Full text of DOI:10.1109/38.814567

Vision 2000
Discovering
Cyb erw orld s
In the late 1960s, he realized the value of raster graph-
ics over the predominant vector graphics for displaying
versatile cyberworlds with colors and textures. This led
him to initiate a project to build raster graphics with
4,096 concurrent colors with a virtual frame buffer.
n this new millennium each of us should When an object’s shape changes, it usually retains the
I
only compose messages designed to reach same colors and textures, and so are invariants. Howev-
one person. Global, or mass, messages often hold only er, to publish his findings, he had to wait for the first Sig-
momentary appeal—how long do you remember long graph to meet in 1974. Even then, he had to go against
speeches on television? A contrasting example is found the pressure of the vector graphics community, which
in the Bible, which has increased in value for two mil- preferred high-resolution graphics to raster graphics.
lennia and which records individuals’ messages meant
for other individuals. What follows here is my person- Th e p ot en t ial of cyb erw orld s
al message to you about the invariants we should rely
Although cyberworlds lie in the information domain,
on to live successfully in our rapidly changing real world they are not always virtual. Pulse motors can turn each
and in cyberworlds.
bit directly into a step movement. Robots with pulse
motors in their joints can turn cyberworld objects direct-
ly into those in the real world. Cyber business and e-
Cyb erw orld s
The real world we live in is complex. Once upon a time business are also real. Another example is the e-business/
on an island in the Far East, there was a grade-school e-financial business, which trades a GDP-equivalent
boy who learned from a book that there were three hun- amount of real-world money in a day.
dred thousand different chemical compounds. He hated
Unless we identify invariants, we cannot quickly
any complexity because it blocked his mental vision. A understand changing cyberworlds, which are out of
bad memory of wartime information blockages had human control and could put the real world into crisis.
made him allergic to the lack of vision. Upon further On computer graphics screens, we can display a crisis
study, he soon found that theories about elementary as a singularity, a discontinuity of the world growth.
particles, atoms, and molecules—the periodic table Often, a robot becomes immobile when its configura-
combined with the octant theory—simplified the mate- tion becomes singular in its workspaces. Singularity
rial world. The octant theory gave eight as an invariant signs designate invariants and play essential roles.
of molecules, indicating eight electrons at the outermost Research provides deep insights into their nature and
molecular orbit would predict a stable molecular for- leads to indexes that characterize cyberworlds.
mation from atoms.
Any world spans time and space, and consists of sub-
In the late 1960s, the boy as a PhD student discovered worlds. Time is an irreversible space. Computer graphics
cyberworlds while synthesizing intermolecular proper- per se is engaged in display. To display images, we need
ties in an effort to understand life. The intermolecular a coordinate and a distance measure. Hence, we rely on
properties were computed from the structural informa- geometry, which inherits topological properties. Com-
tion of molecules assumed to exist. He realized that puter graphics as computers are set theoretical machines
often he had computed the properties of hypothetical using AND, OR, and NOT logic. Topology is built on the
molecules, resulting in the creation of nonexistent mate- AND and OR of power sets. In terms of the abstraction of
rial worlds inside computers. Such information worlds, invariants hierarchically organized from general to spe-
now named cyberworlds,1 attracted him because they cific for realizing the modular and incremental design of
could extend beyond the real world in potential and objects, the following is a reasonable example of an incre-
scale. (The story of the real-world creation in the Bible’s mentally modular abstraction hierarchy:
story of Genesis was attractive enough.) A few years
Tosiyasu L.
Kunii
Hosei
University
later, he initiated the creation of a new Information Sci- ■ a set level;
ence Department to study information worlds further. ■ an extension level, a homotopy level as a special case;
Created first as a research institution with a graduate ■ a topology level, a graph theoretical level as a special
program in 1970, it turned into a department in 1975.
case;
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January/ February 2000
0272-1716/00/$10.00 © 2000 IEEE

■ a cellular structured space level;
■ a geometry level; and
■ a visualization level.
For computer graphics, a cellular structured space
level based on cellular spatial structures such as CW
spaces provides a far more versatile basis than that based
on a graph theoretical level common in conceptual and
data modeling. It allows computer graphics to specify
objects in cognitive and computational spaces as cells
and their boundaries. Cellular modeling also allows cel-
lular composition and decomposition while maintain-
ing cell dimensions and connectivity as invariants.
Object identification is carried out systematically
through identification mapping (often called quotient
mapping).
We can model a cyberworld appropriately as an n-
dimensional cellular space topologically equivalent to
an n-dimensional ball Bn. A space has a dimension as a
degree of freedom. We can construct a one-dimensional
space consistently from a zero-dimensional space by
attaching one-dimensional cells. By repeating cell
attachment, we can inductively compose an n-dimen-
sional space, where an n cell is topologically equivalent
to an n-dimensional ball.2 This stands for the “genesis”
of a cyberworld because we are creating a new world
from scratch.
An exam p le
The boy is now over 60 years old and a professor
emeritus. He has found this cellular space to be very con- 1 A shirt frill folding into a cuff.
venient in extensively modeling varying cyberworlds.
He has also realized the complete lack of cyberworld
research from this general viewpoint.3,4
Designing a new cyberworld from existing cyber-
worlds is exactly the same in its way as designing a gar-
ment frill (pleats in a shirt cuff in Figure 1). A frill design
process involves the cell decomposition of every tuck (a
two cell) being attached to the edge (a one cell) of a shirt
cuff and the rest (a two cell). The next step is cell com-
position via cell attachment to identify the tucked edge
1
1
|_|
i
1
[ (B
B )]tuck edge with a line (∂B2collar that is a one
|_|
|_|
B
i
i
i
cell) on the cuff where every tuck is attached. See Fig-
ure 2. Though the garment changes shape geometrical-
ly while it is being worn, we can still see the frill as the
tucks being attached to the cuff. So the frill is an invari-
ant of the shape change.
2
∂B collar: A frill attaching line
Similarly, we can form a new cyberworld by attach-
ing existing cyberworlds. The way we do this becomes
an invariant of cyberworld shape changes through cell 2 Frill composition: cuff attachment to tucks via the
1
1
1
|_|
composition and decomposition. The garment frill attaching map. f: ∂B2
example helps us understand rapidly changing cyber-
i (B i B i B i)tuck edge.
|_|
|_|
collar
→
worlds and how to display the way cyberworlds are
changing on computer graphics screens.
■
3. T.L. Kunii, “Homotopy Modeling as World Modeling,” Proc.
Computer Graphics Int’l 99 (CGI99), IEEE Computer Soci-
ety Press, Los Alamitos, Calif., 1999, pp. 130-141.
4. T.L. Kunii, “Valid Computational Shape Modeling: Design
And Implementation,” Int’l J. Shape Modeling, World Sci-
entific, Singapore, Dec. 1999.
References
1. Cyberworlds, T.L. Kunii and A. Luciani, eds., Springer,
Tokyo, 1998.
2. H.–J. Baues, Homotopy Types and Homology, Oxford Univ.
Press, Oxford, 1996.
Contact Kunii at tosi@kunii.com
IEEE Computer Graphics and Applications
65