Polyhedron Man Spreading the Word About the Wonders of Crystal-Like Geometric Forms

Polyhedron Man Spreading the word about the wonders of crystal-like geometric forms

Ivars Peterson

A towering stack of 642 compact disks could represent an impressive music or software collection. The same number of CDs strung together into iridescent arcs can be a mathematical sculpture.

Such a glittering assemblage, more than 6 feet in diameter, hangs in an atrium of the computer science building at the University of California, Berkeley. Titled "Rainbow Bits," the sculpture slowly rotates in response to air currents. Viewers experience a kaleidoscope of fleeting glints, dazzling colors, and shifting shadows.

Created by George W. Hart of Northport, N.Y., the sculpture is more than an intriguing visual effect. Its strands of recycled CDs trace the edges of a novel mathematical solid—a polyhedron George Hart's "Rainbow Bits". made up of 20 equilateral triangles and 60 kite-shaped Séquin quadrilaterals.

This particular solid, which Hart dubs a propellorized icosahedron, is one example of a newly defined class of polyhedra. "As far as I know, these forms have never previously been described by mathematicians," Hart says.

Hart's Berkeley sculpture is just one of his many creations based on polyhedra. Hart has also collaborated with mathematicians and computer scientists to define and visualize a variety of new mathematical shapes.

"As a sculptor of constructive geometric forms, I try to create Computer-generated illustration of engaging works that are enriched by an underlying geometrical the geometric solid—a propellor depth," Hart says. "I share with many artists the idea that a pure form icosahedron—that served as the is a worthy object." basis for the sculpture shown above. Hart Such forms tantalize artists and scientists alike. "George Hart is a very special individual, a gifted mathematician, a creative artist, and an inspiring teacher," says Berkeley computer scientist Carlo H. Séquin. "He is among the very best of a rather small group of people who successfully integrate art and mathematics. Hart's geometrical constructions are intriguing, educational, and artistic at the same time."

Passion for polyhedra

The sculptor's passion for polyhedra was already evident by the time he had reached his teen years. He enjoyed making large structures, including mathematical solids, by painstakingly George Hart's "Twisted Rivers." gluing together hundreds upon hundreds of toothpicks. Inspired Hart by a book about mathematical models, he later took to constructing intricate polyhedra out of paper.

Hart majored in mathematics, then earned graduate degrees in linguistics and in electrical engineering and computer science. He taught computer and engineering courses at Columbia University and Hofstra University in Hempstead, N.Y. Along the way, he wrote a textbook on multidimensional analysis, a branch of mathematics that relates physical measurements to one another. He also developed and patented methods for determining the energy consumption of individual appliances from measurements of total power used by a household.

Hart is currently a visiting scholar with the computational geometry group at the applied mathematics and statistics department of the State University of New York at Stony Brook. He devotes the bulk of his time, however, to his varied polyhedral pursuits.

"Polyhedra have an enormous aesthetic appeal," Hart contends. Moreover, "the subject is fun and easy to learn on one's own," he notes.

Since 1996, Hart has been building a remarkable Web site devoted to all things polyhedral (http://www.georgehart.com). Its enormous catalog of polyhedra now includes many solids never previously illustrated in any publication.

Hart's Web pages "provide important information for my university courses on geometric modeling," Séquin says. Furthermore, he adds, "they provide inspiration for me as a part-time artist, and they provide sheer visual pleasure for geometrically inclined minds." George Hart's "72 Pencils." Linking polygons Hart

Polyhedra are made by linking triangles, squares, hexagons, and other polygons to form closed, three-dimensional objects. Different rules for linking various polygons generate different types of polyhedra.

The Platonic solids, known to the ancient Greeks, consist entirely of identical regular polygons, which are defined as having equal sides and equal angles. There are precisely five such objects: the tetrahedron (made up of four equilateral triangles), cube (six squares), octahedron (eight equilateral triangles), dodecahedron (12 regular pentagons), and icosahedron (20 equilateral triangles). If you relax the conditions for generating the Platonic solids and specify that faces must be regular but not necessarily identical polygons, there are 13 such polyhedra, known as Archimedean solids. One example is the truncated icosahedron, familiar as the pattern on a soccer ball, which consists of 12 pentagons and 20 hexagons.

Polyhedra also can be constructed from polygons that don't necessarily have equal sides and equal angles, and they may be indented or spiky, like three-dimensional stars. The great stellated dodecahedron, first described by 17th-century mathematician Johannes Kepler, is one example of a spiky polyhedron. Nonindented polyhedra, such as the A reconstruction in cherry wood Platonic solids, are called convex. of Leonardo da Vinci's drawing of a truncated icosahedron. Hart has delved deeply into the history of the discovery and Hart representation of polyhedra. During the Renaissance, he notes by way of example, Leonardo da Vinci invented a way to show the front and back of a polyhedron without confusion by depicting edges as if they were made from wooden laths and leaving the polyhedron's faces open.

Inspired by Leonardo's drawings of wooden models, Hart has crafted his own beautiful models of these designs. He has also programmed computers that guide stereolithography machines to produce, layer by layer, three-dimensional plastic replicas of the models.

With a vast palette of geometric solids to choose from, Hart has fashioned polyhedral models and sculptures from all sorts of materials and objects, including Plexiglas, brass, papier-mâché, pipe cleaners, pencils, floppy disks, toothbrushes, paper clips, and A plastic stereolithography model, 8 plastic cutlery. centimeters in diameter, of an "elevated rhombicuboctahedron," He "brings the ideal forms of the polyhedra into the material world originally depicted by Leonardo da with wit and humor," says computer programmer and artist Bob Vinci. Brill of Ann Arbor, Mich. "The austere beauty of the polyhedra is Hart thus made accessible and familiar."

New possibilities

Given the lengthy history associated with polyhedra, it may seem astonishing that there are still mathematical discoveries to be made. Hart has proved particularly adept at identifying rules and variants that lead to new possibilities. For example, his discovery of propellorized polyhedra came out of a new notation for describing polyhedra, proposed by mathematician John H. Conway of Princeton University. In Conway's scheme, a capital letter specifies a "seed" polyhedron. For example, the Platonic solids are denoted T, C, O, D, and I. A lower-case letter then specifies an operation. The combination tC means truncate a cube; that is, cut off the corners of a cube to form a new solid with six octagonal and eight triangular faces. Conway's simple notation can be used to derive the Archimedean solids and infinitely many other symmetric polyhedra.

In the course of writing a computer program to implement Conway's A wooden model of a type of notation, Hart tried to think of a simple operator that wasn't yet symmetrohedron. This particular included in Conway's list. He came up with what he called the geometric solid consists of 20 propellor [sic] operation: Start with a polyhedron, then spread apart regular enneagons (nine-sided polygons), 12 regular pentagons, the faces and introduce quadrilaterals so that each face is surrounded and 60 isosceles triangles. by a ring of these forms. Hart "The [propellor] operation can be applied to any convex polyhedron and combined with other operations to produce a variety of new forms," Hart says. "These provide a storehouse of intriguing structures, some of which have inspired artworks."

Hart based his design for "Rainbow Bits" on an icosahedron transformed via his propellor operator into a three-dimensional mosaic of 20 equilateral triangles and 60 kite-shaped quadrilaterals. The resulting sculpture required one CD at each vertex, five CDs along each of the longer edges, and three CDs along each of the shorter edges. The positions and angles of slots cut into the CDs determined their placement.

"For many visitors, this stunning sculpture is a main attraction of our building, particularly when it is hit by rays of sunlight and [sprays] intense colored spots over the atrium walls," Séquin says.

Generating novel forms

Working with Craig S. Kaplan of the University of Washington in Seattle, Hart has recently invented a procedure for generating many polyhedral forms, including new types of polyhedra. Dubbed symmetrohedra, these geometric solids result from the symmetric placement of regular polygons (see illustration below).

The new technique generates a variety of known polyhedra, including most of the Archimedean solids, and infinitely many new ones. One particularly striking product is a polyhedron made up of 20 regular nine-sided polygons (called enneagons), 12 regular pentagons, and 60 isosceles triangles. "You rarely see enneagons in a polyhedron," Hart comments.

In another recent effort, Hart and computer scientist Douglas Zongker of the University of Washington developed Steps showing the construction of a cuboctahedral symmetrohedron a novel method for constructing from a set of octagons (left) and a set of nine-sided enneagons polyhedra by blending together two or (middle left). These polygons slide inward until they meet (middle more existing polyhedra. right). Trapezoidal faces fill in the holes between the mated polygons (right) to create a novel polyhedron. Kaplan

Hart, Zongker, and Kaplan all described their new visualization methods at a meeting that highlighted mathematical connections in art, music, and science, held last July at Southwestern College in Winfield, Kan. Blending an octahedron (top left) and a dodecahedron (bottom left) produces another sort of polyhedron (right). Zongker Beauty and joy

Whether assembling an intricate polyhedral design, leading a model-building workshop, or describing his latest ventures, Hart brings an infectious enthusiasm to his task. To help spread the word about the beauty and joy of polyhedra, he recently coauthored a book on building models using Zome plastic components, produced by Zometool of Denver. Embodying mathematical ratios such as the golden mean, the system's balls and sticks lend themselves to polyhedron building. On another communications front, Hart is now writing a history of polyhedral geometry in art.

"The best way to learn about polyhedra is to make your own paper models," Hart insists. For those who prefer to skip the spilled glue, misshapen corners, ill-fitting edges, and hours of George Hart (right) with high-school student Thomas Engdahl (left), David work, Hart's Web site provides interactive, three-dimensional Richter (middle) of Southeast Missouri virtual models that can be observed from different perspectives. State University, and an elaborate model constructed from Zome Brill says about Hart, "His energy, his curiosity, his components. inventiveness appear to be boundless." Those qualities come in D. Richter handy when exploring the infinite realm of polyhedra.

References:

Hart, G.W. In press. In the palm of Leonardo's hand. Nexus Network Journal.

______. 2001. Rapid prototyping of geometric models. Available at http://www.georgehart.com/cccg/rpgm.html.

______. 2000. Solid-segment sculptures. Available at http://www.georgehart.com/solid-edge/solid- edge.html.

______. 2000. Sculpture based on propellorized polyhedra. Available at http://www.georgehart.com/propello/propello.html.

Hart, G.W., and H. Picciotto. 2001. Zome Geometry: Hands-on Learning with Zome Models. Emeryville, Calif.: Key Curriculum Press. See http://www.georgehart.com/zomebook/zomebook.html.

Kaplan, C.S., and G.W. Hart. 2001. Symmetrohedra: Polyhedra from symmetric placement of regular polygons. In Bridges: Mathematical Connections in Art, Music, and Science Conference Proceedings, R. Sarhangi and S. Jablan, eds. See http://www.sckans.edu/~bridges/.

Zongker, D., and G.W. Hart. 2001. Blending polyhedra with overlays. In Bridges: Mathematical Connections in Art, Music, and Science Conference Proceedings,>, R. Sarhangi and S. Jablan, eds. See http://www.sckans.edu/~bridges/.

Further Readings:

Demaine, E.D., et al. Preprint. Vertex-unfoldings of simplicial manifolds. Available at http://xxx.lanl.gov/abs/cs.CG/0110054.

Hart, G.W. 1999. Zonohedrification. Mathematica Journal 7(No. 3):374-389.

______. 1995. Multdimensional Analysis: Algebras and Systems for Science and Engineering. New York: Springer-Verlag. See http://www.georgehart.com/research/multanal.html.

Images of some of George Hart's geometric sculptures are available at http://www.georgehart.com/sculpture/sculpture.html.

Learn more about George Hart's "Rainbow Bits" at http://www.cs.berkeley.edu/~sequin/SCULPTS/RAINBOW/index.html and http://www.georgehart.com/sculpture/rainbow-bits.html.

Information about the "Bridges: Mathematical Connections in Art, Music, and Science" conference can be found at http://www.sckans.edu/~bridges/.

Information about the Conway notation for polyhedra can be found at http://www.georgehart.com/virtual-polyhedra/conway_notation.html. Zometool has a Web site http://www.zometool.com/.

Additional information about the Zome model of a rectified 600-cell, constructed by David Richter and his coworkers, can be found at http://cstl-cst.semo.edu/richter/bridgeszome.htm.

Sources:

Bob Brill Web site: http://users.migate.net/~bobbrill/

George W. Hart Web site: http://www.georgehart.com/.

David Richter Department of Mathematics Southeast Missouri State University Cape Girardeau, MO 63701 Web site: http://cstl-cst.semo.edu/richter/.

Carlo H. Séquin University of California, Berkeley College of Engineering EECS Computer Science Division Berkeley, CA 94720-1776 Web site: http://www.cs.berkeley.edu/~sequin/.

From Science News, Vol. 160, No. 25, Dec. 22, 2001, p. 396.

x 11 m

- t,- tt PolyhedronMan ` Spreading the word about the wonders of crystal-like

I..v- geometric forms I

By IVARSPETERSON George Hart's 'Twisted Rivers.'

A towering stack of 642 ing large structures, including compact disks could rep- mathematicalsolids, by painstak- resent an impressive ingly gluing together hundreds music or softwarecollection. The upon hundreds of toothpicks. same number of CDs strung Inspiredby a book about mathe- together into iridescent arcs can matical models, he later took to be a mathematicalsculpture. 0 constructing intricate polyhedra Such a glittering assemblage, out of paper. more than 6 feet in diameter, Hart majored in mathematics, hangs in an atriumof the comput- then earned graduate degrees in er science buildingat the Univer- linguistics and in electrical engi- sity of California,Berkeley. Titled neering and computer science. "Rainbow Bits," the sculpture He taught computer and engi- slowly rotates in response to air neering courses at ColumbiaUni- currents. Viewers experience a versity and Hofstra University in kaleidoscope of fleeting glints, Hempstead, N.Y.Along the way, dazzling colors, and shifting he wrote a textbook on multidi- shadows. mensional analysis, a branch of Created by George W Hart of mathematics that relates physi- Northport, N.Y.,the sculpture is cal measurements to one anoth- more than an intriguingvisual ef- er. He also developed and patent- fect. Its strands of recycled CDs ed methods for determining the trace the edges of a novel mathe- energy consumption of individual matical solid-a polyhedron appliances from measurements made up of 20 equilateral trian- George Hart(right) with high-school student Thomas of total power used by a house- gles and 60 kite-shapedquadrilat- Engdahl(left), David Richter (middle) of Southeast hold. erals. MissouriState Universityand an elaboratemodel Hart is currently a visiting This particular solid, which constructedfrom Zome components. scholar with the computational Hartdubs a propellorized icosa- geometrv grouD at the aDDlied hedron, is one example of a newly de- "Asa sculptor of constructive geomet- mathematics and statistics department fined class of polyhedra. "As far as I ric forms, I try to create engagingworks of the State University of New York at know, these that are enriched by an underlyinggeo- Stony Brook.He devotes the bulk of his forms have never metrical depth,"Hart says. "Ishare with time, however, to his varied polyhedral previously been many artists the idea that a pure form is pursuits. described by a worthy object." "Polyhedrahave an enormous aesthet- mathematicians," Such forms tantalize artists and scien- ic appeal,"Hart contends. Moreover,"the Hartsays. tists alike. "GeorgeHart is a very special subject is fun and easy to learn on one's Hart's Berke- individual,a gifted mathematician,a cre- own,"he notes. ley sculpture is ative artist, and an inspiring teacher," Since 1996, Hart has been building a just one of his says Berkeley computer scientist Carlo remarkable Web site devoted to all many creations H. Seciuin. "He is among the very best of things polyhedral (http://www.george- based on polyhe- a rathersmall groupof peo- hart.com). Its enormous dra. Hart has ple who successfully inte- catalog of polyhedra also collaborated with grate art and mathematics. now includes many mathematicians and Hart's geometrical con- solids never previously computer scientists to structions are intriguing, illustrated in any publi- define and visualize a educational, and artistic at cation. variety of new mathe- the same time." Hart'sWeb pages "pro- maticalshapes. vide important information for my university George Hart's "Rainbow Bits". (top) he sculptor's passion courses on geometric Computer-generated illustrationof the for polyhedra was modeling,"Sequin says. geometric solid-a propellor icosahedron alreadyevident by the Furthermore, he adds, (bottom)-that served as the basis for the time he had reached his "theyprovide inspiration sculpture. teen years. He enjoyedmak- George Hart's "72 Pencils.." for me as a part-time

396 SCIENCE NEWS, VOL. 160 DECEMBER22 & 29, 2001

This content downloaded from 220.156.171.76 on Sat, 2 Nov 2013 00:24:57 AM All use subject to JSTOR Terms and Conditions artist, and they provide sheer visual dra,such as the Platonicsolids, are called cian John H. Conwayof PrincetonUniver- pleasure for geometrically inclined convex. sity. In Conway'sscheme, a capital letter minds." Harthas delved deeply into the history specifies a "seed"polyhedron. For exam- of the discovery and representation of ple, the Platonic solids are denoted T, C, polyhedra. During the Renaissance, he 0, D, and I.A lower-caseletter then speci- Polyhedra are made by linkingtrian- notes by way of example, Leonardo da fies an operation. The combination tC gles, squares, hexagons, and other Vinci invented a way to show the front means truncatea cube; that is, cut off the polygons to form closed, three- and back of a polyhedronwithout confu- corners of a cube to form a new solid dimensional objects. Different rules for sion by depicting edges as if they were with six octagonal and eight triangular linkingvarious polygons gen- made fromwooden laths faces. Conway'ssimple notation can be erate differenttypes of poly- and leaving the polyhe- used to derive the Archimedean solids hedra. dron'sfaces open. and infinitely many other symmetric The Platonic solids, known Inspired by Leonar- polyhedra. to the ancient Greeks,consist do's drawingsof wooden Inthe course of writinga computerpro- entirely of identical regular models, Harthas crafted gram to implement Conway's notation, polygons, which are defined his own beautifulmodels Hart tried to think of a simple operator as having equal sides and of these designs. He has that wasn't yet included in Conway'slist. equal angles. There are pre- also programmed com- He came up with what he called the pro- cisely five such objects: the puters that guide stere- pellor [sic] operation:Start with a polyhe- tetrahedron(made up of four A reconstruction in cherry olithography machines dron, then spread apart the faces and in- equilateral triangles), cube wood of Leonardo to produce, layer by lay- troducequadrilaterals so that each face is (six squares), octahedron da Vinci's drawing of a er, three-dimensional surroundedby a ringof these forms. (eight equilateral triangles), truncated icosahedron. plastic replicas of the "The[propellor] operation can be applied dodecahedron (12 regular models. to any convex polyhedronand combined pentagons),and icosahedron (20 equilat- Witha vast palette of geometric solids with other operationsto producea variety eral triangles). to choose from,Hart has fashioned poly- of new forms,"Hart says. "Theseprovide a Ifyou relax the conditions for generat- hedral models and sculptures from all storehouseof intriguingstructures, some of ing the Platonic solids and specify that sorts of materialsand objects, including whichhave inspired artworks." faces must be regularbut not necessarily plexiglass, brass, papier-mache, pipe Hart based his design for "Rainbow identical polygons, there are 13 such cleaners, pencils, floppy disks, tooth- Bits"on an icosahedron transformedvia polyhedra, known as Archimedean brushes,paper clips, and plasticcutlery. his propellor operator into a three-di- solids. One example is the truncated He "bringsthe ideal forms of the poly- mensional mosaic of 20 eouilateraltrian- icosahedron,familiar as the pattern on a hedra into the materialworld gles and 60 kite- soccer ball, which consists of 12 pen- with wit and humor," says shaped quadrilaterals. tagons and 20 hexagons. computer programmer and The resulting sculp- Polyhedra also artist Bob Brill of Ann Arbor, ture required one CD can be construct- Mich. "Theaustere beauty of at each vertex, five ed from polygons the polyhedrais thus madeac- CDs along each of the that don't neces- cessible and familiar." longer edges, and sarily have equal three CDs along each sides and equal of the shorter edges. angles, and they G iven the lengthy histo- The positions and an- may be indented ry associated with gles of slots cut into or spiky,like three- polyhedra,it may seem A wooden model of a type the CDs determined dimensional stars. astonishingthat there are still of symmetrohedron. This their placement. The great stellated mathematical discoveries to particular geometric solid "For many visitors, dodecahedron, be made.Hart has provedpar- consists of 20 regular this stunning sculp- first described by ticularly adept at identifying enneagons (nine-sided ture is a main attrac- A plastic stereolithography 17thcentury mathe- rules and variantsthat lead to polygons), 12 regular tion of our building, model, 8 centimeters in matician Johannes new possibilities. pentagons, and 60 particularlywhen it is diameter, of an "elevated Kepler, is one For example, his discovery isosceles triangles. hit by rays of sunlight rhombicuboctahedron," example of a spiky of propellorized oolvhedra and [sprays] intense originally depicted by polyhedron. Non- came out of a new notation for describ- colored spots over the atrium walls," Leonardo da Vinci. indented polyhe- ing polyhedra, proposed by mathemati- Sequinsays.

Steps showing the construction of a cuboctahedral symmetrohedron from a set of octagons (left) and a set of nine-sided enneagons (middle left). These polygons slide inward until they meet (middle right). Trapezoidalfaces fill in the holes between the mated polygons (right)to create a novel polyhedron.

DECEMBER22 & 29, 2001 SCIENCE NEWS, VOL. 160 397

This content downloaded from 220.156.171.76 on Sat, 2 Nov 2013 00:24:57 AM All use subject to JSTOR Terms and Conditions W5~orking with Craig S. Kaplan of MV the Universityof Washingtonin WW Seattle,Hart has recentlyinvent- ed a procedurefor generatingmany poly- hedralforms, including new types of polyhedra. Dubbed symmetrohedra, these geometricsolids result fromthe symmetric placement of regular polygons (see illustrationon page 397). The new technique generates a variety of knownpolyhedra, including most of the Archimedeansolids, and infinitelymany new ones. One particularlystriking product is a polyhedronmade up of 20 regular nine-sided polygons (called enneagons), 12 regularpentagons, and 60 isosceles tri- Blendingan octohedron(top left)and a dodecahedron(bottom left) produces angles. "Yourarely see enneagons in a anothersort of polyhedron(right). polyhedron,"Hart comments. In another recent effort,Hart and com- at a meeting that highlightedmathemati- themselves to polyhedron building. On puter scientist Douglas Zongker of the cal connections in art, music, and sci- another communications front, Hart is -Aart University of Washington de- ence, held last July at SouthwesternCol- now writing a history of polyhedral veloped a novel method lege in Winfield,Kan. geometryin art. for constructingpolyhe- "Thebest way to learn about polyhedra by blendingto- dra is to make your own paper models," gether two or W ^ ^ hether assembling an intricate Hart insists. For those who prefer to more existing polyhedral design, leading a skip the spilled glue, misshapen cor- polyhedra. model-building workshop, or ners, ill-fittingedges, and hours of work, describing his latest ventures, Hart Hart's Web site provides interactive, Z o n g k e r, brings an infectious enthusiasm to his three-dimensional virtual models that and Kaplan task. To help spread the word about the can be observed from differentperspec- all described beauty and joy of polyhedra,he recently tives. their new visu- coauthored a book on building models Brill says about Hart, "Hisenergy, his alization methods using Zomeplastic components,product- curiosity,his inventiveness appear to be ed by Zometool of Denver. Embodying boundless." Those qualities come in A wooden model of a blended mathematicalratios such as the golden handy when exploring the infinite realm polyhedron created from five tetrahedra. mean, the system's balls and sticks lend of polyhedra.y

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