Geometreks in New Orleans

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 Museum of Art. The garden is free to the public and well worth visiting.

The largest sculpture in the garden is Kenneth Snelson's tensegrity structure, titled Virlaine Tower. Set in a lagoon and rising 45 feet into the air, this gravity-defying construct consists of stainless steel tubes held together and supported by cables (see "Tensegrity Tower in New Orleans").

Visitors intrigued by curious or striking geometries may also be interested in seeing the following artworks.

Joel Shapiro's untitled work is a striking assemblage of angular blocks that resembles a contorted torso with straining limbs.

Untitled by Joel Shapiro.

In Castle of the Eye, II, Minoru Niizuma created a set of four stacked cubes, with a repeated square-within-square pattern on four sides of each block, to produce a structure reminiscent of a medieval Japanese castle.

Castle of the Eye, II by Minoru Niizuma.

Menashe Kadishman's massive but seemingly unstable sculpture, Open Suspense, features a precariously balanced collection of geometric forms fashioned from distinctively colored Cor-Ten steel.

Open Suspense by Menashe Kadishman.

Sunyatta, by Linda Howard, is made from arrays of aluminum strips fanning out from a central vertical axis to create a grid of light and shadow embodying the transformation of matter into energy (light).

Sunyatta by Linda Howard.

Outside the garden, New Orleans artist Arthur Silverman has based more than 400 sculptures on the tetrahedron, stretching, slicing, skewing, and assembling copies of this form in myriad ways (see "Three Sentinels," "Art of the Tetrahedron," "Art of the Tetrahedron, Revisited," and "Four Corners, Four Faces").

About 20 of Silverman's sculptures are on public display throughout New Orleans, many within walking distance in the downtown portion of the city.

Located at the corner of Poydras and Loyola, Echo consists of a pair of elongated tetrahedra that rise dramatically 60 feet into the air.

Interlocking Boxes, Closed stands outside the front entrance of City Hall. Near the rear entrance, on Poydras Street, another of Silverman's sculptures consists of welded tubing that delineates the edges of stacks of interlocking tetrahedra.

A sculpture near the corner of Poydras and Magazine, titled Painted Trio, consists of three colorful tetrahedra standing on edge.

Painted Trio by Arthur Silverman.

The lobby of a building called Place St. Charles, near Canal Street, features a Silverman sculpture based on the notion of removing tetrahedra from a rectangular block.

The office building at 1555 Poydras Street (directly across from the Superdome) has a large relief on the far wall as you enter. Its basic elements are sections made through a group of tetrahedra attached to each other. A slowly changing light plays over the piece, continually highlighting different areas of the relief.

Further uptown, a streetcar ride away, Silverman has two sculptures on the campus of Tulane University, one in front of the Tulane Law School and another at the A.B. Freeman School of Business. He also created a large outdoor menorah for Temple Sinai, at 6227 St. Charles Avenue.

Sculptor Clement Meadmore is known for his twisted rectangular prisms (see "Bending a Square Prism"). A prime example of his work, titled Out of There, can be found in front of the Hale Boggs Federal Building in New Orleans.

A median park along Diamond Street, near the New Orleans Convention Center, has a number of interesting sculptures. Giro Naito's Diamond is a massive polyhedron with identical facets. Terry Weldon's River Stones features five distorted, intriguingly fingerprinted spheres.

Diamond by Giro Naito.

Looking down as you walk along the streets of New Orleans, you might notice the distinctive tiling of hexagons and rhombuses that decorates local manhole covers.

It's also worth noting the mathematical significance of some curious signs around the city. The Tulane campus has a sign that specifies a speed limit of 23 (a prime number) miles per hour.  If you take the St. Charles streetcar to the end of the line, you'll find an intersection where south meets south. And the warehouse district has a pizza and pasta restaurant that, perhaps inevitably, is named πie.

For those attending JMM, the meeting will itself host a juried exhibition of mathematical art.

Photos by I. Peterson

Martin Gardner’s Möbius Surprise

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 96^th 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 when I recently came across a letter  that he had written to me about 10 years ago, when I was pursuing some investigations into Möbius strips as the basis of artworks.

In replying to my query, Martin remarked that he had, by coincidence, just completed an article about Möbius strips for a children's science magazine.  He noted that the article contained nothing that hadn't been in his Scientific American column on the subject in December 1968. "Except for the following," he added, and he had drawn a diagram (below) to illustrate the construction.

Martin concluded with the following instructions: "Trisect the twisted band, then bisect the untwisted one, and open up for a big surprise!"

Of course, he didn't reveal what the surprise was, so I had to try it for myself.

Initial configuration, with one twisted band and one untwisted band, made from a cross-shaped sheet of paper.

Tangled configuration after cutting.

Final configuration, revealing a small Möbius strip interlocked with a rectangular loop.

As Martin had suggested, the result was a delightful surprise.

Photos by I. Peterson

Möbius in Toronto

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, twisted band that stretches skyward in a wide, stiff loop. I quickly confirmed that the band does, indeed, have the one edge and one side characteristic of a Möbius strip.

I later discovered that the sculpture, created by Toronto artist Lilly Otasevic, is titled Möbius. Her own description of the sculpture acknowledges its mathematical roots and notes the Möbius strip's present-day ubiquity as the underlying form of the three-arrow recycling symbol.

Otasevic says that her sculpture represents "transformation and timeless continuity of natural processes," symbolizing balance and "our unity with nature." Many of her other artworks have also been inspired by nature and natural processes, especially the interplay between light and shadow and interrelationships between organic and inorganic matter, as seen in crystals, Fibonacci spirals, cellular structures, and elsewhere.

Otasevic joins a growing number of artists who have found inspiration in the wonderful mathematical discovery of August Ferdinand Möbius, a list that includes Max Bill, Charles Perry, and others.

Photos by I. Peterson

Palace of Mirrors

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 sparkles in the sunlight. At night, a single candle, reflected multiple times, is enough to illuminate the room.

Photos by I. Peterson

Golden Circles

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 desert, near the border between India and Pakistan, the town's strategic position brought it wealth, and its merchants and government officials built elaborate mansions (havelis) out of this yellow sandstone.

Elaborate mansions (havelis) with overhanging balconies flank narrow streets within Jaisalmer Fort.

The sandstone is relatively soft, so it can be carved easily into elaborate patterns and intricate latticework, evident throughout the town.

Balcony designs can be highly intricate, with both relief carvings and latticework.

Set on either side of narrow, winding lanes within Jaisalmer Fort, these mansions stand as monuments to the anonymous stone carvers who covered seemingly every square inch with exquisite patterns. One mansion features 38 balconies, each one with a different design.

Intricate circular designs are common features of the sandstone carvings in Jaisalmer.

Latticed friezes provided ventilation and privacy for women, who could peek out without anyone seeing them.

Photos by I. Peterson

Tilings at the Taj Mahal

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 were designed around principles of reflection and repetition. The tomb itself is essentially a cube with chamfered corners, to give it an octagonal cross section (see "Octagons and Squares"). The four sides are identical, each one featuring a huge vaulted archway. Four minarets frame the tomb.

The Taj Mahal's symmetrical structure is evident in this northward view of the tomb.

Reflection symmetries also abound in the decorations, made from precious and semiprecious stones inlaid on white, translucent marble (below).

Even the reflecting pools of water add to the sense of exquisite symmetry throughout the site.

At dawn, the eastern entryway to the Taj Mahal complex. 

But it's also worth looking down—at the intriguing tiling patterns of the paving stones that cover the ground around the Taj Mahal.

Next to the tomb, the stones lie in a distinctive pattern of four-pointed stars (red sandstone) and diamonds (marble).

Reflection symmetries characterize the pattern of paving stones surrounding the Taj Mahal.

Farther away, the tiling pattern consists of four-pointed stars and elongated hexagons (above).

Even the drainage holes in some of the stones have a striking hexagonal pattern.

In other locations, the tiling pattern combines regular hexagons with six-pointed stars (above).

And amid the symmetrical gardens in front of the Taj Mahal, walkway stones are laid in a pattern that combines squares and elongated hexagons to create regular octagons.

All in all, the Taj Mahal is surely one of the world's most impressive and beautiful examples of the use of symmetry in architecture and design.

Photos by I. Peterson

Giant Sundial

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 capital and fabricated out of masonry, marble, and bronze between 1727 and 1734, the dozen or so instruments that constitute the collection were used for making remarkably precise determinations of astronomical position without the aid of telescopes.

Together, the devices could be used to measure time, predict eclipses, track star locations, ascertain declinations of planets, and determine celestial altitudes.

The largest instrument (samrat yantra) casts a shadow that tells the time of day. Its triangular gnomon, 90 feet high, is angled at 27°, Jaipur's latitude. The triangle's hypotenuse rests parallel to Earth's axis. A quadrant of a circle lies on either side of the gnomon, parallel to the plane of the equator.

The small cupola at the top of the samrat yantra was used as a platform for announcing eclipses and the arrival of the monsoon season.

Another astronomical instrument (jai prakash yantra) consists of a pair of large hemispherical bowls, which serve as a reflection of the sky above. Crossing wires stretched across each bowl hold a centered metal ring so that every point in the sky can be reflected to a corresponding point on the bowl through the ring.

The two bowls complement each other. The open spaces in one correspond to surfaces in the other. The cutouts allow observers to view positions without inadvertently blocking the light. When a shadow falls within a cutout in one bowl, an observer simply moves to the other bowl.

Bronze devices related to astrolabes allowed the measurement of time and the positions of celestial objects. 

Telling time from the smaller of the two giant sundials in the Jantar Mantar.

In July, the Jantar Mantar in Jaipur was added to the UNESCO World Heritage List as an "expression of the astronomical skills and cosmological concepts of the court of a scholarly prince at the end of the Mughal period" in Indian history.

Photos by I. Peterson

Octagons and Squares

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 elongated hexagons.

A variant (below) of this design, with stretched octagons and rectangles, can be seen at the Red Fort in Agra.

Octagons and squares are also an important feature of the Taj Mahal and associated structures.

A structure near the Taj Mahal in Agra features an alternative combination of octagons and squares (above). Note that the interior of each octagon consists of four squares and eight pentagons.

The Taj Mahal itself (above) is essentially a square with cut-off corners to create an octagonal cross section with alternating long and short sides. The inner tomb chamber is a regular octagon.

Two overlapping squares create an eight-pointed star.

The same eight-pointed star can be seen in the Taj Mahal's gardens, surrounding by a pathway tiled with overlapping octagons made from squares and hexagons.

Amer Fort in Jaipur has a pool in the shape of an eight-pointed star, surrounding an octagonal island.

A complex octagonal design (above) carved out of red sandstone at Fatehpur Sikri.

A floor tiling combines regular octagons with octagons having alternating long and short sides.

In North America, we are used to seeing octagons in the guise of stop signs. Curiously, in India, stop signs are usually circular.

A circular stop sign in Udaipur, India.

Photos by I. Peterson