Find this document at http://new.math.uiuc.edu/calculart/

This website collects the mathematics that is embodied in CALCUL*RT, a 5 month-long celebration of art and mathematics in the 21st Century Gallery at the Krannert Art Museum (KAM) of the University of Illinois.

The show opened 9 March with a rich program of events, and will close
in July.

You are cordially invited to come to the Krannert Art
Museum on Thursday, 4 May 2006, for an

At 8pm the audience is invited to join show contributors Donna Cox, George Francis, Hank Kaczamarski, Rose Marshack and Rick Powers in a free flowing discussion on the subject:

In the Collaborative Advanced Navigation Virtual Art Studio (CANVAS) there are ten real-time interactive computer animatiions (RTICA). On the random access DVD player next to the CANVAS are videos which provide brief explanations of what the RTICA are about. We add a partial bibliography of papers for additional reference.

---------------------------------------------------- Venus (The Etruscan Venus, Venus and Milo) This surface is a Whitney-stable 3D projection of a nonorientable surface with Euler characteristic 0 (a Kleinbottle) embedded in 4-space. Like Steiner's Roman Surface, the Etruscan Venus loses its singularities under Francois Apery's Romboy homotopy and becomes an immersed surface equivalent to the connected sum of two copies of Boy's Surface. ---------------------------------------------------- Snail (Post-Euclidean Walkabout) These figures are all 3-D shadows of surfaces embedded in spherical (positively curved) space. The Snail begins as a Moebius band whose boundary is a planar circle. Next, the circle is drawn together to form a closed crosscap surface. Fly inside to see the pinchpoints and their Whitney umbrella neighborhoods. Rotating this projetive plane in 4 space morphs its 3 shadow from the crosscap to Steiner's Roman surface. Next we give the ribbon a second half twist, which expands to form a Clifford torus. Note the Hopf circles, every pair link each other once. It's fun to fly inside. It finishes with Brehm's Knotbox. To the topologist, the Knotbox is a standard spine for the complement of the trefoil knot in the 3-sphere. Think of a bicycle inner tube, but tied in a knot; blow it up until there is no place left in the universe to expand it into. The rubber walls will merge into a 2-dimensional shape. It could be the Knotbox. ---------------------------------------------------- Hspace (Post-Euclidean Walkabout) ---------------------------------------------------- Tangle (Air on the Dirac Strings) ---------------------------------------------------- Optiverse (The Optiverse) ---------------------------------------------------- StarEvert (The Optiverse) ---------------------------------------------------- NotKnot (Knot Energies) ---------------------------------------------------- Borromean (Knot Energies) ---------------------------------------------------- Lorenz ---------------------------------------------------- Cosmos ---------------------------------------------------- Atlantis ----------------------------------------------------

Etruscan Venus (Venus)

by George Francis, Donna Cox, and Ray Idaszak, NCSA, 1989.

3 min SIGGRAPH video.

Venus and Milo (Venus)

Donna Cox, Chris Landreth, et al, NCSA 198?.

Post-Euclidean Walkabout (Hspace, Snail)

by George Francis, Chris Hartman, Glenn Chappell, Ulrike Axen, Paul McCreary, and Alma Arias, NCSA, 1994.

3 min SIGGRAPH video, This real-time interactive CAVE application takes you on a visit to the post-Euclidean geometry of Gauss, Riemann, Klein, Poincare, and Thurston. Here you can walk into a rectangular dodecahedron, a shape which is possible only in negatively curved hyperbolic space. With a wand, you can summon and play with the snail-shaped 3D shadows of soap films in positively curved elliptic space. You can see how to sew the edges of hyperbolic octagons together into the surface of a 2-holed donut. The CAVE becomes a spaceship you can navigate with the wand, as it glides through the phantasmic shapes that populate the 3-sphere.

The purpose of this project is to perfect persuasive visual and sonic environments in which to exhibit geometrical wonders and their startling metamorphoses, which interest research geometers. Convincing visualizations of multi-dimensional, time-varying geometrical structures are equally useful in applied and pure mathematics.

Air on the Dirac Strings (Tangle)

by Lou Kauffman, George Francis and Dan Sandin, EVL 1993.

http://www.evl.uic.edu/hypercomplex/html/dirac.html

The Optiverse (Optiverse, StarEvert)

by John Sullivan, George Francis, Stuart Levy, Camille Goudeseune, NCSA, 1998.

2.5 min SIGGRAPH video.

The Optiverse is a 6.5-minute computer-animated video showing an entirely new way to turn a sphere inside out. The video captures scenes that can also be viewed as real-time interactive computer animations, on a workstation console or in immersive virtual environment. The narration is accompanied by parambiences, which are novel experiments in scientific sonification.

The Optiverse was premiered at the VideoMath festival at ICM'98, the International Congress of Mathematicians, August 1998 in Berlin. A special two-minute cut was shown in the Electronic Theater at SIGGRAPH 98, July 1998 in Orlando.

http://new.math.uiuc.edu/optiverse/

Knot Energies (NotKnot)

by John Sullivan and Stuart Levy, NCSA 1998. 3 min video at ICM, Berlin 1998.

http://torus.math.uiuc.edu/jms/Videos/ke/

Minimal Flower 3 by John Sullivan and Ben Grosser (Time lapse video of its making in the 3D printer) --------------------------------------------------- Phscholograms by Donna Cox and Ellen Sandor (Barrier strip holograms with a Venus theme) --------------------------------------------------- Umbilic Torus by Helaman Ferguson (Non-Euclidean geometry, as in Snail and Hspace) A toroid with cross-section a cusped trefoil sweeping out a Moebius ribbons decorated with a surface filling Hilbert curve. 27" silicon bronze with antique verde patina, 1994. http://helascultp.com, and http://www.philsoc.org/1994Fall/2036minutes.htm. --------------------------------------------------- Untitled by Brent Collins (Knot Energies) From his "Early Spiral Models Series" of abstract mathematical surfaces of high genus and knotted edges and with one and two sides. 6' wood carvings 1990. http://http.cs.berkeley.edu/~sequin/SCULPTS/collins.html. ---------------------------------------------------

As for operating the real-time interactive CANVAS animations (RICA), here are the essentials. If we label the six buttons (MORPH) (FLY) (ZAP) (TRAIL) (STAY) (CYCLE) Then the important ones are ZAP = returns to the initial state from anywhere MORPH = toggles the homotopy on/off CYCLE = modes through MUSEUM > GESTURE > JOYSTICK see below The currently unimportant ones are FLY = meaningful only if tracking is working, then toggles fulcrum (center of turning between object center and (tracked) head. STAY = meaningful only if handtracking is working, then each press resets to current hand placement. TRAIL = meaningful only in the Snail, so far, then cycles between NOTRAIL>LEAVETRAIL>SEETRAIL. Note, I never got the chance switch the (MORPH) button in the zound and zound2 rticas from lower-left to upper-left. Footnotes: MUSEUM MODE = this isolates the navigation from any and all input from both the joystick (avoid joystick drift) and arbitrary tracking data. Instead it provides a TUMBLE motion. The tumble motion is an (approximate) grand-tour of all viewpoints of the object. GRAND-TOUR, as definded by Dan Asimov, is path of a space (in this case, the sphere) that passes every point with equal probability, equally often statistically speaking. On a torus, think of as a square with top and bottom identified, and sides identified, just take a diagonal line with an irrational slope ratio and continue it. My line is 1.618:1, approximately the golden section. The torus parametrizes the sphere by the two angles: latitude and longitude. This gives an approximate grand-tour, but one that spends way too much time looking at north and south poles, looking for bears and penguins. JOYSTICK MODE = uses the default David Pape navigation controlled by the joystick. Stick fore/aft moves you forward/backward, and stick side-to-side turns you right-left of the direction the wand is currently pointing (if tracked). This is the simplest (and least satisfactory) navigator that can get you to every place in the scene. If hand tracking does not work, then you can travel only in the horizontal plane. GESTURE MODE = is any navigation mode that (1) depends on some or all six degrees of freedom of one or both hand and head tracking, and (2) has a fluid, natural attenuation at the start and stop of the gestural motion. The default gesture tracker in the CANVAS apps (now crippled to avoid bad tracking) works like this: When toggled ON by the (CYCLE) button, rotating your wrist influences the rotational heading of the object about its center (TURN MODE) or about the navigators head (FLY MODE). The xyz displacement of the hand similarly influences linear motion in that direction. The influence is incremental. If I turn my wrist slightly, the object will continue turning proportional to the angular displacement from the initial position. The (STAY) button resets the position of displacement to the curren hand. So, it has the effect of freezing any motion, as long as the hand is held still.