November 22, 2020

Scrambled Grids

Amazingly simple mathematical operations can lead to intriguingly complex results. Repeated operations, in particular, often produce surprising images.

Consider, for instance, the iterative geometric process that resembles the making of flaky pastry dough. Flatten and stretch the dough, then fold it over on top of itself. Do it again and again and again. Repeating the pair of operations—stretch and fold—just 10 times produces 1,024 layers; 20 times, more than a million layers.

In dynamical systems theory, the so-called baker's map, or transformation, does nearly the same thing. Here's one special case of that transformation: Start with a square. Stretch it to twice its original length while making it half as wide. Cut the result in half, and stack one half on top of the other to return the combination to the square's original dimensions.

In the so-called baker's transformation, an image is repeatedly flattened and stretched horizontally, then sliced in half and stacked vertically. These diagrams show just the first stage in the mixing process.

Notice that at each step the square's area is preserved, but its components are rearranged. Indeed, applying this transformation repeatedly has the effect of jumbling whatever image may be drawn within the square. If that image happens to be a grinning feline, you end up with a madly scrambled cat.

During the mixing process, it occasionally happens that some of the square's points come back close to their initial locations within the square. For an instant, some form of the original image would flash into view before vanishing again into a murky, nearly homogeneous sea.

This surprising reappearance is an example of a phenomenon known as Poincaré recurrence, named for the French mathematician Henri Poincaré (1854-1912). In essence, if a transformation is applied repeatedly to a mathematical system and the system cannot leave a bounded region, it must return infinitely often to states near its original state.

In 1986 for an article on chaos in Scientific American, Bill Sanderson and James P. Crutchfield created a striking illustration of Poincaré recurrence. They applied a certain stretching operation (sometimes called Arnold's cat map) to a digitized portrait of Poincaré.

At each step, the computer stretched the image diagonally as if it were printed on a sheet of rubber. Any portion of the image that extended past the original border was chopped off and reinserted elsewhere. Most of the time, successive iterations of the stretching process produced diagonal streaks or seemingly random splotches of color. Once in a while, however, the original image reappeared.

When Bob Brill first saw this sequence of images, he was fascinated. "Stirring an image into a soup only to see that more of the same treatment restored it was like seeing the dead come to life," he observed.

A professional computer programmer, Brill decided to explore this phenomenon for its potential in creating artful images. Although he hadn't started out as an artist, he had been teaching himself how to turn a computer into an artist's tool ever since he had bought his first personal computer in 1983.

Two articles by A.K. Dewdney in Scientific American—one featuring vivid depictions of the Mandelbrot set and another describing computer algorithms and formulas for generating "wallpaper for the mind"—ignited Brill's interest in algorithmic art. "That got me started, and I've been hooked ever since," he said.

To create an image, Brill would write a simple program in a special computer language he developed and named E in honor of Dutch graphic artist M.C. Escher (1898-1972). His DOS-based, menu-driven "E-Run" software executed the program and generated the picture.

One of Brill's favorite strategies involved applying area-preserving transformations to grids, where each box of the grid can be as small as a single pixel on a computer monitor. His algorithm specified where all the boxes go with each application of a given transformation.

For example, suppose you decide to use a 4-by-3 grid, with its boxes numbered as follows:

One possible transformation shuffles the boxes so that the numbers now start in the lower right corner, increasing as you go up and to the left. So 1 goes to where 12 was, 2 goes to where 8 was, and so on, as shown in the table below.

Notice that boxes 1 and 12 simply exchange places with each application of the transformation. The remaining boxes each cycle through 2, 8, 6, 3, 4, 11, 5, 7, 10, and 9. In this particular instance, the original image reappears after 10 iterations. Such a reappearance is a characteristic of the application of any area-preserving transformation to a finite grid.

The idea is to color the boxes to create some sort of pattern, then apply the transformation again and again to see what happens.

Suppose the starting configuration consists of three horizontal bars, each one a different color (below, left). Each successive application of the transformation shuffles the boxes to create a new pattern.

Suppose the starting pattern consists of three horizontal bars, each one a different color. Successive applications of Brill's transformation to this four-by-three grid shuffle the boxes to create new patterns.

Applying a transformation to a bull's-eye pattern (above) scrambles the original image more and more until the pattern becomes a sea of mush (below: top row right).

Distinctive patterns can suddenly appear amid the mush after enough applications of the transformation. In this case the original bull's-eye appears after 6,580 applications of the transformation. The number with each image indicates how many cycles were required to reach that stage.

The majority of steps produce boring, nearly structureless images. "To me, the images are interesting when they show structure, especially when there is some echo of the original image," Brill noted. Finding the interesting ones without having to check each of the thousands of possibilities, however, can be tricky and time-consuming.

In the following sequence of images (below) from his artistic creation Transformation, Brill applied the same transformation repeatedly to a starting pattern (top left) in a 1,024-by-1,022 rectangle.

The images above result from 171, 254, 256, 341, and 512 applications of the transformation. In this case the repeat cycle is 1,023.

Brill tried out a variety of grid-scrambling transformations to see what happens. Aesthetic sensibilities govern which transformation to use, what size the rectangular grid should be, and which iteration to look at, he said. "Once a fruitful transformation, rectangle size, and iteration number have been found, the artist is in a position to create compelling imagery."

The choice of starting pattern and color scheme can also strongly influence the appeal of the various gridscapes that arise. "Color is a particularly interesting problem," Brill said. When pixels of different color appear side by side, especially when the combination occurs repeatedly over a region large enough to be visible to the eye, the eye averages them to generate a new color—an effect familiar to pointillist painter Georges Seurat (1859-1891) and other artists.

In computer graphics, this visual effect is known as dithering. "With Poincaré recurrence, the dithering is done entirely by the transformation," Brill said. "You might think that the results would be muddy, but not so, if one selects colors with an eye toward their mixing potential and does not choose too many."

"The transformations are not random," he added. "The dance of the pixels is quite orderly, so then is the appearance of new colors."

In the sequence of images shown above, Brill applied the same transformation repeatedly to a starting pattern (top) in a 403-by-399 rectangle. In this case, the repeat cycle was 6,580.

Brill envisioned using his algorithms for fabric design—perhaps for drapes and bedspreads.

"There are worlds of order and beauty lying dormant in our various mathematical systems that these simple algorithmic processes are able to make visible," Brill concluded. "Mathematics, more than any other human activity, seems to offer connections to the underlying order of the world. This is a great inspiration for an artist and a great challenge."

Originally posted August 28, 2000

November 18, 2020

Lucky Numbers

Hunting for prime numbers, those evenly divisible only by themselves and 1, requires a sieve to separate them from the rest. For example, the sieve of Eratosthenes, named for a Greek mathematician of the third century B.C., generates a list of prime numbers by the process of elimination.

To find all prime numbers less than, say, 100, the hunter writes down all the integers from 2 to 100 in order (1 doesn't count as a prime). First, 2 is circled, and all multiples of 2 (4, 6, 8, and so on) are struck from the list. That eliminates composite numbers that have 2 as a factor.

The next unmarked number is 3. That number is circled, and all multiples of 3 are crossed out. The number 4 is already crossed out, and its multiples have also been eliminated. Five is the next unmarked integer. The procedure continues in this way until only prime numbers are left on the list. Though the sieving process is slow and tedious, it can be continued to infinity to identify every prime number.

Other types of sieves isolate different sequences of numbers. Around 1955, the mathematician Stanislaw Ulam (1909-1984) identified a particular sequence made up of what he called "lucky numbers," and mathematicians have been playing with them ever since.

Starting with a list of integers, including 1, the first step is to cross out every second number: 2, 4, 6, 8, and so on, leaving only the odd integers. The second integer not crossed out is 3. Cross out every third number not yet eliminated. This gets rid of 5, 11, 17, 23, and so on.

The third surviving number from the left is 7; cross out every seventh integer not yet eliminated: 19, 39,…. Now, the fourth number from the beginning is 9. Cross out every ninth number not yet eliminated, starting with 27.

This particular sieving process yields certain numbers that permanently escape getting "killed." That's why Ulam called them "lucky."

Lucky numbers less than 200: 1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, 43, 49, 51, 63, 67, 69, 73, 75, 79, 87, 93, 99, 105, 111, 115, 127, 129, 133, 135, 141, 151, 159, 163, 169, 171, 189, 193, 195.

What's remarkable is that the "luckies," though generated by a sieve based entirely on a number's position in an ordered list, share many properties with primes. For example, there are 25 primes less than 100, and 23 luckies less than 100.

Indeed, it turns out that primes and luckies come up about equally often within given ranges of integers. The distances between successive primes and the distances between successive luckies also keep increasing as the numbers increase. In addition, the number of twin primes—primes that differ by 2—is close to the number of twin luckies.

Perhaps the most famous problem involving primes still unsolved is Goldbach's conjecture, which states that every even number greater than 2 is the sum of two primes. Luckies are featured in a similar conjecture, also unsolved: Every even number is the sum of two luckies. Computer searches have so far not found an exception.

Martin Gardner described many more features of lucky numbers in a delightful article, "Lucky numbers and 2187," in a 1997 issue of The Mathematical Intelligencer. "There is a classic proof by Euclid that there is an infinity of primes," he wrote. "Although it is easy to show there is also an infinity of lucky numbers, the question of whether an infinite number of luckies are primes remains, as far as I know, unproved."

How did the topic of lucky numbers happen to come up? The house where Gardner grew up in Tulsa, Okla., had the address 2187 S. Owasso. "Of course I never forgot this number," he said. It also happens to be one of the lucky numbers.

Gardner's imaginary friend, the noted numerologist Dr. Irving Joshua Matrix, can readily find additional remarkable properties associated with that number. Exchange the last two digits of 2187 to make 2178, multiply by 4, and you get 8712, the second number backward.

Take 2187 from 9999 and the result is 7812, the number in reverse. Moreover, the first four digits of the constant e, 2718, and the number of cubic inches in a cubic foot, 123 = 1728, are each permutations of 2187!

However, to those inclined to seeing meaning in certain numbers, Dr. Matrix issued the following warning: "Every number has endless unusual properties."

Originally posted September 8, 1997

November 16, 2020

Cutting Corners

Go to just about any college campus or public park and you're bound to see two kinds of trails: "official" paths defined by paved or gravel surfaces and "unofficial" routes trodden into the grass or dirt marking where pedestrians have preferred to walk or runners to jog.

Such collective path breaking is an example of a human social phenomenon that involves some form of self-organization. In 1997, a team of researchers developed a mathematical model of pedestrian motion to explore the evolution of trails in urban green spaces.

In the article "Modelling the evolution of human trail systems," published in Nature, Dirk Helbing, Joachim Keltsch, and Peter Molnar reported that their model was "able to reproduce many of the observed large-scale spatial features of trail systems."

The paths pedestrians forge can be quite complex. "In many cases, the pedestrians' desire to take the shortest way and the specific properties of the terrain are insufficient for an explanation of the trail characteristics," the researchers observed. In other words, the trails generated by walkers don't always reflect the most direct routes between entry and exit points.

Several factors come into play. People usually want to take the most direct route, but they also prefer to walk on worn, existing trails rather than tramp over rougher terrain to blaze new ones. Moreover, the nearer a segment of an existing trail is to a walker, the greater an attraction it holds.

Helbing and his colleagues built these assumptions about human behavior into a set of equations that describes how the routes taken by pedestrians would change as time passes. They could then observe what happens when a crowd of "virtual" walkers with given entry points and destinations venture across the park at random times. Because not every route is used by the same number of walkers, some paths typically start to become more pronounced than others.

The type of trail pattern that emerges depends largely on a parameter the researchers called attractiveness (or comfort of walking). If attractiveness is low, a direct-route system develops. If attractiveness is high, a minimal-route system forms. Otherwise, a compromise between the two extremes appears, as is often observed in real trail systems.

The attractiveness parameter determines whether a direct-route (left) or a minimal-route (right) system of trails forms to connect three park gates.

In one simulation, the researchers started with a square, level, grassy park having a gate at each corner. Initially, randomly oriented walkers tend to use direct paths from gate to gate. However, because frequently used trails become more comfortable, the different routes begin to blend together in such a way that the overall length of the trail system decreases.

The evolution of a path system in a square park. Direct paths are prominent in a new park (left) whereas a compromise path system develops in older parks (right).

"The resulting [trail] system…could serve as a planning guideline," the researchers said. "It provides a suitable compromise between minimal construction costs and maximal comfort. Moreover, it balances the relative detours of all walkers."

Self-organization is evident in a wide variety of human social phenomena, from the evolution of cooperation to the development of traffic patterns and the growth of settlements. Mathematical models help elucidate the factors that influence such processes.

"There is a wealth of possible applications and extensions of such models, not only to human but to animal populations, and to social as well as physical spaces," commented geographer Michael Batty. "These models naturally extend to problems where the goal is to optimize, or design 'best' paths, as is characteristic of many planning problems where the task is to design high-quality urban environments."

May the path be with you!

Originally posted July 28, 1997

November 15, 2020

Mountain Trail


Dale Ball Trails North (Sierra del Norte), Santa Fe, New Mexico, 2020.

Photo by I. Peterson

November 6, 2020

Tilt-A-Whirl Chaos

Much of the fun of an amusement park ride results from its stomach-churning, mind-jangling unpredictability. The Tilt-A-Whirl, for example, spins its passengers in one direction, then another, sometimes hesitating between forays and sometimes swinging abruptly from one motion to another. A rider never knows exactly what to expect next.

Yet these complicated, surprising movements arise from a remarkably simple geometry. A passenger rides in one of seven cars, each mounted near the edge of its own circular platform but free to pivot about the center. The platforms, in turn, move at a constant speed along an undulating circular track that consists of three identical hills separated by valleys, which tilt the platforms.

The platform movements are perfectly regular, but the cars whirl around independently in an irregular manner. Moreover, there is essentially just one adjustable parameter—the rate at which the platforms move around the track.

When the platforms travel at very low speeds, the cars complete one backward revolution as their platforms go over each hill. In contrast, at high speeds a car gets slammed to its platform’s outer edge and stays locked in that position. In both cases, the motion is predictable.

What happens at intermediate speeds?

To model dynamical systems like the Tilt-A-Whirl, mathematicians, scientists, and engineers use equations that describe how the positions and velocities of a system and its components change over time in response to certain forces.

It's convenient to characterize a system's dynamics by plotting how its position and velocity evolve over time. Each plotted point represents the system’s state of motion at a particular instant, and successive points generate a winding line through an imaginary mathematical space (known as phase space) representing all possible motions. Different starting points generally initiate different curves.

A simple, repeating motion, like the to-and-fro oscillations of a swinging pendulum, appears as a circle or some other closed curve. Such a plot shows that the system cycles through precisely the same state of motion again and again at regular intervals.

More complicated sequences of movements produce tangled paths that wander through phase space, sometimes never forming a closed loop.

Often, it helps to examine such complicated movements not at every moment but at predetermined, regular intervals. In other words, you start with a point representing the system’s initial state, then wait a given time and plot a second point to give the system’s new state, and so on.

In the case of a simple pendulum, selecting an interval equal to the time it takes the pendulum to complete one oscillation produces a plot that consists of a single point. The pendulum is always back in its initial state at every repeated glimpse of its motion.

When the motion is chaotic, however, there is no characteristic period. The resulting plot, known as a Poincaré map, shows points scattered across the plane—like bullets puncturing a sheet of paper. In a sense, the system is continually shifting from one unstable periodic motion to another, giving the appearance of great irregularity.

To describe the Tilt-A-Whirl's dynamics, physicists Bret M. Huggard and Richard L. Kautz developed a mathematical equation that approximates the motion of an idealized Tilt-A-Whirl.

In essence, the movements of an individual car resemble those of a friction-impaired pendulum hanging from a support that is both rotating and being rocked back and forth while the pendulum swings. Solving the equation determines how a Tilt-A-Whirl car would behave under various conditions.

To find out what happens at intermediate Tilt-A-Whirl speeds, Kautz and Huggard plotted a set of points representing the velocity and angle of a car at the beginning of each of 100,000 tilt cycles. They found that the values never repeated themselves but were scattered in a distinctive swirling pattern confined to a portion of the plane.

For these platform velocities, even slight changes in starting point lead to radically different sequences of points. At the same time, it becomes virtually impossible to predict several steps ahead of time precisely what will happen. Such sensitive dependence on initial conditions (Butterfly effect) stands as a hallmark property of chaos.

Hence, what happens to an individual Tilt-A-Whirl car is highly dependent upon the weight of its passengers and where they sit. The resulting jumbled mixture of car rotations never repeats itself exactly, which gives the Tilt-A-Whirl its lively and unpredictable character. Indeed, no two trips are ever likely to produce exactly the same thrills and chills.

At the same time, the mathematical model used by Kautz and Huggard predicts that chaotic motion would occur at a speed close to the 6.5 revolutions per minute at which the ride is normally operated.

Interestingly, Tilt-A-Whirl fanatics know by experience that they can actually take advantage of this sensitivity. They can affect the motion of a car by throwing their weight from side to side at crucial moments, turning cycles with little or no action into thrilling whirls.

"Thus, it would seem that aficionados of the Tilt-A-Whirl have known for some time that chaotic systems can be controlled using small perturbations," Huggard and Kautz observed.

It turns out that Tilt-A-Whirl operators can also take advantage of this sensitivity. Software engineer Dave Boll described his experience one summer running a Tilt-A-Whirl, "which is easily the most entertaining carnival ride to operate."

Why? The operator can actually orchestrate the movement of individual cars. A single lever controls the ride's speed, so an operator can slightly retard or accelerate the ring of platforms at any moment. By applying just the right amount of velocity change at exactly the right time, it's possible to spin a particular car. For example, if a car is currently not spinning, is about to go uphill, and is positioned toward the inside, accelerating the platform will send the car into a very fast spin.

That's what makes the ride so attractive to operate, Boll said.

Of course, any adjustment in speed affects all cars. What happens to particular cars depends on its current spin, its position with respect to the ride's hills and valleys, whether the car is on the inside or outside of its platform, and the velocity change applied by the operator. Very fast spins occur in the same direction as the platform is rotating, and slower spins are in the opposite direction.

Superior spins also provide a bonus for the alert operator. Such whirling inevitably shakes loose coins out of the pockets of passengers—"tips" that can be gathered up after a ride is over and the riders have stepped away!

"There is an art to giving good rides," Boll remarked. "A good operator can sustain a spin on any car, in any direction. After some practice, it is possible to control two cars at once."

"What I usually did was to focus on one particular car, and if I could get another to spin with it, I would try to time it so that both cars maintained their spins as long as possible," he added.

"A walk around an amusement park suggests that several other common rides display chaotic behavior similar to that of the Tilt-A-Whirl," Huggard and Kautz noted. Typically, rides that fit this category have cars that are free to rotate or shift back and forth as they follow a fixed track.

The Tilt-A-Whirl first operated in 1926 at an amusement park in White Bear Lake, Minnesota. Most likely, the ride's inventor, Herbert W. Sellner, discovered its unpredictable dynamics not through mathematical analysis but by building one, trying it out, and making trial-and-error adjustments.

"Ride designers have been fairly adept at finding chaos without appreciating the mathematics underpinning what they’re doing," Kautz noted. The situation is changing, however. To fine-tune the thrills, manufacturers can take advantage of mathematical analyses and computer simulations to help build chaotic motion deliberately into amusement park rides.


Kautz, R.L., and B.M. Huggard. 1994. Chaos at the amusement park: Dynamics of the Tilt-A-Whirl. American Journal of Physics 62(January):59.

November 5, 2020

Minesweeper Logic

The game Minesweeper was one of the more insidious time wasters once readily available to users of Microsoft Windows-based computers. It was easy to become addicted to this seemingly simple solitaire game, in which strategy, logic, and luck added up to a potent, mind-teasing mixture.

In the basic version of the game, the player faces a gray landscape of 64 blank squares in an eight-by-eight grid. The object is to locate 10 randomly placed "land mines" hidden among the playing field's squares, working as quickly as possible without detonating any of them. Larger boards are also available.

To uncover a square, the player clicks on it with the left mouse button. If the square harbors a mine, the player loses. If instead a number (from 1 to 8) appears on the square, it specifies how many mines are present in the eight squares surrounding the numbered square.

It's clear, for example, that finding a square numbered 8 means that all eight adjacent squares contain mines. If the square remains blank, there are no mines in the surrounding squares.

To mark a square suspected of containing a mine, the player clicks on it with the right mouse button, putting a little flag on the square.

The first few moves require one or more wild, potentially fatal guesses. Given that uncertainty, is it better to start in a corner, on the side, or in the middle? The choice depends on the sorts of things that can happen on subsequent moves, which are not the same for every starting position. Experts recommend beginning somewhere in the middle.

If you survive the initial phase, you can then rely on logic to carry you the rest of the way—most of the time. There are a few situations in which you may be forced to guess a mine's location to complete a game.

It doesn't take long to develop strategies for playing and beating the game. In general, clearing a board rapidly requires a form of pattern recognition. For example, if an uncovered square is labeled 1, and there is only one covered square touching it, that covered square must be a mine.

Finding a square with the number 1 next to a suspected mine means that all other unknown squares surrounding the numbered square can be safely uncovered (assuming you marked the mine correctly). In fact, if you have found all the mines around a given numbered square, you can uncover the remaining squares by clicking the numbered square with the left and right mouse buttons simultaneously.

You can also take advantage of a little quirk. In some versions of Minesweeper, Microsoft programmers designed the game so that the first click never hits a mine. If the square originally contained a mine, the computer moves it to the top left-hand corner of the board or to the first available square to its right.

A while ago, mathematicians Patti Frazer Lock and Allan A. Struthers took the business of developing Minesweeper strategies one step further. They used the game in the classroom to introduce students to formal mathematical proofs.

In Lock's sophomore courses, students played a few games, then tried to evaluate various types of positions to determine which squares are definitely safe and which ones are definitely mined. The exercise gave them a sense of what, given the rules and the evidence, to conjecture, then prove or test.

"Putting a flag on a square is a theorem—you know there's a bomb there," Struthers explained.

Students ended up learning reasoning techniques, such as proof by contradiction or the role of counterexamples, that not only apply to Minesweeper but also were useful for proving theorems later in their mathematics courses.

Originally posted May 3, 1999


Lock, P.F., and A.A. Struthers. 1999. Using the game Minesweeper to introduce students to proofs. Abstracts of Papers Presented to the American Mathematical Society 20(No. 1):189.

Mackenzie, D. 1999. Addicted to logic. American Scientist 87(May-June):217.

November 4, 2020

Water Clock


Working model of Su Song's water-powered astronomical clock tower. Science Museum, London, England, 1975.

Photo by I. Peterson

November 3, 2020

From Microfilm to World Brain

"A new way of duplicating records, manuscripts, books, and illustrations is being developed, and scientists predict that it is destined to play a large part in the scholarly research of the future," a news item declared.

The article went on to say that this revolutionary means of storing and distributing information will have two important effects. It will make library materials more accessible by allowing librarians, in effect, to loan a book or document while, at the same time, keeping it on the shelf. It will also permit the publishing of voluminous, technical, or highly illustrated manuscripts, theses, and other material that now languish in laboratories and offices because no journal has funds or space to publish them.

When I first read these words, I couldn't help thinking about the current explosion of activity in electronic publishing, the Internet, and the World Wide Web. But the words were actually written in 1937, and the article, published in Science News Letter (now Science News), heralded the arrival of microfilm into the scholarly domain.

Watson Davis, 1942.

The article's author was Watson Davis, editor of Science News Letter and director of Science Service (now Society for Science & the Public), which was then described as "the institution for the popularization of science." Some accounts credit Davis with coining the term "microfilm."

"Soon, it is predicted, this word will be as common as 'book' or 'journal' in library, educational, and scientific circles," Davis announced.

The technology had originally been introduced in the 1920s for copying bank checks in clearinghouses by photographing them on 16-millimeter movie film. It didn't take long for this application to be extended to business records and legal documents.

Science Service, along with the Library of Congress, the Works Progress Administration, the U.S. Department of Agriculture Library, the U.S. Bureau of the Census, and other organizations, played a central role in sponsoring the development of this novel information technology for use in research and education.

In fact, Science Service became, for a number of years, the repository of articles, tables of data, illustrations, and other material that couldn't be published by conventional means. These documents came from the editors of more than 25 major scientific journals, and Science Service staff assigned each item a number and set a price for which the organization would furnish a microfilm copy. The scientific journal would publish a short version of the paper together with an announcement of the availability of the microfilm.

Microfilm represented an attractive solution to the growing urgency of providing for more and easier publication of specialized scientific information, including "the details of cosmic ray observations, the cryptic derivations of mathematical formulae, the intricacies of chemical determinations, the delving of a scholar of language into a tongue long dead, [and] the columns of statistics compiled to chart the course of population."

"In this way, important but specialized material can be made accessible without burdening libraries and individuals with material that they may never need," Davis wrote.

What were the issues back then? Technical concerns centered on the stability of cellulose acetate film for long-term preservation, increasing the level of miniaturization to pack even more information onto film, and the development of reliable copiers and viewers, including portable units for the traveling researcher.

"In the future, those engaged in scholarly research will think of a microfilm reading machine as they do of a typewriter," Davis predicted, "and studies, libraries, and laboratories will be equipped with them as commonly as with typewriters."

The big project was to apply microfilm to the problem of compiling a massive index of all scientific literature. "Heretofore, scientists have not dared to contemplate such an undertaking because of the millions of cards that would need to be classified and filed, to say nothing of the cost of printing," Davis commented. "If an 'electric eye' were perfected to select from the rolls of microfilm the references a scientist might desire, then the building and use of such a great guide to the world's knowledge might be contemplated."

Articles in subsequent issues of Science News Letter promised additional benefits, explored new developments, and raised new concerns. One was the development of national and international standards for uniformly preparing articles for technical and scientific periodicals and for classifying and indexing books and other documents.

Davis himself became president of the newly founded American Documentation Institute (the forerunner of the Association for Information Science and Technology), which addressed some of these needs and took over the repository function that Science Service had held. He also led the U.S. delegation at the August 1937 World Congress of Universal Documentation, held in Paris.

Occurring just two years before the horror and devastation of World War II, the Congress was a stellar affair, radiating an optimistic spirit the belied the events to come. Watson Davis addressed the meeting, as did writer H.G. Wells. Wells described the contemplated documentation effort as the beginning of the creation of a "world brain."

"What you are making, we realize, is a sort of cerebrum for humanity, a cerebral cortex, which, when it is completely developed, will constitute a memory and also a perception of current reality for the entire human race," Wells told the Congress. "This is exciting the imagination of some of us very greatly."

"We begin most easily with the documentation of concrete facts," he went on. "But ideas are also facts, and I do not see how this new and great encyclopedia, this race brain whose foundations you are laying, can fail to develop into anything but a mighty structure for comparison, reconciliation, and synthesis of common guiding ideas for the whole world."

These are mindful words to ponder, more than 60 years later, as I type this column on my word-processing computer. The World Wide Web is only a few keystrokes away. I can communicate nearly instantly with colleagues anywhere in the world. I have access to far more information than I can ever assimilate and use.

We have come far, but we also have a long way to go toward fulfilling Wells's vision.

Originally posted June 3, 1996


Anon. 1936. Urges photographing books as preservation measure. Science News Letter 30(Nov. 21): 327.

______. 1937. Publishing scholarly data is vital world problem. Science News Letter 32(July 3): 3.

______. 1937. American delegation goes to world documentation meet. Science News Letter 32(Aug. 14): 102.

______. 1937. Scholars form American Documentation Institute. Science News Letter 32(Aug. 14): 102-103.

______. 1937. Unpublished manuscript to be accessible to scholars. Science News Letter 32(Aug. 21): 124.

______. 1937. Documentation Congress step toward making "world brain." Science News Letter 32(Oct. 9): 228-229.

______. 1938. Translations and microfilm unlock science from abroad. Science News Letter 33(Mar. 5): 158-159.

______. 1938. New methods vastly increase usefulness of libraries. Science News Letter 33(Mar. 12): 166.

______. 1938. Translation and microfilm unlock science from abroad. Science News Letter 33(Apr. 23): 272.

______. 1938. Scientists can now publish research upon microfilm. Science News Letter 33(June 18): 402-403.

Davis, Watson. 1937. Microfilms hailed as new way to duplicate books, pictures. Science News Letter 31(Mar. 20): 179-180.

______. 1937. How documentation promotes intellectual world progress. Science News Letter 32(Oct. 9): 229-231.

Wells, H.G. 1937. Today's distress and horrors basically intellectual—Wells. Science News Letter 32(Oct. 9): 229.

November 2, 2020

Digits, Squares, and Cycles

Fascinating patterns lurk among the digits of whole numbers.

Pick a positive integer, say 57. Square each of the digits, then add the squares together: 52 + 72 = 25 + 49 = 74. Do the same thing with the digits of 74: 72 + 42 = 49 + 16 = 65. Keep repeating the procedure, using successive sums of squares.

You end up with the following sequence of numbers: 57, 74, 65, 61, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, and so on. Notice that after a few steps, the numbers begin repeating themselves in a definite cycle (or loop): 37, 58, 89, 145, 42, 20, 4, and 16.

Try another number, say 88. The following sequence arises: 88, 128, 69, 117, 51, 26, 40, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, and so on. It takes a little longer to get there and the cycle's entry point (16) is different, but the same repeating set of numbers comes up.

Amazingly, it turns out that for a given positive integer, this procedure always leads to one of just two cycles: either the cycle of numbers noted above or a string of 1s. For instance, starting with 19 produces the sequence 82, 68, 100, 1, 1, 1, and so on. Beginning with 123,457 leads to the repeating set of numbers 4, 16, 37, 58, 89, 145, 42, and 20.

Mathematician Eugene D. Nichols first encountered a mathematical proof establishing this cyclic digital behavior in a Polish book of problems by Hugo Steinhaus, published in 1958. However, Steinhaus proved the theorem only for squaring the digits. "I got curious about what would happen when you did it to the third power, fourth power, and so on," Nichols said.

For example, suppose you use 3 as the exponent and start with the positive integer 57. You get 53 + 73 = 125 + 343 = 468. Continuing the procedure generates the following numbers: 792 (43 + 63 + 83), 1080, 513, 153, 153, 153, 153, and so on. In general, this procedure always leads to some sort of cycle, and there are nine possible cycles. The longest cycles contain only three numbers: 55, 250, and 133; 160, 217, and 352; and 160, 217, and 352.

"We wanted to find out whether there is any relationship between the number of loops we get and the power to which the digits are raised," Nichols said. Collecting data up to the fifteenth power, "we couldn't discern any kind of relationship, except one. For the odd powers, there are more loops than for the even powers."

Jerry Glynn and Theodore W. Gray devoted a section of their book The Beginner's Guide to Mathematica Version 3 to the problem and suggested ways of calculating the necessary sequences and searching for patterns.

Glynn had learned about the problem from Nichols during a lull at a mathematics meeting. "He started me off on the problem with a conspiratorial tone and a gleam in his eye," Glynn recalled. "He also started very, very slowly so I followed the steps easily until I could move ahead on my own. I was trapped and no longer bored."

In the Beginner's Guide, Gray noted, "Readers should be made aware that Jerry is having some sort of religious experience. He seems awed by these numbers. It is quite remarkable that the cycle lengths are so short, given how large the numbers are."

There's still lots to explore in this niche of number theory, and there's no proper proof yet that all numbers and all exponents always cycle.

Originally posted February 2, 1998

November 1, 2020