Copper Reflections:Don Mitchell,Pat Hanrahan:Special_Topics,Modeling_and_Simulation:Special_Topics/Modeling_and_Simulation/copper2.tiff::Produced by ray tracing software written by Pat Hanrahan (Princeton) and Don Mitchell (AT&T). Their software uses Fermat's principle to find reflections from an implicitly-defined surface.:Dec. 1991:807828514 Crystal Symmetry:Olaf Holt:Special_Topics,Modeling_and_Simulation:symmetry/eucsyms.tiff:Snapshot of Olaf Holt's "eucsyms" program, which displays any of the 230 crystallographic groups in Euclidean 3-space.:Eucsyms can be found as an external module of geomview, and once the picture looks right pssnapshot can record the pixels.:Dec. 1991:809020813 Escher Fish:Silvio Levy:Special_Topics,Hyperbolic_Geometry:Special_Topics/Hyperbolic_Geometry/escher.tiff:Silvio Levy's tesselation of the Poincare model of the hyperbolic plane by fish in M.C. Escher's style.:Created with Mathematica.:Dec. 1991:809020673 Antoine's Necklace:Matt Grayson,Charlie Gunn:Special_Topics,Topology;General_Interest,Fractals:Special_Topics/Topology/antoine.tiff:A stage in the construction of Antoine's Necklace. Beginning with a single torus, the necklace is the limit of a sequence where each step replaces a torus with N linked tori (here N=10). It's known that a function exists whose critical set is exactly the necklace.::Feb 1992:807828514 Hyperbolic Dirichlet Domains:Charlie Gunn:Special_Topics,Hyperbolic_Geometry:Special_Topics/Hyperbolic_Geometry/dirdoms.tiff:Snapshot of 3-D viewing software under development at the Geometry Center. This view shows Dirichlet domains in H^3.::1992:809020697 Sudanese Moebius Band:George Francis:Special_Topics,Differential_Geometry:Special_Topics/Differential_Geometry/illiview.tiff:Snapshot of George Francis' "Illiview" viewer, displaying the Sudanese Moebius band, whose boundary is a circle.::Jan 1992:807828514 2-d Crystal Growth:Andy Roosen:Special_Topics,Modeling_and_Simulation:crystals/roosen.tiff:Simulation of crystal growth in 2-D. Color encodes temperature; white lines are solid-liquid interfaces.::Dec 1991:809020117 Borromean Rings:Toby Orloff:Video_Productions,Not_Knot:Video_Productions/Not_Knot/BorRings.tiff:Shows the Borromean Rings in a symmetric configuration, on a stand.::1990:809020273 2-D Cone Space:Stuart Levy:Video_Productions,Not_Knot:Video_Productions/Not_Knot/Cones.tiff:Gives three outside views of 2 dimensional cone space. The first shows a cone from the side, the second from above, and the third several fundamental domains glued together.::1990:809020708 3-D Cone Space:Stuart Levy:Video_Productions,Not_Knot:Video_Productions/Not_Knot/Cones2.tiff:Beginning of the sequence showing how to build a 3-D cone space from 2-D cone spaces.::1990:809020724 1-D Euclidean Tiling:Charlie Gunn:Video_Productions,Not_Knot:Video_Productions/Not_Knot/EucTiling.tiff:A cube with a figure in it is replicated according to a set of symmetries which result in an infinite line of these cubes.::1990?:809020738 2-D Euclidean Tiling:Charlie Gunn:Video_Productions,Not_Knot:Video_Productions/Not_Knot/EucTiling2.tiff:A cube with a figure in it is replicated to fill a plane.::1990?:809020756 3-D Euclidean Tiling:Charlie Gunn:Video_Productions,Not_Knot:Video_Productions/Not_Knot/EucTiling3.tiff:A cube with a figure in it tiles 3-d space by repeated application of rotational symmetries.::1990?:809020777 Figure 8 Knot:Toby Orloff:Video_Productions,Not_Knot:Video_Productions/Not_Knot/Fig8.tiff:A figure 8 knot shown on a pedestal.::1990:809020298 Hyperbolic Dodecahedron:Charlie Gunn:Video_Productions,Not_Knot;Special_Topics,Hyperbolic_Geometry:Video_Productions/Not_Knot/HDodec.tiff:A regular dodecahedron in hyperbolic space, with ninety degree dihedral angles. (This is not possible in our usual, Euclidean three dimensional space.)::1990:807828514 Hyperbolic Space Tiled with Dodecahedra, 1:Charlie Gunn:Video_Productions,Not_Knot;Special_Topics,Hyperbolic_Geometry:Video_Productions/Not_Knot/HSpace.tiff:Three dimensional hyperbolic space, as seen by an inside observer, tiled with regular dodecahedra with 90 degree dihedral angles.::1990:807828514 Hyperbolic Space Tiled with Dodecahedra, 2:Charlie Gunn:Video_Productions,Not_Knot;Special_Topics,Hyperbolic_Geometry:Video_Productions/Not_Knot/HSpace2.tiff:Three dimensional hyperbolic space, as seen by an inside observer, tiled with regular dodecahedra with 90 degree dihedral angles.::1990:807828514 Life in 3-D Cone Space:Stuart Levy:Video_Productions,Not_Knot:Video_Productions/Not_Knot/InfCone.tiff:A view of what space looks like when the cone point is pulled to infinity, for an inside observer.::1990:809020788 Borromean Rings in 3-D:Toby Orloff,Delle Maxwell:Video_Productions,Not_Knot:Video_Productions/Not_Knot/Rings.tiff:The borromean rings, arranged to show how the three rings are equivalent to eachother, on a stand.::1990:838935025 Cube with Borromean Rings Cut Out:Charlie Gunn:Video_Productions,Not_Knot;Special_Topics,Modeling_and_Simulation:Video_Productions/Not_Knot/RingsCube.tiff:A transparent cube with the borromean rings cut out of it, on a floor showing two dimensional hyperbolic space tiled with triangles.::1990:807828514 The Order-7 Borromean Ring Orbifold:Charlie Gunn:Video_Productions,Not_Knot;Special_Topics,Hyperbolic_Geometry:Video_Productions/Not_Knot/ShrinkAxis.tiff:Three dimensional hyperbolic space, as seen by an inside observer, tiled with dodecahedra. This is the covering space of the order-7 Borromean ring orbifold.::1990:807828514 The Borromean Ring Complement Manifold:Charlie Gunn:Video_Productions,Not_Knot;Special_Topics,Hyperbolic_Geometry:Video_Productions/Not_Knot/ShrinkAxis2.tiff:Three dimensional hyperbolic space, as seen by an inside observer, tiled with dodecahedra some of whose vertices are on the sphere at infinity. An arrow points to one of these ideal vertices.::1990:807828514 Borromean Ring Complement Manifold 1:Charlie Gunn:Video_Productions,Not_Knot;Special_Topics,Hyperbolic_Geometry:Video_Productions/Not_Knot/ShrunkAxis.tiff:Three dimensional hyperbolic space, as seen by an inside observer close to the sphere at infinity, tiled with rhombic dodecahedra.::1990:807828514 Borromean Ring Complement Manifold 2:Charlie Gunn:Video_Productions,Not_Knot;Special_Topics,Hyperbolic_Geometry:Video_Productions/Not_Knot/ShrunkAxis2.tiff:Three dimensional hyperbolic space, as seen by an inside observer close to the sphere at infinity, tiled with rhombic dodecahedra.::1990:807828514 Borromean Ring Complement Manifold 3:Charlie Gunn:Video_Productions,Not_Knot;Special_Topics,Hyperbolic_Geometry:Video_Productions/Not_Knot/ShrunkAxis3.tiff:Three dimensional hyperbolic space, as seen by an inside observer close a vertex on the sphere at infinity, tiled with rhombic dodecahedra.::1990:807828514 4 Dodecahedra in Hyperbolic Space:Charlie Gunn:Video_Productions,Not_Knot;Special_Topics,Hyperbolic_Geometry:Video_Productions/Not_Knot/Tiling.tiff:One regular dodecahedron has been rotated three times around one of its edges, and the original and the replicas fit together perfectly since the dihedral angle is 90 degrees.::1990:807828514 Title, "Not Knot":Scott Kim:Video_Productions,Not_Knot;General_Interest,Digital_Art:Video_Productions/Not_Knot/Title.tiff:Title frame of the film.::1990:807828514 Cube with 3 Pair Colored Axes:Stuart Levy:Video_Productions,Not_Knot;Special_Topics,Topology:Video_Productions/Not_Knot/WrapAxis.tiff:An ordinary cube is shown with axes symmetrically placed, one per face.::1990:807828514 Cube with 1st Pair Glued:Stuart Levy:Video_Productions,Not_Knot;Special_Topics,Topology:Video_Productions/Not_Knot/WrapAxis2.tiff:The first stage in identifying sides of the cube according to half turns around the colored axes.::1990:807828514 Cube with 2nd Pair Glued:Stuart Levy:Video_Productions,Not_Knot;Special_Topics,Topology:Video_Productions/Not_Knot/WrapAxis3.tiff:The second stage in identifying sides of the cube according to half turns around the colored axes.::1990:807828514 Cube with 3rd Pair Glued:Stuart Levy:Video_Productions,Not_Knot;Special_Topics,Topology:Video_Productions/Not_Knot/WrapAxis4.tiff:The third stage in identifying sides of the cube according to half turns around the colored axes. Actually, it is not quite complete in this view, since the remaining faces are identified "at infinity".::1990:807828514 4-d Map:Eduardo Tabacman:General_Interest,Higher_Dimensional_Objects;Special_Topics,Dynamical_Systems:General_Interest/Higher_Dimensional_Objects/tabacman2.tiff:Two 2-dimensional surfaces in 4D, projected to 3D. (The color, from green to blue, represents the fourth coordinate). The surfaces are points in R^4 that converge (for stable manifold) to a fixed point (roughly in the middle of the surfaces) under iteration of a symplectic map, or diverge (for unstable one).:Made by iteration of a (small) grid around the fixed point, in the directions of the eigenvectors of the Jacobian of the map. Viewed with geomview (4d-module).:4/92:807828514 4-d Map II:Eduardo Tabacman:General_Interest,Higher_Dimensional_Objects;Special_Topics,Dynamical_Systems:General_Interest/Higher_Dimensional_Objects/tabacman3.tiff:Same as above, only that the two surfaces are together, as they really are in R^4.:Same as above.:4/92:807828514 Everting the Sphere:David Ben-Zvi,Nathaniel Thurston:Video_Productions,Outside_In:Video_Productions/Outside_In/evertseq.tiff:Nine stages in a sphere eversion, in William Thurston's style. We see 1/16th of the sphere (a 45-degree sector from pole to equator) as opaque, while the remainder is nearly transparent. A more detailed description appears in /u/gcg/pix/incoming/evert.note.:"make" in data/sphere/evertseq. It depends on running the "sphere" program in /u/sphere/bin..:Oct 26, 1992:807828514 Hoops in R3:Mark Phillips:Special_Topics,Topology:Special_Topics/Topology/hoops.tiff:A hoop is a circle of radius 1 and the interesting thing about them is that you can completely fill up the interior of a torus with a disjoint union of them. This material is from a presentation that Dan Asimov gave at the October 1992 MSRI workshop.:Mark generated the pictures interactively using Celeste Fowler's "sweep" geomview module. A short film can be found on S-61 from 0\-51\-00 to 0\-56\-30.:Oct 27, 1992:809040970 Square Stretched Over Sphere:Paul Burchard:Special_Topics,Complex_Analysis;Special_Topics,Topology:Special_Topics/Differential_Equations/PlaneMap.tiff:The picture shows an intermediate step in the process of stretching a tiny, thin, flat, square sheet of rubber onto the surface of a globe (first placing it near the north pole, and then stretching it down around the globe until all the edges are near the south pole). The cloverleaf-shaped hole is the part of the sphere not covered by the square.:This picture is derived from the application /u/burchard/Apps/PlaneToSphere.app which demonstrates the formation of bubbling-off singularities in the harmonic map heat flow.:Oct 5, 1992:808776063 237 Tiling of the Hyperbolic Plane:Anonymous:Special_Topics,Hyperbolic_Geometry:Special_Topics/Hyperbolic_Geometry/237conformal.tiff:A black and white tiling with lines that connect to give a fiery appearance.::1991:807828514 Sphere Scribble:Millie Niss:Special_Topics,Differential_Geometry:spherical/spherescribble.tiff:As described in the Summer Institute 1993 report, Niss designed and implemented a drawing program called Sphere Scribble which draws on the surface of a sphere. It was designed to work similarly to a typical drawing program, where the user selects any of several drawing tools on the user interface panel, and then draws objects with the mouse. This picture is one such drawing.::summer, 1993:808778254 Plants on Island in Water:Kate Jenkins:Special_Topics,Modeling_and_Simulation;General_Interest,Fractals;General_Interest,Digital_Art:L-systems/all.tiff:These plants were created with L-systems, which also simulate the impact of outside forces on the plant such as gravity, wind, and sunlight.:The L-systems were created in Mathematica to output the initial data, which was then displayed by geomview (developed here at the Geometry Center) or rayshade (a raytracer by Craig Kolb).:summer 1992:809022042 Variety of Flowering Plants:Kate Jenkins:Special_Topics,Modeling_and_Simulation;General_Interest,Fractals;General_Interest,Digital_Art:L-systems/bunch.tiff:These plants were created with L-systems, which also simulate the impact of outside forces on the plant such as gravity, wind, and sunlight.:The L-systems were created in Mathematica to output the initial data, which was then displayed by geomview (developed here at the Geometry Center) or rayshade (a raytracer by Craig Kolb).:summer 1992:809022052 Garden at Midday:Kate Jenkins,Daeron Meyer:Special_Topics,Modeling_and_Simulation;General_Interest,Fractals;General_Interest,Digital_Art:L-systems/lightgarden.tiff:These plants were created with L-systems, which also simulate the impact of outside forces on the plant such as gravity, wind, and sunlight.:The L-systems were created in Mathematica to output the initial data, which was then displayed by geomview (developed here at the Geometry Center) or rayshade (a raytracer by Craig Kolb).:summer 1992:809022064 Garden at Sunset:Kate Jenkins:Special_Topics,Modeling_and_Simulation;General_Interest,Fractals;General_Interest,Digital_Art:L-systems/sunset.tiff:These plants were created with L-systems, which also simulate the impact of outside forces on the plant such as gravity, wind, and sunlight.:The L-systems were created in Mathematica to output the initial data, which was then displayed by geomview (developed here at the Geometry Center) or rayshade (a raytracer by Craig Kolb).:summer 1992:809022076 Glowing Contour Surface:Nelson Max,Rob Almgren:Special_Topics,Modeling_and_Simulation:crystals/bintest.tiff:Nelson Max's visualization of Rob Almgren's crystallization simulation. Contour surface of crystallization function. Surface color, and also volume glow density and color, are determined by temperature.::August 1992:809020149 School of Ellipsoidal Fish:Maria Nagan:General_Interest,Digital_Art:educational/school.tiff:Maria Nagan's school of fish made during 1991-1992 school year as her mentoring project.:snapshot of geomview window.:spring 1992:808777404 Henon Map (Main):Brian Meloon:General_Interest,Fractals,Henon:General_Interest/Fractals/henon.tiff:The Henon map is the two dimensional analogue of the logistic equation H(x,y) = (x^2 - a*y + c, x), where a and c are constants. The picture shows a portion of the locus of points in the real plane where the rate of escape of a point under iteration is the same going forwards and backwards (using the inverse map). The picture is described further in Meloons Summer Institute report, 1993.::summer, 1993:807841863 Penrose by Quasitiler:Eugenio Durand:General_Interest,Higher_Dimensional_Objects;Special_Topics,Tilings:General_Interest/Higher_Dimensional_Objects/quasitiler.tiff:This figure shows a Penrose tiling obtained by projecting a slice of a regular lattice in 5-space onto 3-space, and the one-skeleton of a five-cube projected onto 3-space (evoking the complementary dimensions). The tiling was generated using QuasiTiler, a program whose interesting history exemplifies the advantages of the Center's mode of operation. The program was written by graduate student programmer Eugenio Durand for sabbatical visitor Marjorie Senechal of Smith College, who was writing a book on quasicrystals. Postdocs Paul Burchard and Leonidas Palios advised Durand. Together with apprentice Daeron Meyer, Burchard and Durand adapted the program for the World-Wide Web, where it became widely known. (See also Quasitiler.):Geomview and its module NDview were used for the three dimensional composition. RenderMan was used to texture-map the tiling onto the ground plane and to render the whole.:May, 1994:809025776 Transparent Red-Blue Eversion:David Ben-Zvi,Nathaniel Thurston:Video_Productions,Outside_In:Video_Productions/Outside_In/blue-red-alpha.tiff:These are frames from a new animation of the sphere everting made by David Ben-Zvi and Nathaniel Thurston using an algorithm of Bill Thurston. The video is being made at The Geometry Center by a team including Silvio Levy, Delle Maxwell, and Tamara Munzner.::Summer 1992:807828514 Red-Blue Eversion:David Ben-Zvi,Nathaniel Thurston:Video_Productions,Outside_In:Video_Productions/Outside_In/blue-red.tiff:These are frames from a new animation of the sphere everting made by David Ben-Zvi and Nathaniel Thurston using an algorithm of Bill Thurston. The video is being made at The Geometry Center by a team including Silvio Levy, Delle Maxwell, and Tamara Munzner.::Summer 1992:807828514 Transparent Peach Eversion:David Ben-Zvi,Nathaniel Thurston:Video_Productions,Outside_In:Video_Productions/Outside_In/peach-alpha.tiff:These are frames from a new animation of the sphere everting made by David Ben-Zvi and Nathaniel Thurston using an algorithm of Bill Thurston. The video is being made at The Geometry Center by a team including Silvio Levy, Delle Maxwell, and Tamara Munzner.::Summer 1992:807828514 8snails:George Francis:Special_Topics,Differential_Geometry:Special_Topics/Differential_Geometry/gkf.8snails.tiff:Blaine Lawson's class \tau of ruled, minimal surfaces on S^3 have this quaternionic parameterization\- u,v -> e^{2iu}cos v + j e^{biu}sin v which can yield the untwisted annulus, spherical bead, half-twisted Moebius band, Sudanese Moebius band, once-twisted annulus, Clifford torus, three-half-twisted Moebius band, and Brehm's smooth knotbox.:based on the soloRTICA called illiSnail in /u/gfrancis.:January 1993:807828514 4snails:George Francis:Special_Topics,Differential_Geometry:Special_Topics/Differential_Geometry/gkf.4snails.tiff:Blaine Lawson's class \tau of ruled, minimal surfaces on S^3 have this quaternionic parameterization\- u,v -> e^{2iu}cos v + j e^{biu}sin v which can yield the once-twisted annulus, Clifford torus, three-half-twisted Moebius band, and Brehm's smooth knotbox.:based on the soloRTICA called illiSnail in /u/gfrancis.:January 1993:807828514 Wild Complex Function:Olaf Holt,Daeron Meyer:General_Interest,Higher_Dimensional_Objects;Special_Topics,Complex_Analysis:General_Interest/Higher_Dimensional_Objects/cplxfn2.tiff:view of an inverse cosine function early in the development of the complex function viewer. Color indicates the imaginary component of the graph.:Using the external module cplxview of Geomview:September 1993:808778354 Crystal Dendrites:Andrew Roosen:Special_Topics,Modeling_and_Simulation:crystals/2d-dendrites.tiff:An example of one phenomenon being studied is the effect of noise of various types on crystal growth. Shown here are patterns which can arise when the tips of growing dendrites are periodically subjected to short small pulses of heat, such as might be introduced by a laser. Each pulse appears to momentarily slow the growth of the tip and encourage the development of side branches. When the frequency is too high, surface tension prevents any growth of side branches (in the absence of additional noise). At lower frequencies, natural instabilities in the growth are triggered, as shown, and at much lower frequencies, as shown, the growth looks very unnatural as the growth laws cope with this (highly nonlinear) perturbation.:The mathematical and computational model developed to make these simulations was developed by team member Andrew Roosen, working with senior team member Jean Taylor. A related model has been developed by Robert Almgren, working with senior team member Fred Almgren; Roosen in fact used part of the code written by R. Almgren.:1991:809020070 Vibrating Minimal Surface:Jeremy Ackerman:Special_Topics,Modeling_and_Simulation:Special_Topics/Modeling_and_Simulation/vibration.tiff:Several steps in the vibration of a minimal surface, in which a catenoid collapses during dynamic simulation. Each point on the surface was given a velocity to the right, while the rings remained fixed\- a)the initial surface, b) surface shifts to the right, c) surface shifts to the left, d) surface returns to center, e) neck narrows, and f) neck closes and surface separates into two disks which continue vibrating. This is Figure 3 in Ackerman's Summer Institute report, 1993.:This simulation created using Ken Brakke's evolver, modified to allow inertia.:summer, 1993:809378812 Circulatory System:Kirsten Bancroft:Special_Topics,Modeling_and_Simulation:Special_Topics/Modeling_and_Simulation/circulatory.tiff:This is a frame from one of Bancroft's simulations of the circulatory system. It depicts the proportional blood volumes in the different parts of the circulatory system. The pressure levels are depicted in bar graphs at the side of the corresponding arterial and venous divisions. This simulation is described in Bancroft's Summer Institute report, 1993, and based on the mathematical model given by Charles Peskin in Mathematics in Medicine and the Life Sciences, Springer-Verlag, New York 1992.:This is one of a series of VECT animation files created for Geomview.:summer, 1993:807828514 Diving Simulation:Elizabeth Callaghan:Special_Topics,Modeling_and_Simulation:Special_Topics/Modeling_and_Simulation/diver.tiff:Several stages in a dive are shown here using Callaghan's diving simulator. The simulator allows the user to choose the type of dive (forward, backward, inward), various parameters including the height and weight of diver, takeoff angle, takeoff speed, and times at which the diver begins and ends the moment-of-intertia changing maneuvers (tuck, pike, straight, lineup, and save). It then solves the differential equations and a geomview mannequin executes the actual dive. More information in found in Callaghan's Summer Institute report, 1993.:the diver was modelled using Geomview.:summer, 1993:807828514 Weierstrass P-function:Sang H. Chin:Special_Topics,Complex_Analysis:Special_Topics/Complex_Analysis/weierstrass.tiff:This shows the image of the Weiestrass P-function using CRSolver. CRSolver lets users draw a boundary on the complex plane and shows the image of an analytic map from the 2x2 square centered on the origin, to the region defined by the boundary. CRSolver was first developed by Paul Burchard, then he and Sang Chin implemented an improved algorithm which allowed it to handle boundaries with negative turning numbers.:Using CRSolver, find a conformal map from the domain square which approximates 1/z^2 by drawing a boundary curve with index -2. Then choose the doubly periodic boundary condition, which has the effect of identifying the opposite sides of the square.:summer, 1993:807828514 Two Neurons:Brendan Dunn:Special_Topics,Modeling_and_Simulation:Special_Topics/Modeling_and_Simulation/neuron.tiff:This picture was created by Newneuron, a 3-dimensional graphic system for the comparitive diplay and connection of neurons, written by Brendan Dunn. It is an extension of the Dispneuron program by Loren Frank, compatible with and dependant on the Makecyl programs by Loren Frank and the Network and Neuron programs by Jude Mitchell.::summer, 1993:807828514 Clouds 1:Danek Duvall:General_Interest,Fractals,Clouds:General_Interest/Fractals/cloud1.tiff:This picture is one of four created using spectral synthesis, as outlined by Dietmar Saupe and Richard Voss in The Science of Fractal Images, Springer-Verlag, 1988. Comparison of this method with texture modelling and a description of the actual generation of the image can be found in the Summer Institute 1993 report, "Cloud Generation".::summer, 1993:808761801 Clouds 2:Danek Duvall:General_Interest,Fractals,Clouds:General_Interest/Fractals/cloud2.tiff:This picture is one of four created using spectral synthesis, as outlined by Dietmar Saupe and Richard Voss in The Science of Fractal Images, Springer-Verlag, 1988. Comparison of this method with texture modelling and a description of the actual generation of the image can be found in the Summer Institute 1993 report, "Cloud Generation".::summer, 1993:808761841 Clouds 3:Danek Duvall:General_Interest,Fractals,Clouds:General_Interest/Fractals/cloud3.tiff:This picture is one of four created using spectral synthesis, as outlined by Dietmar Saupe and Richard Voss in The Science of Fractal Images, Springer-Verlag, 1988. Comparison of this method with texture modelling and a description of the actual generation of the image can be found in the Summer Institute 1993 report, "Cloud Generation".::summer, 1993:808761857 Clouds 4:Danek Duvall:General_Interest,Fractals,Clouds:General_Interest/Fractals/cloud4.tiff:This picture is one of four created using spectral synthesis, as outlined by Dietmar Saupe and Richard Voss in The Science of Fractal Images, Springer-Verlag, 1988. Comparison of this method with texture modelling and a description of the actual generation of the image can be found in the Summer Institute 1993 report, "Cloud Generation".::summer, 1993:808761872 Meanders:Patrick Friel:Special_Topics,Combinatorics:Special_Topics/Combinatorics/meander.tiff:A meander is an alternative representation of an acceptable row sequence for a simple alternating transit maze, where the numbers indicate the order of the crossings of a curve through a line. Friel worked on the problem of counting meanders, and created a program to visualize meanders and test integer sequences for acceptability as meander patterns. The user interface is shown here. More information is available in the Summer Institute 1993 reports.::summer, 1993:807828514 Amoeba Fractal:Christine Heitsch:General_Interest,Fractals:General_Interest/Fractals/amoeba.tiff:This is a two-dimensional slice of the cubic connectedness locus, a higher dimensional analogue of the Mandelbrot set. Heitsch extended the program Brot to be able to compute these intersections for her summer project, described in the Summer Institute 1993 reports.::summer, 1993:807828514 Brot Cheese Fractal:Christine Heitsch:General_Interest,Fractals:General_Interest/Fractals/brot_cheese.tiff:This is a two-dimensional slice of the cubic connectedness locus, a higher dimensional analogue of the Mandelbrot set. Heitsch extended the program Brot to be able to compute these intersections for her summer project, described in the Summer Institute 1993 reports.::summer, 1993:807828514 Nebula Fractal:Christine Heitsch:General_Interest,Fractals:General_Interest/Fractals/nebula.tiff:This is a two-dimensional slice of the cubic connectedness locus, a higher dimensional analogue of the Mandelbrot set. Heitsch extended the program Brot to be able to compute these intersections for her summer project, described in the Summer Institute 1993 reports.::summer, 1993:807828514 BSP Trees:Kate Jenkins:Special_Topics,Computational_Geometry:Special_Topics/Modeling_and_Simulation/bsptrees.tiff:This picture shows the interface for an interactive graphics application that demonstrates binary space partitioning (bsp) trees and some of their applications in computer graphics, such as for ray-tracing algorithms. It was written as part of Jenkins summer project and is described in the Summer Institute 1993 reports.::summer, 1993:808778584 Tonality Circles:Jing Li:General_Interest,Digital_Art:General_Interest/Digital_Art/tonalitycircles.tiff:Musical notes and their arrangement in a score are reminiscent of sequences of numbers. Harmony in particular can be analysed by the computer because it is mostly defined by intervals between neighboring notes, which correspond to intervals between numbers. The picture shows the interface to program Li wrote during Summer Institute, 1993 which graphically depicts the harmonies in a piece of music as it is playing. The algorithm is described in the Summer Institute 1993 reports.::summer, 1993:807828514 Face Drawn on Sphere:Millie Niss:Special_Topics,Differential_Geometry:spherical/face.tiff:As described in the Summer Institute 1993 report, Niss designed and implemented a drawing program called Sphere Scribble which draws on the surface of a sphere. It was designed to work similarly to a typical drawing program, where the user selects any of several drawing tools on the user interface panel, and then draws objects with the mouse. This picture is one such drawing.::summer, 1993:808778219 Henon Isthmus:Brian Meloon:General_Interest,Fractals,Henon:General_Interest/Fractals/isthmus.tiff:The Henon map is the two dimensional analogue of the logistic equation H(x,y) = (x^2 - a*y + c, x), where a and c are constants. The picture shows a portion of the locus of points where the rate of escape of a point under iteration is the same going forwards and backwards (using the inverse map). The picture is described further in Meloon's Summer Institute report, 1993.::summer, 1993:807841819 Henon Chains:Brian Meloon:General_Interest,Fractals,Henon:General_Interest/Fractals/chains.tiff:The Henon map is the two dimensional analogue of the logistic equation H(x,y) = (x^2 - a*y + c, x), where a and c are constants. The picture shows a portion of the locus of points where the rate of escape of a point under iteration is the same going forwards and backwards (using the inverse map). The picture is described further in Meloon's Summer Institute report, 1993.::summer, 1993:807841734 Henon Cometogether:Brian Meloon:General_Interest,Fractals,Henon:General_Interest/Fractals/cometogether.tiff:The Henon map is the two dimensional analogue of the logistic equation H(x,y) = (x^2 - a*y + c, x), where a and c are constants. The picture shows a portion of the locus of points where the rate of escape of a point under iteration is the same going forwards and backwards (using the inverse map). The picture is described further in Meloon's Summer Institute report, 1993.::summer, 1993:807841751 Henon Crucified:Brian Meloon:General_Interest,Fractals,Henon:General_Interest/Fractals/crucified.tiff:The Henon map is the two dimensional analogue of the logistic equation H(x,y) = (x^2 - a*y + c, x), where a and c are constants. The picture shows a portion of the locus of points where the rate of escape of a point under iteration is the same going forwards and backwards (using the inverse map). The picture is described further in Meloon's Summer Institute report, 1993.::summer, 1993:807841762 Henon Curious:Brian Meloon:General_Interest,Fractals,Henon:General_Interest/Fractals/curious.tiff:The Henon map is the two dimensional analogue of the logistic equation H(x,y) = (x^2 - a*y + c, x), where a and c are constants. The picture shows a portion of the locus of points where the rate of escape of a point under iteration is the same going forwards and backwards (using the inverse map). The picture is described further in Meloon's Summer Institute report, 1993.::summer, 1993:807841773 Henon Jaws:Brian Meloon:General_Interest,Fractals,Henon:General_Interest/Fractals/jaws.tiff:The Henon map is the two dimensional analogue of the logistic equation H(x,y) = (x^2 - a*y + c, x), where a and c are constants. The picture shows a portion of the locus of points where the rate of escape of a point under iteration is the same going forwards and backwards (using the inverse map). The picture is described further in Meloon's Summer Institute report, 1993.::summer, 1993:807841840 Henon Overlap:Brian Meloon:General_Interest,Fractals,Henon:General_Interest/Fractals/overlap.tiff:The Henon map is the two dimensional analogue of the logistic equation H(x,y) = (x^2 - a*y + c, x), where a and c are constants. The picture shows a portion of the locus of points where the rate of escape of a point under iteration is the same going forwards and backwards (using the inverse map). The picture is described further in Meloon's Summer Institute report, 1993.::summer, 1993:807841882 Henon Ring:Brian Meloon:General_Interest,Fractals,Henon:General_Interest/Fractals/ring.tiff:The Henon map is the two dimensional analogue of the logistic equation H(x,y) = (x^2 - a*y + c, x), where a and c are constants. The picture shows a portion of the locus of points where the rate of escape of a point under iteration is the same going forwards and backwards (using the inverse map). The picture is described further in Meloon's Summer Institute report, 1993.::summer, 1993:807841895 Henon Strange:Brian Meloon:General_Interest,Fractals,Henon:General_Interest/Fractals/strange.tiff:The Henon map is the two dimensional analogue of the logistic equation H(x,y) = (x^2 - a*y + c, x), where a and c are constants. The picture shows a portion of the locus of points where the rate of escape of a point under iteration is the same going forwards and backwards (using the inverse map). The picture is described further in Meloon's Summer Institute report, 1993.::summer, 1993:807841931 Henon Stadium:Brian Meloon:General_Interest,Fractals,Henon:General_Interest/Fractals/stadium.tiff:The Henon map is the two dimensional analogue of the logistic equation H(x,y) = (x^2 - a*y + c, x), where a and c are constants. The picture shows a portion of the locus of points where the rate of escape of a point under iteration is the same going forwards and backwards (using the inverse map). The picture is described further in Meloon's Summer Institute report, 1993.::summer, 1993:807841919 Henon Happy:Brian Meloon:General_Interest,Fractals,Henon:General_Interest/Fractals/happy.tiff:The Henon map is the two dimensional analogue of the logistic equation H(x,y) = (x^2 - a*y + c, x), where a and c are constants. The picture shows a portion of the locus of points where the rate of escape of a point under iteration is the same going forwards and backwards (using the inverse map). The picture is described further in Meloon's Summer Institute report, 1993.::summer, 1993:807841786 Henon Interesting:Brian Meloon:General_Interest,Fractals,Henon:General_Interest/Fractals/interesting.tiff:The Henon map is the two dimensional analogue of the logistic equation H(x,y) = (x^2 - a*y + c, x), where a and c are constants. The picture shows a portion of the locus of points where the rate of escape of a point under iteration is the same going forwards and backwards (using the inverse map). The picture is described further in Meloon's Summer Institute report, 1993.::summer, 1993:807841798 Henon Mixing:Brian Meloon:General_Interest,Fractals,Henon:General_Interest/Fractals/mixing.tiff:The Henon map is the two dimensional analogue of the logistic equation H(x,y) = (x^2 - a*y + c, x), where a and c are constants. The picture shows a portion of the locus of points where the rate of escape of a point under iteration is the same going forwards and backwards (using the inverse map). The picture is described further in Meloon's Summer Institute report, 1993.::summer, 1993:807841873 Flat Moebius Strip:Henry Rowley:Special_Topics,Differential_Geometry;Special_Topics,Topology:Special_Topics/Differential_Geometry/moebius_strip.tiff:This moebius strip is isometric to a flat rectangle, which differs from the standard parametrization. The steps involved in its creation are found in Rowley's Summer Institute 1991 report.:Mathematica was used to obtain the parametrization, and MinneView (the precursor to Geomview) was used to view it.:summer, 1991:808778654 Three Ring Circus:Jacques Friedman:Special_Topics,Modeling_and_Simulation;General_Interest,Fractals:Special_Topics/Modeling_and_Simulation/three_ring_circus.tiff:Freidman's exploration of the dynamics of a system of three interconnected rings resulted in the visualization of a three-torus. Following a suggestion of Bill Thurston, he ray traced an object in this three-torus by allowing light rays to follow geodesics specified by the configuration space metric. This project is described in detail in the Summer Institute 1992 reports.::summer, 1992:807828514 Penumbral Shadows:Adrian Mariano,Linus Upson:Special_Topics,Modeling_and_Simulation;Special_Topics,Summer_Institute,1992;General_Interest,Higher_Dimensional_Objects:Special_Topics/Modeling_and_Simulation/penumbral.tiff:Mariano and Upson, undergraduates in the 1992 Geometry Center Summer Institute, produced a video that was accepted for the 1994 ACM Symposium on Computational Geometry video review. This frame illustrates their original observation that the shadows cast by a polygonal light source shining on a single polygon in a three-dimensional scene are the projections of four-dimensional polytopes. They used a direct simulation of the illumination process for the very complex light intensity distribution. This project is interesting both as an example of undergraduate research fostered by the unique Geometry Center environment, and as perhaps the first example of a research result communicated primarily by video rather than text. Their report is included in the Summer Institute 1992 reports.::summer, 1992:812108680 Sunrise on Io:David Ben-Zvi,Paul Burchard:General_Interest,Fractals,CCL:General_Interest/Fractals/sunrise_on_io.tiff:This is an image of a two dimensional slices of the cubic connectedness locus. The locus (CCL for short) is a four-dimensional analog of the Mandelbrot set. This picture appeared in the October 1993 issue of Scientific American, along with "Black Hole". (See also the long description.):This picture was created using the program "Brot", written at the Geometry Center by Linus Upson and Christine Heitsch.:summer, 1993:807831403 Black Hole:David Ben-Zvi,Paul Burchard:General_Interest,Fractals,CCL:General_Interest/Fractals/black_hole.tiff:This is an image of a two dimensional slices of the cubic connectedness locus. The locus (CCL for short) is a four-dimensional analog of the Mandelbrot set. This picture appeared in the October 1993 issue of Scientific American, along with "Sunrise on Io". (See also the long description.):This picture was created using the program "Brot", written at the Geometry Center by Linus Upson and Christine Heitsch.:summer, 1993:807829877 Cantor Whirlpool:David Ben-Zvi,Paul Burchard:General_Interest,Fractals,CCL:General_Interest/Fractals/cantor_whirlpool.tiff:This is an image of a two dimensional slices of the cubic connectedness locus. The locus (CCL for short) is a four-dimensional analog of the Mandelbrot set. Note the overlapping spiral configuration, reminiscent of the Mandelbrot set, and the Cantor set comb, a typical feature of real dynamical systems. (See also the long description.):This picture was created using the program "Brot", written at the Geometry Center by Linus Upson and Christine Heitsch.:summer, 1993:807829892 Cell Growth:David Ben-Zvi,Paul Burchard:General_Interest,Fractals,CCL:General_Interest/Fractals/cell_growth.tiff:This is an image of a two dimensional slices of the cubic connectedness locus. The locus (CCL for short) is a four-dimensional analog of the Mandelbrot set. (See also the long description.):This picture was created using the program "Brot", written at the Geometry Center by Linus Upson and Christine Heitsch.:summer, 1993:807829902 Circles:David Ben-Zvi,Paul Burchard:General_Interest,Fractals,CCL:General_Interest/Fractals/circles.tiff:This is an image of a two dimensional slices of the cubic connectedness locus. The locus (CCL for short) is a four-dimensional analog of the Mandelbrot set. (See also the long description.):This picture was created using the program "Brot", written at the Geometry Center by Linus Upson and Christine Heitsch.:summer, 1993:807829942 Comb of Doom:David Ben-Zvi,Paul Burchard:General_Interest,Fractals,CCL:General_Interest/Fractals/comb_of_doom.tiff:This is an image of a two dimensional slices of the cubic connectedness locus. The locus (CCL for short) is a four-dimensional analog of the Mandelbrot set. This configuration is a good depiction of the "topologist's comb", a typical example of a non-locally connected set. (See also the long description.):This picture was created using the program "Brot", written at the Geometry Center by Linus Upson and Christine Heitsch.:summer, 1993:807829950 Crabs:David Ben-Zvi,Paul Burchard:General_Interest,Fractals,CCL:General_Interest/Fractals/crabs.tiff:This is an image of a two dimensional slices of the cubic connectedness locus. The locus (CCL for short) is a four-dimensional analog of the Mandelbrot set. (See also the long description.):This picture was created using the program "Brot", written at the Geometry Center by Linus Upson and Christine Heitsch.:summer, 1993:807829969 Eko:David Ben-Zvi,Paul Burchard:General_Interest,Fractals,CCL:General_Interest/Fractals/eko.tiff:This is an image of a two dimensional slices of the cubic connectedness locus. The locus (CCL for short) is a four-dimensional analog of the Mandelbrot set. (See also the long description.):This picture was created using the program "Brot", written at the Geometry Center by Linus Upson and Christine Heitsch.:summer, 1993:807831374 Electric Storm:David Ben-Zvi,Paul Burchard:General_Interest,Fractals,CCL:General_Interest/Fractals/electric_storm.tiff:This is an image of a two dimensional slices of the cubic connectedness locus. The locus (CCL for short) is a four-dimensional analog of the Mandelbrot set. The electric storm depicts a slice of the CCL very similar in spirit to familiar images of the Mandelbrot set. (See also the long description.):This picture was created using the program "Brot", written at the Geometry Center by Linus Upson and Christine Heitsch.:summer, 1993:807831382 Eyes Watching:David Ben-Zvi,Paul Burchard:General_Interest,Fractals,CCL:General_Interest/Fractals/eyes_watching.tiff:This is an image of a two dimensional slices of the cubic connectedness locus. The locus (CCL for short) is a four-dimensional analog of the Mandelbrot set. (See also the long description.):This picture was created using the program "Brot", written at the Geometry Center by Linus Upson and Christine Heitsch.:summer, 1993:807829926 Fearful Symmetry:David Ben-Zvi,Paul Burchard:General_Interest,Fractals,CCL:General_Interest/Fractals/fearful_symmetry.tiff:This is an image of a two dimensional slices of the cubic connectedness locus. The locus (CCL for short) is a four-dimensional analog of the Mandelbrot set. (See also the long description.):This picture was created using the program "Brot", written at the Geometry Center by Linus Upson and Christine Heitsch.:summer, 1993:807829916 Firewheel:David Ben-Zvi,Paul Burchard:General_Interest,Fractals,CCL:General_Interest/Fractals/firewheel.tiff:This is an image of a two dimensional slices of the cubic connectedness locus. The locus (CCL for short) is a four-dimensional analog of the Mandelbrot set. (See also the long description.):This picture was created using the program "Brot", written at the Geometry Center by Linus Upson and Christine Heitsch.:summer, 1993:807829865 Frosted Pane:David Ben-Zvi,Paul Burchard:General_Interest,Fractals,CCL:General_Interest/Fractals/frosted_pane.tiff:This is an image of a two dimensional slices of the cubic connectedness locus. The locus (CCL for short) is a four-dimensional analog of the Mandelbrot set. (See also the long description.):This picture was created using the program "Brot", written at the Geometry Center by Linus Upson and Christine Heitsch.:summer, 1993:807829850 Harriet:David Ben-Zvi,Paul Burchard:General_Interest,Fractals,CCL:General_Interest/Fractals/harriet.tiff:This is an image of a two dimensional slices of the cubic connectedness locus. The locus (CCL for short) is a four-dimensional analog of the Mandelbrot set. (See also the long description.):This picture was created using the program "Brot", written at the Geometry Center by Linus Upson and Christine Heitsch.:summer, 1993:807829840 Henon Lookalike:David Ben-Zvi,Paul Burchard:General_Interest,Fractals,CCL:General_Interest/Fractals/henon_lookalike.tiff:This is an image of a two dimensional slices of the cubic connectedness locus. The locus (CCL for short) is a four-dimensional analog of the Mandelbrot set. The configuration in this image closely resembles images of the Henon attractor and its parameter space. (See also the long description.):This picture was created using the program "Brot", written at the Geometry Center by Linus Upson and Christine Heitsch.:summer, 1993:807829829 Jekyll and Hyde:David Ben-Zvi,Paul Burchard:General_Interest,Fractals,CCL:General_Interest/Fractals/jekyll_and_hyde.tiff:This is an image of a two dimensional slices of the cubic connectedness locus. The locus (CCL for short) is a four-dimensional analog of the Mandelbrot set. This picture portrays a sinister intermingling of the real and the complex aspects of the CCL. (See also the long description.):This picture was created using the program "Brot", written at the Geometry Center by Linus Upson and Christine Heitsch.:summer, 1993:807829814 Waves:David Ben-Zvi,Paul Burchard:General_Interest,Fractals,CCL:General_Interest/Fractals/waves.tiff:This is an image of a two dimensional slices of the cubic connectedness locus. The locus (CCL for short) is a four-dimensional analog of the Mandelbrot set. (See also the long description.):This picture was created using the program "Brot", written at the Geometry Center by Linus Upson and Christine Heitsch.:summer, 1993:807829801 Torrential Storm:David Ben-Zvi,Paul Burchard:General_Interest,Fractals,CCL:General_Interest/Fractals/torrential_storm.tiff:This is an image of a two dimensional slices of the cubic connectedness locus. The locus (CCL for short) is a four-dimensional analog of the Mandelbrot set. (See also the long description.):This picture was created using the program "Brot", written at the Geometry Center by Linus Upson and Christine Heitsch.:summer, 1993:807829790 Peacock:David Ben-Zvi,Paul Burchard:General_Interest,Fractals,CCL:General_Interest/Fractals/peacock.tiff:This is an image of a two dimensional slices of the cubic connectedness locus. The locus (CCL for short) is a four-dimensional analog of the Mandelbrot set. (See also the long description.):This picture was created using the program "Brot", written at the Geometry Center by Linus Upson and Christine Heitsch.:summer, 1993:807829511 Red Eyes:David Ben-Zvi,Paul Burchard:General_Interest,Fractals,CCL:General_Interest/Fractals/red_eyes.tiff:This is an image of a two dimensional slices of the cubic connectedness locus. The locus (CCL for short) is a four-dimensional analog of the Mandelbrot set. (See also the long description.):This picture was created using the program "Brot", written at the Geometry Center by Linus Upson and Christine Heitsch.:summer, 1993:807831396 River Current:David Ben-Zvi,Paul Burchard:General_Interest,Fractals,CCL:General_Interest/Fractals/river_current.tiff:This is an image of a two dimensional slices of the cubic connectedness locus. The locus (CCL for short) is a four-dimensional analog of the Mandelbrot set. (See also the long description.):This picture was created using the program "Brot", written at the Geometry Center by Linus Upson and Christine Heitsch.:summer, 1993:807830043 Samurai:David Ben-Zvi,Paul Burchard:General_Interest,Fractals,CCL:General_Interest/Fractals/samurai.tiff:This is an image of a two dimensional slices of the cubic connectedness locus. The locus (CCL for short) is a four-dimensional analog of the Mandelbrot set. (See also the long description.):This picture was created using the program "Brot", written at the Geometry Center by Linus Upson and Christine Heitsch.:summer, 1993:807830033 Sea of Mandelbrot:David Ben-Zvi,Paul Burchard:General_Interest,Fractals,CCL:General_Interest/Fractals/sea_of_mandelbrot3.tiff:This is an image of a two dimensional slices of the cubic connectedness locus. The locus (CCL for short) is a four-dimensional analog of the Mandelbrot set. Note the accumulation of little arms along the main body, illustrating tha lack of local connectivity of the CCL. (See also the long description.):This picture was created using the program "Brot", written at the Geometry Center by Linus Upson and Christine Heitsch.:summer, 1993:807830021 Ships on the Sea:David Ben-Zvi,Paul Burchard:General_Interest,Fractals,CCL:General_Interest/Fractals/ships_upon_the_sea.tiff:This is an image of a two dimensional slices of the cubic connectedness locus. The locus (CCL for short) is a four-dimensional analog of the Mandelbrot set. (See also the long description.):This picture was created using the program "Brot", written at the Geometry Center by Linus Upson and Christine Heitsch.:summer, 1993:807830011 Spiral Galaxy 28948:David Ben-Zvi,Paul Burchard:General_Interest,Fractals,CCL:General_Interest/Fractals/spiral_galaxy_28948.tiff:This is an image of a two dimensional slices of the cubic connectedness locus. The locus (CCL for short) is a four-dimensional analog of the Mandelbrot set. (See also the long description.):This picture was created using the program "Brot", written at the Geometry Center by Linus Upson and Christine Heitsch.:summer, 1993:807829998 Stromatolite:David Ben-Zvi,Paul Burchard:General_Interest,Fractals,CCL:General_Interest/Fractals/stromatolite.tiff:This is an image of a two dimensional slices of the cubic connectedness locus. The locus (CCL for short) is a four-dimensional analog of the Mandelbrot set. (See also the long description.):This picture was created using the program "Brot", written at the Geometry Center by Linus Upson and Christine Heitsch.:summer, 1993:807829988 Symmetry:David Ben-Zvi,Paul Burchard:General_Interest,Fractals,CCL:General_Interest/Fractals/symmetry.tiff:This is an image of a two dimensional slices of the cubic connectedness locus. The locus (CCL for short) is a four-dimensional analog of the Mandelbrot set. (See also the long description.):This picture was created using the program "Brot", written at the Geometry Center by Linus Upson and Christine Heitsch.:summer, 1993:807829978 Outside In Overhead Sequence:Geometry Center Staff:Video_Productions,Outside_In:Video_Productions/Outside_In/postcard.comp.tiff:Overhead view of fifteen stages of the sphere eversion illustrated in the Geometry Center's video Outside In. It illustrates an amazing mathematical discovery made in 1957\- you can turn the surface of a sphere inside out without making a hole, if you think of the surface as being made of an elastic material that can pass through itself. Computer graphics helps to explain as well as present the visual elegance of this process.::April, 1994:807828514 Museum Mathematics\- Polyhedron:Charlie Gunn,Stuart Levy,Tamara Munzner,Olaf Holt:Special_Topics,Tilings:educational/museumscreen.tiff:Every week over 3000 museum visitors use the Triangle Tiling exhibit developed by the Geometry Center in collaboration with the Science Museum of Minnesota. The program features mathematical concepts such as the relationship between Platonic and Archimedean solids, and the dual of a polyhedron. The program is also used extensively at the Center itself during interactive tours, and will be on display at the SIGGRAPH computer graphics conference in July 1994. This picture is a snapshot of the screen when the tiling is a closed polyhedron.:Triangle Tiling is an external module of Geomview that runs on SGI workstations.:Spring, 1994:808777064 Museum Mathematics\- Plane:Charlie Gunn,Stuart Levy,Tamara Munzner,Olaf Holt:Special_Topics,Tilings:educational/2Dmuseumscreen.tiff:Every week over 3000 museum visitors use the Triangle Tiling exhibit developed by the Geometry Center in collaboration with the Science Museum of Minnesota. The program features mathematical concepts such as the relationship between Platonic and Archimedean solids, and the dual of a polyhedron. The program is also used extensively at the Center itself during interactive tours, and will be on display at the SIGGRAPH computer graphics conference in July 1994. This picture is a snapshot of the screen when the tiling is a flat plane.:Triangle Tiling is an external module of Geomview that runs on SGI workstations.:Spring, 1994:808777073 Museum Mathematics\- Platonic and Archimedean Solids:Charlie Gunn,Stuart Levy,Tamara Munzner,Olaf Holt:Special_Topics,Tilings:educational/allmuseumsolids.tiff:Every week over 3000 museum visitors use the Triangle Tiling exhibit developed by the Geometry Center in collaboration with the Science Museum of Minnesota. The program features mathematical concepts such as the relationship between Platonic and Archimedean solids, and the dual of a polyhedron. The program is also used extensively at the Center itself during interactive tours, and will be on display at the SIGGRAPH computer graphics conference in July 1994. This picture is a collage of the 5 Platonic solids and 11 of the 13 Archimedean solids. The names, starting at the top left, are\- dodecahedron, truncated dodecahedron, icosidodecahedron, truncated icosahedron, icosahedron, rhombicosidodecahedron, rhombitruncated icosidodecahedron, cube, truncated cube, cuboctahedron, truncated octahedron, octahedron, rhombicuboctahedron, rhombitruncated cuboctahedron, tetrahedron, truncated tetrahedron.:Triangle Tiling is an external module of Geomview that runs on SGI workstations.:Spring, 1994:808777085 120 Cell Soap Bubble:John Sullivan:Special_Topics,Modeling_and_Simulation;General_Interest,Digital_Art:optimal_geometries/sullivan-120cell.tiff:This image, produced by postdoc John Sullivan of the Geometry Center, represents tessellation of the 3-sphere by 120 regular dodecahedra. Three dodecahedra meet along each edge, at 120 degree angles, so the projection can be thought of as a cluster of soap bubbles. This is the first rendering of soap bubbles made from the basic laws of optics for thin films. For more information, see Sullivan's paper "Generating and rendering four-dimensional polytopes", The Mathematica Journal, 1(3) \-76--85, 1991. This image was the cover picture (Winter 1991).:This picture was rendered using a custom shader in RenderMan which simulates thin-film interference of light, thus producing the characteristic rainbow patterns of soap films.:Winter 1991:808778485 Not Knot Poster:Charlie Gunn:Video_Productions,Not_Knot:Video_Productions/Not_Knot/NKposter.1500.tiff:Knot complement. Fundamental domains. Hyperbolic space. These are some of the mathematical terms introduced to viewers of ``Not Knot'', the award winning, computer-generated, mathematical video produced at the Geometry Center. This scene depicts a flight through a hyperbolic space tiled by right-angle dodecahedra. This picture is available as a 2'x3' poster from The Geometry Center.::1990:807828514 Lord Kelvin's Conjecture Disproved (I):Stuart Levy,John Sullivan,Ken Brakke:Special_Topics,Tilings:optimal_geometries/kelvin-9tile.tiff:In 1887, Lord Kelvin posed the problem of finding the partition of space into equal volume cells minimizing the interface area. He suggested a partition which is basically the Voronoi cell for a BCC lattice. Robert Phelan and Denis Weaire of Trinity College, Dublin, have found a structure using two types of cells that has 0.3% less area than Kelvin's. This picture shows 9 opaque cells.:The Surface Evolver, a program written by Ken Brakke with support from The Geometry Center, was used to compute the areas involved. Geomview was used to set up the appearance and point of view, and RenderMan was used to create the final image.:February, 1994:808778516 Lord Kelvin's Conjecture Disproved (II):Stuart Levy,John Sullivan,Ken Brakke:Special_Topics,Tilings:optimal_geometries/kelvin-9tile-transp.tiff:In 1887, Lord Kelvin posed the problem of finding the partition of space into equal volume cells minimizing the interface area. He suggested a partition which is basically the Voronoi cell for a BCC lattice. Robert Phelan and Denis Weaire of Trinity College, Dublin, have found a structure using two types of cells that has 0.3% less area than Kelvin's. This picture shows 9 transparent cells.:The Surface Evolver, a program written by Ken Brakke with support from The Geometry Center, was used to compute the areas involved. Geomview was used to set up the appearance and point of view, and RenderMan was used to create the final image.:February, 1994:808778532 Insider's View of 3D Klein Bottle:Celeste Fowler,Charlie Gunn:Special_Topics,Topology:symmetry/klein2.tiff:Using Maniview, an external module of Geomview, the user can fly around in the insider's view of the 3D Klein bottle. Note that spaceships in alternating columns are mirror reversed. This is a frame from the video "The Shape of Space". For more information, see Gunn's paper\- ``Discrete Groups and Visualization of Three-Dimensional Manifolds,'' in Proceedings of SIGGRAPH 93 (Anaheim, CA, August 1--6, 1993), pp. 255--262. In Computer Graphics Proceedings, Annual Conference Series (ACM SIGGRAPH, New York, 1993).:Maniview, an external module of Geomview, written by Charlie Gunn.:March 1993:809020868 Qhull cone:Brad Barber:Special_Topics,Computational_Geometry:Special_Topics/Computational_Geometry/cone.tiff:

The convex hull is the smallest convex set containing a set of points. For example, a cube is the convex hull of the cube's vertices. This cone is the convex hull of two 20-sided polygons and a cospherical point. It was generated by Qhull.

This cone illustrates a unique feature of Qhull. The program handles roundoff errors due to floating point arithmetic. If adjacent facets are non-convex, Qhull merges one of the facets into a neighbor. For example, the 20-sided facet and squares are merged facets. The result is an inner and outer approximation to the convex hull.

If exact arithmetic was used instead of floating point arithmetic, all of the facets would be non-coplanar triangles. This is because the point coordinates are floating point numbers. There were numeric errors in creating the regular polygons and additional errors in rotating the points.

:rbox r s 20 Z1 G0.2 | qhull s C-0 QR1 G >a:April 1995:807828514 Qhull with random perturbations:Brad Barber:Special_Topics,Computational_Geometry:Special_Topics/Computational_Geometry/fixed.tiff:

This sphere is an approximate convex hull of 200 cospherical points. During its construction, every computation was randomly perturbed by a number between -0.01 and 0.01. Despite the perturbations, all pairs of neighboring facets are convex. This demonstrates that the program, Qhull, can handle arbitrary amounts of round-off error.

Each approximate facet consists of an inner plane and an outer plane. The outer planes are above all of the points (red-yellow lines) while the inner planes are below the vertices (white spheres). The exact convex hull lies between the inner and outer planes.

:rbox 200 s B1 | qhull C-0 Qc R0.01 Gpav >a:April 1995:807828514 Delaunay triangulation with Qhull:Brad Barber:Special_Topics,Computational_Geometry:Special_Topics/Computational_Geometry/delaunay.tiff:

This picture shows how the Delaunay triangulation (in light green) is derived from a convex hull (the 3-d object). Each vertex of the triangulation corresponds to a vertex of the convex hull. Each edge of the triangulation corresponds to an edge of the convex hull.

The Delaunay triangulation is the triangulation with empty circumcircles for each triangle. The dual of the Delaunay triangulation is the Voronoi diagram. Both structures have many applications in science, engineering, and mathematics.

To compute the Delaunay triangulation, project the points to a paraboloid by summing the squares of their coordinates. Then take the convex hull of the projected points. Remove the vertical dimension from the lower envelope of the convex hull. The projected edges are the Delaunay triangulation of the original points.

Notice the 6 co-circular points near the center of the triangulation. They correspond to a flat facet of the convex hull. The program Qhull constructed a hexagon instead of three triangles. The hexagon has an empty circumcircle.

The same process can be used for points in 3-d or higher. In 3-d, each tetrahedron of the Delaunay triangulation has an empty circumsphere, and the corresponding paraboloid sits in 4-d.

:qhull < eg.data.17 C-0 d Ga >a (q_eg in qhull):April 1995:807828514 Qhull 4-d cube:Brad Barber:Special_Topics,Computational_Geometry:Special_Topics/Computational_Geometry/4dcube.tiff:

This is the approximate convex hull of 10,000 points that are randomly distributed in a 4-d hypercube. Each polygon in the picture corresponds to a ridge or 4-d edge of the convex hull. The black lines are Geomview's bounding box. Geomview is displaying this 4-d object as if it were in 3-d.

A hypercube has eight facets, two for each coordinate axis. Each facet is a cube. A ridge is the intersection of two facets. Each ridge of the hypercube is a square.

A hypercube is the same as a cube, but in one dimension higher. A cube has six facets, two for each coordinate axis. Each facet is square.

The convex hull of all the points in the 4-d hypercube would be the hypercube. So if an approximate convex hull of 10,000 points in the hypercube has eight facets, it should approximate the hypercube.

An approximate convex hull has "thick" facets whose ridges are "close" to the intersection between the facets. This picture projects the vertices of the ridges to the intersection of the facets' hyperplanes. The vertices define the polygons.

This picture shows four of the eight facets in the approximate convex hull. You can see a small cube in the center. The edges do not meet up because of the approximation. The cube is the facet of the approximate convex hull that is furthest away from you. You can't see the facet that is closest because Geomview's camera is inside this facet.

You are looking at the small cube from three of the facets that are adjacent to the cube. You can see hints of the three other facets that are attached to the back side of the cube. These facets look trapezoidal because they were projected "at an angle" from 4-d. The center facet looks like a cube because it was projected "straight down" from 4-d.

The picture was generated with Qhull and Geomview. Qhull is a program for computing convex hulls, Delaunay triangulations, Voronoi vertices, and halfspace intersections.

:rbox 10000 D4 | qhull s Gh C-0.03 C0.1 Qc >a:April 1995:807828514 Halfspace intersection with Qhull:Brad Barber:Special_Topics,Computational_Geometry:Special_Topics/Computational_Geometry/half.tiff:

A halfspace is all points to one side of a plane. The intersection of a set of halfspaces is a convex set. A related area is linear programming. The output of a linear program is a vertex of a halfspace intersection.

This Geomview picture from Qhull illustrates how to compute a halfspace intersection with the convex hull. The cone with the small square facets is the convex hull of two cospherical polygons and an apex. The other cone is the dual of the first. Each vertex of the second cone corresponds to a facet or halfspace of the first. Each facet of the second cone corresponds to a vertex or halfspace intersection of the first. For example, a vertex with four edges corresponds to a facet with four neighbors.

Assume the origin is inside the cone and let the first cone's facets define a set of halfspaces. To compute a dual point divide a facet's coefficients by its offset. The second cone is the convex hull of the dual points. The dual points for the second cone's facets are the vertices of the first cone. The vertices are the intersection of the first cone's facets.

Qhull uses floating point arithmetic. The non-triangular facets were merged by Qhull. For halfspace intersection, this is needed when more than three planes meet at the same point.

:rbox 10 r s Z1 G0.3 | qhull Qx FV n | qhull H Qx G >a:April 1995:807828514 Linked Triangles:Heidi Burgiel:Special_Topics,Topology:symmetry/linkedtriangles.tiff:At the center of this picture are three triangles. The link they form is equivalent to the Boromean Rings featured in Not Knot. The smaller pictures in the upper left and right hand corners show two views of a set of four linked triangles. This picture was inspired by the sculptures of John Robinson and George Odom. The mathematical properties of these sculputures were brought to the artist's attention by H.S.M. Coxeter.:Mathematica and Geomview:March 1996:825961073 Arbitrary substitution tiling:Chaim Goodman-Strauss:General_Interest,Fractals;Special_Topics,Tilings:General_Interest/Fractals/Substitution.tiff:All these squares are congruent. Yes, its true, by tautology, because we can define our space of congruences ourselves. This is a substitution tiling, using these weird congruences. Amazingly, there are matching rules on these tiles, in this strange space, so that ANY tiling with these tiles satisfying the matching rules looks basically like this.:MacPaint, of all things.:January 1995:839275424 DodecaFoam!:Chaim Goodman-Strauss,Daniel Krech:General_Interest,Fractals;Special_Topics,Tilings:General_Interest/Fractals/dodecafoam.tiff:This dodecahedron is filled with tiny dodecahedra, giving recipe for filling all of space with dodecas! This symmetry is closely related to that of the three-dimensional Penrose tiles, and generalizes to dimension 4 (but no higher).:Mathematica wrote the production rules, a program by Dan Krech iterated the rules, GeomView set up the pictures and RenderMan rendered the final result.:June 1995:839277175 DodecaFoam! First Stellation:Chaim Goodman-Strauss,Daniel Krech:General_Interest,Fractals;Special_Topics,Tilings:General_Interest/Fractals/first_stell.tiff:The recipe for filling a large dodecahedron with small dodecas involves examing the three stellations of the dodecahedron. Here, the first stellation of the dodecahedron is filled with tiny dodecas. This symmetry is closely related to that of the three-dimensional Penrose tiles, and generalizes to dimension 4 (but no higher).:Mathematica wrote the production rules, a program by Dan Krech iterated the rules, GeomView set up the pictures and RenderMan rendered the final result.:June 1995:839277184 DodecaFoam! Second Stellation:Chaim Goodman-Strauss,Daniel Krech:General_Interest,Fractals;Special_Topics,Tilings:General_Interest/Fractals/second_stell.tiff:The recipe for filling a large dodecahedron with small dodecas involves examing the three stellations of the dodecahedron. Here, the second stellation of the dodecahedron is filled with tiny dodecas. This symmetry is closely related to that of the three-dimensional Penrose tiles, and generalizes to dimension 4 (but no higher).:Mathematica wrote the production rules, a program by Dan Krech iterated the rules, GeomView set up the pictures and RenderMan rendered the final result.:June 1995:839277193 DodecaFoam! Third Stellation:Chaim Goodman-Strauss,Daniel Krech:General_Interest,Fractals;Special_Topics,Tilings:General_Interest/Fractals/third_stell.tiff:The recipe for filling a large dodecahedron with small dodecas involves examing the three stellations of the dodecahedron. Here, the third stellation of the dodecahedron is filled with tiny dodecas. Note the vertices of this figure lie on the vertices of a large dodecahedron. This symmetry is closely related to that of the three-dimensional Penrose tiles, and generalizes to dimension 4 (but no higher).:Mathematica wrote the production rules, a program by Dan Krech iterated the rules, GeomView set up the pictures and RenderMan rendered the final result.:June 1995:839277205 Lattice of conics:Chaim Goodman-Strauss:Special_Topics,Topology:Special_Topics/Topology/proj.tiff:These are general conics on the projective plane. A lattice of circles, lying on a plane tangent to the unit sphere has been normalized to lie on the unit projective plane. Regardless of the conics chosen in the original plane, similar images (spherical ellipses) would be derived by this procedure. Note that the circles seem to be in perspective, at least if you only look at a small portion of the image. This is far from coincidental.:Mathematica produced GeomView output, rendered by RenderMan.:June 1995:839275443 Hyperbolic Kaleidoscope\- 2-3-5:Chaim Goodman-Strauss:Special_Topics,Hyperbolic_Geometry:Special_Topics/Hyperbolic_Geometry/2.3.5.gif:This image is from a Kaleidoscope you can actually build! Here the hyperbolic wall paper group *22222 is illustrated. Any 2- dimensional discrete hyperbolic group of reflections can be physically rendered with one of these scopes. Unfortunately, one needs cylindrical mirrors, which are hard to find and cut up. By the way, reflection in a circle or sphere is not inversion but some strange map that preserves the combinatorics. For more information, see Unusual Kaleidoscopes:Drafted and raytraced in SoftImage.:April 1995:839275456 Hyperbolic Kaleidoscope\- 3-4-4:Chaim Goodman-Strauss:Special_Topics,Hyperbolic_Geometry:Special_Topics/Hyperbolic_Geometry/3.4.4.gif:This image is from a Kaleidoscope you can actually build! Here a symmetry of hyperbolic three-space is illustrated. This image is not successful, for parts of the space are hidden. Still, its neat to look at. Any 2-dimensional discrete hyperbolic group of reflections can be physically rendered with one of these scopes. Unfortunately, one needs cylindrical mirrors, which are hard to find and cut up. By the way, reflection in a circle or sphere is not inversion but some strange map that preserves the combinatorics. For more information, see Unusual Kaleidoscopes:Drafted and raytraced in SoftImage.:April 1995:839275473 Hyperbolic Kaleidoscope\- S2:Chaim Goodman-Strauss:Special_Topics,Hyperbolic_Geometry:Special_Topics/Hyperbolic_Geometry/s2.tiff:This image is from a Kaleidoscope you can actually build! Here a symmetry of s2xs1 is illustrated. For more information, see Unusual Kaleidoscopes:Drafted and raytraced in SoftImage.:April 1995:839275485 An aperiodic tiling:Chaim Goodman-Strauss:General_Interest,Fractals;Special_Topics,Tilings:Special_Topics/Tilings/tribs.tiff:This is a tiling of the plane with just two tiles, the "trilobite" and the "cross". Any such tiling must be aperiodic. The construction generalizes to any dimension Euclidean space.:Drafted in Adobe Illustrator.:February 1995:839281838 Aperiodic Puzzle I:Chaim Goodman-Strauss:Special_Topics,Tilings:Special_Topics/Tilings/trib_puzzle1.tiff:Any tiling of the plane with these two tiles must be aperiodic and the construction generalizes to any dimension Euclidean space. Here, a puzzle is presented\- get the tiles into the outlined region. It's not that hard but may be fun!:Drafted in Adobe Illustrator.:April 1995:839281898 Aperiodic Puzzle II:Chaim Goodman-Strauss:Special_Topics,Tilings:Special_Topics/Tilings/trib_puzzle2.tiff:Print these and have fun!:Drafted in Adobe Illustrator.:April 1995:839281944 Threebolites I:Chaim Goodman-Strauss:Special_Topics,Tilings:Special_Topics/Tilings/threeb_cassis.tiff:Any tiling with these three tiles must be aperiodic. The construction is based on inflations of the Schwarz surface and an underlying body centered lattice, and generalizes to all dimensions greater than one. For fuller description, see Goodman-Strauss' papers.:Drafted and ray-traced in SoftImage.:April 1995:839285039 Threebolites II:Chaim Goodman-Strauss:Special_Topics,Tilings:Special_Topics/Tilings/threeb_cube.tiff:This is a cluster of tiles that force aperiodicity based on inflations of the Schwarz surface and an underlying body centered lattice. The construction generalizes to all dimensions greater than one. For fuller description, see Goodman-Strauss' papers.:Drafted and ray-traced in SoftImage.:April 1995:839285343 Threebolites III:Chaim Goodman-Strauss:Special_Topics,Tilings:Special_Topics/Tilings/schwarz.tiff:Here are three inflations of a piece-wise linear equivalent to the Schwarz surface. If we keep inflating, the resulting complex is aperiodic. We can use this to construct aperiodic tilings of space. The construction generalizes to all dimensions greater than one. For fuller description, see Goodman-Strauss' papers.:Drafted and ray-traced in SoftImage.:April 1995:839285352 Threebolites (smooth):Chaim Goodman-Strauss:Special_Topics,Tilings:Special_Topics/Tilings/threeb_smooth.tiff:This is a smooth version of the threebolite tiles. The construction is based on inflations of the Schwarz surface and an underlying body centered lattice, and generalizes to all dimensions greater than one. For fuller description, see Goodman-Strauss' papers.:Drafted and ray-traced in SoftImage.:April 1995:839285367