Sea Shell Spirals

The chambered nautilus is a sea creature that belongs in the same class as the octopus. Unlike the octopus, it has a hard shell that's divided into chambers. As the nautilus matures and grows, it periodically seals off the shell behind it and creates a new, larger living chamber. The shells of adults may have as many as 30 such chambers.

This growth process yields an elegant spiral structure, visible when the shell is sliced to reveal the individual chambers. Many accounts describe this pattern as a logarithmic (or equiangular) spiral and link it to a number known as the golden ratio.

This cutaway of a nautilus shell shows its chambers and reveals an elegant spiral structure.

A logarithmic spiral follows the rule that, for a given rotation angle (such as one revolution), the distance from the pole (spiral origin) is multiplied by a fixed amount.

An example of a logarithmic spiral.

When this fixed amount is the golden ratio, (1 + √5)/2, or 1.6180339887…, you get a particular type of logarithmic spiral. Such a logarithmic spiral can be inscribed in a rectangle whose sides have lengths defined by the golden ratio.

Does the spiral of a chambered nautilus shell actually fit such a model?

In the 2005 article "The Golden Ratio—A Contrary Viewpoint," published in the College Mathematics Journal, mathematician Clement Falbo contended that the nautilus shell does not have a spiral shape based on the golden ratio.

In 1999, when Falbo measured nautilus shells in a collection at the California Academy of Sciences in San Francisco, he found that the spirals of these shells could be inscribed within rectangles with sides in the ratio of about 1.33—not 1.618…, as they would be if a spiral based on the golden ratio matched the shell shape.

Roughly speaking, the spiral of the chambered nautilus triples in radius with each full turn whereas a golden-ratio spiral grows by a factor of about 6.85 per full turn.

The measured ratios ranged from 1.24 to 1.43. "It seems highly unlikely that there exists any nautilus shell that is within 2 percent of the golden ratio, and even if one were to be found, I think it would be rare rather than typical," Falbo concluded.

In a 2002 article, "Spirals and the Golden Section," in the online Nexus Network Journal, John Sharp pointed out the same problem. Starting with the observation that shell spirals are logarithmic spirals, many people automatically assume that, because the golden ratio can be used to draw a logarithmic spiral, all shell spirals are related to the golden ratio, when, in fact, they are not.

Sharp's own measurements of nautilus shells also confirmed that the golden ratio rectangular spiral and the nautilus spiral "simply do not match."

Nonetheless, many accounts still insist that a cross section of nautilus shell shows a growth pattern of chambers governed by the golden ratio.

"One of the amazing things about such misconceptions is that it is so widespread, even by mathematicians who should know better," Sharp observed. "It is a prime example of why geometry needs to be taught more widely and not only geometry, but the visual appreciation of shape and proportion."

And it's always useful to check things out in the real world.

Originally posted April 4, 2005

Folding Maps

Anyone trying to refold an opened road map is wrestling with the same sort of challenges confronted by origami designers and sheet metal benders.

The problem of returning a creased sheet to its neatly folded state gets tougher when you're not sure if the sheet can be folded into a flat packet and when you're not permitted to change the crease directions. Such conundrums can arise, for example, when designers specify how to bend sheet metal to produce, say, car doors, airplane parts, or heating ducts.

Folding a 2 x 4 map via a sequence of three simple folds.

In 1991, Erik D. Demaine and his coworkers developed an efficient method for recognizing when a creased sheet indeed is foldable into a flat package. The researchers reported their results in the paper "When Can You Fold a Map?"

"This represents an initial step toward developing an understanding of the three-dimensional, sheet-metal-folding problem," applied mathematician Joseph Mitchell commented. "We need better mathematical tools for dealing with problems in going from a design to a manufactured part."

Demaine and his collaborators started with the one-dimensional case of the folding problem: When is it possible to refold a line segment, which had been creased upward to form mountain creases and downward to form valley creases, into a compact configuration resembling the cross section of a neatly folded map?

The researchers focused on recognizing flat-folding crease patterns that arise as the result of so-called simple foldings. "In this model, a flat folding is made by a sequence of simple folds, each of which folds one or more layers of paper along a single line segment," they remarked.

Sample flat foldings in one and two dimensions. M = mountain folds, V = valley folds.

The researchers discovered that certain mixes of two configurations—a zigzag fold, called a crimp, and a doubled-back fold, or hem—permit a creased segment to be folded into a flat profile. This finding enabled the researchers to develop criteria and an efficient algorithm for recognizing a one-dimensional crease pattern that can be folded flat.

"The two-dimensional case is more complicated," Demaine said. However, if a rectangular sheet is creased only along vertical and horizontal lines to form a grid, it's possible to analyze the resulting mountain-valley crease pattern in terms of the criteria developed for the one-dimensional case. As a result, there's an efficient method for deciding whether a creased sheet can fold flat. Adding complications, such as diagonal creases, makes solving the problem considerably more time-consuming, Demaine noted.

Demaine's interest in foldability arose out of his hobby, origami. Mathematicians and others have been studying ways to systematize origami design by developing rules that would enable a computer to calculate what sequence of creases in a square a paper will produce a desired figure. Important to this task is a determination from a crease pattern whether the resulting three-dimensional figure can collapse neatly into a flat form, as required in traditional origami.

Beyond the mathematics of origami, "our study is motivated by applications in sheet metal and paper product manufacturing, where one is interested in determining if a given structure can be manufactured using a specified creasing machine, which is typically restricted to performing simple folds," the researchers noted.

"While origamists can develop particular skill in performing non-simple folds to make beautiful artwork, practical problems of manufacturing sheet goods require simple and constrained folding operations," they added. "Our goal is to develop a first suite of results that may be helpful towards a fuller understanding of the several manufacturing problems that arise, for example, in making three-dimensional cardboard and sheet-metal structures."

Demaine said his work has also yielded insights into refolding road maps. One trick is to start with the fold that serves as a border between sequences of mountain and valley creases that mirror each other on either side of the border.

Whether anyone would have the patience to do such a careful analysis while in the throes of a refolding adventure is another matter, however. The whole business calls to mind the old saw: The easiest way to refold a road map is differently.

Originally posted January 15, 2001

Melrose Abbey

St. Mary's Abbey, Melrose, Scotland, 1975.

See also "Gargoyle Pig."

Photos by I. Peterson

Prime Listening

The human ear has remarkable capabilities—so remarkable that we generally accept as routine such formidable tasks as recognizing a voice, picking out a single word in a cacophony of cocktail-party chatter, or hearing a flute's sweet tone in the midst of an orchestral romp.

Our sense of hearing can integrate disparate sounds into a harmonious whole or detect subtle nuances buried in noise. It has amazing powers of pattern recognition. Indeed, by using sound to represent masses of data or jumbles of numbers, we can take advantage of that capability to identify regularities or make subtle distinctions.

Mathematician Chris K. Caldwell has developed a scheme for listening to sequences of primes—to hear both simple patterns and perplexing irregularities found among those numbers.

"Multimedia allows the use of sight and sound, so why not use sound?" Caldwell asks. "After all, don't we use our ears to detect patterns as we listen to the car motor for a problem or shake a box to determine its contents?"

The MIDI (Musical Instrument Digital Interface) specification often used in computer programs to represent musical notes assigns a number to each note on a keyboard. According to this recipe, middle C is 60, C-sharp is 61, D is 62, and so on, for a total of 128 notes. One could use that correspondence directly to "play" primes. For example, 67 would then be G.

There are infinitely many primes, however, so Caldwell's strategy is to divide each prime by a fixed number, then play just the remainder—a novel application of that standard number-theory tool known as modular arithmetic.

For example, if the divisor, or modulus, is 7, then for the primes {2, 3, 5, 7, 11, 13, 17, 19, 23. . .}, one would play {2, 3, 5, 0, 4, 6, 3, 5, 2,…}. Because those particular notes on the MIDI scale would be too low in frequency to be audible, Caldwell adds a constant, such as 55. Hence, the first prime, 2, is played as the note A. There are seven possible notes.

Primes modulo seven

In listening to the primes modulo 7, you find that all seven notes are played, but the lowest note occurs just once. That lowest note is the prime number 7. Any other number that leaves a remainder of 0 when divided by 7 must be a multiple of 7, so it can't be a prime. A theorem of Peter Gustav Lejeune Dirichlet (1805-1859) on primes in arithmetic progression guarantees that all the other notes are heard infinitely often when one plays all the primes.

Trying various prime and composite divisors reveals other patterns, and one can check which notes are played infinitely often, which ones are played just once, and which ones are never played for each modulus.

It's possible to use the duration of notes to represent the gaps between prime numbers. There's no gap between the first two primes, 2 and 3. There's a gap of one between 3 and 5, a gap of one between 5 and 7, and a gap of three between 7 and 11. According to the prime number theorem, the average gap length up to a certain prime equals the value of the natural logarithm of that prime.

Caldwell makes the note representing a particular prime last a time proportional to the size of the gap to the next prime in the sequence, divided by the natural logarithm of that prime.

Caldwell's "Prime Number Listening Guide" offers his weirdly tuneful "primes modulo 41 with prime gap tempo" and other primal sounds and even a way to make your own prime music.

There may be more to come. Caldwell has started to develop a program to play the factorizations of sequences of numbers. "The low primes are used to determine what percussion instrument to play, the next larger primes are mapped to one instrument, and the larger ones (modulo some base) are mapped to a second," he says. "If the integer is divisible by a power (greater than 1) of the prime, then that prime's 'note' is played louder."

You may one day hear more of the concert theorems that mathematicians have so painstakingly gleaned from their studies of integers and the primes.

Originally posted June 22, 1998

Warwick Castle

Warwick Castle, Warwick, England, 1975.

Photos by I.Peterson

Euler Bricks and Perfect Polyhedra

There's something about integers that makes them perfectly irresistible to many mathematicians, both amateur and professional.

Number theorists have the advantage that they can indulge their pleasure without feeling overly guilty, whether it's in the connection between Fermat's last theorem and elliptic curves or the link between random matrices and the distribution of prime numbers (see "The Mark of Zeta").

The wide use of computers has also brought attention to the realm of the discrete. Indeed, that trend was noted more than 50 years ago. In his 1963 book Combinatorial Mathematics, H.J. Ryser remarked, "Our new technology with its vital concern with the discrete has given the recreational mathematics of the past a new seriousness of purpose."

The recreational aspect is alive and well, as seen, for example, in the continuing fascination with magic squares, magic cubes, and magic tesseracts.

In a 1999 article titled "Integer Antiprisms and Integer Octahedra," published in Mathematics Magazine, Blake E. Peterson and James H. Jordan drew attention to perfect boxes and polyhedra.

Their starting point was the problem of finding a rectangular box with integer dimensions and all diagonals of integer length. Such a figure is known as a perfect box. Whether it exists is an unsolved problem.

Leonhard Euler (1707-1783) described the smallest solution for the special case when the sides and face diagonals are all integers, but not the space diagonal passing through the box's center from one corner to its opposite. (Though Euler is often credited with its discovery, the German mathematician Paul Halcke mentioned this solution first in 1719.)

An Euler brick has integer dimensions.

Euler's "almost" perfect brick has the following dimensions: a = 240, b = 117, and c = 44. The face diagonals are 244, 125, and 267. The space diagonal is 5 times the square root of 2929.

Peterson and Jordan focused on other three-dimensional figures with integer edges and diagonals, particularly pyramids and prisms.

You can construct a pyramid by drawing a polygon (to serve as the base), then joining each vertex of the polygon to a point not in the plane of the polygon. A triangular pyramid, or tetrahedron, has a triangular base and four faces, counting the bottom.

An integer polyhedron is one in which the distance between each pair of vertices is an integer. Because the faces of an integer polyhedron must themselves be integer polygons, it's natural to use integer polygons as the building blocks of integer polyhedra, Peterson and Jordan remarked. Octahedral pyramids are a good starting point.

An octahedron has eight faces. In its most familiar form as one of the Platonic solids, each face is an equilateral triangle.

A regular octahedron, in which each face is an equilateral triangle.

An octahedral pyramid has a seven-sided heptagon as its base.

An octahedral pyramid with a heptagonal base.

To get an integer heptagon, adjacent vertices of the heptagon must lie on a circle and be separated by the following distances: 10, 16, 16, 10, 16, 16, and 16. In this case, all of the points lying along a line through this circle's center and perpendicular to the plane of the heptagon are equidistant from the polygon's vertices. You can then choose the lateral edges of the pyramid to be 17.

You can readily extend the same approach to other polyhedra, such as prisms and antiprisms. An antiprism consists of two identical polygons in parallel planes joined in such a way that all the other faces are isosceles triangles.

Example of an antiprism.

Peterson and Jordan went on to investigate interesting links between integer octahedra and integer antiprisms.

Originally posted October 25, 1999

Glastonbury Tor

Glastonbury Tor with a view of St, Michael's Tower. Glastonbury, Somerset, England, 1975.

Photos by I. Peterson

Coloring Penrose Tiles

In 1976, mathematicians Kenneth Appel and Wolfgang Haken proved the four-color theorem: Four colors are sufficient to color any map so that regions sharing a common border receive different colors.

There are, however, special cases in which fewer than four colors suffice. For example, it takes only two colors to fill in a checkerboard pattern. In fact, any planar map in which intersecting lines run from edge to edge, requires only two colors.

Example of a map that requires only two colors.

Placing ceramic tiles so that adjacent tiles have different colors suggests similar issues. It is certainly possible, for example, to use just two colors when setting square tiles in a checkerboard pattern. Three colors are needed for a honeycomb pattern of hexagonal tiles.

One particularly intriguing case involves so-called Penrose tilings. In 1974, mathematical physicist Roger Penrose discovered a set of two tiles that, when used together, cover a surface without forming a regularly repeating pattern. One tile resembles an arrowhead and is described as a dart, and the other tile looks like a diamond with one foreshortened end and is known as a kite. The two pieces fit together to form a rhombus.

A portion of a kite-and-dart Penrose tiling of the plane.

It turns out there are many different pairs of quadrilateral shapes that form a aperiodic tiling pattern, though all are related in some way to the original kite-and-dart pair. One particularly striking set consists of a pair of diamond-shaped figures—one fat and one skinny.

A portion of a diamond-based (or rhomb-based) Penrose tiling of the plane.

Reports of attempts to color such Penrose diamond tilings led John H. Conway to conjecture that three colors suffice. In 1999, mathematicians Tom Sibley and Stan Wagon proved that to be the case. They described their results in the article "Rhombic Penrose Tilings Can be 3-Colored," published in the American Mathematical Monthly.

Sibley and Wagon generalized the result to any map (or tiling) made up of parallelograms, as long as two adjacent countries (or tiles) meet in a single point or along a complete edge of the constituent pieces. The mathematicians described such a map or pattern as "tidy."

Example of a three-colored Penrose diamond (or rhomb) tiling. 

The proof involved showing that, given a tidy finite map, a country has at most two neighbors. The results, however, do not hold for all possible quadrilateral shapes and configurations.

In 2001, Robert Babilon proved that tilings by kites and darts are three-colorable. Mark McClure than found an algorithm to three-color tilings by kites and darts and by rhombs.

Originally posted May 17, 1999

Tintagel Castle

Location of the ruins of Tintagel Castle, Cornwall, England, 1975.

Photos by I. Peterson

The EKG Sequence

Sequences of numbers have long fascinated both amateur and professional mathematicians. Many people are familiar with the Fibonacci sequence, in which each new term is the sum of the previous two terms: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, and so on. More than 330,000 other sequences of interest to mathematicians are cataloged in Neil J.A. Sloane's On-Line Encyclopedia of Integer Sequences.

Here's an example discovered in 2001 that has prompted some serious mathematical investigation.

The first term is 1 and the second is 2. Each succeeding term is the smallest number not already used that shares a factor with the preceding term. So, the third term must be 4 (2 and 4 have the common factor 2), the fourth term 6 (4 and 6 have the common factor 2), the fifth term 3 (6 and 3 have the common factor 3), and the sixth term 9 (6 is already used, so the next available number that shares a factor with 3 is 9).

SEQUENCE: 1, 2, 4, 6, 3, 9, 12, 8, 10, 5, 15, 18, 14, 7, 21, 24, 16, 20, 22, 11, 33, 27, 30, 25, 35, 28, 26, 13, 39, 36, 32, 34, 17, 51, 42, 38, 19, 57, 45, 40, 44, 46, 23, 69, 48, 50, 52, 54, 56, 49, 63, 60, 55, 65, 70, 58, 29, 87, 66, 62, 31, 93, 72, 64, 68, 74, 37, 111, 75, 78, 76, 80, 82, and so on.

The sequence was discovered by Jonathan Ayres, and it appears as entry A064413 in Sloane's encyclopedia.

When plotted, the sequence of numbers looks somewhat like an electrocardiogram, or EKG, so it was dubbed the EKG sequence. Interestingly, although relatively short segments of the sequence appear to behave erratically, a plot of the first 10,000 or so terms shows considerable regularity.

A plot of the first 72 terms of the EKG sequence.

Intrigued by this combination of disorder and regularity, Sloane, Jeffrey C. Lagarias, and Eric M. Rains decided to see what they could find out about the sequence's mathematical properties.

"The EKG sequence has a simple recursive definition, yet seems surprisingly difficult to analyze," the researchers noted in a 2002 paper describing their findings. "Its definition combines both additive and multiplicative aspects of the integers, and the greedy property of its definition produces a complicated dependence on the earlier terms of the sequence."

Despite such difficulties, the mathematicians succeeded in demonstrating that the sequence contains all positive integers—something that is not immediately obvious. Moreover, every number appears exactly once, so the EKG sequence is a permutation of the positive integers. At the same time, the primes appear in ascending order.

Lagarias and his colleagues worked out an efficient way to generate the EKG sequence and computed 10 million terms. Trends apparent in these data suggested several conjectures. For instance, whenever a prime p occurs in the sequence, it is preceded by 2p and followed by 3p.

"Although it is theoretically possible that some other multiple of p occurs before p (for example, we might have seen…, 3p, p, 2p,…), this does not appear in the first 107 terms," the researchers observed.

In general, however, "there remains a large gap between what is conjectured and what is proved," the mathematicians cautioned. And there remains much territory for numerical exploration.

Originally posted April 8, 2002

Jaipur Deco

Amer Fort, Jaipur, India, 2010.

Photos by I. Peterson

Buffon's Needling Ants

The classic probability experiment known as Buffon's needle produces a statistical estimate of the value of pi (π), the ratio of a circle's circumference to its diameter.

The experiment consists of randomly dropping a needle over and over again onto a wooden floor made up of parallel planks. If the needle's length is no greater than the width of the boards, the probability of the needle meeting or crossing a seam between boards is twice the needle's length, l, divided by the plank width, d, times pi: 2l/dπ.

The idea of estimating pi by randomly casting a needle onto an infinite plane ruled with parallel lines was first proposed by the naturalist and mathematician Georges-Louis Leclerc, Comte de Buffon (1707-1788). He himself apparently tried to measure pi by throwing sticklike loaves of French bread over his shoulder onto a tiled floor and counting the number of times the loaves fell across the lines between the tiles.

In 1901, the Italian mathematician Mario Lazzarini claimed to have tossed a needle 3,408 times and obtained a value of pi equal to 355/113, or 3.1415929, which differs from the exact value by less than 0.0000003. How he managed to ensure truly random needle casting and got his remarkably accurate pi estimate isn't clear, though mathematicians have argued that cheating must have been involved.

Subsequent experiments by other investigators typically produced less accurate values of pi. In recent years, computer simulations have taken over—with results modulated by quirks of the random number generators involved in the computations.

In the year 2000, researchers reported that an ant species appears to use a Buffon's needle algorithm to measure the size of potential nest sites. Eamonn B. Mallon and Nigel R. Franks described their findings in the article "Ants estimate area using Buffon's needle," published in the Proceedings of the Royal Society, London (B).

Ants of the species Leptothorax albipennis inhabit small, flat crevices in rocks. A colony typically consists of a single queen, her brood, and 50 to 100 workers. When a nest happens to get destroyed, the colony sends out scouts to assess potential new nest sites.

Given a choice, a colony's preference is for a nest of a certain "standard" size (related to the number of ants in the colony), which suggests that these ants can measure area. How do they do it?

Mallon and Franks collected ants from areas near the Dorset coast of England and cultured them in the laboratory. They then transferred colonies to large, square Petri dishes and offered the colonies choices of various cavity habitats, made from pairs of microscope slides with cardboard walls spanning the narrow gap between glass floor and glass ceiling.

"We used such microscope slide nests with nest cavities of different sizes, shapes, and configurations in order to examine preferences," the researchers noted.

Experiments involving individually marked workers demonstrated that a scout generally spends about 2 minutes scurrying within any particular candidate cavity. Moreover, scouts typically end up making two visits to an acceptable nest site before recruiting followers.

When a scout initially explores a potential nest site, it lays down a pheromone-laced track. On its second visit, it follows a different track, repeatedly crossing its original path.

Mallon and Franks argued that a scout can obtain an estimate of the potential nest's area by detecting the number of intersections between the first and second set of tracks. In effect, an ant scout applies a variant of Buffon's needle theorem: The estimated area, A, of a flat surface is inversely proportional to the number of intersections, N, between two sets of lines, of total lengths S and L, randomly scattered across the surface, or A = π.

"There is evidence that individual scouts recognize and respond to intersections between their second visit path and their first visit path," the researchers said. "Scouts briefly but significantly slowed down during their second visit when they intersected their first visit path."

Additional observations bolster the plausibility of the claim that these ants assess nest size using a Buffon's needle algorithm. Moreover, the method is relatively insensitive to the shape of the area to be assessed and to the exact pattern of the tracks (as long as the tracks are not concentrated within just one region). In addition, it will work in complete darkness.

"Our findings, that individual ants can make accurate assessments of nest areas based on a rule of thumb, show in a unique way how animals use robust algorithms to make well-informed quantitative decisions," Mallon and Franks concluded. The results demonstrated how information gathering by individual workers can contribute to crucial collective decisions.

Originally posted May 15, 2000

Golconda Arches

Golconda Fort, Hyderabad, India, 2010.

Photos by I. Peterson

The Wacky Math of Abbott and Costello

The comedy team of Bud Abbott and Lou Costello, probably most famous for their "Who's on first" baseball routine, also made arithmetic shenanigans the basis of several of their comic dialogues in their movies and television programs.

Abbott and Costello became a comedy team in 1936, becoming popular on the burlesque stage, then achieving wider acclaim by performing on radio. Released in 1941, their second movie, Buck Privates, was a box office hit. In this World War II comedy, Abbott and Costello play tie salesmen who accidentally enlist in the army to avoid getting arrested.

One mathematical bit in Buck Privates involves a word problem reminiscent of those spoofed by John Scieszka and Lane Smith in the delightful children's picture book Math Curse.

Abbott: You're 40 years old and you're in love with a little girl that's 10 years old. You're four times as old as that girl. You couldn't marry that girl, could you?

…

Abbott: So you wait 5 years. Now the little girl is 15, and you're 45. You're only three times as old as that girl. So you wait 15 years more. Now the little girl is 30, and you're 60. You're only twice as old as that little girl.

Costello: She's catching up.

Abbott: Here's the question. How long do you have to wait before you and that little girl are the same age?

Costello: Well…What kind of question is that? That's ridiculous. If I keep waiting for that girl, she'll pass me up. She'll wind up older than I am. Then she'll have to wait for me!

Fast-talking, inveterate con man Abbott had a sneaky way with numbers (aided by mangled logic), especially when they had dollar signs next to them. Here's another encounter from Buck Privates, one echoed in several later movies and shows.

Abbott: Do me a favor. Loan me $50.

Costello: I can't lend you $50. All I've got is $40.

Abbott: That's okay. Give me the $40, and you'll owe me $10.
Costello: How come I owe you $10?
Abbott: What did I ask you for?
Costello: $50.
Abbott: What did you give me?
Costello; $40.
Abbott: So you owe me $10.
Costello: That's right. But you owe me $40. Give me my $40 back.
Abbott: There's your $40. Now give me the $10 you owe me. That's the last time I'll ever ask you for the loan of $50.
Costello: How can I loan you $50 now? All I have is $30.
Abbott: Give me the $30, and you’ll owe me $20.
Costello: This is getting worse all the time. First I owe you $10, and now I owe you $20!
Abbott: So you owe me $20. Twenty and 30 is 50.
Costello; Nope! Twenty-five and 25 is 50.
Abbott: Here's your $30. Give me back my $20.
Costello: All I've got now is $10!

Abbott then entices Costello into a silly, double-or-nothing number game.

Abbott: Take a number, any number at all from 1 to 10, and don't tell me.
Costello: I got it.
Abbott: Is the number odd or even?
Costello: Even.
Abbott: Is the number between 1 and 3?
Costello: No.
Abbott: Between 3 and 5?
Costello: No. I think I got him.
Abbott: Between 5 and 7?
Costello: Yes.
Abbott: Number six?
Costello: Right… . How did he do that?

Toward the end of the movie, during a boxing match, Costello is knocked to the canvas, and the biased referee gives a quick count: 2, 4, 6, 8, 10.

Costello: What's this? 2, 4, 6, 8, 10? What happened to 1, 3, 5, 7, and 9?

Ref: I don't like them numbers. They're odd.

Such nuggets of mathematical fun turn up not only in Buck Privates but also in other Abbott and Costello movies and later in sketches from their television programs. The 1941 film In the Navy, for example, features a hilarious episode in creative counting. Costello comes up with three different ways to prove that 7 times 13 equals 28.

If you look and listen closely, you'll find an amazing amount of number play in those old Abbott and Costello movies and TV programs.

See also "The Cow in the Classroom."

Qutb Shahi Tombs

Qutb Shahi Tombs, Hyderabad, India, 2010.

Photos by I. Peterson

Acoustic Residues

There's a surprising mathematical ingredient in the sound of many performing artists and recording stars. It manifests itself in the form of clusters of panels hanging on the walls of recording studios, concert halls, nightclubs, and other venues. Sculpted from wooden strips separated by thin aluminum dividers, each panel consists of an array of wells of equal width but different depths.

Called reflection phase gratings, these panels scatter sound waves. The result is a richer, livelier sound with an enhanced sense of space. Listeners claim that the panels seem to make the walls disappear. A small room takes on the air of a great hall.

The secret lies in the varying depths of a panel's wells. With depths based on specific sequences of numbers rooted in number theory, the wells scatter a broad range of frequencies evenly over a wide angle.

The scientist who pioneered the ideas responsible for this development was Manfred R. Schroeder (1926-2009). In the 1970s, Schroeder and two collaborators undertook a major acoustical study of more than 20 famous European concert halls. One of their findings was that listeners like the sound of long, narrow halls better than that of wide halls. Perhaps the reason for this, Schroeder reasoned, is related to another finding that listeners prefer to hear somewhat different signals at each of their two ears.

In a wide hall, the first strong sound to arrive at a listener's ears, after sound traveling directly from the stage, is the reflection from the ceiling. Ceiling reflections produce very similar signals at each ear. In narrow halls, however, the first reflections reach the listener from the left and right walls, and the two reflections are generally different.

This may be one reason why many modern halls are acoustically unpopular. Economic constraints dictate construction of wide halls to accommodate more seats, and modern air conditioning systems allow lower ceilings. To improve the acoustics of such halls, sound must be redirected from the ceiling toward the walls.

A flat surface by itself can't do the job. It reflects sound in only one direction, according to the same rules that govern light reflecting from a mirror. The ceiling must have carefully orchestrated corrugations that scatter sound so that roughly the same amount of energy goes in every direction.

Schroeder discovered that number theory can be used to determine the ideal depth of the notches, resulting in an acoustic grating that's analogous to diffraction gratings used to scatter light.

One effective acoustic grating is based on quadratic-residue sequences. Such a sequence consists of the remainders, or residues, after squaring consecutive whole numbers, then dividing them by a given prime number.

Suppose, for example, the given prime number is 17. The first sequence member is the remainder, or residue, after the first number, 1, is squared and divided by 17. The answer is 1. Squaring all the numbers from 1 to 16, then dividing by 17 and determining the residue, produces the sequence: 1, 4, 9, 16, 8, 2, 15, 13, 13, 15, 2, 8, 16, 9, 4, 1. For larger numbers, the pattern simply repeats itself.

Finding the depth of a given grating well involves multiplying the appropriate number in the sequence by the longest wavelength for which the grating is designed to scatter sound efficiently and then dividing by a factor that depends on the well's numerical position. Mathematical analysis shows that for such an arrangement, the spectrum of energies scattered into different directions is essentially flat, meaning that roughly equal amounts of energy go in all directions.

Why does number theory work so well? The answer is in the way waves cancel or reinforce each other, depending on whether the crest of one wave meets the trough or crest of another wave. For perfectly periodic waves, destructive interference occurs whenever one wave lags behind the other by half a wavelength, one-and-a-half wavelengths, two-and-a-half wavelengths, and so on. In each case, it's the extra half wavelength that decides when waves cancel each other out.

So, in wave interference, it's not the total path difference between two waves that determines the resulting pattern but the residue after dividing by the wavelength. Hence, modular-arithmetic techniques and quadratic residues are relevant to acoustics.

Architectural acoustics designers have only three ingredients they can use to conjure up every imaginable type of acoustic environment; namely, absorption, reflection, and diffusion. Sound-absorbing surfaces made of foam or fiberglass and sound-reflecting surfaces, such as flat or curved panels, are widely used. Until reflection phase gratings came along, there really were no surfaces designed to spread sound around in both space and time. For designers, it was like trying to type a paragraph without using, say, the letter "d."

Inspired by Schroeder's work, Peter D'Antonio started RPG Diffusor Systems (now RPG Acoustical Systems) in 1983 to bring reflection phase gratings based on quadratic residues and other mathematical constructs to the acoustic marketplace. In recent years, the company has developed novel diffusor designs based on such mathematical concepts as primitive roots and fractals.

Now that improved digital recordings, electronic instruments, and home theater systems are readily available, demand has increased for superior acoustic surroundings for making and listening to recordings. The use of reflection phase gratings to diffuse sound helps create a listening environment in the home and elsewhere that allows a listener to experience an old-fashioned concert-hall ambiance.

Number theory makes an important contribution to the sound of music.

Originally posted July 9, 2001

Wall Eyes

Golconda Fort, Hyderabad, India, 2010.

Photos by I. Peterson

Tolstoy's Calculus

"Absolute continuity of motion is not comprehensible to the human mind. Laws of motion of any kind become comprehensible to man only when he examines arbitrarily selected elements of that motion; but at the same time, a large proportion of human error comes from the arbitrary division of continuous motion into discontinuous elements."

This striking (and perhaps cryptic) passage contrasting the continuous and the discrete started off the 11th book of the epic novel War and Peace by Leo Tolstoy (1828-1920). Tolstoy's saga concerned the tribulations of a group of Russian aristocrats during the turbulent period of Napoleon's campaign in Russia.

To reinforce his point, Tolstoy then referred to the ancient story of Achilles and the tortoise. Achilles could travel 10 times faster than a tortoise that he was following. By the time Achilles covered the distance that separated him from the tortoise, the tortoise would have covered one-tenth of the distance ahead of it. When Achilles had covered that tenth, the tortoise would have covered an additional one-hundredth, and so on. Hence, it would appear that you could come to the absurd conclusion that Achilles would never overtake the tortoise.

"By adopting smaller and smaller elements we only approach a solution of the problem, but never reach it," Tolstoy declared. "Only when we have admitted the conception of the infinitely small, and the resulting geometrical progression with a common ratio of one tenth, and have found the sum of this progression to infinity, do we reach the solution of the problem."

Building on this analogy, Tolstoy turned to the calculus as a model of how to apprehend history. "A modern branch of mathematics having achieved the art of dealing with the infinitely small can now yield solutions in other more complex problems of motion which used to appear insoluble," he wrote.

One such problem is the perceived progress of history. "Only by taking infinitesimally small units for observation (the differential of history, that is, the individual tendencies of men) and attaining to the art of integrating them (that is, finding the sum of these infinitesimals) can we hope to arrive at the laws of history," Tolstoy argued.

It's likely that Tolstoy was familiar with the work of Pierre-Simon Laplace (1749-1827), computer scientist Paul M.B. Vitányi noted in his article "Tolstoy's Mathematics in War and Peace," published in The Mathematical Intelligencer.

The success of Newton's laws of motion made it possible for Laplace to envision a completely transparent, deterministic world in which the entire past and future lay within reach. In principle, everything was predictable, and the finest detail accessible to calculation. You could construct yesterday's or tomorrow's world from what you knew today.

At the same time, Laplace imagined the world as a mechanistic ensemble of moving and colliding particles that by their combined microscopic actions produce macroscopic effects.

In assessing the role of probability in understanding such a world, Laplace wrote, "I am particularly concerned to determine the probability of causes and results, as exhibited in events that occur in large numbers, and to investigate the laws according to which that probability approaches a limit in proportion to the repetition of events."

"The investigation is one that deserves the attention of philosophers in showing how in the final analysis there is a regularity underlying the very things that seem to us to pertain entirely to chance, and in unveiling the hidden but constant causes on which that regularity depends," Laplace added.

Echoing Laplace, Tolstoy applied an analogous notion to understanding history. "To study the laws of history we must completely change the subject of our observation, must leave aside kings, ministers, and generals, and study the common, infinitesimally small elements by which the masses are moved," Tolstoy wrote in War and Peace.

From Vitányi's perspective, however, Tolstoy was not really seeking in the calculus and Laplacian thought a usable model of history as much as he was trying to demonstrate the futility of the quest for explanations of wars' causes and outcomes.

"[Tolstoy's] conclusion is not that the future is in the laps of the gods," Vitányi contended in his paper,"but rather that it is decided deterministically and precisely, but is practically (and possibly in principle) unknowable by humans."

"…the imperfect human mind cannot access all information about the present at once," he emphasized, "and so is reduced to ignorance or at best probabilistic reasoning."

Originally posted October 29, 2001

Four Minarets

Charminar, Hyderabad, India, 2010.

Photos by I. Peterson

Algebra, Philosophy, and Fun

I don't often encounter the words "philosophy" and "fun" right next to the term "algebra." Nowadays, these words don't seem to fit together comfortably. However, the three terms do appear in the title of an engaging little book called Philosophy & Fun of Algebra, written by Mary Everest Boole (1832-1916) and published in 1909.

Boole's gentle, conversational introduction to algebra was meant for children. It now also serves as a window on math education—as it was perceived in some circles more than a century ago.

"Arithmetic means dealing logically with facts which we know (about questions of number)," Boole began. And she immediately launched into a discussion of what "logically" means and the scope and primacy of the laws of logic.

No Parliament can pass a law to make an answer come out right, she insisted. "…governments have grown wiser by experience and found out that, as far as arithmetic goes, there is no use in ordering people to go contrary to the laws of the Logos [hidden wisdom], because the Logos has the whip hand, and knows its own business, and is master of the situation."

Why bother with algebra if you already know arithmetic? Boole answered: "When people had only arithmetic and not algebra, they found out a surprising amount of things about number and quantities. But there remained problems which they very much needed to solve and could not. They had to guess the answer; and, of course, they usually guessed wrong. And I am inclined to think they disagreed… . Probably they quarreled, and got nervous and overstrained and miserable, and said things which hurt the feelings of their friends… ."

That impasse led to the birth of x. "Instead of guessing whether we are to call it nine, or seven, or a hundred and twenty, or a thousand and fifty, let us agree to call it x, and let us always remember that x stands for the Unknown," she declared.

In 17 short chapters, Boole presented the basic concepts of algebra, with a variety of examples and snippets of mathematical theory, all seasoned with anecdotal pinches of history and philosophy. She frequently referred to biblical events and characters. One chapter on the question of choosing the proper working hypothesis focused on Macbeth's tragic mistake in the play by William Shakespeare of failing to distinguish between the real and the imagined.

In the book's final chapters, Boole confronted the perplexities of the square root of −1 and the unlimited vistas of the infinite. As it did elsewhere in the book, her tone verged on the mystical.

"A story is told of a man at Cambridge who was expected to be Senior Wrangler [top mathematics student]; but he got thinking about the square root of minus one as if it were a reality, till he lost his sleep and dreamed that he was the square root of minus one and could not extract himself; and he became so ill that he could not go to his examination at all," Boole recounted.

"Angels, and the square roots of negative quantities, and the other things that have no existence in three dimensions, do not come to us to gossip about themselves; or the place they came from; or where they are going to; or where we are going to in the far future," she continued. "They are messengers from the As-Yet-Unknown; and come to tell us where we are to go next; and the shortest road to get there; and where we ought not to go just at present."

Boole's reference to the "As-Yet-Unknown" is sometimes quoted in present-day discussions of pantheism and other systems of belief.

"When square root of minus one comes to you, behave reasonably about him," Boole advised. "Treat him logically, exactly as if he were six or nine; only always remember to keep well in front of you the fact of your own ignorance. You may never find out any more about him than you know now; but if you treat him sensibly he will tell you plenty of truths about your x's and y's, and other unknown things."

Boole herself is a fascinating character. She was born in England but raised in France, where her father, a minister, sought a cure for a serious, lengthy illness that afflicted him. Her uncle, George Everest, had made the family name famous, leading a surveying party up to the mountain that now bears his name.

At the age of 11, Mary returned with her family to England. She was taken out of school to assist her father, teaching Sunday School and helping with sermons. She did not abandon her studies, however, and used books in her father's library to teach herself calculus. Through another uncle, John Ryall, she met the famous mathematician George Boole (1815-1864). He ended up tutoring her in mathematics, and she helped edit Boole's epochal 1854 book The Laws of Thought. Even though Mary was 17 years younger, the two were married not long after her father died.

Tragically, George Boole died of pneumonia in 1864, leaving Mary to take care of five daughters, the youngest only 6 months old. She accepted a job as a librarian at Queen's College, London. Her real love was teaching, and, when she got the chance, she proved very good at it. Her fascination with the spirit world, however, led to her eventual resignation from the college, when controversy dogged the publication of her book on the message of "psychic science" for mothers and nurses.

Many years later, Mary's 1904 book, The Preparation of the Child for Science, had a lasting effect on the move toward progressive schools in England and the United States during the early part of the 20th century. She followed up that initial success with Philosophy & Fun of Algebra; Logic Taught by Love: Rhythm in Nature and in Education; and The Mathematical Psychology of Gratry and Boole. She often wrote about and encouraged the use of hands-an activities and items that are now often called "manipulatives" as a crucial part of math education.

Mary Everest Boole's eldest daughter, Mary Ellen, married Charles Howard Hinton (1853-1907), who devised methods for visualizing the fourth dimension, invented the word tesseract to describe a hypercube, and wrote a story that is said to have inspired Edwin A. Abbott's Flatland. Another daughter, Alicia, developed an amazing feel for four-dimensional geometry as a child, became a mathematician, and introduced into English the word "polytope" to describe a four- or higher-dimensional convex solid. She later worked with the famed geometer H.S.M. Coxeter (1907-2003).

So, I started with an intriguing book title, and I ended up immersed in four-dimensional geometry!

Originally posted January 17, 2000