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Science News, October 27, 1987 By Ivars Peterson.
Twists of space: an artist, a computer programmer and a mathematician
work together to visualize exotic geometric forms.
Link to this page TWISTS OF SPACE
The Mobius strip is one of the more familiar yet intriguing objects in
mathematics. Discovered in 1858 by German astronomer and mathematician
August F. Mobius, it can be constructed simply by gluing together the
two ends of a long, narrow strip of paper after one end has been given
a half twist. The surprising result is a twisted three-dimensional
form that has only one side and one edge.
The Mobius strip (or band) is one example of a wide variety of
geometrical forms that play important roles in the mathematical
field of topology. Topologists emphasize the properties of shapes
that remain unchanged, no matter how much the shapes are stretched,
twisted or molded so long as they aren't torn or cut. For example,
a doughnut and a coffee mug are topologically equivalent. Because
both forms have exactly one hole, one can imagine smoothly deforming
a doughnut-shaped piece of clay to produce a mug with a single handle.
Like a Mobius strip, a hemispherical bowl has a single edge. If a
disk, also having just one edge, is sewn to a hemisphere, the result
is a closed, two-sided surface that is topologically equivalent to a
sphere. That stitched object can be easily inflated into the shape of
a ball.
What happens if a Mobius strip is sewn to the edge of a disk? The
possible answers to this question have, over the last century, carried
mathematicians deep into the strange and twisted world of four- and
higher-dimensional forms. Most recently, this question has formed
the basis for a unique collaboration involving an artist, a computer
programmer and a mathematician working at the National Center for
Supercomputing Applications in Urbana, Ill.
One of the earliest answers to the Mobius-strip stitching problem was
found among geometrical constructions made by German mathematician
Jacob Steiner during the early part of the 19th century. His "Roman
surface,' named in memory of a particularly productive stay in Rome,
fits the requirements of a Mobius strip sewn to a disk. Steiner's Roman
surface, at least from one viewpoint, looks like a severely deformed
bowl with a fat, pinched lip.
In 1900, Werner Boy discovered a "simpler' surface that also meets the
same criteria. His convoluted surface looks like a pretzel that has
stepped out of "The Twilight Zone.' However, Boy couldn't find the
algebraic equations that would specify the location of every point
defining its shape. All he could do was to describe cross sections
through the surface. That was enough to construct a wire framework
or sculpt a plaster model but not enough to find a formula for the
surface. French mathematician Francois Apery, a student of topologist
Bernard Morin at the University of Strasbourg, finally worked out the
equations in 1984.
Because both the Roman surface and the Boy surface are closely
related, it was natural for mathematicians to look for orderly ways of
transforming one surface into the other. Such a procedure, known as
a homotopy, involves a careful cancellation of singularities--places
where the surface twists into itself to form a double curve or where
the surface is pinched and abruptly changes direction.
Singularities can often be removed by thinking of the figure
as a higher-dimensional form. For example, the two-dimensional
shadow of a bent wire loop sometimes appears to cross itself,
although the three-dimensional loop itself doesn't actually intersect
anywhere. Likewise, what appears in three dimensions to be a pinch point
could very well be a perfectly regular feature in four dimensions. The
pinched three-dimensional form is merely a particular shadow cast by
the four-dimensional surface.
The Romboy homotopy--the name for the transformation from the Roman
to the Boy surface--is based on the idea that both the Roman and
Boy surfaces can be generated by moving an oval or ellipse through
space. A circle, for example, when rotated through 180 degrees,
generates a sphere. In the same way, an ellipse, its motion governed
by well-defined constraints, can stretch and contract as it sweeps
out a wobbly path to produce a particular surface. This bouquet of
ellipses defines the shape.
Apery, starting with the Roman surface, discovered that by smoothly
altering the parameters governing the requisite motion of an ellipse,
he could gradually transform the Roman surface into the Boy surface. In
fact, the equations governing this procedure show that both surfaces
can be considered three-dimensional "shadows' of a higher-dimensional
form viewed from different vantage points.
The existence of Apery's equations made it possible to program
a computer to display the two figures and the homotopy linking
them. Mathematician George K. Francis of the University of Illinois
at Urbana-Champaign, who has long been interested in methods for
visualizing geometrical forms, programmed an Apple computer to
generate a rough version of the homotopy. Such computer sketches are
the equivalent of the wire frameworks and hand drawings frequently
employed by mathematicians a century ago to study mathematical forms.
When artist Donna J. Cox, who was working with computer programmer
Ray Idaszak, came to Francis looking for a mathematically oriented
project to do at the university's new supercomputing center, Francis
suggested they try recreating the Romboy homotopy. Francis, Cox and
Idaszak became what Cox terms a "Renaissance' team, with each person
making important contributions to the project.
It took less than two weeks for the group to convert the original
Apple program into one usable on a supercomputer. They added shading
and color to the computer drawings, putting a "skin' on what originally
were little more than skeletons of the surfaces. The result was a short,
600-frame, computer-drawn film that shows a smooth deformation from the
Roman to the Boy surface, then back to the original. "This deformation
had never been realized before as an animation,' says Cox.
In the course of fiddling with the computer program, Francis also
discovered a simpler way of performing the homotopy. Instead of
generating the surfaces by letting a wobbling ellipse sweep through
space the whole time, Francis also used a mathematical figure known
as a limacon. This figure looks like a closed loop coiled so that
it crosses itself once to form a double loop. It can also take on a
roughly heart-shaped form.
By following this alternative homotopy, the three researchers stumbled
upon a new surface that appears along the way. From one point of view it
looks somewhat like an owl, and from another, like a female torso. The
new surface was dubbed the "Etruscan Venus.' The term Etruscan reflects
the idea that the homotopy used is simpler or more primitive than the
original Apery homotopy, just as the ancient Etruscan civilization in
Italy predated the Roman Empire.
The Etruscan Venus turns out to be a form known as a topological Klein
bottle. It can be created by gluing together two Mobius strips along
their edges, something that can be done only in the roominess of four
dimensions. Just as a disk is the shadwo of a three-dimensional sphere,
the Etruscan Venus is the three-dimensional shadow of a four-dimensional
Klein bottle.
Francis and his team also applied his version of the Romboy homotopy to
the Boy surface generated by Apery's homotopy. Again, they discovered
a new mathematical shape. Because Idaszak was the first to see it,
this one came to be called "Ida.' Like the Etruscan Venus, Ida is also
the shadow of a four-dimensional Klein bottle.
All three participants got something worthwhile out of the project. Cox
now has a kind of sculpture machine--an interactive computer program,
strongly rooted in the mathematics of topology, which is proving to
be a rich source of dramatic and beautiful shapes. By manipulating
the 10 parameters (or dimensions) in the equations defining the Romboy
homotopy, she can create an endless array of entirely new forms. "It's
fascinating to watch and work with on the screen,' she says. "You
can tug on one dimension and transform things radically. You can see
surfaces that are really quite surprising.'
"The artistic aspect [of the project] now has a life of its own,'
says Francis. "By manipulating formulas, entirely new shapes that have
never been seen before can be created.'
Recently, Cox has been working with two artists in Chicago to create
special three-dimensional images of some of these newly discovered
surfaces. Several will be on display later this fall at Chicago's
Museum of Science and Industry.
Cox is also interested in exploring how artists can help researchers
find better ways of conveying the information represented by vast
amounts of data. Her studies of the role of color have already
helped an astronomer and an entomologist (SN: 1/10/87, p.20) present
their results graphically. "I'm interested in using color to get
more information out of a system,' says Cox, "for demystifying and
clarifying images.' Pictures of "Ida,' for example, have been colored
to indicate the orientation of the ovals used to create the surface.
Meanwhile, Idaszak has developed new computer graphics techniques that
make it easier to picture surfaces that abruptly reverse their direction
or penetrate themselves. Such features often confuse conventional
computer programs designed for coloring in and shading geometric
figures. Idaszak's methods work equally well on a microcomputer and
a supercomputer. The main difference is a matter of scale.
For Francis, the project has generated some new mathematics and
effectively demonstrated the important role that computer graphics can
play in mathematical research. Inspired by Morin, who, though blind,
has been one of the strongest proponents of visualization in topology,
Francis has spent much of his career promoting the use of drawings
and models to aid in formulating definitions and proofs in mathematics.
In his new book, "A Topological Picturebook' (Springer-Verlag,
1987), Francis writes: "It is . . . in the making of the model,
in the act of drawing a recognizable picture of it, or nowadays, of
programming some interactive graphics on a microcomputer, that real
spatial understanding comes about. It does this by showing how the
model is generated by simpler, more familiar objects, for example,
how curves generate surfaces.'
Francis is particularly interested in developing computer tools that
make it easy and quick for students and mathematicians to "sketch'
geometrical figures and explore their properties. "We're talking about
computer graphics, first of all, as a way of illustrating theorems that
are already known, and secondly, as an experimental tool for finding
new conjectures,' he says. What's needed is "an inexpensive graphics
tool for producing recognizable images, in huge quantities and fast
enough to keep from wasting time.' The first steps toward that goal
have already been taken.
The Illinois collaboration showed how well a disparate group of people
can work together. "The project worked better than we had any reason
to believe,' says Francis. The team is now interested in producing a
definitive version of the Romboy homotopy, seen from angles useful to
mathematicians and with careful use of color and shading to illuminate
important features.
"The common thread throughout this collaborative research has been
the use of advanced technology and color techniques to make visible
the multidimensional layers of abstract information,' writes Cox in an
article to appear in LEONARDO. "Supercomputers, graphics, and creative
human beings have the power to bring about visual enlightenment with
regard to much in this universe that was formerly abstruse mathematics.'
Photo: This transparent, computer-generated image of the topological
form known as the Etruscan Venus allows a viewer to peer into the
figure's convoluted interior.
Photo: The topological transition from Steiner's Roman surface (far
left) to Boy's surface (far right) is based on the cancellation of
pinch points. The second and third diagrams in the sequence show
cross sections of the Roman surface as one pinch point is cancelled
out. Further alterations lead to Boy's surface, sliced open to provide
a clearer picture of its complex shape.
Photo: A computer-generated incarnation of Boy's surface, discovered by
mathematician Werner Boy in 1900, casts a shadow across a surrealistic
landscape.
Photo: This selection of frames from the videotape "Metamorphosis:
Shadows From Higher Dimensions' illustrates steps in the Romboy
homotopy. The Etruscan Venus (A) is transformed first into the Roman
surface (C). Next, the Roman surface is turned into the Boy surface
(D), which in turn evolves into a new surface called Ida (G). Finally,
Ida turns into the Etruscan Venus (I or A), completing the cycle.