Leonardo

No.2. (above) Brent CoIIins, Two-SidedSurfacewith Cruciform Pattern,wood, 47 x 16 x 6 in, 1991. The fifth piece in a series, this sculpture preserves. the symmetries of the ellipsoid. Its six border curves are nearly planar and could be regarded as the rims of transparent windows into this closed, orientable surface of genus six. (See article by George K. Francis.) COLOR PLATE A No.1. (above) Branko Griinbaum and G. C. Shephard, example of a colored interlace pattern. Six colors are used systematically, and one strand is thickened and shown in black for better visibility. Some colored interlace patterns appear in Islamic ornamentation, but are not very common. The coloring used is said to be 'perfect'; this means that every symmetry of the interlace pattern changes the colors of the strands in a consistent way. No.3. (right) Lucio Saffaro, PlatonicForms, computer graphics, 1986. A first-order dodecahedron is shown on the left, made of six regular dodecahedra rotated 36° with respect to axes passing through the middle points of a starting dodecahedron . A cross-section of the same dodecahedron, showing its interior, is shown on the right. COLOR PLATE B No.1. (top left) David Brisson, Hypercube, watercolor, 10 x 15 in, 1967. This representation of the hypercube is similar to the one drawn by W. I. Stringham in 1880. This representation incorporates color to assist in fonn definition. (See article by Harriet E. Brisson.) No.2. (top right) David Brisson, Hyperanaglyph, red, blue and white painted metal rods on rotating turntable, 18 x 18 x 18 in, 1975. This is a 3D drawing of the two equations z2 + w2 =1 (red) and z2 + w2 =-1 (blue), plotted for complex values of z and w in a 4D space of two real and two imaginary coordinates. (See article by Harriet E. Brisson.) No.3. (left) Harriet E. Brisson, Truncated ClosePacking Octahedra, Rhombidodecahedm and Cubes, tensegrity structure of plexiglass, aluminum tubes and nylon cord, 28 x 28 x 28 in, 1976. This form differs from other space fillings in that the closepacking octahedron is not a Platonic or an Archimedean Solid. No.4. (left) Harriet E. Brisson, Magic Box, truncated octahedra of neon tubes in two-waymirrored cube on four vertical sides with mirrored top and bottom, 1978. This is one of the self-packing polyhedra-this single fonn will fill space completely . Here it is reflected indefinitely in a cubic kaleidoscope of two-way mirrors, which are opaque when behind the fonn but become transparent when in front of it. COLOR PlATE C No.1. (top left) Fred Almgren andJohn Sullivan, computer graphics. A large soap bubble cluster illustrating the computergraphics techniques that accurately model the Fresnel effect of decreased transparency at oblique angles and the colored interference patterns in a thin Him. This is the most intricate soap bubble geometry that has been rendered by such techniques. It is the stereographic projection into tbree-space of a regular 4D polytope called the dodecaplex. The soap films in this cluster are all pieces of spheres and meet at the correct angles for a physical soap film. No.2. (top right) Fred Almgren andJohn Sullivan, computer graphics. A hypercube, when stereographically projected to tbree-space, forms this cluster of seven bubbles. The center bubble is a cube with spherical faces, meeting at 120· angles as sheets of soap film must. It was rendered by the techniques the authors describe, modeling the optics of thin fllms. No.3. (center) Attilio Pierelli, 62~3 Ipercubo1, stainless steel, 8 x 8 x 8 in, 1974. This sculpture depicts a central projection of the hypercube from a point in four-space. (See article by Thomas Banchoff and Davide P. Cervone.) No.4. (bottom) Thomas Banchoff and Nicholas Thompson, TorusStereographically Projectedfrom FourSpace ,computer-graphics, 1989. This image shows a banded torus projected from the hypersphere in fourspace , stretching to infinity and separating space into two congruent pieces. COLOR PlATE D No.1. (top left) Charles O. Perry, Eclipse, aluminum sculpture, 35 ft, 1973. This detail of a sculpture installed in the Hyatt Regency, San Francisco , allows a close-up view of the helical explosion of its faces, rotating from a dodecahedron through an...