June 30, 2010

Pythagoras at the Plate

Baseball is played on a field of geometric regularity. The baseball "diamond," for instance, is properly a square, 30 yards on each side.


Official league rules also specify the size and shape of home plate: Home base shall be marked by a five-sided slab of whitened rubber. It shall be a 12-inch square with two of the corners filled in so that one edge is 17 inches long, two are 8 1/2 inches and two are 12 inches.


But something isn't quite right. The diagram implies the existence of a right triangle with sides 12, 12, and 17. If it were truly a right triangle, the Pythagorean theorem would hold, and 122 + 122 would be the same as 172. It's not: 122 + 122 = 288 and 172 = 289.

So, the dimensions of home plate (an irregular pentagon) are not mathematically correct.

But there's a difference between measured numbers (accurate to a certain number of significant digits) and purely mathematical numbers. To the degree of accuracy required to construct a workable home plate, 17 is as good as (and certainly more measurable than) the more exact value of 12 times the square root of 2.

The history of baseball sheds some light on how the dimensions of home plate came about.

The playing field has been the same shape and size since the rules of baseball were first published more than 140 years ago. The size, placement, and shape of the bases, however, have changed over the years.

Initially, the rules insisted that bases be 1 square foot in area (most simply, a 1 foot by 1 foot square). Out on the field, the center of each base sat directly over a corner of the infield square. Home plate started as a circular iron plate, painted white, with a diameter not less than 9 inches. By the 1870s, however, home plate had become a square just like the other bases.

In 1877, the width of the bases was increased to 15 inches but home plate stayed at 12 inches. First and third base were moved to their present positions, where they fit snugly inside the corners of the square that defines the infield. This change was made so that umpires could call foul balls more easily. Second base, however, still stuck out of the square, where it remains to this day.

The year 1900 saw the introduction of the five-sided home plate, with a flat side rather than a point facing the pitcher. The extra rubber made it easier for both umpires and pitchers to judge when a ball "cut the corner," especially when dirt happened to cover the corners of home plate.

Original version posted March 25, 1996.
Updated July 12, 2004; June 30, 2010.

References:

Bradley, M.J. 1996. Building home plate: Field of dreams of reality. Mathematics Magazine 69 (February): 44-45.

Kreutzer, P., and T. Kerley. 1990. Little League's Official How-to-Play Baseball Book. New York: Doubleday.

Peterson, I. 2002. Pythagoras plays ball. In Mathematical Treks: From Surreal Numbers to Magic Circles. Mathematical Association of America.

Thorp, J., and P. Palmer, eds. 1995. Total Baseball, 4th ed. New York: Viking.

June 29, 2010

A Faulty Sign


Who gets the faulty offices?

Photo by I. Peterson

June 28, 2010

Fire Hydrant Pentagons

Fire hydrants are an inescapable part of the urban landscape. They are so commonplace that it's easy to take them for granted or to overlook them—except perhaps in the case of an emergency.


These devices for delivering water, however, are worth a second look. If you examine the operating nuts that control the flow of water, you'll see that their shape is a regular pentagon. Most nuts that you might encounter in home building projects or car repairs are hexagonal or sometimes even square, but rarely pentagonal.


You can't use a standard wrench to tighten or loosen one of these pentagonal nuts. Indeed, firefighters have special wrenches to do the job.


One example of a tool for turning a fire hydrant's pentagonal nut.

But the novel geometry presumably deters others from tampering with the hydrants.


Notice that although the operating nut at the top of the hydrant (along with others on the side) to control water flow is pentagonal, the nuts holding the hydrant together are hexagonal.

Fire hydrants with pentagonal operating nuts are common in the United States. I have heard, however, that this is not the case in Canada or in other countries. I'll have to check that out the next time I travel outside the United States.

Photos by I. Peterson

June 27, 2010

The Washington Right Triangle

Washington, D.C., was planned around a large right triangle, with the White House at the triangle's northern vertex and the U.S. Capitol at its eastern vertex, linked by Pennsylvania Avenue (as the hypotenuse).


The White House stands at one vertex of a large right triangle around which the city of Washington was built.

A 1793 survey established the location of the triangle's 90° vertex, and Thomas Jefferson, when he was Secretary of State, had a wooden post installed to mark the spot. This post was replaced in 1804 by a more substantial marker, which came to be known as the Jefferson Pier.


A granite marker now fills in for the original Jefferson Pier, which stood at the southwest corner of the right triangle defining the city of Washington. The lines on top of the stone represent the north-south (longitudinal) and east-west (latitudinal) directions.

Pierre L'Enfant's original design for the city had called for an equestrian statue of George Washington at the triangle's southwest corner, but it was never commissioned. A later design for the memorial featured a massive obelisk, but the ground was too unstable for it to be located at the intended spot. Instead, construction of the Washington Monument began at a site a short distance to the south and east of the Jefferson Pier.


The U.S. Capitol (left) stands due east of the Jefferson Pier, which is located a short distance to the north and west of the Washington Monument (right).

A memorial marker has now replaced the original Jefferson Pier. The White House is due north of the marker, and the Jefferson Memorial, completed in 1943, is due south.


The view southward from the Jefferson Pier to the Jefferson Memorial.

The Jefferson Memorial, Jefferson Pier, and White House all lie on what was once considered a potential prime meridian of the United States.


An equestrian statue of Andrew Jackson stands on the meridian north of the White House.

See "American Meridian" for information on the official U.S. prime meridian, established in 1850.

Photos by I. Peterson

June 26, 2010

Pi Places

The ratio of the circumference of a circle to its diameter is an irrational number, so there's no way to express its decimal digits explicitly without using an approximation: 3.14159 . . . . Hence, it's handy to have a special symbol to represent the number in all its glory.

Welsh mathematician William Jones (1675-1749) was apparently the first one, in 1706, to use the Greek letter π (pi) in connection with this number. Jones was influential, a friend of Isaac Newton (1643-1727) and Edmond Halley (1656-1742), and later an official of the Royal Society. Leonhard Euler (1707-1783) adopted the practice in 1737, and π became a staple of the mathematical literature.

The number and its symbol have now achieved such renown that Pi Day is celebrated on March 14 (3/14) in more and more places each year. And puns associated with π, pi, and pie abound.

It’s probably not surprising to see members of a student math club wear T-shirts referring to pi.


Student T-shirt from Southwest Texas Junior College.


Math club T-shirt at Winona State University, Minnesota.

But you could also argue that familiarity with pi in the mathematical sense has contributed to the use of pi and its symbol in other contexts, such as restaurant signs and menu items.

In New Orleans:


Pie, Pizza & Pastas is in the Warehouse District of New Orleans.

In Austin, Texas:


Featured at Amy's Ice Creams in Austin, this flavor is delicious!

In Washington, D.C.:


Digits of pi have also figured into a subway mural in Toronto (see "Sliding Pi in Toronto") and in the design of math jewelry, including an apple pi necklace.

Photos by I. Peterson

June 22, 2010

The Mathematical Vocabulary Problem

The language of mathematics can throw up barriers to broad dissemination of information about mathematics.

Mathematical statements are supposed to be precise, devoid of the ambiguities of ordinary speech. The language is unusually dense and relies heavily on a specialized vocabulary. The meaning and position of every word and symbol make a difference.

Mathematician William Thurston once expressed the difference between reading mathematics and reading other subject matter in this way: "Mathematicians attach meaning to the exact phrasing of a sentence, much more than is conventional. The meanings of words are more precisely delimited. When I read articles or listen to speeches in the style of the humanities . . . I find I have great trouble concentrating and comprehending: I think I try to read more into the phrases and sentences than is meant to be there, because of habits developed in reading mathematics."

Such habits can add to the difficulties that mathematicians face in trying to communicate with the public, when they have to surrender the clarity and economy of their usual modes of expression to the messiness of ordinary language. Comfortable with their specialized vocabulary, mathematicians too often fall into the trap of assuming their listeners or readers have equal facility, or at least some familiarity, with the language.

To complicate the situation, at least in English, mathematicians have appropriated simple, everyday words for their own purposes, using them in unexpected ways or assigning them specific, technical meanings to express abstract concepts.

Consider, for example, the term "function," a notion fundamental to mathematics. The American Heritage Dictionary of the English Language offers the following definitions:

1. The action for which a person or thing is particularly fitted or employed.
2.            a. Assigned duty of activity.
               b. A specific occupation or role: in my function as chief editor.
3. An official ceremony or a formal social occasion.
4. Something closely related to another thing and dependent on it for existence, value, or significance. Growth is a function of nutrition.

The mathematical meaning comes next:

5. Mathematics
a. A variable so related to another that for each value assumed by one there is a value determined for the other.
b. A rule of correspondence between two sets such that there is a unique element in the second set assigned to each element of the first set.

It is followed by three more definitions:

6. Biology The physiological activity of an organ or body part.
7. Chemistry The characteristic behavior of a chemical compound, resulting from the presence of a specific functional group.
8. Computer Science A procedure within an application.

That's a hefty load for one word to carry. Readers or listeners encountering the word "function" may understandably have difficulties sorting through so many definitions to ascertain the word's meaning in a particular context. Even when such a word is properly defined near the beginning and the context is clear, a reader unfamiliar with the notion may later revert to other, more familiar meanings of the word, potentially creating confusion in the reader's mind.

When I was a writer for Science News magazine, I could only on rare occasions get away with using the word "function" in my mathematics news articles without offering some sort of definition of the concept, expressed in words. My editors were there to ensure that my articles were accessible to as broad a range of readers as possible, and this meant keeping in mind that a reader’s notion of what a word means could differ enormously from the author's intended meaning.

In the same way, mathematicians should realize that words they use routinely can echo in unexpected ways in the minds of their listeners or readers, particularly in ways that reflect different experiences and contexts. Such words include acute, base, chaos, chord, composite, concurrent, coordinate, degree, dimension, domain, exponent, factor, graph, group, linear, matrix, mean, network, obtuse, order, power, prism, proof, radical, range, relation, root, series, set, vector, and volume. Each has a precise mathematical meaning; each also has multiple alternative meanings.

On the other hand, the word "fractal," coined by mathematician Benoit Mandelbrot, is a noteworthy example of a term that works in both a mathematical and a popular context. Mathematics could use more such words.

People are genuinely curious about mathematics, despite the overwhelming fear of the subject that many may feel. Mathematicians who pay particular attention to how they express themselves and connect with their audiences through a common, nontechnical language can make important contributions to the public understanding of mathematics.

This article is part of a contribution by I. Peterson to the Proceedings, International Congress of Mathematicians, Hyderabad, India, Aug. 25, 2010. For more, see “Communicating Mathematics.”

References:

Gowers, T., editor. 2008. The Princeton Companion to Mathematics. Princeton University Press.

Peterson, I. 1991. Searching for new mathematics. SIAM Review 13(March):37-42.

June 19, 2010

Tiling by Isosceles Right Triangles

The Blanton Museum of Art at the University of Texas is one of the largest university art museums in the United States, housing more than 17,000 artworks. The current complex, completed in 2006, consists of an expansive gallery building, an auxiliary auditorium, classroom, and office building, and a public plaza and garden.


The impressive, 124,000-square-foot gallery building features a roof of red Spanish tiles and vast, limestone walls. The outer walls are more than plain sheets or blocks of limestone. They incorporate subtly shaded patterns: large squares made up of triangular tiles that fit together in a symmetric design.


The basic units are isosceles right triangles (angles 45, 45, and 90 degrees). Two such triangles joined hypotenuse to hypotenuse form a square; eight combine to form a larger square, and so on. Indeed, squares of different sizes show up in several distinct orientations.

You can readily see a visual proof of the Pythagorean theorem in the tiling by looking at the squares corresponding to each of the sides of the triangular units.

Photos by I. Peterson

June 18, 2010

Light Rings

The ceiling light fixtures in the Texas Ballroom at the Hyatt Regency Austin have an interesting geometry. Each one consists of four concentric cylinders.


The configuration suggests a variety of mathematical questions concerning, say, relative area or volume.

Even more intriguing, however, are the rings of lights between the cylinders. One light sits in the center circle, eight in the first ring, ten in the second ring, and sixteen in the third. Is there a pattern?


We have the sequence: 1, 8, 10, 16, which suggests the following question: If the fixture were to have a fourth ring, how many lights would it contain?

Neil Sloane's On-:Line Encyclopedia of Integer Sequences is one source of possible answers. Querying the database, however, produces just one result for the string 1, 8, 10, 16. These are four consecutive numbers representing the absolute difference between the sum of the odd digits and the sum of the even digits of the nth prime, starting at n = 19. The next number in this sequence is 5, and that doesn't make sense in the context of the light fixtures.

So, is there a pattern, beyond the notion that the number of lights should increase as you move away from the center? What would be a plausible number for the fourth ring?

Photos by I. Peterson

June 16, 2010

Tensegrity Tower in New Orleans

Sprouting from a lagoon, a slim framework of stainless-steel tubes and cables seems to defy gravity as it extends 45 feet into the air. The spindly column looks fragile, ethereal.


The sculpture's dramatic setting adds to its allure. Surrounded by water and a wooded shoreline, it reaches for the sky, irresistibly drawing the eye.


Created by Kenneth Snelson and titled Virlaine Tower, this structure is the largest artwork in the Besthoff Sculpture Garden in New Orleans. It is an example of a tensegrity structure. The tubes do not connect to each other. Instead, a single cable winds through the tubes to hold the structure in place.


In his sculptures, Snelson sought to portray in a visible way the interplay of physical forces in space. See "Needle Tower" for another example of Snelson's tensegrity structures.

Photos by I. Peterson

June 15, 2010

Three Sentinels

Three brightly colored tetrahedra, stretched tall, stand erect and in close formation in front of an office building in downtown New Orleans.


Created by local sculptor Arthur Silverman, the artwork is titled Painted Trio. As you walk around it, the sculpture looks startlingly different from different angles, as one sharp edge gives way to another and the sculpture's colored faces—red, green, black, and white—appear in turn.


The tetrahedron is the simplest of all polyhedra—solids bounded by polygons. Any four points in space not all on the same plane mark the corners of four triangles, which serve as the faces of a tetrahedron.

To Silverman, this seemingly humble form is special. "The tetrahedron is very exciting visually," he says. "It's very difficult to anticipate what you are going to see."

We are accustomed to thinking about orientation in space in terms of three perpendicular axes defining left and right, up and down, and forward and backward. A tetrahedron has no right angles, so a tetrahedral structure jars us out of spatial complacency. It has so few faces compared to other polyhedra that its aspect changes abruptly as an observer moves around to view it from different angles.

Silverman has been using tetrahedra as the basic building blocks of his three-dimensional designs for more than two decades.  A number of his sculptures are on display in public spaces throughout New Orleans.


Painted Trio is located at 400 Poydras. Silverman's signature piece, dubbed Echo, is a few blocks farther up the street.

To create Echo, Silverman elongated several of a tetrahedron's six edges to create a slim, stainless-steel tower sixty feet high, then twinned it with an identical tower. Standing in the middle of a plaza fountain, this glistening pair seems to soar in formation into the sky.


Arthur Silverman's Echo features a pair of elongated tetrahedra, each balanced on one edge. The sculpture is 60 feet tall and rests on a foundation that extends 20 feet into the ground.

Two more of Silverman's sculptures stand in the grounds around City Hall.


Interlocking Boxes, Closed, is located in front of City Hall in New Orleans.


In this skeletal sculpture near City Hall, Silverman focused on the edges of tetrahedral structures.

"I find that the unique geometric relations intrinsic to the tetrahedron persist in the final sculpture, notwithstanding all the manipulations I carry out," Silverman says.

At the same time, he notes, "photographs do not do these works justice. One must actually see, feel, and walk around these works in order to experience them in their reality."

For more about Silverman's tetrahedral art, see "Art of the Tetrahedron," "Art of the Tetrahedron, Revisited," and "Four Corners, Four Faces."

References:

Peterson, I. 2001. Fragments of Infinity: A Kaleidoscope of Math and Art. Wiley.

Photos by I. Peterson

June 13, 2010

South Meets South

The signs are at right angles, as are the streets. One sign says S Carrollton Av, and the other says S Claiborne.


Normally, you would expect streets that are both labeled "South" to run parallel to each other rather than at right angles. But this is New Orleans—a city nestled in the bends of the Mississippi River.


Imposing a traditional rectilinear grid on this geography doesn't work very well. Within a bend, streets would more logically follow a wheel-and-spoke pattern.


A wheel-and-spoke pattern surrounds a sidewalk tree in New Orleans.

A curious hybrid leads to streets that start out parallel but end up crossing.

Photos by I. Peterson

June 12, 2010

Hexagons, Pentagons, and Geodesic Domes

Expo '67 in Montreal was a wonderland of architecture. One of the most striking structures at this world's fair was the U.S. pavilion, designed by R. Buckminster Fuller. This gigantic geodesic dome looked like a glistening bubble, barely tethered to the ground.


Looking out from inside the pavilion, you see row after row of the hexagonal units that make up the structure's skin. At the same time, you know that the skin cannot consist entirely of regular hexagons. Regular hexagons fit together to cover a flat surface. Where does the curvature come from?


The secret is in the pentagon. Twelve regular pentagons combine to form a regular solid known as a dodecahedron.


You can enlarge this structure and make it more spherical by adding a ring of regular hexagons around each pentagon. The first iteration produces a truncated icosahedron—the traditional pattern on a soccer ball. This structure has twelve pentagonal and twenty hexagonal faces.


Adding more rings of hexagons produces structures even closer to a sphere. Remarkably, each of these structures contains precisely twelve pentagons, and it is these pentagons that force the curvature. Fuller made such configurations the basis for his geodesic domes.


It's easy to spot the pentagons in small geodesic structures.


Finding them in structures as large as the U.S. pavilion, now the Biosphère, can be very difficult. I know the pentagons are there, and I have tried to find them, but I have had very little success in locating even one.

I was at Expo '67 during the summer that the fair opened, and I have visited Montreal several times since. I still don't have a photo of any of the elusive pentagons that must be present.


My eye is continually fooled as it tries to make sense of the array of metal struts that define the structure, and all I see are hexagons.

Photos by I. Peterson

June 9, 2010

Skylight Fractal

Looking up through a skylight can open up new dimensions in geometry, especially when the straight lines framing the skylight contrast sharply with the fuzzy, irregular boundaries of puffy clouds visible through the glass.


Clouds seen through a skylight at Hood College in Frederick, Md.

As mathematician Benoit Mandelbrot famously noted in his book The Fractal Geometry of Nature, "Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line."

Instead, a fragment of a rock may look like the mountain from which it was fractured. Clouds keep their distinctive wispiness whether viewed distantly from the ground or close up from an airplane window. A tree's twigs often have the same branching pattern seen near its trunk.


Rugged coastlines, ragged clouds, breaking waves, and fractured rocks are examples of natural forms that resemble self-similar mathematical objects described as fractals.

So clouds, mountains, and trees wear their irregularity in an unexpectedly orderly fashion. Indeed, nature is full of shapes that repeat themselves on different scales within the same object. In all these examples, zooming in for a closer view doesn’t smooth out the irregularities. Objects tend to show the same degree of roughness at different levels of magnification.

Mandelbrot coined the word “fractal” as a convenient label for these self-similar shapes—those structures that look the same on different scales.


The straight lines of the Washington Monument (background) contrast sharply with the branching geometry of a fir tree on the National Mall in Washington, D.C.

References:

Mandelbrot, B.B. 1982. The Fractal Geometry of Nature. W.H. Freeman.

Peterson, I. 2001. Fragments of Infinity: A Kaleidoscope of Math and Art. Wiley.

______. 1998. The Mathematical Tourist: New and Updated Snapshots of Modern Mathematics. W.H. Freeman.

Photos by I. Peterson

June 8, 2010

Sliding Pi in Toronto

The entrance to the 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, now a resident of Calgary, Alberta. 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 Downsview 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 says. "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 remarks.


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 says.


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 notes. "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 says. "But surely some must wonder how such a seemingly random, ever-changing pattern of color happenedbecause 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 sortthat 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.

References:

Peterson, I. 2002. Tiling with pi. Math Horizons 10(November):11-14.

______. 2001. Fragments of Infinity: A Kaleidoscope of Math and Art. Wiley.

Photos by I. Peterson