tag:blogger.com,1999:blog-362699732012-01-09T21:51:17.771-05:00Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.comBlogger1541100tag:blogger.com,1999:blog-36269973.post-1174255968821088262012-01-09T20:42:00.000-05:002012-01-09T20:42:08.850-05:00

Courtesy of Frank Farris, Santa Clara University. Published in Mathematics Magazine, Vol. 84, No. 4 (October 2009), p. 254. We need more articles! Suggest titles and/or authors.

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com1tag:blogger.com,1999:blog-36269973.post-43604281860869372322012-01-08T21:18:00.000-05:002012-01-08T21:18:19.559-05:00

This year's display of mathematical art at the Joint Mathematics Meetings in Boston featured an eye-popping array of three-dimensional structures. One of my favorites was a crinkled surface—an intricately pleated cone—crafted from a single sheet of blue paper. Titled Pleated Multi-sliced Cone, this beautiful example of geometric origami was the product of a collaboration. Origami artist Robert J

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-61954828750311553572011-12-06T11:40:00.000-05:002011-12-06T11:40:48.764-05:00

  Boston skyline as seen from Cambridge, Mass., across the Charles River. The tallest building in the photo is the Prudential Tower at 800 Boylston Street. The skyscraper to its left is 111 Huntington Avenue, sometimes called the "R2-D2" building. Boston will host the Joint Mathematics Meetings (JMM) on January 4-7, 2012, bringing more than 5,000 mathematicians to the city. One of the more

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-21217557985183909082011-12-05T16:24:00.000-05:002011-12-05T16:24:45.508-05:00

Some years ago, David Wells surveyed readers of the Mathematical Intelligencer about what they consider to be beautiful in mathematics. The respondents ranked Euler's identity linking e, π, and i as the most beautiful expression in mathematics. See "Euler's Beauties." In the intriguing new book Beautiful Mathematics (MAA, 2011), Martin Erickson expands on this theme, contemplating the esthetic

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-19632745808053208662011-10-27T20:56:00.000-05:002011-10-27T20:56:26.109-05:00

The puzzle known as TopSpin consists of 20 circular pieces, numbered 1 to 20, filling and sliding along an oval track. TopSpin was introduced by ThinkFun (formerly Binary Arts) in 1988. Pieces can be moved around the track in either direction, keeping their order. Or any four consecutive pieces can be maneuvered into reverse order. For example, consecutive pieces labeled 1, 2, 3, and 4, can be

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"Möbius bands (or strips) are beautiful as objects of art, and their mysterious qualities fascinate those who discover or encounter them," sculptor Larry Frazier wrote in an article titled "Möbius strips of wood and alabaster," published in the Journal of Mathematics and the Arts. "I'm not a mathematician, but as a sculptor, I have been fascinated by the myriad forms that a Möbius band can take,

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-44655847272769248402011-10-07T21:04:00.000-05:002011-10-07T21:04:46.756-05:00

To bring a trace of irregularity to his pristine geometric structures, conceptual and minimalist artist Sol LeWitt (1928-2007) started with a polygon sketched on a flat surface. He placed dots at various positions within the polygon. These points were then elevated to different heights, dictating the edges of the resulting three-dimensional, faceted object that eschewed the right angles typical

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-68482159021173905202011-09-25T10:38:00.000-05:002011-09-25T10:38:55.483-05:00

Examples of Sol LeWitt's Incomplete Open Cubes on display at City Hall Park in New York City. A cube has six faces, eight vertices, and twelve edges. In his series titled Incomplete Open Cubes, conceptual and minimalist artist Sol LeWitt (1928-2007) chose to work with cubes represented as frameworks. LeWitt started by removing one edge from an open cube, then two edges, and so on, as an

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-12637371500064233222011-09-16T20:51:00.001-05:002011-09-16T20:53:13.105-05:00

THE FERRY BOATS Two ferry boats ply back and forth across a river with constant speeds, turning at the banks without loss of time. They leave opposite shores at the same instant, meet for the first rime some 700 feet from one shore, continue on their way to the banks, return and meet for the second time 400 feet from the opposite shore. [Without using pencil and paper] determine the width of the

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-15628198282666981442011-09-15T20:58:00.001-05:002011-09-16T20:52:36.941-05:00

The American Mathematical Monthly has a long tradition of publishing problems, going all the way back to its first issue in 1894. In a letter that appeared in the debut issue, Monthly coeditors B.F. Finkel and J.M. Colaw argued the value of posing and solving mathematical problems. "While realizing that the solution of problems is one of the lowest forms of Mathematical research . . . its

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-86267159515144659512011-09-13T20:23:00.000-05:002011-09-13T20:23:12.344-05:00

A neon-framed model of a regular tetrahedron hangs in an entrance to the Milan Central Train Station (Stazione di Milano Centrale). Defined by four triangular faces, the tetrahedron is the simplest of all polyhedra. Any four points in space that are not in the same plane mark its corners. Despite its apparent simplicity, a variety of artists have used the tetrahedron as the inspiration for

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-22188201391581811422011-09-11T10:58:00.000-05:002011-09-11T10:58:18.845-05:00

Block quilt pattern: Double Irish Chain. The Charles Hotel in Cambridge, Mass., prides itself on its extensive collection of early American quilts, many of which are on display throughout the hotel. One particularly striking array is a set of nine blue-and-white quilts by the hotel's grand staircase. Created in the 1880s and 90s, these hand-crafted quilts feature striking traditional designs

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com1tag:blogger.com,1999:blog-36269973.post-44445397200661248782011-09-08T19:34:00.000-05:002011-09-08T19:34:09.764-05:00

Pythagorean Fractal Tree was designed by Koos Verhoeff, cast in bronze by Anton Bakker and  Kevin Gallup, and displayed at the first art and mathematics conference in Albany, New York, in 1992. Born in Holland in 1927, Verhoeff studied mathematics and computer science. He worked for a time at the Mathematical Center in Amsterdam, where he encountered the Dutch artist M.C. Escher, who often came

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-75648552589478685112011-08-05T21:45:00.001-05:002011-08-05T21:45:53.518-05:00

Ragged spiral arms, enclosed within a square, define a large metal sculpture standing in front of the College of Education at the University of Kentucky (Dickey Hall at 251 Scott Street). Titled "Exponential Symmetry," the artwork was created by Michael Martinez, a University of Kentucky alumnus. The two-sided, boxed structure references the spiral geometry of a chambered nautilus shell, which

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com1tag:blogger.com,1999:blog-36269973.post-10113719431210243482011-07-28T09:46:00.000-05:002011-07-28T09:46:56.425-05:00

MAA MathFest will be held Aug. 4-6 in Lexington, Kentucky, bringing more than 1,400 mathematicians and students to the city. This will be my first visit to Lexington since the early 1980s, and I am looking forward to exploring and photographing its mathematical and architectural sights. My main memory of my previous visit is of an excursion to the Kentucky Horse Park (I still have the souvenir

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-24106715750941980722011-07-09T14:55:00.001-05:002011-07-09T15:25:45.512-05:00

One of my favorite demonstrations of the one-sidedness of a Möbius strip is a model train that loops endlessly along a track bisecting a Möbius strip. Such a demonstration is one of the prime displays in the "Mathematica: A World of Numbers. . . and Beyond" exhibit currently at the Boston Museum of Science. In this case, the model train is a segmented arrow. Commissioned by IBM and created by

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com1tag:blogger.com,1999:blog-36269973.post-14774680078247530562011-07-07T21:47:00.001-05:002011-07-08T10:29:18.437-05:00

Ordinary dice—those sold in novelty stores and with many board games—have rounded edges and little hollows for each of the pips denoting the numbers from one to six. But the hollows mean that such ordinary dice are somewhat biased. A little more material has been removed from sides with a larger number of pips, so a die with six hollows on one side and only one hollow on the opposite would have

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-12565158767867378472011-07-04T11:06:00.000-05:002011-07-04T11:06:38.477-05:00

An intricate geometric framework of linked pentagons hangs in the atrium of the Fields Institute for Research in Mathematical Sciences in Toronto. The five-foot-diameter construction resembles a giant soccer ball stripped of its skin to reveal an elaborate supporting structure. Created by Marc Pelletier, the stainless-steel sculpture represents a mathematical object known as the 120-cell. It is

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-19442489668418114222011-07-03T12:15:00.001-05:002011-07-03T12:15:29.792-05:00

One of the stranger sights in downtown Toronto is a massive rectangular box, seemingly hovering in the air amid older, somewhat more conventional brethren. It looks like an alien import seeking a place to land. This curious structure houses the Sharp Centre for Design at the Ontario College of Art and Design (OCAD University). Designed by Will Alsop (Alsop Architects), the structure is a

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-59204935834065737362011-06-28T20:48:00.003-05:002011-07-03T12:21:07.412-05:00

The Vessel, by Ilan Sandler, stands near the corner of a small park in Toronto. From some angles, with its hollow handle, the sculpture looks like a wire-frame model of the standard representation of a Klein bottle. You can almost imagine the inside connecting with the outside as the handle curves into the body of the giant pitcher. The sculpture, which was officially unveiled earlier this

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-19450716802825691062011-06-27T20:54:00.000-05:002011-06-27T20:54:10.082-05:00

Intuition, by sculptor John Robinson, consists of three, interwoven equilateral triangles presented in a symmetrical arrangement. Immersed in greenery, the four-foot, stainless-steel sculpture sits near the entrance to the Fields Institute for Research in the Mathematical Sciences at 222 College Street in Toronto. Robinson's construction is an angular version of the famous Borromean rings.

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-78786826099096231892011-05-25T05:00:00.004-05:002011-06-01T19:13:16.439-05:00

No visit to Macalester College can be complete without a quick spin on a square-wheeled trike. It's a weird contraption, but you can ride it quite smoothly, without the sequence of jarring bumps that you might expect. The secret is in the shape of the road over which the square wheels roll. The author aboard the Macalester square-wheeled tricycle. Photo by P. Zorn. A square wheel can roll

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-61353181287292330122011-05-22T11:04:00.000-05:002011-05-22T11:04:58.114-05:00

Shooting graceful arcs of water into the air, fountains can offer lessons in geometric spectacle. The fountain at the National Gallery of Art Sculpture Garden in Washington, D.C., is a notable example. The circular fountain at the National Gallery of Art Sculpture Garden shoots parabolic streams of water from its circumference toward its center. One factor that makes some fountains more

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com3tag:blogger.com,1999:blog-36269973.post-10419883283469016752011-03-13T21:21:00.001-05:002011-03-13T21:22:58.097-05:00

California sculptor William Wareham is known for constructions made up of sections of spheres and segments of rings, put together in fanciful—even whimsical—arrangements. Made from polished steel, these assemblages represent an exploration of the relationship between a two-dimension form (circle) and a three-dimensional shape (sphere). One striking example is on the campus of Santa Rosa Junior

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-86897675212225726402011-03-06T15:18:00.001-05:002011-03-06T15:33:19.122-05:00

Galileo's Vision is a new outdoor sculpture on the campus of Missouri Western State University in St. Joseph. Created by Jim Estes, retired professor of art at Missouri Western, the steel sculpture evokes Galileo's quest to understand the heavens and his famous treatise The Starry Messenger (Sidereus Nuncius). Curiously, the top of the sculpture also bears a resemblance to the Greek letter pi

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-89415313692117913712011-02-28T21:42:00.000-05:002011-02-28T21:42:06.422-05:00

From one viewpoint, the sculpture looks like an ellipse with a thick rim. In reality, it is a tilted, circular ring, 19 feet across. More than that, it ingeniously tracks the sun, casting distinctive shadows at different times of the year. Titled Annual Ring, the  weathered steel (COR-TEN) artwork was created by Northern California sculptor Roger Berry in 1987 and installed on the campus of

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com1tag:blogger.com,1999:blog-36269973.post-31538088281663635812011-02-21T21:35:00.000-05:002011-02-21T21:35:29.590-05:00

The plaza in front of the Lawrence Hall of Science in Berkeley, Calif., features not only a model of a fin whale, which attracts clamberers of all ages, but also something more mundane—a fountain that simply  spouts water into the air. Nonetheless, on a windy day, the fountain can produce an intriguing splash pattern (below) that is well worth pondering. Notice that a fairly smooth curve bounds

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-56107386393954400322011-02-20T21:40:00.000-05:002011-02-20T21:40:16.134-05:00

Perched high on the steep slopes of a hill overlooking Berkeley, Calif., the Mathematical Sciences Research Institute (MSRI) has a spectacular site, with a view across the bay to the Golden Gate Bridge and beyond. Visitors to MSRI are also greeted by a fascinating tile mural that stretches across a wall beside the entrance to the building. Designed by Linda Vanderkolk and Scott Frankenberger 

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I recently found a new U.S. penny in my loose change. The "heads" side still shows Abraham Lincoln's profile, but the "tails" side now features the Union Shield instead of the Lincoln Memorial. I had to wonder whether this new coin has the same distinctive bias that older Lincoln pennies display. Starting in 2010, U.S. pennies featured the Union Shield instead of the Lincoln Memorial on one

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com2tag:blogger.com,1999:blog-36269973.post-45527680841783415262011-01-10T19:07:00.000-05:002011-01-10T19:07:42.401-05:00

Long trips often give me a chance to catch up with my reading. I spent much of my journey home from the 2011 Joint Mathematics Meetings (JMM) in New Orleans reading the Dec. 13, 2010, issue of The New Yorker. The issue proved a winner, with several interesting articles and, best of all, two mentions of Möbius strips. I have been collecting references to Möbius strips in articles, stories, and

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com1tag:blogger.com,1999:blog-36269973.post-46041199677285039682010-12-24T15:37:00.001-05:002010-12-24T21:47:37.336-05:00

The Joint Mathematics Meetings (JMM) will be held Jan. 6-9, 2011, in New Orleans, bringing together nearly 6,000 mathematicians. Famous for its French Quarter, jazz, food, and more, the city also has a claim to fame in the realm of public art, some of it mathematical in nature. River Stones by Terry Weldon. One noteworthy site is the Sydney and Walda Besthoff Sculpture Garden at the New Orleans

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com1tag:blogger.com,1999:blog-36269973.post-76347722802798954842010-10-20T20:50:00.000-05:002010-10-20T20:50:03.459-05:00

A prolific writer on a wide variety of topics, Martin Gardner died earlier this year. On Oct. 21, 2010, on what would have been his 96th birthday, groups all over the world will gather to celebrate Martin's work and continue his pursuit of a playful approach to mathematics, magic, science, art, literature, and much more. I was reminded again of Martin's generosity and of his delight in surprises

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com3tag:blogger.com,1999:blog-36269973.post-25410610069832015302010-10-12T21:02:00.000-05:002010-10-12T21:02:51.567-05:00

A glistening, sinuous shape at the other end of a courtyard caught my eye as I was walking down Yonge Street during a recent visit to Toronto. I had to take a closer look. The sculpture sits in the Anne Johnston Courtyard, between two high-rise towers (named Quantum and Quantum 2) at 2181 and 2191 Yonge Street, in Toronto, Canada. About 8 feet tall, the stainless-steel sculpture forms a giant,

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-40806395011416270742010-09-21T20:39:00.000-05:002010-09-21T20:39:10.767-05:00

The palace complex at Amer Fort near Jaipur, India, includes a dazzling chamber—the Sheesh Mahal, or palace of mirrors—that once housed the private rooms of the maharaja and his queen. Its walls are decorated with intricate mosaics fashioned out of thousands of tiny mirrors and shards of colored glass, arranged in patterns that display all sorts of symmetries. During the day, the chamber

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-52929676548585935792010-09-19T09:24:00.000-05:002010-09-19T09:24:58.148-05:00

Jaisalmer is known as the "Golden City," after the distinctive yellow sandstone, found locally, that is the area's main construction material. The sandstone's yellow color comes from its high sulfur content. Constructed from yellow sandstone, Jaisalmer Fort sprawls across a hilltop in the Thar desert. More than 2,000 people live within the fort. Located on an ancient trade route in the Thar

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com1tag:blogger.com,1999:blog-36269973.post-52442559407111236282010-09-12T08:29:00.003-05:002010-09-19T09:29:16.896-05:00

The breathtaking, glowingly extravagant Taj Mahal in Agra, India, is not only a memorial to Mumtaz Mahal, third wife of Mughal emperor Shah Jahan, but also a tribute to mirror symmetry. The Taj Mahal's translucent white marble has a warm glow in the early morning light. Click on photos to see more detail. Oriented on north-south and east-west axes, the mausoleum and its associated structures

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com1tag:blogger.com,1999:blog-36269973.post-4946279980290690342010-09-11T09:54:00.002-05:002010-09-19T09:30:40.879-05:00

The largest sundial I have ever seen is in the Jantar Mantar in Jaipur, India. It looms over a remarkable collection of naked-eye astronomical instruments, where large scale and geometrical ingenuity make up for the absence of optical magnification. The term "jantar" means "instrument" and "mantar" may be interpreted as "formula" or "calculation." Constructed for Maharaja Jai Singh II at his new

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-55934780074842497642010-09-09T20:54:00.004-05:002010-09-19T09:30:59.532-05:00

Combinations of regular octagons and squares are often a feature of Indo-Islamic design. Such tilings appear in a variety of settings, particularly grills or screens fashioned from metal or cut out of marble or sandstone. This doorway (above) at Golconda Fort in Hyderabad is protected by a wire grill with a distinctive pattern of overlapping octagons (below), constructed from squares and

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com1tag:blogger.com,1999:blog-36269973.post-7897864426043831472010-08-29T08:52:00.001-05:002010-09-19T09:31:11.856-05:00

When I travel, I have a habit of photographing numbers—telephone numbers, license plate numbers, street address numbers, identification numbers, room numbers, and so on. I then use the photos as illustrations for the daily entries in the MAA NumberADay blog. I am always struck by how many numbers we live by. You can find them just about anywhere, signifying or representing one thing or another.

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-71946234196343596042010-08-26T07:37:00.001-05:002010-09-19T09:31:27.456-05:00

Hyderabad, India. The exhibits area at the International Congress of Mathematicians (ICM) had only about two dozen booths, but they included a variety of societies, publishers, institutions, and retailers. The Cambridge University Press exhibit included an array of books published by the Mathematical Association of America. Congress attendees could purchase autographed copies of the book

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com2tag:blogger.com,1999:blog-36269973.post-81310218227221027412010-08-26T03:35:00.005-05:002010-09-19T09:31:47.230-05:00

Hyderabad, India, The panel discussion on "Communicating Mathematics to Society at Large" at the International Congress of Mathematicians attracted a large, attentive crowd. The presentations and audience comments all served to illustrate the many opportunities available for reaching out to the general public while acknowledging the diversity of that audience. Of particular interest was a

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-24426020408156795512010-08-24T10:55:00.002-05:002010-09-19T09:32:18.697-05:00

Hyderabad, India. The International Congress of Mathematicians has a tradition of breaking up its proceedings by offering a day with no programmed activities. So, after three intense, packed days of lectures, discussions, and other events, attendees got a chance to relax, explore Hyderabad, dine in exotic settings, go souvenir hunting, or pursue other interests. For many, the break meant getting

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com0tag:blogger.com,1999:blog-36269973.post-23446143172612011252010-08-23T22:16:00.003-05:002010-09-19T09:32:29.639-05:00

Hyderabad, India. The International Congress of Mathematicians is taking place at the Hyderabad International  Convention Centre (HICC, below). Note the concentric circles that mark the plaza in front of the Hyderabad International Convention Centre. A sign (below) outside the centre welcomes delegates from all over the world to the meeting. The new Fields Medalists each get a chance to

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Hyderabad, India. One of the highlights of the International Congress of Mathematicians was the chance to see the remarkable play A Disappearing Number, presented by the London-based theatre company Complicite. Like the plays Proof and Arcadia, it deals with mathematical genius; in this case, with Srinivasa Ramanujan, his relationship with Cambridge mathematician G.H. Hardy, and his unique

Math Touristhttp://www.blogger.com/profile/00014397210725962876noreply@blogger.com1tag:blogger.com,1999:blog-36269973.post-9185181342958333382010-08-18T20:51:00.000-05:002010-08-18T20:51:35.678-05:00

Among the tall buildings of Pittsburgh, the United Steelworkers Building stands out not for its height (only 13 stories) but for its dramatic diamond-lattice façade. The diamond grid is more than just decorative. The steel exterior walls represent the building's supporting structure; they serve as a load-bearing exoskeleton. Designed by the architectural firm Curtis and Davis and constructed in

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The triangle of three bent arrows that signifies recycling is a fixture of the world in which we live. This recycling symbol appears in newspapers and magazines and on signs, bottles, envelopes, cardboard cartons, trash receptacles, and many other containers. If you look closely, however, you'll see all sorts of variants of this ubiquitous symbol. In the version of the recycling symbol on this

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Why is the cover of a manhole nearly always round? Why isn't it oval or square? Manhole covers are nearly always circular, but they may feature distinctive geometric designs, like the ring-and-spoke pattern shown here.. One answer is that a circular lid, unlike a square or an oval cover, won't fall through the opening. There's no way to position a round lid so that it can slip through a slightly

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The swooping roof of the David L. Lawrence Convention Center in Pittsburgh hangs from a series of 15 enormous cables tied to tall masts. Designed by architect Rafael Viñoly, the structure mimics the curves of Pittsburgh's many suspension bridges. The cable-suspended roof owes its graceful shape to the natural curve formed by a flexible cable or chain hanging between its fixed ends—a catenary.

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As you walk along the streets in Washington, D.C., you'll notice that signs at corners give not only the street name but also a numerical indicator representing the numbers assigned to a given block—typically in multiples of 100. So, as you approach the U.S. Capitol, you're likely to see signs saying 500, 400, 300, 200, 100. Then what happens? Do you get 000 or 00? Curiously, the chosen

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My father was a draftsman and, as a child, I was fascinated by the drawing tools that he had around the house. I remember being particularly intrigued by his set of French curves—curlicued plastic templates for piecing together arbitrary curves from appropriate fragments. In these days of computer-aided design, there isn't much need for French curves or the other physical templates that were once

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The grand building that once housed Pennsylvania Railroad's Union Station in Pittsburgh features a majestic entrance rotunda lit during the day by a central dome skylight. The skylight's rings of glass are divided into segments. The outer rings each have 28 identical sections. The innermost ring has a twelvefold symmetry. That combination is a bit puzzling: 28 and 12 don't appear to fit together

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Expressed as Roman numerals, the first twelve numbers are usually given as I, II, III, IV, V, VI, VII, VIII, IX, X, XI, XII. However, on many clock faces, when the numbers on the dial are in Roman numerals, IIII replaces IV. In Pittsburgh, for example, the ornate Kaufmann's clock (above) at the corner of Smithfield Street and Fifth Avenue, shows IIII instead of IV. There are many stories about

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Thirteen simple geometric forms, each about 9 feet tall, line a marble wall of the mezzanine at the Wood Street T Station in Pittsburgh. Cut from sheets of dark slate, the figures include a square, circle, cross, rhombus, trapezoid, quarter circle, equilateral triangle, and right triangle, several in more than one orientation or combination. The assemblage stretches more than 200 feet, extending

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The Sydney Opera House in Australia is a spectacular urban sculpture, perched on the tip of a narrow peninsula jutting into Sydney Harbor. Designed by Danish architect Jørn Utzon and inaugurated in 1973, the structure consists of three groups of interlocking, vaulted "shells," which cover two main performance areas and a restaurant, set on a vast platform. Interestingly, the precise geometry of

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The card game known as SET® is deceptively simple. Its object is to identify, as quickly as possible, a grouping (SET) of three cards, selected from 12 cards laid out face up on a table. A SET deck has 81 (34) cards. Each card displays a design with four attributes: shape, number, shading, and color. Each attribute has three possible values. Attribute Values Shape {oval, diamond,

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The crazily crinkled structure, framed within a cube, is an impressive sight. Suspended above the stairs on the sixth floor of the mathematics building at St. Olaf College in Northfield, Minn., it is also the solution to a mathematical puzzle. Painstakingly crafted from wood by retired St. Olaf mathematician Loren Larson, the sculpture embodies a path that a knight could take on a

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A drum vibrates at characteristic frequencies, depending mainly on the size, shape, tension, and composition of its sound-generating drumhead. This spectrum of frequencies—the set of pure tones produced by a vibrating membrane stretched across a frame—gives a drum its particular voice. Physicists and mathematicians have long recognized that the shape of the boundary enclosing a membrane plays a

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The law of reflection, combined with a simple geometry, can lead to visual wonders of surprising intricacy and complexity. Cloud Gate, a sculpture by Anish Kapoor, is a particularly striking example of such a spectacle. The large structure—66 feet long, 42 feet wide, 33 feet high—seems barely tethered to the ground of AT&T Plaza in Chicago's Millennium Park. It has no color of its own; it

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Artist Sol LeWitt (1928-2007) often featured geometric and combinatorial themes in his numerous creations (see "LeWitt's Pyramid"). Indeed, many of his earlier artworks were explicitly combinatorial. In 1973, for example, he composed Straight Lines in Four Directions and All Their Possible Combinations. This study consisted of 15 square etchings, each inscribed with one or more horizontal,

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The summer meeting of the Mathematical Association of America, MathFest 2010, will be held Aug. 5-7 in Pittsburgh. The city has a wonderful array of public art and some spectacular architecture, making it an apt venue for excursions into mathematical art and more. Four rectangular blocks add a colorful touch to the stark entrance to Wean Hall, Carnegie Mellon University, Pittsburgh. With a

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Looking up can present you with remarkable geometric vistas in a variety of settings. One of my most vivid memories is of the interior of a spire that consisted of a sequence of stacked, rotated hexagons of diminishing size, producing a convincing illusion of infinite extent. The spectacular, soaring spire of the Man in the Community pavilion at Expo '67 in Montreal consisted of a sequence of

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The sculptures of Clement Meadmore (1929-2005) are based on simple geometric elements—often a square prism, wrought into an evocative form. Meadmore's Slew (1971) is in front of the Esther Raushenbush Library, Sarah Lawrence College, Bronxville, N.Y. In a typical sculpture, a single rectangular volume twists and turns upon itself before lunging into space, suggesting a forceful release of

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Standing near the southwest corner of Philadelphia's City Hall, the weathered bronze sculpture has an edge that seems to wind around into a mathematical knot. Its curved surfaces, dedined by the twisty edge, resemble patches of soap film—pieces of minimal surfaces. Titled The Triune, this bronze is the work of Robert Engman, long a professor of sculpture at the University of Pennsylvania. It is

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As a mathematician, Paul R. Halmos (1916-2006) made fundamental contributions to probability theory, statistics, functional analysis, mathematical logic, and other areas of mathematics. He was also known and widely recognized as a masterly mathematical expositor. And he served as editor (1981-1985) of the American Mathematical Monthly. Halmos described his approach to writing in an essay

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The Albany campus of the State University of New York features an "academic podium" of 13 buildings on a common platform, all connected by a continuous roof and long colonnades. Designed by Edward Durrell Stone and completed in the mid-1960s, this carefully arranged collection of structures can itself be considered a mathematical artwork, built on precise, repeating relationships among the

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"'There is a multilevel space,' [architect David] Rockwell told me," Rebecca Mead writes in the July 5 New Yorker in an article on innovative playground designs. "He gestured to a raised walkway that resembled a Möbius strip." That caught my attention. I knew of a climber based on a three-dimensional analog of a Möbius strip at a playground in Florida, a Möbius-strip jungle gym at the Burning Man

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The Statue of Liberty stands on a tiny island in New York Harbor, facing out to sea. It rises more 150 feet into the air, from its base on a tall pedestal to the tip of its torch. As envisioned by sculptor Frédéric-Auguste Bartholdi and engineer Alexandre-Gustave Eiffel, the statue combines ancient materials with an innovative structure. An iron pylon serves as the Statue of Liberty's core.

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The carved stone decorations on the Bulfinch Gatehouse, located near the White House in Washington, D.C., look like mathematical knots. Mathematically, a knot is a one-dimensional curve that winds through itself in three-dimensional space and catches its own tail to form a loop. If the loop has no knot in it and can be manipulated and made tangle-free to look like a circle, mathematicians call

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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

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Who gets the faulty offices? Photo by I. Peterson

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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

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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

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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

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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

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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

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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

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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

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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,

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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.

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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,

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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

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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

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Some years ago, at a conference that brought together mathematicians and science writers to discuss ways to inform the public more effectively about mathematics and discoveries in mathematics, a prominent mathematician presented a talk on exciting new findings linking the Riemann Hypothesis, quantum physics, and random matrix models. Though carefully prepared and relying on little more than basic

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The sculpture standing in the plaza near the main entrance to the National Gallery of Canada in Ottawa consists of four rectangular pillars, each one a different color. The steel pillars have a square cross section, and the four pillars mark the corners of a square. Created by prominent Canadian artist Guido Molinari (1933-2004), the sculpture is titled Homage to Samuel Beckett. Some years ago,

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Housed in historic buildings in Washington, D.C., the headquarters of the Mathematical Association of America features an unusual floor tiling in its entry foyer. The tiles are identical pentagons, which fit together to cover the surface. This particular tiling of congruent pentagons had been discovered by Marjorie Rice in 1995, and mathematician Doris Schattschneider adapted the tiling for use

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The Prime Meridian runs through the Royal Observatory, Greenwich, England. Half of a great circle that divides Earth into the Eastern and Western Hemispheres, it is the line of longitude defined to be 0°. Astride the Prime Meridian, Royal Observatory, Greenwich. Unlike the equator and the corresponding parallels of latitude, which are set by Earth's rotational axis, the Prime Meridian has no

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Möbius strips can show up in some unexpected places—including the lyrics of a pop song. Canadian singer and songwriter Nelly Furtado showcases this amazing, one-sided, one-edged object in her song "Hey, Man!" The song leads off her debut album Whoa, Nelly!, which was released in 2000. The surprise comes in the following portion of the chorus, repeated several times throughout the song: We are

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The campus of George Washington University in Washington, D.C., is home to a tall, propeller-shaped object mounted within an elliptical ring. Created by John Safer and titled Limits of Infinity III, this bronze sculpture rises to a height of 15 feet. "I carved the interior shape while groping . . . for a shape which would seem fulfilling, internally harmonious, and perhaps inspiring," Safer

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Many years ago, while I was in high school, I got my first broad view of mathematics from a beautifully illustrated book in the LIFE Science Library. One photograph from this book that has remained lodged in my mind is of the Kresge Auditorium at the Massachusetts Institute of Technology. Kresge Auditorium (designed by Eero Saarinen), Massachusetts Institute of Technology, Cambridge, Mass. The

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A complicated, twisty form stands guard at the entrance to the National Air and Space Museum on the National Mall in Washington, D.C. Created by architect and sculptor Charles O. Perry and installed in 1976, the sculpture is called Continuum. Perry describes his artwork as a Möbius surface of seven saddles, designed to represent the continuum of the universe. In effect, despite its confounding

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Based on a trapezoid split into two triangles, the National Gallery of Art's East Building features walls that meet at odd angles, eschewing the right angles of more conventional structures. One particularly sharp corner, visible to the right of the East Building's H-shaped, west-facing façade, has attracted a lot of attention. There, two walls meet at 19 degrees to form the apex of a narrow

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In Washington, D.C., the National Gallery of Art's East Building, which opened to the public in 1978, features a façade that teases the eye. Designed by architect I.M. Pei, the massive structure is a regimented assemblage of vast walls, skewed polygons, and sharp edges. Walls unexpectedly meet at acute or obtuse angles instead of commonplace right angles. To someone used to the relentlessly

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Sol LeWitt's Four-Sided Pyramid, installed in 1999 at the National Gallery of Art Sculpture Garden in Washington, D.C., consists of concrete blocks precisely stacked to form a stark, eye-catching pyramid. In bright sunlight, the blindingly white blocks and shadows play curious visual tricks on the eye as you view the structure from different angles. Although the blocks are rectangular, each one

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I was saddened to hear this morning that Martin Gardner had died, at the age of 95. I first encountered Martin's writings when I was in the ninth grade, and I bought a copy of the September 1962 issue of Scientific American. This particular issue featured a series of articles on Antarctica, a topic of study in my geography class. But it was Martin's discussion of divisibility tests that really

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Slim and graceful, Kenneth Snelson's Needle Tower stretches 60 feet into the sky. The structure looks too delicate to stand so tall, but it's strong enough to withstand severe storms. Erected in 1968 beside the Hirshhorn Museum and Sculpture Garden on the National Mall in Washington, D.C., this tapered framework of aluminum tubes and stainless-steel cables is an example of a tensegrity

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From one angle, the sculpture looks like a giant, swooping loop. From another angle, it outlines a pair of wings boldly sketched across the sky. Eight feet tall and sixteen feet wide, this stainless-steel sculpture slowly rotates atop a black granite pillar in front of the National Museum of American History in Washington, D.C. As it completes each eight-minute revolution, the glistening loop

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In one famous episode of Homer's Odyssey, the wily Greek hero of the Trojan War Odysseus recounts the tale of how he tricked a one-eyed Cyclops named Polyphemus. Trapped in the Cyclops' cave, Odysseus offers the giant three bowls of fine wine from his ship's stores. The Cyclops downs the wine greedily and asks Odysseus his name so that he can thank Odysseus properly. Odysseus replies that his

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It looks like a spiral staircase that had lost its way before finally winding back to its starting point. The looped sculpture, titled Révolutions, stands in a little park near the entrance to the Papineau Métro station in Montreal. Created by Michel de Broin, the artwork pays tribute to the curving iron stairs that serve as entries to many city residences. Constructed from aluminum, the

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Regularly compiling the NumberADay blog for the MAA website has made me more aware than ever of how much we are surrounded by numbers, from street addresses and highway signs to serial numbers and other identifiers. Curious sequences of numbers may also pop up unexpectedly. While visiting Madison, Wis., for example, I happened to notice that each bus shelter lists, in numerical order, the

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The setting is a morgue. Attendants wheel in two draped bodies. Investigators examine the corpses, painstakingly seeking a link between two murders that appear unrelated. This scene is part of an audacious experiment in mathematical exposition. The vehicle is a screenplay, written by number theorist Andrew Granville and his sister Jennifer Granville, an actor, playwright, and producer. In this

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