Quilting Pi

When John Sims contemplates a number, he sees color and shape. And an intriguing, enigmatic number such as pi, the ratio of a circle's circumference to its diameter, conjures up vivid patterns that belong on quilts.

Starting with 3.14159265, the decimal digits of pi run on forever, and there's no discernible pattern to ease the task of compiling (or memorizing) these digits. Computer scientists have so far succeeded in computing 50 trillion decimal digits of pi.

Both a mathematician and an artist, Sims taught for many years at the Ringling College of Art + Design in Sarasota, Fla. He's passionately interested in the collision of mathematical ideas and visual culture.

Pi is one of the few mathematical constants that have successfully entered the pop-culture psyche, Sims noted. Pi has appeared as the title of movie, for instance, and as the name of a perfume.

A while ago, Sims created a visualization of pi's digits in a digital video format—with music by Frank Rothkamm and the participation of Paul D. Miller, who was better known on the New York City scene and elsewhere as DJ Spooky. In this visualization, each of the digits from 0 to 9 is represented by its own color on a vast grid of squares.

Seeing Pi by John Sims.

Working in base 2 and using the colors black and white, Sims then created Black White Pi. In base 3, using red, white, and blue, he made American Pi.

From left to right, Seeing Pi, American Pi, and Black White Pi. Courtesy of John Sims.

A second pi-based project involved a collaboration with conceptual artist Sol LeWitt (1928-2007). LeWitt's instructions were to put 1,000 straight lines inside a square. Sims achieved that result by dividing each side of the square into 10 parts (like the axes of a graph), labeling the divisions from 0 to 9, and drawing lines from a division on one side to a division on an adjacent side. The lines followed successive digits of pi from side to side, starting at the top and moving in a clockwise direction until the wall drawing had 1,000 lines.

Sims' former student, Brandon Styza, drew the lines. The result formed the basis for a LeWitt wall drawing in the math lounge at Wesleyan University.

In 2006, before heading for New York City, Sims completed a number of pi works, including several quilts that were constructed by an Amish quilting group in Sarasota. These artworks were on display at Sarasota's mack b gallery.

John Sims working with a group of Amish women to create a pi-based quilt. Courtesy of John Sims.

Sims started out with a drawing of pi's decimal digits on a square grid, with successive digits forming a clockwise spiral from the center.

A square spiral of the digits of pi. Photo by Tobey Albright.

In the gallery, this drawing was displayed with a phonograph that played a recording of Sims reciting the digits of pi in order. A second track presented the digits in German.

With each digit from 0 to 9 mapped to a different color (but not black or white), the central portion of the drawing was then converted into a striking, square quilt of colored patches, with a black border. Sims called the creation Pi sans Salt and Pepper.

Pi sans Salt and Pepper by John Sims. The square quilt is 8 feet wide. Photo by Tobey Albright.

In a variation on this pi-based theme, another quilt designed by Sims featured several, differently color-coded representations of pi. It was called Civil Pi Movement.

In this pi-based quilt, called Civil Pi Movement, the upper left unit shows the first 36 binary digits of pi (0 = white and 1 = black) and the lower right unit reverses the color scheme. The upper right unit shows the first 36 ternary digits of pi (0 = dark blue, 1 = red, and 2 = white) and the lower left unit uses a different color scheme (0 = green, 1 = red, and 2 = black). The center unit matches the center of Pi sans Salt and Pepper. The square quilt is 8 feet wide. Photo by Tobey Albright.

"The mathematical art that I seek to develop combines mathematical language and analysis with the expressiveness and creativity of the process to make expressive visual theorems," Sims said. "To see mathematically, one draws from creativity and intuition, as in the case with the art process itself."

See also "Fractal Roots and Artful Math" and "Form Plus Function."

Originally posted May 8, 2006

The Limits of Mathematics

At the beginning of the 20th century, the German mathematician David Hilbert (1862–1943) advocated an ambitious program to formulate a system of axioms and rules of inference that would encompass all mathematics, from basic arithmetic to advanced calculus. His dream was to codify the methods of mathematical reasoning and put them within a single framework.

David Hilbert. MAA Convergence Portrait Gallery.

Hilbert insisted that such a formal system of axioms and rules should be consistent, meaning that you can't prove an assertion and its opposite at the same time. He also wanted a scheme that is complete, meaning that you can always prove a given assertion either true or false. He argued that there had to be a clear-cut mechanical procedure for deciding whether a certain proposition follows from a given set of axioms.

Hence, it would be possible, though not actually practical, to run through all possible propositions, starting with the shortest sequences of symbols, and check which ones are valid. In principle, such a decision procedure would automatically generate all possible theorems in mathematics.

What Hilbert was saying is that "we can solve a problem if we are clever enough and work at it long enough," mathematician Gregory J. Chaitin wrote in his 1998 book The Limits of Mathematics: A Course on Information Theory and the Limits of Formal Reasoning. "He didn't believe that in principle there was any limit to what mathematics could achieve."

In the 1930s, Kurt Gödel (1906–1978), followed by Alan Turing (1912–1954) and others, proved that no such decision procedure is possible for any system of logic made up of axioms and propositions sufficiently sophisticated to encompass the kinds of problems that mathematicians work on every day.

"More precisely, what Gödel discovered was that the plan fails even if you just try to deal with elementary arithmetic, with the numbers 0, 1, 2, 3,… and with multiplication and addition," Chaitin wrote in his 2004 paper META MATH! The Quest for Omega. "Any formal system that tries to contain the whole truth and nothing but the truth about addition, multiplication, and the numbers 0, 1, 2, 3,… will have to be incomplete."

In Gödel's realm, no matter what the system of axioms or rules is, there will always be some assertion that can be neither proved nor invalidated within the system. Indeed, mathematics is full of conjectures—assertions awaiting proof—with no assurance that definitive answers even exist.

Turing's argument involved mathematical entities known as real numbers. Suppose you happen upon the number 1.6180339887. It looks vaguely familiar, but you can't quite place it. You would like to find out whether this particular sequence of digits is special in some way, perhaps as the output of a specific formula or the value of a familiar mathematical constant.

It turns out that the given number is the value, rounded off, of the so-called golden ratio, which can also be written as (1 + √5)/2, an example of a real number. Given that expression, which represents an infinite number of decimal digits, you can compute its value to any number of decimal places. Going in the opposite direction from the given rounded-off number to the expression, however, is much more difficult and problematic.

For example, it's possible that if the mystery number were available to an extra decimal place, the final digit would no longer match the decimal digits of the golden ratio. You would have to conclude that the given number is not an approximation of the golden ratio. Indeed, the extended string of digits could represent the output of a completely different expression or formula, or even part of a random sequence. It's impossible to tell for sure. There isn't enough information available.

To sort through the relationship between random sequences and the types of numbers that mathematicians and scientists use in their work, Chaitin defined the "complexity" of a number as the length of the shortest computer program (or set of instructions) that would spew out the number.

"The minimum number of bits—what size string of zeros and ones—needed to store the program is called the algorithmic information content of the data," Chaitin wrote in the article "The Limits of Reason," published in the March 2006 Scientific American. "Thus, the infinite sequence of numbers 1, 2, 3,… has very little algorithmic information; a very short computer program can generate all those numbers."

"It does not matter how long the program must take to do the computation or how much memory it must use—just the length of the program in bits counts," he added.

Similarly, suppose a given number consists of 100 1s. The instruction to the computer would be simply "print 1, 100 times." Because the program is substantially shorter than the sequence of 100 1s that it generates, the sequence is not considered random. If a sequence is disorderly enough that any program for printing it out cannot be shorter than the sequence itself, the sequence counts as algorithmically random. Hence, an algorithmically random sequence is one for which there is no compact description.

Interestingly, the number pi (the ratio of a circle's circumference to its diameter), which is expressed by an infinite number of digits, has little algorithmic information content because a computer can use a relatively small program to generate the number, digit by digit: 3.14159…. On the other hand, a random number with merely 1 million digits has a much larger amount of algorithmic information.

Chaitin proved that no program can generate a number more complex than itself. In other words, "a 1-pound theory can no more produce a 10-pound theorem than a 100-pound pregnant woman can birth a 200-pound child," he liked to say.

Conversely, Chaitin also showed that it is impossible for a program to prove that a number more complex than the program is random. Hence, to the extent that the human mind is a kind of computer, there may be a type of complexity so deep and subtle that the intellect could never grasp it. Whatever order may lie in the depths would be inaccessible, and it would always appear to us as random.

At the same time, proving that a sequence is random presents insurmountable difficulties. There's no way to be sure that we haven't overlooked a hint of order that would allow even a small compression in the computer program that produces the sequence.

From a mathematical point of view, Chaitin's result suggests that we are far more likely to find randomness than order within certain domains of mathematics. Indeed, his complexity version of Gödel's theorem states: Although almost all numbers are random, there is no formal axiomatic system that will allow us to prove this fact.

Chaitin's work indicates that there is an infinite number of mathematical statements that you can make about, say, arithmetic that can't be reduced to the axioms of arithmetic. So there's no way to prove whether the statements are true or false by using arithmetic. In Chaitin's view, that's practically the same as saying that the structure of arithmetic is random.

"What I've constructed and exhibited are mathematical facts that are true… by accident," he said. "They're mathematical facts which are analogous to the outcome of a coin toss…. You can never actually prove logically whether they're true."

This doesn't mean that anarchy reigns in mathematics, only that mathematical laws of a different kind might apply in certain situations. In such cases, statistical laws hold sway and probabilities describe the answers that come out of equations. Such problems arise when you ask whether an equation involving only whole numbers has an infinite number of whole-number solutions, a finite number, or none at all.

"In the same way that it is impossible to predict the exact moment at which an individual atom undergoes radioactive decay, mathematics is powerless to answer particular questions," Chaitin stated. "Nevertheless, physicists can still make reliable predictions about averages over large ensembles of atoms. Mathematicians may in some cases be limited to a similar approach."

That makes mathematics much more of an experimental science than many mathematicians would like to admit.

Chaitin went further. Human creativity is absolutely necessary for mathematical work, he argued, and "intuition cannot be eliminated from mathematics."

Originally posted March 6, 2006

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

A Song About Pi

MSRI Journal
The Mark of Zeta
The Return of Zeta
Solitaire-y Sequences

A Song About Pi

In 1999, at the age of 82, Irving Kaplansky (1917-2006) remained actively engaged in mathematical research.

Then Director Emeritus of the Mathematical Sciences Research Institute (MSRI) in Berkeley, Calif., Kaplansky spent much of his time in the library, poking into various nooks and crannies of mathematical history. Tidying up loose ends and filling in unaccountable gaps in the mathematical literature, he patiently worked through mathematical arguments, proved theorems, and prepared papers for publication. His remarkably wide-ranging efforts belied the oft-repeated notion that mathematicians are most productive when they are young.

A distinguished mathematician who made major contributions to algebra and other fields, Kaplansky was born in Toronto, Ontario, several years after his parents had emigrated from Poland. In the beginning, his parents thought he was going to become a concert pianist. By the time he was five years old, he was taking piano lessons. That lasted for about 11 years, until he finally realized that he was never going to be a pianist of distinction.

Nonetheless, Kaplansky loved playing the piano, and music remained one of his passions. "I sometimes say that God intended me to be the perfect accompanist—the perfect rehearsal pianist might be a better way of saying it," he said. "I play loud, I play in time, but I don't play very well."

While in high school (Harbord Collegiate in Toronto), Kaplansky started to play in dance bands. During his graduate studies at Harvard University, he was a member of a small combo that performed in local night clubs. For a while, he hosted a regular radio program, where he played imitations of popular artists of the day and commented on their music. A little later, when Kaplansky became a math instructor at Harvard, one of his students was Tom Lehrer, later to become famous for his witty ditties about science and math.

In 1945, Kaplansky moved to the University of Chicago, where he remained until 1984, when he retired, then became MSRI director.

Songs had always interested him, particularly those of the period from 1920 to 1950. These songs tended to have a particular structure: the form AABA, where the A theme is repeated, followed by a contrasting B theme, then a return to the original A theme.

Early on, Kaplansky noticed that certain songs have a more subtle, complex structure. This alternative form can be described as AA'BAA'(B/2)A", where A is a four-bar phrase, A' and A" are variants, and B is a contrasting eight-bar phrase. "I don't think anyone had noticed that before," he remarked. Kaplansky's discovery is noted in a book about the American musical by University of Chicago film scholar Gerald Mast (1940-1988).

Kaplansky argues that the second structure is really a superior form for songs. To demonstrate his point, he once used it to turn an unpromising source of thematic material—the first 14 decimal digits of pi—into a passable tune. In essence, each note of the song's chorus corresponds to a particular decimal digit. When Chicago colleague Enid Rieser heard the melody at Kaplansky's debut lecture on the subject in 1971, she was inspired to write lyrics for the chorus.

A SONG ABOUT PI

Through all the bygone ages,
Philosophers and sages
Have meditated on the circle's mysteries.
From Euclid to Pythagoras,
From Gauss to Anaxag'ras,
Their thoughts have filled the libr'ies bulging histories.
And yet there was elation
Throughout the whole Greek nation
When Archimedes did his mighty computation!
He said:

Chorus

3 1 4 1 Oh (5) my (9), here's (2) a (6) song (5) to (3) sing (5) about (8,9) pi (7).
Not a sigma or mu but a well-known Greek letter too.
You can have your alphas and your great phi-bates, and omegas for a friend,
But that's just what a circle doesn't have—a beginning or an end.
3 1 4 1 5 9 is a ratio we don't define;
Two pi times radii gives circumf'rence you can rely;
If you square the radius times the pi, you will get the circle's space.
Here's a song about pi, fit for a mathematician's embrace.

The chorus is in the key of C major, and the musical note C corresponds to 1, D to 2, and so on, in the decimal digits of pi.

The music and lyrics are unpublished. However, singer-songwriter Lucy Kaplansky (Irving Kaplansky's daughter) sometimes includes a rendition of "A Song About Pi" in her programs. Although she has her own distinctive style, she doesn't mind occasionally showcasing her father's old-fashioned tunesmanship. Video.

In 1993, Irving Kaplansky wrote new lyrics for the venerable song "That's Entertainment" (video) to celebrate his enthusiasm for mathematics. He dedicated the verses to Tom Lehrer.

THAT'S MATHEMATICS

The fun when two parallels meet
Or a group with an action discrete
Or the thrill when some decimals repeat,
That's mathematics.
A nova, incredibly bright,
Or the speed of a photon of light,
Andrew Wiles, proving Fermat was right,
That's mathematics.
The odds of a bet when you're rolling two dice,
The marvelous fact that four colors suffice,
Slick software setting a price,
And the square on the hypotenuse
Will bring us a lot o' news.
In genes a double helix we see
And we cheer when an algebra's free
And in fact life's a big PDE.
We'll be on the go
When we learn to grow with mathematics.

With Lagrange everyone of us swears
That all things are the sums of four squares,
Like as not, three will do but who cares.
That's mathematics.
Sporadic groups are the ultimate bricks,
Finding them took some devilish tricks.
Now we know—there are just 26.
That's mathematics.
The function of Riemann is looking just fine,
It may have its zeros on one special line.
This thought is yours and it's mine.
We may soon learn about it
But somehow I doubt it.
Don't waste time asking whether or why
A good theorem is worth a real try,
Go ahead—prove transcendence of pi;
Of science the queen
We're all of us keen on mathematics.

Originally posted July 12, 1999.

Row Your Boat
Juggling By Design
Averting Instant Insanity
Matrices, Circles, and Eigenthings
Lunar Shadows
MSRI Reflections

Pi Path

10,001 Digits of Pi by John Snow. Mathematical Art Gallery, Joint Mathematics Meetings, Baltimore, Maryland, 2019.

Photo by I. Peterson

Infinite Pi

Pi by David Reimann. Displayed at Bridges 2012, Towson University, Towson, Maryland, 2012.

Photo by I. Peterson

Exploratorium Pi

Pi plaque. Exploratorium, San Francisco, California, 2014.

For more pi places, see "Pizza, Beer, and Pi," Portland Pi," "Square Root of Pi," "Pi Doors," "DC Pi," "Pi Places," "Sliding Pi in Toronto," "Segments of Circles, Sections of Spheres," and "Galileo's Vision in Missouri."

Photo by I. Peterson

Pizza, Beer, and Pi

Pi Pizzeria. Bethesda, Maryland, 2016.

Photos by I. Peterson

Portland Pi

Composition by Michihiro Kosuge. Portland Art Museum, Portland, Oregon, 2014.

For more pi sightings, see "Square Root of Pi," "Pi Doors," "DC Pi," "Pi Places," "Sliding Pi in Toronto," "Segments of Circles, Sections of Spheres," and "Galileo's Vision in Missouri."

Photos by I. Peterson

Square Root of Pi

Symbols for the square root of pi are among the adornments on a stainless steel trash container, Euclid Corridor, Cleveland, 2011. "Visual trash" artwork, inspired by a bad computer printout, by Mark Howard.

For other appearances of pi, see "Pi Doors" and "Pi Places."

Photos by I. Peterson

Pi Doors

Pi door handles, National Museum of Mathematics, New York City.

For other appearances of pi, see "Pi Places" and "DC Pi."

Photos by I. Peterson

DC Pi

Sighting of pi on Wisconsin Avenue in Washington, D.C.

For more on pi sites, see "Pi Places" and "Sliding Pi in Toronto."

Photo by I. Peterson

Sliding Pi in Toronto

The entrance to the Sheppard West (formerly Downsview) subway station in Toronto presents visitors with a stunning vista. A vast mosaic of small square tiles sweeps across a curved wall, inviting travelers to trace its lines and ponder intriguing irregularities in its color scheme of blues, greens, roses, and other hues.

Of the hordes of people who hurry through the station, perhaps only a few take a moment or two to contemplate the curious blend of the regular and the seemingly random in the bands of this cryptic spectrum writ large. Far fewer people even suspect that the artist responsible for this mosaic based her remarkable design on the decimal digits of the number pi (π), the ratio of a circle's circumference to its diameter.

The artist is Arlene Stamp. In 1993, at the height of her interest in non-repeating patterns, she was intrigued by the possibility of using designs with an element of unpredictability to enliven large areas in public spaces, such as floors and walls, which are usually covered with simple, repeating motifs.

To win the commission to decorate the newly built subway station, Stamp had to come up with an interesting design that would work equally well on various surfaces in different parts of a large station yet stay within a tight budget.

"It became clear that any kind of design based on…modules would be very repetitive within such a restrictive budget," Stamp said. "So I came up with a design that was rich in information but inexpensive in…realization."

The basic unit in Stamp's design was a rectangle, 10 units wide, made up of square tiles. Instead of placing these rectangles side by side along a baseline to create a regular, linear pattern, she overlapped adjoining rectangles. The amount of overlap was governed by successive decimal digits of pi.

"I used pi as a source for a string of unpredictable digits because the circle and curved walls were a design feature of this station," Stamp remarked.

The amount of overlap between adjacent rectangles is governed by the decimal digits of pi. For example, the first digit of pi after the decimal is 1, so the first two rectangles from the left (black and red) overlap by one unit, just as if the second rectangle were sliding over to the left to rest partly on top of the first. The second digit is 4, so the third rectangle (green) overlaps those already in place by four units.

Stamp used four sets of eight colors for the project, deploying them in different ways on surfaces in various parts of the station. Each set of colors had a different cast: yellowish green, bluish green, reddish blue, and bluish red.

"I assigned colors by thinking of the first layer as the lightest, with the colors deepening according to the number of layers of overlap," she said.

Describing a non-repeating design spread over a very large area presented an additional difficulty. "One of the big challenges of this commission was to find a way of encoding the color information for the tile installer in some kind of space-efficient way," Stamp noted. "With the help of an architect and a number-coded color system, we got all the information on a single sheet of the architect's plans."

Stamp's design doesn't repeat itself anywhere in the station. Indeed, it's fun to look for all the different "pieces of pi" scattered throughout the structure.

Stamp titled her artwork Sliding Pi.

"It's true that few visitors to the station would have any idea that there is a connection between the patterns there and a math concept," Stamp said. "But surely some must wonder how such a seemingly random, ever-changing pattern of color happened—because it is such a rarity in public design."

Moreover, upon glimpsing the various manifestations of Stamp's pi-based mosaic, a few may even sense an underlying plan of some sort—that there is method to the randomness.

For another example of Stamp’s fascination with numbers and patterns, see "Binary Frieze."

Original version posted June 5, 2000.
Updated June 8, 2010.

Photos by I. Peterson