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Hyperseeingnovember 2006 the Journal of the International Society of the Arts, Mathematics, and Architecture HYPERSEEINGNovember 2006 The Journal of the International Society of the Arts, Mathematics, and Architecture www.isama.org Articles Exhibits ISAMA’07ISAMA’07 Resources atat TexasTexas A&MA&M Cartoons May 18-21 May 18-21 Books 20072007 News Ilustrations Announcements Communications ForFor fourfour daysdays TexasTexas A&MA&M willwill bebe TexasTexas ArtsArts andand MathematicsMathematics HYPERSEEING Editors. Ergun Akleman, Nat Friedman. Associate Editors. Javier Barrallo, Anna Campbell Bliss, Claude Bruter, Benigna Chilla, Michael Field, Slavik Jablan, Steve Luecking, Elizabeth Whiteley. NOVEMBER, 2006 Cover Photo: Robert Longhurst’s Arabesque 29, Bubinga, 12”h x 10 ½ “w x 9½ “d. Articles Exhibitions Robert Longhurst:Arabesque 29. Nat Friedman An Exhibition of Mathematical Art, Claude Bruter. Basketball is an Octahedron: Intriguing Structures of Balls, Ergun Akleman Communications Two Americans in Paris: George Rickey and Kenneth Books Snelson, Claude Bruter Illustrations Space, Nat Friedman Illustrations by Robert Kauffmann Cartoons Resources Cartoons from Flatland by Friedman & Akleman Cartoons by Tayfun Akgul ____________________________________________ News Article Submission Journal of Mathematics and the Arts, Gary Greenfield. For inclusion in Hyperseeing, members of ISAMA are invited to email material for the preceding categories A Coxeter Colloquium to [email protected] Announcements Note that articles should be a maximum of three pages. ISAMA’07 Joint Meeting of the MAA and the AMS. Mathematics and Culture, 2007. Bridges 2007. Nexus V11 2008. ROBert LONGHURST’S ARABESQUE 29. Nat FRIEDMAN Robert Longhurst is a sculptor who lives in Chestertown, NY about a 1½ hour drive north of Albany. I have known Bob for about fifteen years. He carves absolutely beautiful wood sculptures gener- ally in Bubinga wood. These sculptures are extremely thin surfaces Three Views of Robert Longhurst, Arabesque 29, Bubinga, 12”h x 10½ “w x 9½” d. that often have hyper- bolic geometry. The cover sculpture Arabesque 29 see all the possible front views. is an example. This sculpture is actually based on an Enneper’s Surface. The computer image of the usual Note that the left photo has 1/3 turn rotational sym- position of an Enneper surface is shown below. metry. This implies that rotating the Usual Position 1/3 turn horizontally results in hyperseeing the sculpture Bob has rotated the usual position ¼ turn counter- completely in the Usual Position. Additional photos of clockwise to obtain the position in the center photo Longhurst sculptures can be seen at above. In this case the sculpture is mounted on a www.robertlonghurst.com. visible rod that is inserted into a thickened edge. If Computer images are from the excellent website the sculpture in the center photo is rotated on the rod rsp.math.brandeis.edu/3D-XplorMath/Surface/gallery_m.html n about a 1/8th turn to the right, the view in the right photo is obtained. Another 1/8th turn to the right Alfred Enneper is a German mathematician who earned his PhD will result in the view in the left photo, which is quite from the Georg-August-Universität Göttingen in 1856 for his disser- striking and would correspond to the Top view of the tation about functions with complex arguments. He studied minimal surfaces and parametrized Enneper’s minimal surfaces in 1863. A usual position. Another ¼ turn to the right will return contemporary of Karl Weierstrass, the two created a whole class of to the center view. Thus ½ turn to the right returns parameterizations. the sculpture back to its initial view, which implies From Wkipedia the sculpture has ½ turn symmetry from above. This can be seen in the computer image of the top view of Enneper’s surface corresponding to the above photos. The top view has ½ turn rotational symmetry. This implies that for any front view, as in the above photos, the view is identical to the view directly opposite (1/2 way around). That is, if we consider any of the three photos as a front view, then the back view will be the same. Thus the half-turn symmetry of the top view helps to hypersee the sculpture. In particular, we only need to rotate the sculpture ½ turn on the rod to hyper- Usual Position Top View Computer Generated Enneper’s Surface BASKETBALL IS AN OCtaheDRON: ErgUN AKLEMAN INTRIGUING STRUCTURES OF BALLS Each sport has its own easily none of the courses I took dealt identifiable ball. The major differ- with geometry. It is interesting that ence between the balls comes not I learned geometry by myself when from their sizes or colors but from I worked on my thesis. their mesh structures. Each sport’s ball consists of a set of spherical I first realized the importance of polygons. For instance, the current mesh structures of balls during our mesh structure of a soccer ball is visit to Germany in the Summer a spherical version of a truncated 2004. We stayed with my wife’s icosahedron that came from Buck- sister. It was the time of the Eu- minster Fuller’s famous geodesic ropean Soccer Cup and as a rabid Dodecahedral ball dome designs. Soccer balls consist soccer fan I loved watching soccer of 12 spherical pentagons and 20 every night with my brother in-law spherical hexagons. At each vertex, who was was an amateur soccer three polygons meet. This particu- coach. Naturally, in their backyard lar design of a soccer ball was first there were a wide variety of soccer marketed in the 1950’s and used balls. I realized that some of them in the 1970 world cup. Before the were unusual. So, I took the pho- 1950’s, soccer balls were simply tographs of these unusual soccer spherical versions of the football balls. A soccer ball in the backyard used in American Football. For a was particularly interesting to me detailed history of soccer balls see as a researcher. In this soccer ball, www.soccerballworld.com/. 2004 European Cup official ball the designers drew hexagons on the surfaces. Obviously, they thought Good understanding of mesh struc- that if they drew hexagons small tures of the balls is important for enough, they can cover a sphere illustrators. Any student of illustra- using only hexagons. Interest- tion must learn geometry to be able ingly enough, I had just written a to observe the structures that exist paper on pentagonal subdivision in nature. Unfortunately, geometry (a remeshing algorithm that can is not a subject that is taught in art convert any given mesh to a pen- schools. In fact, geometry is not tagonal mesh) and showed that even really taught in engineering a pentagonal subdivision is the schools. In 1978, I was a profes- Soccerball with honecomb texture only one beyond quadrilateral and sional cartoonist for Girgir maga- a cartoon. I knew that there had to triangular subdivisions. Using the zine and I was also an engineering be some pentagonal patches. That Euler-Poincare equation it is easy students. I had already taken cours- was all I knew. I drew a soccer ball to prove that a hexagonal subdivi- es in calculus, linear algebra, ordi- but that soccer ball looked strange. sion does not exist. In other words, nary and partial differential equa- The cartoon was published but it is not possible to cover a sphere tions, discrete mathematics and I could not figure out what was with even non-regular hexagons even engineering drawing, but I did wrong with my drawing. During regardless of how small they are. not know anything about platonic my PhD on Computer Graphics, Ian Stewart had a very nice article solids. I clearly remember that one I had a minor in mathematics but on this subject. Obviously, the week I had to draw a soccer ball for designers of that ball did not know about it. basketball has its own beauty. So, this is a clear example of the whole The most interesting soccer ball being more than its parts. in the backyard was a hand-sewn dodecahedral ball. Although a do- Another interesting shape turned decahedron is not as good approxi- out to be the tennis ball. It is not mation of a sphere as a truncated known who designed the ten- icosahedron, the sole existence of nis ball, but, Charles Goodyear’s this ball suggested to me that the Spherical Triangle of Basketball vulcanization process allowed mass dodecahedral structure have been production of rubber balls in the used for soccer balls until Buck- 1850’s. I do not know how similar minster Fuller’s design won over those balls in the 1850’s were with any other rival structure. today’s tennis balls. However, the current tennis ball consists of two Another observation from the trip spherical polygons drawn on the was the design of the official ball of sphere as if they are 3D yin and the 2004 European Cup. Obviously, Two Spherical Triangles yang symbols. I also cut a tennis the designers of the ball wanted to ball to clearly see the shapes of make the ball more interesting and these 3D yin and yang symbols. drew brush strokes that create an Since the ball is pressurized, it was octahedral structure on the surface. very hard to cut it, but it is really They did not complete the tri- worth it to see the shapes. A base- angles but the octahedral structure ball shares the same mesh structure was clearly visible from the brush with the tennis ball. strokes. So, I wondered what are other mesh structures on the surfac- On the other hand, a volleyball es of balls that are used in sports. Hemisphere with four Spherical Triangles seems to be the most uninteresting ball. But, if you look at one care- When I returned back to the US, fully you will see 6 quadrilaterals I started to look at other balls.
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