The Mathematical Tourist Dirk Huylebrouck, Editor

New YorkMathematicalTouristsJOE HAMMER

Does your hometown have any

mathematical tourists attractions such

as statues, plaques, graves, the cafe

where the famous conjecture was

made, the desk where the famous

initials are scratched, birthplaces,

houses, or memorials? Have you

encountered a mathematical sight on

your travels? If so, we invite you to

submit an essay to this column. Be

sure to include a picture, a

description of its mathematical

significance, and either a map or

directions so that others may follow

in your tracks.

Please send all submissions to

Mathematical Tourist Editor,

Dirk Huylebrouck, Aartshertogstraat 42,

8400 Oostende, Belgium

e-mail: dirk.huylebrouck@gmail.com

TTourists sometimes claim thatwhatever they seek in any citywill be found in New York.

The city is known for its museums, artscenters, finance and corporate head-quarters. There are numerous ‘‘mustsee’’ sites, including the Statue ofLiberty, Empire State Building, TimesSquare and the Broadway theaterscene. In this article, the mathematicaltourist will virtually visit two sciencemuseums and an aviary, all featuringinteresting architectural geometry.Their exhibits are complementary, pro-viding a vista of the sciences of today.

The Rose CenterFirst we visit the Rose Center, a wing ofthe American Museum of Natural His-tory on the west side of Central Parkbetween 77th and 81st Streets. It is thelargest institution of its kind, boast-ing more than 36 million artifacts andspecimens. Twenty-five interconnectedbuildings in an area of 1.25 million sq ft(about 116,000 m2) include 46 exhibi-tion halls, several research labs and aninvaluable library. In its 140-year his-tory, it has undergone numerous devel-opments, extensions, alterations andmodernizations, so it is not surprisingthat the buildings exhibit hybrid styles.

The Rose Center for Earth and Spaceis a recent addition to the museum,opened in 2002. Its main purpose is tomake astronomy, astrophysics andearth sciences accessible to the public.Architects James S. Polshek and ToddH. Schliemann succeeded in designinga landmark complex, adding a 350,000sq ft space to the museum, with exhi-bition areas designed by Ralph Appel-baum and Associates.

The centerpiece of the complex isthe 87 ft diameter sphere, the HaydenSphere, clad with spherical trapezoid-shaped aluminum panels (Fig. 1). Thefabrication of these panels requiredprecision to achieve seamless jointsalong both longitudinal and latitudinaldivision lines on the sphere (Fig. 2).Because of spherical symmetry, only a

small number of different sized panelswere needed, making mass productionpossible. The skeleton of the sphereis a longitudinal and latitudinal steelframe grid on which are laid the pan-els. The skeleton is held together by amassive ring truss built along its‘‘equator.’’

The giant sphere is encased in a 120ft glass cube. Its huge walls are madefrom totally transparent clear water-white glass that creates the illusion thatthe sphere is floating weightlessly inspace. That look is further enhanced byplacing the supporting stainless steeltruss structure of the walls at a distancefrom them so as not to obstruct thetransparency of the glass. Moreoverthere are no visible supporting columnsor suspending cables to the sphere atany level of the complex.

Suspended from the ceiling of thecube are 13 smaller colored spheressurrounding the Sphere. One depictsSaturn, easily recognized by its systemof rings. Another is apparently Jupiterwith its colored bands. The impressionis that the Sphere (the ‘‘sun’’) with itssurrounding ‘‘planets’’ represents oursolar system or perhaps some otherpart of the universe, or it might be amonumental geometric sculpture withcelestial themes.

The Sphere is lit at strategic pointswith fluctuating blue fluorescent lightscreating the illusion that it rotates in thecompany of the rest of the stars in thesky. It is a nighttime excursion site notto be missed. All this is best seen fromthe adjacent Terrace Garden, itself acosmic-related exhibition area. How-ever, the silver sheen Sphere with itssurrounding planets and the envel-oping cube are not just architecturalbravado. They have multitudinousfunctions to facilitate the exhibitions ofthe Center. Along the above mentionedring truss, the Hayden Sphere is dividedinto two hemispheres. The upper andbigger portion houses a Space Theaterincorporating the Hayden Planetar-ium (named after the founder of the

40 THE MATHEMATICAL INTELLIGENCER � 2009 Springer Science+Business Media, LLC

previous planetarium). It has a 68 ftspherical dome screen. The planetar-ium is equipped with a one-of-a-kindcustom-made Zeiss Star Projector,capable of projecting over 9,000 stars ina night sky. The lower hemispherehouses the 46 ft diameter Big BangTheater. At the bottom of this theater isan 8 ft deep, 36 ft diameter concavespherical screen on which the birth ofthe universe is simulated.

Emanating from the Big Bang The-ater is a 364 ft long, 8 ft wide spiral rampcalled the Cosmic Pathway, on which a13 billion year linearly scaled history ofthe universe is chronicled, from thetime of the Big Bang to the present. The200 computer generated illustrationsare accompanied by explanatory textand diagram panels. The fascinatingfinale of this journey is the length ofhuman history, which is just about thebreadth of a human hair, as one inchcorresponds to about 3 million years.Fabricating the geometry of the rampwas a challenging engineering prob-lem. Its surface bends in two directions:Winding one and a half times aroundthe lower half of the Hayden Sphere,and simultaneously descending gradu-ally with about a 16 ft fall.

An antecedent of the Spiral Ramp,just across Central Park, on the corner ofFifth Avenue and 89th Street, is FrankLloyd Wright’s Guggenheim Museum.The geometry of its main exhibitiongallery is a spiral ramp too, but differingstructurally. It is a 1,320 ft long rampspiraling around six times, creating a

rotunda atrium with a circular skylightdome on the top. The ramp is cantile-vered from massive concrete walls thatare also the walls of the building itself.The second antecedent originates fromgenetics. In 1953, when the doublehelical structure of the DNA was dis-covered, the ‘‘helix’’ became a buzzword and subsequently an inspirationfor art and architecture.

In the Hall of Universe, the bottompart of the Hayden Sphere can be seen23 ft above the ground—its illusion offloating appearing here most vividly. Itssecret lies in the clever placement of theattachment points of the supportingcolumns to the Sphere (Fig. 2). Thesecolumns consist of three V-shaped steeltubes planted on the ground of the Hall.Remarkably, from no point can thethree columns be seen together. Theirsix branches enter the aluminum clad-ding of the Sphere through six holes:There is no direct contact with the alu-minum surface. The supporting tubesare attached to the ring-truss inside theSphere. Additionally, branching fromthe V-tubes, several supplementarytubes support the torsion tube of thespiral pathway. These columns them-selves avert the attention from the mainrole of the V-tubes; it appears as iftheir only role is to support the spiralpathway.

On the corner of Columbus Avenueand 79th Street is the newest addition tothe museum, the Weston Pavilion,opened in 2002. It is an attractive43 ft high, 1,800 sq ft glass box; a

mini-version of the Rose Cube. Itholds a collection of items related toastronomy. The centerpiece is an 18 ftdiameter galactic armillary sphere—astainless steel and aluminum sculpturesuspended from the ceiling of the glasscube (Fig. 3). This is a modern versionof its ancestors—the Chinese and theGreek armillaries of antiquity where theEarth was placed at the center.

At the entry plaza of the Pavilion is astainless steel 5 9 5 9 5 ft sculpture—the New York Times’s Capsule. It is atime capsule in which late twentieth-century artifacts have been stored, col-lected from all over the world. Theplanned opening date is at the end ofthis millennium. The sculpture wasdesigned by the Spanish architect San-tiago Calactrava. Its symmetric concavesurface is meant to be a topologicalimage of a sphere and is an emblem ofthe museum. Bystanders might liken itto a budding flower (Fig. 4). Usually, atime capsule is buried for safety, butthis capsule is visible, touchable andeven climbable.

The New York Hall of ScienceThe second museum in this visit is theNew York Hall of Science, on thegrounds of Flushing Meadows CoronaPark. The architectural icon of thismuseum is the Great Hall (Fig. 5),originally built as a science pavilion forthe 1964 World’s Fair. Architect WallaceK. Harrison is known for designing theUnited Nations Headquarters in New

Figure. 1. The Hayden Sphere encased in the glass cube

showing a V column and some of the smaller spheres (photo:

Tania Hammer).

Figure. 2. Close-up view of a part of the sphere showing

Saturn and Jupiter, a supporting column entering the sphere,

and some aluminum panels (photo:Tania Hammer).

� 2009 Springer Science+Business Media, LLC, Volume 31, Number 4, 2009 41

York, as well as the Metropolitan OperaHouse at Lincoln Center. The Hall is an80 ft high 7,000 sq ft building with afree-form asymmetric floor plan. Thiscolumn-free structure consists of a mereone-foot thick single concrete wall. Theconcrete is formed into 20 verticallysinuous undulations with the two endsoverlapping, the space between pro-viding an unusual entrance doorway.The geometry of the undulations iscomposed by arcs of circles of 12.5 ft

radius. The entire wall is formed into agrid of an amazing 5,400 mini-windowsall fitted with cobalt blue glass. Insidethe Hall, the daylight filters throughthis billowing, multifaceted, blue glass-filled windows creating a feeling ofbeing enveloped by a starry night sky(Fig. 6).

Exhibition galleries were builtunderneath the Hall in a hexagonalshaped area. Years later, a cylindricalentry atrium and an oval shaped

auditorium were added. The decisivebreakthrough in the development ofthe museum was in 2004 when it wasextended to provide 55,000 sq ft moreexhibition space and teaching facilities.Architecturally, the key building—theHall of Light—does not seek to com-pete with the revered Great Hall,instead affording geometric counter-points (Fig. 5). The new building is along, slightly folded, angular, hori-zontal, low-lying form, contrastingwith the towering, vertical curved formof the old building. The galleries houseover 400 constantly updated computeraided do-it-yourself exhibits in biology,chemistry, physics and aerodynamics,and an entire section devoted to math-ematics. Features include topics inprobability theory, projective geome-try, topology, celestial mechanics andthe history of mathematics. The exhi-bition, called ‘‘Mathematica: A Worldof Numbers and Beyond,’’ wasdesigned by the educator and artistcouple Charles and Ray Eames in1961. Some of their work can be seenonline.

One of the numerous interactiveexhibits is in the probability section:The formation of a bell-shaped (ornormal) curve is shown on a devicewhere hundreds of small, identicallysized plastic balls cascade through amaze of steel pegs. At any stage you canstop the cascading to see the shape ofthe developing bell. Another highlightis in the graph theory station, calledRopes and Pulleys, designed by Kayle

Figure 3. The armiliary inside the Weston Cube (photo: Tania

Hammer).

Figure 4. The Times’s capsule (photo: Gael Hammer).

Figure 5. Aerial view of New York Hall of Science. Note in the left at mid distance

the geodesic dome (photo: courtesy and �Steven Turner).

42 THE MATHEMATICAL INTELLIGENCER

Dries. This is a network of over 100pulleys linked by cables. It respondsdynamically to force by rotating wheelssituated in front of the network. Move-ment in one place ripples across theentire network, and you can never seethe same pattern twice. It demonstrateshow sensitive networks can be. Withjust small adjustments, the shapes canchange much.

The SurroundingsAt a five-minute’s walk from themuseum is a ‘‘geodesic dome’’ (Fig. 5)designed by R. Buckminster Fullerfor the 1967 World’s Fair. Today it is anaviary with 50 species of Americanbirds. The dome, 85 ft high and 135 ftin diameter, is an aluminum networkof spherical triangles, the inside spacebeing free of supporting columns.

Subsequently, Buckminster Fullerdesigned several domes, of differenttypes and for different purposes, acrossNorth America. Modified versions havenow been built all over the world, arecent one being the New MariinskyTheater in St Petersburg, Russia, byFrench architect Dominique Perrault. Asurprising analogue of Fuller’s domeswas discovered in 1985 by Harold Crotoin Sussex, England, who found a newform of the element carbon: Since it wasa cluster of 60 carbon atoms making aspherical cage resembling a geodesicdome, he consequently named it ‘‘Ful-lerene 60.’’ Subsequently, more fuller-enes and other new molecules fromclusters of atoms with highly symmet-rical geometries were produced in‘‘molecular architecture.’’

Our last stop is Flushing MeadowsCorona Park, the site of the 1964World’s Fair. A 120 ft-diameter steelterrestrial globe, the ‘‘Unisphere,’’ waserected in the Park (Fig. 7) as a symbolfor world peace. Its dedication reads,‘‘Peace Through Understanding.’’

ACKNOWLEDGMENTS

Thanks are due to V. Guy Maxwell,

AIA, of Polshek Partners, and Ed Mur-

ray of New York Hall of Science for

providing information.

LITERATURE

Ball, P. 2005, A New Kind of Alchemy. New

Scientist 495(2), pp. 30–33.

Buckminster Fuller, R. 1977, Synergetics.

New York: Macmillan.

Castelvecchi, D. 2008, Small but Super.

Science News 173, pp. 14–19.

Futter, E. 2001, Rose Center for Earth and

Space: A Museum for the Twenty-First

Century. A Collection of essays.

Laiserin, J. 2000, Many Models Make the

Rose Center. Architectural Record 118,

pp. 171–179.

Lyal, S. 2002, Masters of Structure. London:

Laurence King Publishing.

Stephens, S. 2005, Polshek Partnerships’ Hall

of Science. Architectural Record 193, pp.

224–229.

School of Mathematics

University of Sydney

Sydney, NSW 2006

Australia

e-mail: joeh@maths.usyd.edu.au

Figure 6. Inside the Great Hall showing the blue glass-filled wall (photo: Gael

Hammer).

Figure 7. Unisphere (photo: Gael Hammer).

� 2009 Springer Science+Business Media, LLC, Volume 31, Number 4, 2009 43