True 3D Computer Modeling: Sculpture of Numerical Abstraction
Stewart Dickson · Leonardo · 1992
ly. The mechanism between the endpoints is a series of translations necessary to implement the mathematics in a machine-executable form and to condense the abstract problem into a spatial representation. New compilers of natural mathematical language make the condensation transparent. A two-dimensional (2D) picture representing threeor higher-dimensional form is a lower-dimensional abstraction of the higher-dimensional actuality. I claim that creating sculpture of numerical abstraction increases the immediacy of the abstraction over what a picture can convey. Computeraided prototyping technologies make a direct projection of three-dimensional (3D) abstraction into physical reality. An interactive computer graphic of a 3D space exists behind the glass front of a cathode-ray tube. Even a hologram or stereo projection is only an illusion in light. A sculpture occupies and shares the physical space occupied by the viewer. A sculpture has its own presence, one that dominates these alternate presentation media. ? 1992 ISAST Pergamon Press Ltd. Printed in Great Britain. 0024-094X/92 $5.00+0.00 EARLY WORK ABSTRACT I saw my first interactive, visual computer workstation in 1984. The author gives an account It was a DEC pdpll/40 with of his experience in progamming a Vector General scope that interactive computer graphics. Observations are made of the logiran Tom DeFanti's GRASS incal connections between a computerteractive graphical operating generated picture and the computer system [5] and programming program that generated the picture. language. The first incarnation The action of condensing abstracof GRASS was known as the tion into visual form as an aid in understanding the abstraction is Graphis Sybosis at Ohio State studied in the context of scientific University. It later became the visualization. The author presents Circle Graphics Habitat at the his work in making sculptures of University of Illinois at Chimathematically defined surfaces via direct, three-dimensional computercago. This is the machine Larry p t 6 7 printing technologies. Cuba used to create the Death Star hologram simulation sequence in the original Star Wars motion picture. My experience in computer graphics up to that time consisted of writing FORTRAN batch programs for an electrostatic plotter at AT&T Bell Labs in Naperville, Illinois. Despite the fact that the pdpl 1/40 had only 32k bytes of magnetic core memory, the Vector General had hardware display capability for interactively manipulating 3D objects in vector-list form in real-time. The GRASS language also had some very interesting features. Among them were character string manipulation tools that allowed me to write primitive compilers, such as those for evaluating Lindenmayer [6] systems and Boolean operations. The experience of GRASS was an immediate acceleration of all the techniques I had been developing for making pictures up to that time. From amidst the creative explosion, there emerged a technique of construction that is becoming increasingly important to me. I discovered that aligning circular arcs with the vertices and edges of a polyhedron yields a shape that can be repeated in the manner of an atomic crystalline lattice, but that can define a smoothed topological surface of many handles. Achieving the surface requires triangulating closed wireframes made of connected circular arcs. In implementing the program to do this, I came up with a linear approxiStewart Dickson (computer-graphics progammer), The Post Group Digital Center, 6335 Homewood Avenue, Hollywood, CA 90028, U.S.A. Fax: (213) 464-1953. E-mail: Received 22 March 1991. LEONARDO, Vol. 25, No. 3/4, pp. 2281-287, 1992 This content downloaded from 157.55.39.55 on Tue, 23 Aug 2016 04:15:58 UTC All use subject to http://about.jstor.org/terms Fig. la. Circles oriented at the edges and vertices of a tetrahedron. Dots show edgeoriented circle centers at the points of intersection of vertex-tangent planes and edge-normal planes. Fig. Id. A single tetrahedral patch unit closed with geodesic hemispheres. The number of triangle edges around the edge of the hemisphere is a multiple of 10. The number of triangle edges around an end of the tetrahedral unit is a multiple of three. Thirty is the smallest number satisfying geodesic continuity from a tetrahedral unit to a geodesic hemisphere. Fig. lb. Circular arcs truncated at their points of intersection. Arcs are grouped to form piecewise-circular boundaries about each face of the polyhedron. Fig. le. Six tetrahedra in the configuration of a benzene molecule. Fig. Ic. Six-arc boundaries triangulated to a depth of six. Fig. If. A smoothed hull homeomorphic to a torus with 12 points removed. Fig. Ig. Ten tetrahedra. The edges of the tetrahedra form a large tetrahedron inscribed in the structure. Fig. lh. A smoothed hull homeomorphic to a sphere with six handles and 16 points removed. 282 Dickson, True 3D Computer Modeling This content downloaded from 157.55.39.55 on Tue, 23 Aug 2016 04:15:58 UTC All use subject to http://about.jstor.org/terms mation for the task of drawing a minimal surface from the boundary. The general problem of finding a minimal surface defined by an arbitrary boundary is known as the Plateau Problem. I call my approximation a pseudo-minimal surface patch solution. The resulting construction has a polygon net with the continuity properties of the geodesic dome [7], but has been extended to a nonspherical topology: a topology of many handles. Figure la shows circles oriented at the vertices and edges of a tetrahedron. Figure 1 b shows circular arcs intersecting at the vertex-tangent and edge-normal planes of the tetrahedron. Figure 1 c shows pseudo-minimal surface patches triangulated to a depth of six. Figure I d shows pseudo-minimal surface patches triangulated to depth 10 and joined to geodesic hemispheres. Figures Ie through lh show a surface with six handles constructed using this method. There exist similar constructions based on a simple cubic and an octet lattice structure [8]. The construction of the cubic case is shown in Fig. 2. An energy-minimizing surface of identical topology to the tetrahedral lattice was independently discovered by David M. Anderson and David Hoffman [9] in 1988. This surface exists in the equiprobability surfaces of the covalent bonding electron orbitals between carbon atoms in block copolymer organic compounds. The octahedral infinite periodic minimal surface (IPMS) was shown by A. Schoen [10], in 1970, who attributed it to H. A. Schwarz [11] in the 1800s. I believe my drawing of geodesics on the surface is unique. Minimal surfaces have a property known as vanishing mean curvature. The principal curvatures of a surface at a particular point on the surface are measured in two orthogonal planes intersecting at the line normal to the surface at the point [12,13]. On a minimal surface, the principle curvatures tend to become equal and opposite. This also means that the geodesic strut pattern of my construction not only has the multiple-arch property that made the geodesic dome important, but also has potential tensegrity properties of equally balanced tensions. In 1985 I ported the programs I wrote in GRASS to a C-language object-modeling facility I have developed to complement the Wavefront Technologies software product. Among the techniques I developed in 1986 was one for creating a 3D object from a luminance contour graph of a digitally captured video image (Fig. 3). There is also, of course, interest in forming abstract representations of the observed or measured physical world. Three-dimensional robot vision is the reciprocal operation to direct 3D computer output. In 1986 I experimented with making milled metal sculpture from objects I designed in the Wavefront interactive modeling environment (Fig. 4). This is an older technology, hence the realization in three-space of the abstract design was not as direct as it can be today. Fig. 2. (top, left to right) (a) Circles oriented at the edges and vertices of an octahedron. (b) Circular arcs truncated at their points of intersection. Arcs are grouped to form piecewise-circular boundaries about each face of the polyhedron. (c) Six-arc boundaries triangulated to a depth of 10. (bottom, left to right) (d) A single octahedral patch unit closed with geodesic hemispheres. The number of triangle edges around the edge of the hemisphere is a multiple of 10. The number of triangle edges around an end of the octahedral unit is a multiple of four. Twenty is the smallest number satisfying geodesic continuity from an octahedral unit to a geodesic hemisphere. (e) Eight octahedra in the configuration of a simple-cubic lattice. (f) A smoothed hull homeomorphic to a sphere with 12 handles and 24 points removed. Dickson, True 3D Computer Modeling 283 This content downloaded from 157.55.39.55 on Tue, 23 Aug 2016 04:15:58 UTC All use subject to http://about.jstor.org/terms Fig. 3a. A colorresampled, digitally captured, videocamera image taken from a live model. Fig. 4a. A wavefront advanced visualizer rendering of the numerical database of a sculpture as proposed for direct machining in metal. Fig. 4b. Numb2.obj., milled aluminum on American black walnut base, 24 x 12 x 12 in, 1986. This object's features were milled from numerical templates and computer-generated images supplied by the artist [26]. Fig. 3b. A wavei?~ ~ /~ ~ front advanced visualizer render*Given: ''t>'^ dimensional numerical database resin that hardens under theinuenof the luminnnce contour graph of printer, an elevthe image in Fig. 3a. COMPUTER-PRINTED SCULPTURE OF MATHEMATICS