HYPERSEEING The Publication of the International Society of the Arts, , and Architecture March 2007 www.isama.org

ISAMA’07 MAY 18-21 BRIDGES DONOSTIA JULY 24-27

Articles Exhibits Resources Cartoons Cartoons Books News Ilustrations Announcements Announcements Communications 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.

Page Layout. Ranjith Perumalil

march, 2007

Cover Photo: Paper Anomaly by Jen Stark

Articles Book Reviews Article Submission

Cut Colored Paper Stack Sculpture Communications For inclusion in Hyperseeing, au- by Jen Stark thors are invited to email articles for Resources the preceding categories to: Intuiting Topology: [email protected] Sculptures of Bruce White Announcements by Stephen Luecking Articles should be a maximum of ISAMA’07 four pages. Form And Space by Nat Friedman

From Geometry to Art by Daina Taimina

Cartoons

Irrationals in Flatland by Friedman & Akleman

Illustrations

Illustrations by Robert Kauffmann

News

Mathématiques and Art Cut Colored PAper Stack Sculpture Jen Stark

Figure 1. Untitled 2005, 11” x 14” Figure 2. Detail of Figure 1 construction paper

I form sculptures by cutting layers grow from there. could be infinite….it looks like it of a stack of colored paper. Figure I like the idea of a stack of paper can always grow. I usually try to 1 is a stack of paper with designs becoming a 3-Dimensional object. manipulate the paper as little as pos- cut out, each revealing the design The sculpture in Figure 3 was my sible. The most interesting part is before it. This sculpture has a top- third try at the stacked paper ex- how simple yet intricate the design ographical aspect to it. Initially I ploding outwards. I gave it a bit can be with additions. This work is didn’t have any idea in mind. I just of a twist each time I made a new a bit time consuming but definitely began with one cut and it started to one. I like the idea that my artwork worth it.

Figure 3. Paper Anomaly, 2007, 12” x 12” construction paper Figure 4. Detail of Paper Anomaly The sculptures in Figures 5 and 6 are titled “Peephole: Anatomical Evolu- tion” and are paper sculptures inside of a peephole box that reflect back at you when you peer into the hole. The idea is I start with a base cut, and each cut that I make after is a slight change on the last. So the de- sign slowly evolves and the end cut is usually completely different from the base cut.

Figure 5. Peephole: Anatomical Evolution I, detail, 2005, 5” x 6” I have always been wood box, mirror, lights, paper. fascinated by the idea of a “nesting doll” and how ob- jects can fit inside of each other. The sculpture in Figure 7 was created from this idea. The cyl- inder on the out- side is the same shape but a larger version of the one before it. As the objects grow, the cylinders are able to nest inside of one another. I made a cut in each one to see the progression and depth.

I had a chance to install a piece of artwork on a 40 ft wall. I decided to make a huge group Figure 6. Peephole: Anatomical Evolution II, detail, 2005, 5” x 6” of these organic wood box, mirror, lights, paper and topographical Figure 7. Cylinder: Paper Cut, 2005, Figure 8. Degrees of Lightness, Installation, 7” x 14” construction paper 2006, detail, sculptures, as shown in Figure 8. The sculptures in Figures 9 and 10 is yet how extraordinary the I love how they share energy with are stacks of cardstock that hang on result. each other and look like poison- the wall. My process is to trace, cut Additional examples are at: ous growths on the wall. In total I and fold. I love how simple the idea www.jenstark.com made forty of them.

Figure 9. Triangle, 2007, 12” x 12” card stock Figure 10. Circle, 2007, 12” x 12” cardstock Intuiting Topology: Sculptures of Bruce White Stephen Luecking

Figure 1: Aurora I, painted steel, 1/2% for Art, State of Illinois, University of Illinois, Engineering Science Building, Urbana, Illinois, 1981

Like most artists Bruce White is an ematics do not influence art, espe- which was reissued in the mid 1950s avid experimenter and reader, al- cially formalist art. At the height and on a regular basis ever since. A ways on the lookout for new forms of interest in formalism, the late classic of popular science writing, and possibilities. His are not the 1960s, there was a tongue-in-cheek Thompson’s book sought to counter carefully constructed experiments truism making the rounds: “Artists the popular re-emergence of vital- of the scientist, but the somewhat read Playboy for the articles and ism in science in the first decades more serendipitous discoveries of they read Scientific American for of the 20th century. Vitalism was the artist. Picasso put it well with the pictures.” The elegance of form a quasi-theological theory of the when he said: “Je ne cherché, je in nature, especially the heretofore forms of life that held to a force in trouve.” I don’t search, I find. This unseen forms unearthed by science, nature that injected life into matter very much characterizes White’s served often as a model and justifi- and governed its development. In idea generation as well. cation of formalism. Passages in Modern Sculpture, Ro- salind Krauss cites the influence of However, this does not mean that One influential text was D’Arcy this on artists such as Arp. Thomp- science and its handmaiden math- Thompson’s “On Growth and Form” son sought to counter these claims cropping up repeat- edly in Bruce White’s sculpture is topology. Topology studies how form modifies as it sheds more and more of the restrictions of geometry. White does not study topology, but, in his countless experiments with pa- per sheets and strips, its impact keeps ap- pearing.

Aurora (Figure 1), built in 1978, is a looping giant of painted steel that demonstrates the beginnings of topo- logical features in the sculptor’s work. The sculpture is actually split lengthwise into a pair of adjoining strips of triangular section, which form parallelo- gram sections where they mate. Each loops both vertically and horizontally, at the same time splitting in the center. Closer in- spection reveals that, in the course of their looping, the inner and outer surfaces reverse and the sculpture turns itself inside out Figure 2: Bruce White at work on Aurora. The dual bands divide from a beam of (Figure 2). rectangular section into looped bands of triangular section. After completing their independent loops, the bands join back into a single rectangular section. A later work Twin Fin I (1994) embeds a classic topological by showing that form emerges from terns in plants, for example, result structure, the Möbius strip (Figure the chemical and physical compo- from the algorithms coded into 3). This figure is easily crafted by nents of metabolic processes fueling DNA and applied repetitively dur- twisting a strip of paper half turn life forms and that relatively simple ing the growth process. and then joining its ends. The re- mathematics will serve to describe sulting object, discovered the 19th how they effect form. Fractal pat- One such category of form science century German mathematician Au- gust Ferdinand Möbius, has the unusual properties of possessing only one surface and one edge. A number of modern artists, most notably Max Bill, have featured this object in their sculpture.

Unlike Bill’s Endless Band (fig- ure 4), Twin Fin is not a literal rendition of the Möbius strip. In- stead it is a rectangular plane that White split and rejoined with a Möbius-like transition, such that it has only one surface. The result is far more complex than Bill’s Ribbon. Twin Fin’s edges are not continuous, though its surface is. By preserving major portions of the original rectangle the surface is orientable, that is, it has a front side and back side. A classic Mö- bius band, however, has no front and back sides or top and bottom edges. In topological terms it is non-orientable.

In Twin Fin I White has accom- plished the enigmatic effect of integrating orientable and non- Figure 3: Twin Fin I, painted aluminum, 5’ x 10’, Indianapolis orientable surfaces. While the Museum, Indianapolis, Indiana, 1981 artist’s effect is mathematically intriguing, this was not his goal. It was an outcome of his investi- gation of visual continuities and discontinuities and of the imple- mentation of his larger theme of integrating the two in one form.

White also applies mathematics in the guise of sophisticated shop geometry. In order to construct the high twining arcs of Delphin (1983), for example, White mod- ified the traditional spline con- struction of shipbuilding (Figures 5 and 6). On the floor of his stu- dio, he scribed a grid and then hung weights from the ceiling, plumbed to the grid intersections Figure 4: Max Bill, granite, Endless Band, version IV, and raised to heights correspond- Pompidou Centre, 1961-62 Figure 5: Delphin, painted aluminum, 52’ x 21’, % for Art, State of Illinois, Willard Ice Revenue Building, Springfield, Illinois, 1983 ing to points along the paths of his curvature, in order to fit these to the ics, or call it spatial logic, enables arcs. With the aid of cable jacks path plotted by the lead weights. The White to engineer material into ex- he flexed strips of aluminum plate curves thus flow in accord with the pression. according to their innate bending nature of his material. Mathemat-

Figure 6: Delphin, in progress at White’s studio Fractal Form Nat Friedman and Space

Figure 1. River and Streams

Fractal Stone Prints : horizontal mid-line. The granite has Homage was first split vertically A fractal stone print is made by a mind of its own so probability is and then the right and left parts were starting with a flat slab of granite introduced in the splitting, result- each split horizontally. The parts about one inch thick that is polished ing in the random fractal edges. The were separated so that the narrow on one side. Lines are drawn on the parts were then separated to obtain spaces, thought of as white bands, unpolished side and a V-shaped spaces representing a central wide were essentially of equal width. wedge and a hammer are used to tap horizontal river and narrow vertical When making the print, as pressure along the lines in order to split the streams entering the river. The ran- is applied, the ink permeates the pa- granite into parts. The parts, with dom fractal geometry of the stones per and a grayish tone appears on the polished side up, are separated reflects the random fractal geometry the upper surface that is visible, as to create space, resulting in a com- of a river and streams as conveyed in the center section of Homage. The position of fractal form and space. in the image. lower surface of the paper is on the Ink is applied on the polished side stone and has the color of the ink, and a thin piece of Japanese paper is The print Homage to Mondrian and which is black in this case. I came laid down on the inked stones. Pres- Newman refers to the art of Piet to like this grayish tone. I could also sure is applied on the upper surface Mondrian and Barnett Newman. apply more pressure along the frac- with a barren in order to transfer Mondrian divided a rectangle with tal edges so more ink came through, ink to the lower surface of the paper straight horizontal and vertical nar- resulting in the fractal edges being that is on the inked stone, resulting row black bands that could intersect outlined in black against the gray. in the print. The print River and and abut. Newman divided a rect- After the print was dry, I folded over Streams started with a long piece angle with vertical narrow colored the underside black surface on the of granite that was split vertically bands, called “zips”, that went from right and left. The fold widths of into seven parts, where the vertical the top edge to the bottom edge re- the resulting black rectangles were lines were drawn equally spaced. sulting in rectangles of full height determined so that the narrow hori- Each part was then split along a but varying widths. The stone in zontal spaces match up where black rangement above. This switching idea is due to Carlo Séquin and came up in an AM 97 work- shop I was presenting on fractal stone prints. Thank you Carlo!! The initial inner fractal edg- es are now outside and the initial outer straight edges are now inside. Surprisingly, the frac- tal edges always match up to form the outer boundary even though the breaks are con- trolled random . (This can be seen by tearing a paper rectan- gle.) The stone pieces

Figure 2. Homage to Mondrian and Newman

meets gray. Thus Homage com- matched up where the black met the bines the gray side and the black gray. This reminded me of the dif- side, which I refer to as a two-sided ferent colored layers of stone that print. There are two fractal horizon- I saw when visiting Ghost Ranch tal band spaces that meet the fractal in New Mexico where Georgia vertical band space “zip”. The two O’Keefe painted the landscape. The outer black rectangles and the inner different colored layers of stone gray rectangle have full height, as in were strikingly colorful. Some a Newman print. The edges of these stone faces had a vertical split. In three rectangles are straight con- Ghost Ranch I also used the upper trasting with the fractal bands that white space as sky. divide the rectangles. Actually each black rectangle is divided into two Rectangle Inside Out starts with a “rectangles” and the gray rectangle rectangle that is taller then wide. is divided into four “rectangles”, It was first split about halfway up where each “rectangle” has fractal along a positive sloping line. The and straight edges. Thus fractal ge- upper part was then split vertically ometry and straight-line geometry about a third of the way from the are combined. left side. The lower part was split vertically about a third of the way Ghost Ranch, below, was made from the right side. The positions with one vertical split and then the of the upper left and lower right two parts were separated to obtain were switched and the positions a narrow vertical space. The lower of the lower left and upper right end was folded up so that the spaces were switched resulting in the ar- Figure 3. Ghost Ranch ting Yourself Back Together With A In other cases the space can have Little More Space. In this sculpture a positive significance when one the fractal geometry of the broken manages to rid oneself of some- stones conveys a real-life traumatic thing or someone that caused one experience. Here the space has cen- to come apart, such as getting off tral significance. It can represent the drugs or ending a painful relation- space left by the death of a loved ship. Thus the space can represent one as in 9/11 or the loss of every- a feeling of release or freedom. Art thing you own as in Katrina or war. is not always about the beautiful. The traumatic experience can cause Fractal Torso is not beautiful, but one to come apart and then one has about life. There are experiences to put oneself back together. Thus that can be conveyed sculpturally the space is the part that will always only by fractal geometry. be missing.

Figure 4. Rectangle Inside Out could be joined with epoxy to make an interesting tabletop. Another piece of stone of contrasting color and larger then the inner rectangular space could be attached underneath the rectangular space to provide a lower surface for the rectangular space.

Fractal Sculpture The sculpture Fractal Torso was made from a piece of granite about two inches thick. The piece was split so that an inner “rectangular” part could be removed. This result- ed in a space when the remaining parts were rejoined with epoxy. The form is an abstract torso with wide shoulders and a narrower waist. The long title for Fractal Torso is Put- Figure 5. Fractal Torso From Geometry to Art Daina Taimina

Figure 1. Euclid’s 5th Postulate does not hold in the hyperbolic planet

I made my first crocheted model of me to explore the hyperbolic plane, models starting with a chain and the hyperbolic plane in 1997 when making them with different radii, in then exponentially adding the num- I needed it for teaching a geometry different materials, and with differ- ber of stitches, trying to get more class at Cornell University. Since ent colors. It is like doing research, surface area I could use for mak- then I have been experimenting with just the main tool is crochet hook ing “hyperbolic pictures”. These this form, eventually making these illustrations can be seen in the ge- objects more like fiber arts sculp- ometry textbook for undergradu- tures. It has been very interesting for For teaching purposes, I made my ates: David W. Henderson, Daina stant negative curvature (necessary requirement for the hyperbolic plane). I will describe the proper description of this process in my forthcoming book

Once this basic shape is finished, the fun part of sculpting can start. The same basic shape can be turned in many differ- ent forms, as shown in Figures 4. It is possible to create sculptures with holes by joining parts of Figure 2. Author with finished work in Cornell Plantations, October, 2006 the surface as in Figures 5-7.

The hyperbolic plane in Taimina Experiencing Geometry: a shape I call the symmetric hy- Figure 8 is made in four shades of Euclidean and Non-Euclidean with perbolic plane, shown in Figure purple. I was making this for an ex- History, 3rd Ed., Pearson Prentice 3. I start just with 3 chain stitches hibit in American Center for Phys- Hall, 2005. joined together and then I make a ics. That is why I decided to make form following a mathematically an experiment – I measured each For exploring different forms of calculated pattern that insures that color yarn to be exactly 100m long, the hyperbolic plane, I start with the resulting surface will have con- so it means that each color covers

Figure 3. Symmetric hyperbolic plane Figure 4. Sculpting symmetric hyperbolic plane (Private collection) (private collection). Figure 5. This shape has two holes Figure 7. Symmetric hyperbolic plane shaped with four holes equal area. Italy, Latvia. This way of crochet- oexhibits/oe1.html#) ing has followers. The Institute For Some of my work will be presented My works have traveled from Itha- Figuring in Los Angeles has sup- in upcoming exhibit “The Hand- ca to art shows in Washington, D.C. ported my work and has involved Making” (April 14 –July 27, 2007) (http://eleveneleven.50webs.com/) many other people in crocheting in Abington Art Center, Jenkintown, and other places in USA , Belgium, Coral Reef (http://www.theiff.org/ PA.

Figure 6. Hyperbolic shape with three holes Figure 8. Four equal areas Nat Friedman irrationals in flatland square root of two & tree & Ergun Akleman

illustrations by ROBERT Kauffmann

I FEEL SO TRAPPED

DISTRAUGHT OVER BEING SUCKED INTO A HYPERSPHERE, SHE DESPERATELY REACHED OVER HER HEAD AND GRASPED THE SOLES OF HER SHOES News Mathématiques and Art

Mathématiques and Art Art and Mathematics 2007

The first exhibit “Mathématiques and Art” (Gazette des There will be an exhibit Art and Mathematics 2007 in Mathématiciens, 10 (2005) 61-64, and November Hy- the Science Library at the University at Albany-SUNY perseeing Newsletter) is now travelling through Greece. (UAlbany), February 25-April 30, 2007. This exhibit A second exhibit will stand inside the nice library of the will celebrate the fifteenth anniversary of the first Art Université Paris 12 (March 5 – April 7). Works which and Mathematics Conference in 1992 (AM 92) held at could not be sent to Greece (like François Apéry’s and UALbany. The exhibit will feature works by Benigna Philippe Charbonneau sculptures) will be displayed Chilla. There will also be an exhibit of posters, books, again, and works by Tom Banchoff-David Cervone, and small sculptures by presenters at previous AM Jean-François Colonna, Patrice Jeener and John Sul- Conferences. A reception will be held during the af- livan will be shown here for the first time. The visitors ternoon of March 30, 2007, with presentations by Be- will also discover recent works by Jean Constant, Bah- nigna Chilla and Nat Friedman. man Kalantari, Jos Leys and Sylvie Pic.

BOOK reviews

Beyond Geometry: Experiments in Form, 1940s-70s. Lynn Zelevansky with contributions by Valerie L. Hillings, Miklos Peternak, Brandon LaBelle, Peter Frank, Ines Katzenstein, and Aleca Le Blanc, MIT Press, 2004.

This book was the catalog for an exhibit at the Los Angeles County Museum of Art, June 13-October 3, 2004: Miami Art Museum, Florida, November 18, 2004-May 1, 2005.

This is an excellent survey of geometric art during the 1940s-70s. The discussion includes Max Bill, Mel Bochner, Naum Gabo, Sol LeWitt, Piet Mondrian, Francois Morellet, Barnett Newman, George Rickey, Bridget Riley, Richard Serra, Tony Smith and many others. communications

This section is for short communications such as recommendations for artist’s websites, links to articles, que- ries, answers, etc.

a sample of WEB REsources

[1] www.kimwilliamsbooks.com [8] www-viz.tamu.edu/faculty/ergun/research/topol- Kim Williams website for previous Nexus publications ogy on architecture and mathematics. Topological mesh modeling page. You can download TopMod. [2] www.mathartfun.com Robert Fathauer’s website for art-math products in- [9] www.georgehart.com cluding previous issues of Bridges. George Hart’s Webpage. One of the best resources.

[3] www.mi.sanu.ac.yu/vismath/ [10] www.cs.berkeley.edu/ The electronic journal Vismath, edited by Slavik Carlo Sequin’s webpage on various subjects related to Jablan, is a rich source of interesting articles, exhibits, Art, Geometry ans Sculpture. and information. [11] www.ics.uci.edu/~eppstein/junkyard/ [4] www.isama.org Geometry Junkyard: David Eppstein’s webpage any- A rich source of links to a variety of works. thing about geometry.

[5] www.kennethsnelson.com [12] www.npar.org/ Kenneth Snelson’s website which is rich in informa- Web Site for the International Symposium on Non- tion. In particular, the discussion in the section Struc- Photorealistic Animation and Rendering ture and Tensegrity is excellent. [13] www.siggraph.org/ [6] www.wholemovement.com/ Website of ACM Siggraph. Bradfrod Hansen-Smith’s webpage on circle folding.

[7] http://www.bridgesmathart.org/ The new webpage of Bridges. isama’07 Sixth Interdisciplinary Conference of The International Society of the Arts, Mathematics, and Architecture College Station, Texas, May 18-21, 2007

IMPORTANT DATES CONFERENCE Submission

ISAMA’07 will be held at Dec.15, 2006 Submission System Open Authors are requested to Texas A&M University, Feb. 22, 2007 Paper and Short paper submission deadline submit papers in PDF for- College of Architecture, in Mar. 15, 2007 Notification of acceptance or Rejection mat, not exceeding 5 MB. College Station, Texas. The Apr. 1, 2007 Deadline for camera-ready copies Papers should be set in purpose of ISAMA’07 is ISAMA Conference Paper to provide a forum for the Format and should not dissemination of new math- exceed 10 pages. LaTeX ematical ideas related to the and Word style files are arts and architecture. We available at: (will be avail- welcome teachers, artists, able). The papers will be mathematicians, architects, published as the Proceed- scientists, and engineers, as ings of ISAMA’07. well as all other interested persons. As in previous RELATED EVENTS conferences, the objective is to share information and Exhibition discuss common interests. There will be an exhibit We have seen that new whose general objective ideas and partnerships is to show the usage of emerge which can enrich mathematics in creating interdisciplinary research art and architecture. In- and education. structions on how to par- ticipate will be posted on the conference website.

FIELDS OF INTEREST Teacher Workshops There will be teacher The focus of ISAMA’07 will workshops whose objec- include the following fields tive is to demonstrate related to mathematics: Ar- methods for teaching chitecture, Computer Design mathematics using related and Fabrication in the Arts art forms. Instructions on and Architecture, Geometric how to participate will be Art, Mathematical Visualiza- posted on the conference tion, Music, , and website. , Arabesque 29, Bubinga and Tilings. These fields include graphics CALL FOR PAPERS interaction, CAD systems, algorithms, fractals, and Paper submissions are encouraged in Fields of Interest stated above. In particular, we specify the graphics within mathematical following and related topics that either explicitly or implicitly refer to mathematics: Painting, Draw- software. There will also be ing, Animation, Sculpture, Storytelling, Musical Analysis and Synthesis, Photography, Knitting and associated teacher work- Weaving, Garment Design, Film Making, Dance and Visualization. Art forms may relate to topology, shops. dynamical systems, algebra, differential equations, approximation theory, statistics, probability, graph theory, discrete math, fractals, chaos, generative and algorithmic methods, and visualization.

For four days archone.tamu.edu/isama07 Texas A&M will be

Sponsored by College of Architecture, Texas A&M University and International Society of the Arts, Mathematics, and ArchitectureTexas Arts and Mathematics!! Isama ‘07 workshops

Hands-on construction of CD sculptures by George Hart

In this workshop, the participants will learn polyhedral shapes and familiarize themselves with the geometrical ideas by constructing a mathematical structure. The mathematical sculpture will be con- structed with CDs by taking advantage of the holes in the centers of the CDs and use cable ties as the assembly mechanism.

For more information, go to: http://www.georgehart.com/UBC/ubc.html

Designing sculptures with Sculpture Generator by Carlo Sequin

In this workshop, the attendees will learn the concepts behind Scherk-Collins sculptures. The attendees learn to use an interac- tive computer program, Sculpture Generator, which has been de- veloped to visualize a large variety of Scherk-Collins sculptures with different configurations of saddle rings with different num- ber of holes and different amounts of twists.

For more information, go to: http://www.cs.berkeley.edu/%7Esequin/SCULPTS/scherk.html

Persian Mosaics and Architecture by Reza Sarhangi

In this one day workshop, the attendees will learn a variety of geometrical ideas behind behind the beauty of Persian mosaics and architecture. The workshop will include Platonic Solids us- ing Zoom tool, understanding of star polygons using Geometer Sketchpad Software and learning using Tessellation Exploration Software.

For more information, go to: http://pages.towson.edu/gsarhang/Persian%20Presentations.html Isama ‘07 workshops

The Art and Mathematics of by Nat Friedman

This workshop that shows how to teach children deep mathemati- cal concepts with fun. In this workshop, the participants will learn various mathematical properties of knots with hands-on applica- tions. The participants will also create soap film minimal surfaces using shaped frames.

To see soap film minimal surfaces created using knot shaped frames, go to: archone.tamu.edu/isama07/data/soapfilm.doc

Topological Smooth Surface Design by Ergun Akleman, Vinod Srinivasan and Jianer Chen

This course introduces topology change operations such as Euler operations, insert and delete edges, splice. The participants will learn to use Euler equation to determine genus of the surface. The participants using the software developed by the organizers will learn how to connect and disconnect surfaces and increase and decrease genus using topology change operations. The software is freely available and can be used in a classroom.

For more information, go to: http://www-viz.tamu.edu/faculty/ergun/research/topology/

Teaching Topological & Geometrical Concepts with Paper Strips by Ergun Akleman and Jianer Chen

This course shows how to teach a wide variety of geometrical and topological concepts using paper stripes. With hands-on and fun projects, it is possible to convincingly illustrate mathematical con- cepts such as Gaussian curvature and vertex defect; Gauss-Bonnet theorem, orientable and non-orientable 2-manifolds, polygons in elliptic and hyperbolic space; projective plane and Klein bottle..

For more information, go to: archone.tamu.edu/isama07/data/smi07.pdf Workshop: Seeing in 3D A one-day course in visual thinking by Geoff Wyvill

Most people, even among technical draftsmen, designers and computer graphics program- mers, find it very difficult to visualize 3D shapes well enough to reason about them. We demonstrate the problem and take participants through a series of exercises whereby they can begin to acquire this important practical skill.

Question: “Stand a cube on its corner. What is the shape of a horizontal cross-section taken at half the height of this object?” About 4% of us can reason about 3D space well enough to answer this question easily and with confidence. Most of us enter a state of panic when confronted Unit Cube Diagonals by 3D problems. Yet it is possible to train yourself to think and visualize in 3D. We take you What is the distance through a series of exercises over one day that will start you thinking in 3D. Once you have between them? It is not 1/sqrt(2) the basic principles you can develop the skill independently.

Prerequisites: You should be familiar with some basic geometric ideas e.g: “two planes meet in a straight line.” It is helpful if you know how to find distances with pythagoras’ theorem, but this is used for only a few exercises and the course can be done without mathematics.

Background In 1982, Bob Parslow discovered that a group of computer graphics students at Brunel university were unable to imagine clearly the shape of a cube. It appears that about 96% of people cannot even find the corners of an imaginary cube standing on one corner. Bob described this phenomenon as “3D blindness” and after testing more than 4000 people, he has shown that even in such professional groups as engineering draftsmen, fewer than 30% have adequate skills at 3D visualization.

Since 1990, Bob and I have been giving a one day course in visualization skills. We have run this for undergrad- uate computer scientists, general public, at conferences in Lausanne, Melbourne, Hasselt and Hanover, at SIG- GRAPH in 2001 and at GRAPHITE in 2003. We have demonstrated by these courses that the ability to imagine and reason about 3D shapes is a skill that can be learned.

Natalie after successfully Geoff demonstrating in Hanover After the course in Hanover we practise our constructing Super Nova 1998 skills on local sculpture Plexus