in their paintings to convey a sense of three-dimensional space and time in a two-dimensional medium.

The result, Friedman said, can be bewildering to our conventionally conditioned eyes.

Experience with hypersculptures, like those created by Silverman and Ginnever, increases appreciation of sculpture in general, Friedman contended. "You learn to look harder and more closely from all angles."

When you hypersee a three-dimensional object, you begin to visualize the form from all sides, including the top—a view of a sculpture that is often neglected or unavailable.

Interestingly, the mathematical field of knot theory provides an ideal source of three-dimensional shapes that have no preferred front, back, top, or bottom. As open forms, mathematical knots are wonderful subjects on which to practice hyperseeing, Friedman said.

In effect, you can see all the points of a knot from any one view except for a finite number of points where the strand crosses itself.

In mathematics, a knot is a curve that winds through itself in three-dimensional space and catches its own tail to form a loop. Typically, mathematicians examine two-dimensional shadows cast by knots rather than actual three-dimensional knots.

Even the most tangled configuration can be shown as a continuous loop whose shadow sprawls across a flat surface. In drawings of knots, tiny breaks in the lines are often used to signify underpasses or overpasses.

Nat Friedman often used copper tubing to create three-dimensional models or sculptures of knots. The continuous loop of such a three-dimensional mathematical knot casts an intriguing shadow, which varies in interesting ways as the orientation of the knot is changed.

However, just as a suspended wire frame caught in a breeze on a sunny day, casts an ever-changing shadow on the ground, so a rigid knot illuminated from different angles can display different projections for a given knot. Usually, that most useful case is the shadow (or diagram) that has the smallest possible number of crossings.

Truly appreciating a knot, however, requires having a three-dimensional model of it. "You make a knot sculpture," Friedman insisted. He used strips of aluminum foil and embedded wires, plastic aquarium tubing, and copper pipe for his artful experiments.

"A knot can look completely different when viewed from different directions," Friedman noted. At the same time, the infinitely malleable shape of a given knot allows you to create innumerable space-form variants of the same knot, which also can serve as raw material for artistic creativity.

A physical model of a mathematical knot can have any one of an infinite variety of possible forms. The artist can select a configuration that has symmetries or other features that may make it particularly pleasing to the eye.

A knotted loop also can serve as the edge of a surface. An example of such a surface appears when a wire model of a knot is dipped in a soap solution and emerges with a soap film clinging to the wire.

Mathematically the result is known as a minimal surface—the surface of least possible area that spans the looped wire. If the wire is in the form of a circle, the resulting minimal surface is a flat disk.

Experiments show that the minimal surfaces associated with different knotted loops can take on a variety of shapes. Some of these surfaces feature the peculiar one-sidedness characteristic of Möbius strips.

Filling in a knot produces interesting surfaces that share properties of mathematical shapes such as Möbius strips, as shown in these models created by Nat Friedman.

"The operative word that unifies art and mathematics is seeing," Friedman wrote in his paper "Hyperseeing, Hypersculptures, and Space Curves." "More precisely, art and mathematics are both about seeing relationships. One can see certain mathematical forms as art forms, and creativity is about seeing from a new viewpoint."

August 29, 2020

Forging Links Between Mathematics and Art

Science News, June 20, 1992

To many people, art and mathematics appear to have very little in common. The seemingly rigid rules and algorithms of mathematics apparently lie far removed from the spontaneity and passion associated with art. However, a small but growing number of artists find inspiration in mathematical form, and a few mathematicians delve into art to appreciate and understand better the patterns and relationships they discover in the course of their mathematical investigations.

To prove the remarkable fruitfulness of such links, more than 100 mathematicians, artists, and educators gathered last week at the Art and Mathematics Conference (AM '92), held in Albany, N.Y. Organized by mathematician and sculptor Nat Friedman of the State University of New York at Albany, the meeting represented his attempt to find people with whom he could share his deep interest in visualizing mathematics, whether in geometry, sculpture, computer art, or architecture.

Attempts to visualize such mind-bending mathematical transformations as turning a sphere inside out without introducing a sharp crease at any point during the operation demonstrates how mathematics and computer graphics can lead to valuable insights that are potentially useful to both scientists and artists.

In 1959, when Stephen Smale, a mathematician at the University of California, Berkeley, first proved this particular operation possible, no one could readily visualize how it happens. By gradually simplifying the steps involved in turning a sphere inside out, mathematicians eventually found ways of picturing the entire process.

François Apéry of the University of Upper Alsace in Mulhouse, France, has now captured the essence of the process, known as a sphere eversion, in a surprisingly simple model. Imagine a globe marked with an equator and lines of longitude, or meridians, that connect the poles. At the start of the sphere eversion, as one pole moves toward the other, the meridians twist sideways more and more.

François Apéry demonstrating the essence of his model of a sphere eversion at the AM '92 conference.

When the poles meet, the meridians twist so much that they flip like a wind-blown umbrella over the coincident poles to double up into a smaller spherical shape having an open end marked by a ring showing the new position of the original sphere's equator. The twisting continues until the equator closes up into a point and the meridians overlap and cross each other. At this stage, the sphere's outside becomes its inside, completing the eversion.

Apéry speculates that the first half of this sphere eversion may serve as a mathematical model of the way an embryo, starting off as a ball of cells, can pull in part of its outer wall to form a cavity among its dividing, differentiating cells. Biologists call the process gastrulation.

August 28, 2020

Nat Friedman (1938-2020)

Mathematician and sculptor Nat Friedman died on May 2, 2020, at the age of 82. He played a seminal role in bringing together and calling attention to a vibrant community of mathematicians who created artworks and artists inspired by mathematical ideas. Tributes to Nat Friedman.

Nat Friedman at the 2007 Joint Mathematics Meetings in New Orleans. He was fascinated by knots, minimal surfaces (and soap films), and Möbius strips and delighted in demonstrating his ideas and discoveries.

I first encountered Nat in 1992 when he invited me to present a plenary talk at a meeting devoted to mathematics and art that he had organized at the University at Albany-State University of New York. My invitation to this pathbreaking conference came about because of articles I had written for Science News magazine highlighting the increasing use of visualization in mathematics.

One important element of that revolution was the burgeoning role of computer graphics in illuminating and exploring mathematical ideas, from soap-film surfaces, fractals, and knots to chaos, hyperbolic space, and topological transformations. One of my articles had focused on Helaman Ferguson, a sculptor and mathematician who not only worked with computers but also carved marble and molded bronze into graceful, sensuous, mathematically inspired artworks.

Nat's lively gathering, the first of a series of art-math conferences that he organized and hosted, introduced me to many more people fascinated by interactions between art and mathematics. Nat was instrumental in broadening my appreciation of the processes of creativity, invention, and discovery intrinsic to both mathematical research and artistic endeavor.

My meeting report for Science News (June 20, 1992): "Forging Links Between Mathematics and Art."

Nat founded The International Society of the Arts, Mathematics, and Architecture (ISAMA) in 1998 to help further interdisciplinary education relating the arts, mathematics, and architecture.

Nat Friedman at his exhibit at the 2009 Joint Mathematics Meetings in Washington, D.C. He was dedicated to the idea of getting children to see the world in the way a sculptor would as well as the way a mathematician would.

I wrote about some of Nat's ideas in my book Fragments of Infinity: A Kaleidoscope of Math and Art (Wiley, 2001). See "Points of View."

For examples of Nat's artworks, see "Knot Shadow," "Spiral Moebius," and "Fractal Torso."

August 27, 2020

Fractals in Pascal's Triangle

Fascinating patterns can arise out of arrays of numbers defined by simple rules.

For example, start with the number 1, and make it the apex of what will become a triangle of numbers. In the second row, write two 1s. For each subsequent line, add together adjacent numbers of the previous row and write the sums in the new row, then place 1s at both ends of the line.

Here's what you get for the first eight rows:

This set of numbers is now widely known as Pascal's triangle, named for French philosopher and mathematician Blaise Pascal (1623-1662), who studied it intensively. Pascal, however, was not the first to identify the pattern.

The mathematician, astronomer, and poet Omar Khayyam (1048-1122) described this number triangle in his writings. It was also well-known to the Chinese. A nine-row version was featured prominently in the introduction to the book Precious Mirror of the Four Elements," which appeared in 1303. The book's author refers to the triangle as a "diagram of the old method for finding eighth and lower powers."

Indeed, the triangle represents a simple way to determine, for example, that (x + 1)4 = 1x4 + 4x3 + 6x2 + 4x1 + 1x0. In other words, the rows represent the binomial coefficients—the multipliers of the powers of x that occur when you multiply out expressions of the type (x + 1)n.

Notice also that the numbers along the diagonals follow certain patterns. The second diagonal running from 1 to 7, for example, consists of consecutive whole numbers. The numbers along the third diagonal are known as triangular numbers.

It's possible to convert this triangle into eye-catching geometric forms. For example, you can replace the odd coefficients with 1 and even coefficients with 0 to get the following array (for up to eight rows):

Continuing the pattern for many more rows reveals an ever-enlarging host of triangles, of varying size, within the initial triangle. In fact, the pattern qualifies as a fractal. The even coefficients occupy triangles much like the holes in a fractal known as the Sierpinski gasket (or triangle).

In other words, the pattern inside any triangle of 1s is similar in design to that of any subtriangle of 1s, though larger in size, Andrew Granville noted in a paper on the arithmetic properties of binomial coefficients, titled "Zaphad Beeblebrox's Brain and the Fifty-ninth Row of Pascal's Triangle."

"If we extend Pascal's triangle to infinitely many rows, and reduce the scale of our picture in half each time that we double the number of rows, then the resulting design is called self-similar—that is, our picture can be reproduced by taking any subtriangle and magnifying it," Granville wrote.

The pattern becomes more evident if the numbers are put in cells and the cells colored according to whether the number is 1 or 0.

Similar, though more complicated, designs appear if you replace each number of the triangle with the remainder after dividing that number by 3. Thus, you get:

This time, you would need three different colors to reveal the patterns of triangles embedded in the array. You can also try other prime numbers as the divisor (or modulus), again writing down only the remainders in each position.

In this example, the modulus is 5.

Originally posted February 10, 1997

August 26, 2020

A Passion for Pi

I consider myself a loyal member of the Ancient and Honorable Society of Pi Watchers. At various times, I've written about the discovery of an algorithm for calculating individual, isolated digits of pi, the computation of the value of pi to billions of decimal digits, the use of the random distribution of bright stars across the sky to calculate an approximate value of pi, and other topics involving this amazing mathematical constant.

Pi (π) is the number you get when you divide a circle's circumference by its diameter—a number that is the same for a circle of any size. Pi can't be expressed exactly as a ratio of whole numbers. Indeed, starting with 3.14159…, the decimal digits of pi go on forever.

Statistically, the digits of pi appear to behave like a sequence of random numbers. Over the years, those digits have been the subject of considerable scrutiny and an astonishing amount of dedicated memory work.

Some of you may be familiar with the sentence: How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics! If you write under each word the number of letters it contains, you end up with 3.14159265358979, the first 15 digits of pi.

This sentence appeared in a short news item printed in the Dec. 14, 1929, issue of Science News-Letter (which later became simply Science News). The weekly magazine, in turn, cited The Observatory, an English astronomical journal, as the sentence's source, where the author was given as J.H.J. The initials happen to be the same as those of James H. Jeans, a leading British astronomer.

This item was not the first mention in Science News-Letter of a mnemonic device for remembering the digits of pi. The Oct.. 9, 1926, issue featured the following poetic tribute to Archimedes. It gives pi to 30 decimals.

Now I, even I, would celebrate
In rhymes inept, the great
Immortal Syracusan, rivaled nevermore
Who, in his wondrous lore,
Passed on before,
Left men his guidance how to circles mensurate.

Readers were then invited to contribute any other such "memory rimes" they had composed or found useful. One contributor sent in a French sentence for remembering the value of pi to 10 digits: Que j'aime à faire apprendre le nombre utile aux sages.

As noted in the Nov. 27, 1926, issue of Science News-Letter, this example is actually the first line of a four-line poem encoding 30 decimal places of pi. The full poem originally appeared in a Belgian mathematics journal in 1879.

Over the years, pi enthusiasts have created mnemonic devices encoding pi in just about any language you can imagine—from ancient Greek to modern Armenian. These sentences, poems, miniature dramas, comic episodes, and so forth reflect not only the digits of pi but also the considerable ingenuity of their authors.

Many go no further than the 30th decimal digit, perhaps because of the first appearance of zero, as the 32nd decimal digit, necessitates some new rule—such as using 10-letter words—to continue the sequence.

Nevertheless, some inventive souls have ventured well beyond the 30th decimal digit of pi. In one astounding effort, software engineer, amateur mathematician, and pi fanatic Mike Keith encoded 740 digits in a lengthy poem modeled on Edgar Allan Poe's "The Raven." He later topped that effort with a complete short story in which the number of letters of each successive word gives the first 3,835 digits of pi.

New mnemonics continue to surface. In the April 1999 issue of Math Horizons, Mimi Cukier suggested the following sentence for remembering the first 22 digits of pi: Wow! I have a great technique to recall those fun, crazy numerals composing perhaps everyone's all-in-all favorite real number—Pi!

Mathematician and magician Arthur T. Benjamin responded to that article with his own suggestion for a better way to memorize pi, which appeared in an article in the February 2000 Math Horizons. Benjamin began his article with the sentence: How I wish I could elucidate to others: there are often superior mnemonics!

Benjamin then went on to suggest how a phonetic code, which replaces digits with consonant sounds, is superior to traditional mnemonic devices for remembering strings of digits.

Benjamin recommended the following phonetic code, which has been around for more than 140 years: 1 = t or th or d; 2 = n; 3 = m; 4 = r; 5 = l; 6 = sh, ch, or  j; 7 = k or hard g (as in goat); 8 = f or v; 9 = p or b; 0 = s or z.

"A quick way to memorize the code was suggested to me by Tony Marloshkovips," Benjamin hinted.

By placing vowel sounds between consonants, numbers can be turned into words. For example, the first 24 digits of pi can be translated into My turtle Pancho will, my love, pick up my new mower Ginger.

"Invest just a little bit of time to master the code…, and soon you will be able to rattle off 60 digits of pi in no time!" Benjamin insisted.

I came across a remarkable memory feat involving pi when I was researching the 1995 discovery by David H. Bailey, Peter Borwein, and Simon Plouffe of a truly fantastic formula for computing any given hexadecimal (base 16) digit or binary digit of pi without being forced to calculate the preceding digits. Plouffe, Borwein, and Bailey then used the novel algorithm to establish that the 400 billionth binary digit of pi is 0.

Plouffe once held the world title for memorizing decimal digits of pi. He managed to commit a total of 4,096 digits to memory, an achievement that was duly recognized in the 1977 French edition of the Guinness Book of World Records.

Actually, Plouffe had memorized 4,400 digits but settled on 4,096 (212) as a nice round number to report to others interested in his feat. Back then, "I was young and I had not much else to do, so I did it," Plouffe recalled. He liked numbers and was fascinated by pi.

To Plouffe, memorizing the digits of pi was close to a mystical experience. He worked with blocks of 100 digits. He started by writing out a block five or six times. He then recited these digits in his head. To preserve the numbers in his long-term memory, he periodically isolated himself in a room—no lights, no noise, no coffee, no cigarettes. "Like a monk," Plouffe said.

As Plouffe recited the digits to himself, they would gradually seep into his mind. After a day or two, he would be ready to go on to the next block. When Plouffe got to 4,400 he decided to stop. "You can continue…forever," he explained. "You stop mainly because it is boring to do that all the time."

Two years later, the person who had held the previous record of 3,025 digits came back with 5,050 memorized digits. "I knew I could beat him, but…I had had enough," Plouffe said. In 2005, the record stood at 67,890 digits!

Having a good memory for numbers and the ability to recognize them by sight proved useful to Plouffe in his mathematical work, which often involved looking for relationships between different mathematical series or among various number sequences.

Plouffe was the coauthor, with Neil J.A. Sloane, of The Encyclopedia of Integer Sequences, which contains nearly 6,000 examples of number sequences, collected from a variety of sources. Mathematicians and other researchers have used the book, now greatly expanded in an online database, as a reference for counting or tabulating things that involve number sequences, from the number of atoms in various molecules to different types of knots.

Plouffe has also developed software for doing automatically the kind of numerical pattern recognition that he himself did so naturally.

Suppose you happen upon the number 1.618033987. It looks vaguely familiar, but you can't quite place it. You can use Plouffe's Inverter (PI) to find whether this particular number is special in some way, perhaps as the output of a specific formula or the value of a familiar mathematical function or constant. In the case of 1.618033987, the PI database search produces a page of formulas and functions that generate the number. The most intriguing possibility is the expression (1 + √5)/2, which represents the golden ratio.

The PI database contained more than 200 million entries, making it possible to identify all kinds of "special" numbers. But there's a catch. Given a formula or expression such as 2 + 2, there's only one answer, 4. But, given the result 4, there are actually lots of different ways to get there besides 2 + 2.

Thus, it can become tricky to sift the "true" formula from a coincidental expression extracted from the database. The hazard is greatest when only a small number of digits is used and the number is truncated or rounded off.

Of all known mathematical constants, however, pi continues to attract the most attention. Indeed, the pi craze can sometimes take on unusual or unexpected forms. A while ago, the fragrance industry discovered "math appeal" when Parfum Givenchy introduced a men's cologne dubbed Pi.

Pi also made it to the big screen as the title of a thriller in which an eccentric mathematician unlocks the secret of the stock market in the digits of pi. The Exploratorium in San Francisco pioneered the celebration of Pi Day on March 14 each year, starting at 1:59 p.m., and continues the tradition.

There is something delightfully irrational about the enduring interest in—or perhaps obsession with—pi.

"Pi is one of the few concepts in mathematics whose mention evokes a response of recognition and interest in those not concerned professionally with the subject," Len Berggren, Jonathan Borwein, and Peter Borwein wrote in Pi: A Source Book. "It has been a part of human culture and the educated imagination for more than twenty-five hundred years."

"The computation of pi is virtually the only topic from the most ancient stratum of mathematics that is still of serious interest to modern mathematical research," the authors continued. "And to pursue this topic as it developed throughout the millennia is to follow a thread through the history of mathematics that winds through geometry, analysis and special functions. numerical analysis, algebra, and number theory.

"It offers a subject which provides mathematicians with examples of many current mathematical techniques as well as a palpable sense of their historical development."

"This is a field of endeavor that has attracted some of the greatest minds of mankind," mathematician Dario Castellanos wrote in a 1998 article in Mathematics Magazine about the never-ending fascination with the number pi. (See Malcolm W. Browne's New York Times report about Castellanos's paper.) 

"The studious pursuer of the many curious and fascinating properties which surround this number," Castellanos said, "will forever meet new results and new algorithms related to 'the mysterious and wonderful pi.'"

See also "Pick a Digit, Any Digit."

Original version posted March 11, 1996

August 25, 2020

Sequence Puzzles

Given a sequence consisting of the whole numbers 1, 4, 9, 16, 25, 36, and 49, what number comes next in the sequence?

The most likely answer is 64—the next number in a sequence of squares of consecutive integers, starting with 1.

Such sequence puzzles are a staple of textbook exercises, brainteaser collections, and various intelligence and aptitude tests. Some sequences are easy to figure out, some have multiple interpretations, and others can require considerable head-scratching before the pattern becomes evident.

Neil A.J. Sloane has been collecting number sequences ever since he was a graduate student at Cornell University in the 1960s (see the Quanta Magazine article "The Connoisseur of Number Sequences" and the Numberphile video "What Number Comes Next?").

Sloane described nearly 6,000 examples in his 1995 book The Encyclopedia of Integer Sequences and has added thousands upon thousands of additional examples to an online extension of the book, The On-Line Encyclopedia of Integer Sequences (OEIS).

One useful feature of Sloane's OEIS compendium is the ability to enter a set of numbers and search for information about that sequence. For example, suppose you enter the numbers 1, 2, 3, 6, 11, 23, 47, 106, 235. Among other things, the results page tells you that the next term is 551, that this sequence is associated with trees having n nodes, and that there is a formula for calculating the sequence's terms.

As a way to demonstrate his online encyclopedia, Sloane had a practice of assembling entertaining pages of sequence puzzles. One set starts off with several simple classic sequences (perfect squares, Fibonacci numbers, and so on), then quickly moves into more perplexing territory.

For example, what do you make of the sequence 1, 2, 3, 7, 43, 1807, 3263443? It turns out that the terms are members of a variant of Sylvester's sequence, and the sequence's next member is 10650056950807.

At the OEIS puzzle page, when you give up, you can just click on a link to see the answer. That's a handy shortcut you don't have available to you when you're taking your SAT's.

Originally posted May 19, 2003

August 24, 2020

Parting Ways

89. Back to Home Base

Parting Ways

At the end of the three bike paths (see "Puzzling Pathways"), you come to two signs. One sign says "Spaceship Earth" and points to a giant-size soccer ball. It's even larger than the space capsule that you have been riding in.

"That's it!" says Anita, pointing to Spaceship Earth. "That's the taxi that brought us here. It goes back and forth between here and Earth, kind of like a ferryboat. It will take us home!"

You begin to feel a little homesick, as Bill wanders over to check out the other sign.

"Wow!" he exclaims. "This sign says 'Buckyball Field!'" Then you realize that the sign is pointing to a grassy field where two teams of kids are playing soccer. One team has black shirts, and the other team has white shirts, like the ones that Anita and Bill are wearing (see "The Buckyball Asteroid").

"I think our team needs us," says Bill, just as the black team scores a goal. "Want to join us?" he asks you.

Feeling too tired to play soccer or even watch the game, you gaze over at Spaceship Earth,

"If you would rather go home, I'm sure that Spaceship Earth will take you," says Anita. "It's a lot like your space capsule. Check it out inside."

"Are you sure you don't want to play just a little bit?" asks Bill.

You can't wait to get home, though, so you bid farewell to your two new friends and head over to Spaceship Earth.

Planet Earth

"THIRTEEN!" blasts a voice from Spaceship Earth. Inside, it looks a lot like the space capsule you have been riding in, only bigger (see "A Consequential Countdown").

"TWENTY-ONE!" it says. You know what the next number will be, of course (see "A Special Sequence").

"THIRTY-FOUR!" You start to realize that the giant soccer ball is sailing through space! Looking out the window, you recognize Earth. You get closer and closer, and a loud motor sounds.

"FIFTY-FIVE!" Suddenly all is peaceful.

You find yourself lying on your bed, looking up at the ceiling (see "A Consequential Countdown"). You sit up and find that your room is no longer a space capsule. It's no longer a mess, either! All of your stuff is neatly put away!

You gaze around and look at the walls, the floor, and the ceiling, and remember when they had holes that brought you into the fourth dimension (see "Hyperspace Hangout").

You think of the tiling pattern on your bathroom floor and realize that its pattern of octagons and squares is like Octagon Square on the Planet of the Shapes (see "The Planet of the Shapes").

You pick up your old soccer ball and start counting the number of pentagons on its surface (see "The Amazing Buckyball").

Through the doorway you begin to sense the delicious aroma of apple pie. You think of apples and remind yourself of the five-fold rotational symmetry at its core (see "Sorting Patterns").

A slice through the core of an apple reveals that the seeds are arranged into five compartments, which have five-fold rotational symmetry.

You recall the first few digits of pi and wonder if you will ever see the Digits again (see "Pi in the Sky").

Your brain is buzzing with questions. Will you ever get to go back into space? What's the difference between "outer" space and physical space and mathematical space?

There is a knock at your door. "Come in," you say, wondering if the Digits may be visiting, or if Anita and Bill are returning.

Your mother opens the door, and you realize you really are back in mundane, earthly reality.

"Great job getting your room in order," she says. "Come and join us for a pie feast!" (or did she say "pi" feast?).

Adapted from Math Trek 2: A Mathematical Space Odyssey by Ivars Peterson and Nancy Henderson (see "A Mathematical Space Odyssey"). Online Bibliography.

August 23, 2020

Puzzling Pathways

89. Back to Home Base

Puzzling Pathways

When you, Anita, and Bill reach the edge of Checkerboard City (see "A Familiar Place"), you find three bicycles and three parallel bike paths (see "The Bumpy Bike Path"). A sign points to Pentagon Plaza.

All of the wheels have about the same diameter, but one bike has square wheels, another has hexagonal wheels, and the third has octagonal wheels.

A bike with a square wheel on a bumpy pathway.

All of the bike paths are bumpy, but one path has long, tall bumps, one has medium-size bumps, and one has short, relatively low bumps.

Anita picks up the square-wheeled bike, Bill takes the hexagonal-wheeled bike, and you grab the octagonal-wheeled bike.

Match each wheel with its bike path.

Which path should Anita take? What about Bill? Which path will give you the smoothest ride on your octagonal wheels (see "Different Roads for Different Wheels")?

As the number of sides on a wheel in the shape of a regular polygon increases, the wheel fits shorter and flatter "bumps," or inverted catenaries. (The size of the wheel's radius also affects what size bumps the wheel fits, but in this case, all three bikes have wheels of roughly the same size.)

Anita's square-wheeled bike fits the path with the longest, tallest bumps. Bill's hexagon-wheeled bike fits the path with medium-size bumps, and your octagonal-wheeled bike will get the smoothest ride on the path with the shortest, smallest bumps.

August 22, 2020

A Familiar Place

89. Back to Home Base

A Familiar Place

"Awesome," says Bill as you show him and Anita the Fibonacci sequence you found in Pascal's triangle (see "Back to Home Base"). Just as you come to the line that adds up to 8, the voice from your radio returns.

"EIGHT!" it blares. A regular octagon appears on the screen, and the loud motor roars as you feel the spacecraft taking off.

Soon, the screen is filled with three-dimensional shapes: cubes, tetrahedrons, octahedrons, dodecahedronsicosahedrons, and a buckyball. You click on the buckyball, and it grows into a bright, multicolored soccer ball. Admiring its glistening surface of pentagons and hexagons, you click on one of the hexagons. Suddenly the screen fills up with a checkerboard pattern.

The loud motor sounds return, and you feel your stomach rising. Then suddenly it's quiet.

You stand up, walk over to the window, and recognize Checkerboard City (see "Planet of the Shapes").

"It's the Planet of the Shapes! We're back!" you exclaim.

You exit the spacecraft with Anita and Bill, and start taking steps in random directions (see "Walking on a Grid").

"I'm getting tired, and it seems like we could go forever, walking from square to square," says Anita.

Can you get to the edge of Checkerboard City by moving randomly?

Take a random walk on a checkerboard to see if you can get to the edge.

You will need:
  • checkerboard
  • die (You can use a regular six-sided die, but a tetrahedral die is even better, because you need only four sides.)
  • checker pieces (or coins or other markers)
What to do:
  1. Find the four center squares on your checkerboard and place a checker piece on one of these squares.
  2. Roll the die and take a "step" by placing a second checker piece on the appropriate square: if you roll a 1, place it on the square above; if you roll a 2, place it on the square to the right; 3, on the square below; 4, on the square to the left. If you roll a 5 or a 6, ignore it and roll again.
  3. Keep on rolling and placing a checker piece on the appropriate square.
  4. How many rolls does it take you to reach the edge of the checkerboard?
  5. What kind of pattern do your checker pieces form?
  6. Now take a "self-avoiding" random walk (see "Walking on a Grid"). Clear the checkerboard, and place a checker piece on one of the four center squares. Roll the die and place a second checker piece on the appropriate square.
  7. Roll again and place a checker piece on the appropriate square bordering the square you just covered. You may not retrace your "steps," so if the square is already occupied, ignore that roll, and roll again.
  8. See if you can reach the edge of the checkerboard before you run out of pieces or get trapped. You will be trapped and unable to move if you end up on a square surrounded by four squares that are already covered.
  9. Which route gets you to the edge of the checkerboard first, the regular random walk (steps 2, 3), or the self-avoiding random walk (steps 6, 7)?
In the regular checkerboard random walk, you will probably reach the edge of the checkerboard in fewer than fifteen rolls. Your checker pieces will form a ragged pattern that resembles a three-dimensional map of a mountain; the highest piles of checker pieces will be toward the center.

In the self-avoiding random walk, you will probably get to the edge in even fewer steps. You will never reach the edge, however, if you happen to get trapped on a square that is surrounded by checker pieces, which prevent you from moving in any direction.

August 21, 2020

Back to Home Base

89. Back to Home Base

"ONE!" a loud voice blares from your radio. "ONE!" it repeats. "TWO! THREE! FIVE!" the voice booms. Then it stops.

"It repeated '1' and it skipped '4,' just like the Fibonacci sequence (see "A Special Sequence")," says Anita.

"It sounds like the countdown that brought me into space (see "A Consequential Countdown")," you say. "Only this time the count went up instead of down."

"Hey, if the Fibonacci sequence brought you into space, maybe it also can bring us back to Earth somehow," says Bill.

"But the voice stopped counting," you say. "What if we are stuck here?"

Meanwhile, Anita is studying the numbers on the cubes that form Pascal's triangle (see "Pascal's Patterns"). "I bet the Fibonacci sequence is somewhere in Pascal's triangle," she says. "Every number pattern in the universe seems to show up in Pascal's triangle. If we could only find it, we might get somewhere."

The three of you start searching the numbered cubes for Fibonacci numbers, but they don't seem to line up together anywhere in Pascal's triangle. Can you find the Fibonacci sequence by adding together certain sets of numbers found in Pascal's triangle?

Look for Fibonacci numbers in Pascal's triangle.

You will need:
  •  copy of Pascal's triangle (above)
  • pencil
  • ruler
What to do:
  1. On your diagram of Pascal's triangle, draw a series of diagonal lines, as shown below.
  2. For each diagonal line, add the numbers on the squares it passes through, and record the sum. Include only the squares it intersects more or less down the middle.

Looking for Fibonacci numbers in Pascal's triangle.

For example, the first line passes only through 1. So does the second line. The third line passes through two 1's, so it has a total of 2. The fourth line passes through 2 and 1, for a total of 3. Do you recognize the numbers?

The fifth line adds up to 5. The next sum is 8, and the Fibonacci sequence continues with 13, 21, and so on.

Parting Ways

August 20, 2020

The Man Behind the Triangle

55. Triangle Tribulations

The Man Behind the Triangle

Blaise Pascal was born in France in 1623. When he was only three years old, his mother died, leaving Blaise and his three sisters with their father, Étienne Pascal.

Blaise Pascal, later in life as an adult. MAA Mathematical Treasures

Étienne taught his children at home rather than sending them to school, because he believed that children should not be pushed to study a subject until they could master it easily. He also thought that children's natural curiosity, not a stern teacher, should determine what they are taught.

Étienne decided that his children should not study mathematics until age sixteen or so, and he removed all math texts from the house.

Like a child who is never allowed to watch TV, Blaise became especially curious about the banned subject, mathematics. At age twelve, he started to work on geometry all by himself. Without the aid of a teacher or text, he figured out that the sum of three angles in any triangle is the same as the sum of two right angles (180 degrees). His father was so impressed that he allowed Blaise to study the classic geometry of Euclid.

Blaise's sister Jacqueline was also exceptionally talented. She had such a flair for writing poetry that the Queen of France often invited her to the palace, and she was the first girl ever to win a local poetry competition.

Their father, Étienne, was an important government official. As part of France's new ambitious, intellectual nobility, he was acquainted with French mathematicians and other prominent thinkers of the time.

When Blaise was fourteen years old, Étienne began bringing him to meetings with Descartes, Fermat, Mersenne, and other prominent mathematicians. At age sixteen, Blaise was the first to prove some new geometry theorems, which he presented at one of these meetings.

Shortly thereafter, Étienne got a job as a tax collector. To help him, Blaise invented a mechanical calculator.

Blaise's mathematical work greatly influenced leading philosophers and scientists, including René Descartes and Isaac Newton. Pascal's work on the mathematical triangle (now called Pascal's triangle) led to other important discoveries in mathematics.

August 19, 2020

Pascal's Fractals

55. Triangle Tribulations

Pascal's Fractals

One of the simplest geometric patterns in Pascal's triangle (see "Pascal's Patterns") turns out to be an example of one of the more important geometric shapes in modern mathematics: a fractal. In a fractal, each part is made up of scaled-down versions of the whole shape.

When you shade the even numbers (multiples of 2) in Pascal's triangle, the resulting design resembles a special type of fractal called a Sierpinski triangle. This fractal consists of triangles within triangles in a pattern such that smaller triangles contain the same pattern as the larger triangles.

Draw a Sierpinski triangle.

You will need:
  • pencil and paper
  • ruler
  • protractor
What to do:
  1. Using your ruler, draw a horizontal line across the page, a few inches from the bottom.
  2. Use your protractor to draw a 60 degree angle from each end of your horizontal line pointing toward the middle. Extend the angle rays to form an equilateral triangle.
  3. Using your ruler, find and mark the midpoint of each side of the triangle.
  4. Connect the three midpoints to form a new set of triangles. Shade the center (upside-down) triangle.
  5. For each of the three unshaded triangles, mark the midpoint of each side.
  6. Repeat steps 4 and 5 until your triangles get too small to divide.
  7. Compare your result with the pattern you got from shading the even numbers in Pascal's triangle (see "Pascal's Patterns").

First two stages in creating a Sierpinski triangle.


The first four stages in creating a Sierpinski triangle.

August 18, 2020

Pascal's Patterns

55. Triangle Tribulations

Pascal's Patterns

The numbered triangle (see "Triangle Magic") is commonly known as Pascal's triangle, named for Blaise Pascal, a French philosopher and mathematician from the seventeenth century.

Pascal studied this numbered triangle extensively, but he was not the first to identify it. The Persian poet and mathematician Omar Khayyam (1048-1131) described it in his writings. It also appears in ancient Chinese manuscripts.

Chinese version of Pascal's triangle. MAA Mathematical Treasures

Pascal's triangle is full of interesting number patterns. If you add up the numbers in each row, you get successive powers of 2 (the sum of each row is double the sum of the previous row).
Row 1: 1 = 20
Row 2: 1 + 1 = 2 = 21
Row 3: 1 + 2 + 1 = 4 = 2 ✕ 2 = 22
Row 4: 1 + 3 + 3 + 1 = 8 = 2 ✕ 2 ✕ 2 = 23

The triangle is also full of geometric patterns. If you shade all the squares numbered with a multiple of 5, for example, you get a pattern of upside-down triangles.

The first nine rows of Pascal's triangle.

Look for patterns in Pascal's triangle.

You will need:
  • several copies of Pascal's triangle (above)
  • pencil
  • colored pencil (optional)
  • calculator (optional)
What to do:
  1. Look for number sequences along the triangle's diagonals. The sequence along the first diagonal is 1, 1, 1, 1, 1,…. The sequence along the second diagonal is 1, 2, 3, 4, 5,…. Write down the third diagonal sequence. Do you recognize the sequence?
  2. Using a regular or colored pencil, shade all the squares numbered with a multiple of 5. What kind of pattern do you get?
  3. Shade all the multiples of 2. How is this pattern different from the pattern in step 2?
  4. Try shading multiples of 3, 4, 6, 7, or other numbers and see what patterns turn up. A calculator may be helpful for dividing very large numbers to see which should be shaded.
  5. On any of the shaded triangles, use a different color to shade all the ones, and a third color to shade all the squares you have not yet shaded. What sort of pattern do you see?
The third diagonal in Pascal's triangle is 1, 3, 6, 10, 15, 21,…, which are the triangular numbers (see "Triangle Tribulations"). The second number is the first number plus 2. The third number is the second number plus 3. The fourth number is the third number plus 4, and so on.

Shading multiples of 5 in Pascal's triangle produces a pattern of triangles in which each shaded triangle is composed of ten numbers.

Pascal's triangle with multiples of 5 shaded in.

Shading multiples of 2 produces triangles of various sizes, which form the special pattern known as the Sierpinski triangle.

Pascal's triangle with multiples of 2 shaded in.

Multiples of 3, 7, and many other numbers also produce triangle patterns. In each case the triangles are "upside-down," meaning they point the opposite way from the original Pascal's triangle.

Shading the squares that are not multiples of a given number often produces a triangle pattern, with the triangles "right-side up," or pointing the same way as the original Pascal's triangle.

August 17, 2020

Triangle Magic

55. Triangle Tribulations

Triangle Magic

"Now what?" says Anita, staring at the triangle the three of you have formed from the cubes (see "Triangle Tribulations"). "We are still stuck in the fourth dimension. This just doesn't add up."

The phone rings again, and you answer.

"I forgot to tell you something," says Three. "You need to number each block."

"No problem. They go from 1 to 36."

"No, no! You have to make everything add up," says Three. "Once they are stacked in a triangle, write '1' on the top cube. Then, on each cube below, write the sum of the cubes above it. Each cube in the second row has a '1' over it, so write '1' on each second-row cube."

"In the third row, write '1' on the first (left) cube," Three continues. "The center cube has two '1's over it, so write '2.' The third cube has only one '1' over it, so write '1.' Keep on going like that, and you will be amazed by what turns up!"

What's so amazing about those numbers?

Label each cube (or square) with the sum of the cubes in the row above and see what pattern turns up.

You will need:
  • pencil
  • a sheet of paper with a drawing of 36 squares arranged in a triangle

Thirty-six square tiles arranged to form a triangle.

What to do:
  1. Write "1" in the top square.
  2. Write "1" in each square in the second row.
  3. Write "1," "2," and "1," in that order, on the cubes in the third row.
  4. Number each square in the fourth row with the sum of the squares above it. You should get "1," "3," "3," and "1," in that order.
  5. Continue numbering the rows in this way until you have numbered all the squares.
  6. When you are finished, look for number patterns along rows and diagonals. Try adding the numbers along each row to see what you get.

August 16, 2020

Triangle Tribulations

55. Triangle Tribulations

You are wandering around in the fourth dimension (see "Hyperspace Hangout"), going in and out of weird holes in the walls, when suddenly a phone rings.

Puzzled, you find the receiver and answer, "Hello."

"This is Digit Three, your old friend from Pi Land (see "Pi in the Sky")."

"Fantastic! Can you help us?" you ask. "We seem to be stuck in the fourth dimension, and we have no idea how to get back into regular space. Will we ever see the stars again? Will we ever return to Earth?"

"That crazy Eight is up to some nasty tricks again," says Three. "When I heard that Eight had sent you guys to the Cubic Grid Galaxy (see "Galactic Gridlock"), I knew that you would have a knotty problem and would need my help to get you out."

"How do we get out of here? We need to be back in three-dimensional space!"

"Check each corner of your cubic space capsule. In one of the corners you will find a pile of thirty-six cubes about the size of children's building blocks. Arrange the blocks into a perfect triangle. Good luck!"

You hear a click and realize that Three has hung up, so you, Bill, and Anita go and find the thirty-six blocks. How can you arrange them in a perfect triangle?

Arrange thirty-six cubes (or square tiles) into a triangular pattern.

The steps below show how to arrange the blocks (or tiles) into a horizontal triangle lying flat on a table or on the floor. If you are using cubes, you may stack them to form a vertical triangle, if you wish.

You will need:
  • Thirty-six cubic blocks (even sugar cubes would work), or thirty-six square tiles (about 1 inch or 2.5 centimeters wide) cut from a sheet of paper or cardboard
  • flat surface
What to do:
  1. Lay one of the cubes or squares on your surface and place two more below it so that the top cube sits on the border between the two lower cubes to form a triangle.
  2. Below the row of two, place a row of three cubes so your cubes form a larger triangle. Note that each cube goes beneath a crack between two cubes in the previous row, and the cubes at each end stick out.
  3. Make a fourth row using four cubes, and continue adding rows of cubes until you have used up all thirty-six cubes.

Six square tiles arranged to form a triangle.

Triangular Numbers

Your thirty-six-cube triangle should have eight rows. The first row has just one cube. Each row that follows has one cube more than the previous row.

If you count the total number of cubes from the top to a certain row below, you get what is known as a triangular number. The first triangular number is 1, followed by 3, 6, 10, 15, 21, 28, 36, and so on.

Counting up the total number of squares required to make up a triangular pattern of a certain number of rows gives a triangular number: 1, 3, 6, 10, 15, 21, 28, and 36.