Leonardo_36-3_175-254 5/9/03 9:48 AM Page 230 TECHNICAL STATEMENT A THREE-DIMENSIONAL particularly since there are no restric- or even a spatially four-dimensional ZOETROPE OF THE CALABI- tions on the direction of parallax in the object cannot be physically seen, it YAU CROSS-SECTION IN CP4 zoetrope. Furthermore, the zoetrope must somehow be cast into our own 3D object appears to be perfectly solid and physical space. For this work, this is Stewart Dickson, Computer Science “real,” as it in fact is. done by taking a lower-dimensional (3) and Mathematics Division, Oak Ridge The object, however, originated as cross-section of the higher-dimensional National Laboratory, P.O. Box 2008, pure, mathematical language before it (6) true object. Oak Ridge, TN 37831-6367, U.S.A. E- was transformed using that abstract The surface is composed of 5 ϫ 5 mail: <[email protected]>. Web: manipulation engine, the digital com- (ϭ 25) patches, each parameterized in a <http://emsh.calarts.edu/~mathart/ puter (see Appendix A). rectangular complex domain. The rec- MathArt_siteMap.html>. tangular patches are pieced together Description of the Object about a point in groups of 10. The sur- Received 8 April 2002. Accepted for publica- The object can be described in several face has five separate boundary edges. tion by Roger F. Malina. ways. It is a mathematical extension by The structure and complexity of the Andrew Hanson of a superquadric surface are characterized by the expo- In 1999, I proposed the first three- surface whose domain lies in the com- nent n ϭ 5. In Hanson’s parameteriza- dimensional zoetrope of the metamor- plex plane [{a, I b}, where I ϭ͌–1] [6] tion, the surface is computed in a space phosis of a simple torus into the Costa (Appendix B). An extension of quadric defined by two real and two imaginary genus 1 3-ended minimal surface [1]. surfaces (3D extension of conic axes. The real axes are mapped to x The zoetrope [2] creates an illusion of curves), superquadrics are normally and y, while the imaginary axes are life by presenting stroboscopic anima- only considered in the domain of real projected into the depth dimension (z) tion frames. In my zoetropes, the illu- coordinates. This mathematical system after rotation by the angle ␣. sion and motion is in three full can also be described as the Complex The angle of rotation in the projec- dimensions. [3,4]. Projective Varieties determined by xn ϩ tion from complex four-space into three In my second proposal, discussed yn ϭ zn (the n ϭ 5 case of Fermat’s Last real dimensions is animated from 45° to here, I intend to depict in physical Theorem). Alternately, the object of 405° over the span of 60 frames. Note three-space the 3D projection of an the zoetrope can be called the 3D that the object has two-fold rotational object rotating in four complex spatial cross-section of the 6-Dimensional symmetry in 4D, because it only has to dimensions. I propose to “sculpt” the Calabi-Yau manifold embedded in four- rotate 180° to arrive at a geometrically individual “frames” of the animation as dimensional complex space described equivalent orientation with respect to generated by computer, in physical in superstring-theory calculations by a the starting point of our projection. materials, in three physical dimensions homogeneous equation in five complex However, the colored parameterization and in directly modeled-in color. To variables. Calabi-Yau spaces may lie at must rotate the full 360° to return to its do this, I propose to employ color the smallest scales of the unseen di- original orientation. computer-aided 3D printing—a “layer- mensions in Cosmological String The- For the purposes of this sculpture, manufacturing” technique [5]. 3D ory [7]. The Calabi-Yau Manifold is a the animated pieces are each printing constructs a physical prototype proposed mechanism by which a 10- constructed with an integral standard- directly from a CAD file. dimensional “P-brane” (space) is ized base. The base contains extruded In this zoetrope, 60 phases of the wrapped onto normal 4-dimensional numerals identifying each unique object to be transformed project from space-time. Because a six-dimensional phase of the animation. The base also the edge of a wheel at the “spokes.” The rotation of the wheel is “frozen” using a stroboscopic white light that is Fig. 1. Steward optically synchronized to the turning Dickson, 60 spokes, from which the objects project physical objects in (Fig. 1). The result is a 3D computer metamorphosis “morph” in physical materials. on a wheel, digital image, 2001. The animating object is in fact fairly (© Stewart Dick- small—roughly 1 inch high—making it son) These ani- an intimate viewing experience. Be- mating objects are cause the objects are in fact whirling proposed “prints” realized by direct around in space, the piece is sealed 3D computer inside a cabinet and is viewed through “printing” of Plexiglas windows. One could ask what CAD data the difference might be between this with computer- piece and a 2-second holographic specified surface coloration. movie of the piece. The zoetrope object can be viewed through a solid angle of 180° in the horizontal direction by 90° in the verti- cal direction—nearly “in the round.” I believe this is a wider viewing angle than that afforded by color holography, 230 Statements Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/002409403321921460 by guest on 25 September 2021 Leonardo_36-3_175-254 5/9/03 9:48 AM Page 231 contains 1/8 -in mounting holes at description file to one of the Open the coordinate axes is [(ax)n ϩ (by)n ϩ 1/2 -in centers, suitable for accepting Inventor file format <http://oss.sgi. (cz)n ϭ k]. 4-40 ϫ 11/4-in machine screws for com/projects/inventor>. See also Alan Barr’s two papers on mounting the piece to the wheel (see superquadrics: A.H. Barr, “Super- Color Plate A No. 1). thicken by Stewart Dickson: <http:// quadrics and Angle-Preserving Trans- As in the first zoetrope, an infrared emsh.calarts.edu/~mathart/ formations,” IEEE Computer Graphics LED and phototransistor pair are lo- sw/objView/thicken.html>. A program and Applications 18, No. 1, 21–30 ( Janu- cated on opposite sides of the wheel written using Open Inventor for creat- ary 1981); W.R. Franklin and A.H. Barr, rim, such that the base of each meta- ing dimensional thickness on a single- “Faster Calculation of Superquadric morphic piece will interrupt the beam sided (open) polygon surface mesh. Shapes,” IEEE Computer Graphics and as it passes. This produces a transistor- Applications 1, No. 3 ( July 1981). transistor-logic (TTL) signal from the catobj by Stewart Dickson: <http:// Note that the preceding forms are phototransistor as the wheel rotates. An emsh.calarts.edu/~mathart/sw/ implicit, i.e. of the form f(x, y, z) ϭ 0. electronic timer circuit is used to tune objView/catobj.html>. A program for Hanson’s development (see Ref. [6]), the duration and phase of the light concatenating several object descrip- on the other hand, is parametric, i.e. {x, pulse. Because it is optically locked to tion files into a single object with indi- y, z} ϭ f(a,b): the wheel, the light timing never drifts, vidual coordinate transformations. u1[a_,b_] :ϭ .5 (E(a ϩ I*b) ϩ E(- a - I*b)) regardless of the speed at which the u2[a_,b_] :ϭ .5 (E(a ϩ I*b) - E(- a - I*b)) wheel rotates. Digital images and movies for this z1k[a_,b_,n_,k_] :ϭ Because the subject of this zoetrope proposal were captured from the Open E(k*2*Pi*I/n)*u1[a,b](2.0/n) itself contains information depicted by Inventor hardware render viewer using z2k[a_,b_,n_,k_] :ϭ the coloration of its surface, the object objView by Stewart Dickson: <http:// E(k*2*Pi*I/n)*u2[a,b](2.0/n) {x, y, z} -> will be stroboscopically illuminated emsh.calarts.edu/~mathart/sw/ {Re[z1k[a,b,n,k1]], with white light. objView/objView.html>. Re[z2k[a,b,n,k2]], Cos[alpha]*Im[z1k[a,b,n,k1]] ϩ Conclusion Appendix B Sin[alpha]*Im[z2k[a,b,n,k2]]} This piece demonstrates the concretiza- A short description of the mathemati- a: (-1.0,1.0); b:(0, Pi/2); tion of natural mathematical language. cal details of Hanson’s construction. k2: (0, n - 1) It transforms and casts an abstract (Note: Constants E and I and func- k1: (0, n - 1) system expressed in four complex spa- tional notation are from the Mathemat- The surface is composed of n ϫ n tial dimensions into three real physical ica system for doing mathematics by patches, each parameterized in the dimensions, and it does so while allow- computer) (see Appendix A). domain (a, b). The structure and com- ing the object to move in time. The plexity of the surface are characterized motion depicts a 3D shadow of a 4D Conics. (See Neil Dodgson, “Advanced by the exponent n. n ϭ 5; alpha ϭ (0.25 object, metamorphosing as it rotates in Graphics,” University of Cambridge, ϩ frame/60) * 2 * Pi. This mathemati- 4-space. This allows us to see more of Computer Laboratory Part II course, cal system can also be described as the the entirety of an object expressed 1998 <http://www.cl.cam. Complex Projective Varieties Deter- mathematically in more than three ac.uk/Teaching/1998/AGraphics/l2a.