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The Mathematics of the Calcul*rt Exhibit

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

Evening of Mathematical Art

From 7pm the curators and contributors of Calcul*rt will guide, explain and demonstrate items in the show.

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:

How can you tell that it's mathematical art?


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.

CANVAS RTICA (starring in videos)

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)





DVD videos (with RTICAs in the CANVAS)

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.

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.

Knot Energies (NotKnot)
by John Sullivan and Stuart Levy, NCSA 1998. 3 min video at ICM, Berlin 1998.

Sculptures (associated RTICA and videos)

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., and
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.

Operating Instructions for the RTICAs

As for operating the real-time interactive CANVAS animations
(RICA), here are the essentials.  If we label the six buttons

(MORPH)   (FLY)    (ZAP)


Then the important ones are

ZAP   = returns to the initial state from anywhere
MORPH = toggles the homotopy on/off
        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.

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.