Infinity, Impossibility, and Art
Michael Davis
4 April 2003
Hons 183: Mind, Art & the Brain
What is infinity?
The quality or state of being
unbounded or unlimited
endless
not finite
Random House College Dict., 1992.
A Historiographic Look at Infinity
Greek – apeiron
connotation was
1. negative and pejorative
2. unordered
3. undefined
A Historiographic Look at Infinity
beginning with the Greeks
Pythagoras (582 – 507 BCE)
natural numbers
Plato (427? – 347 BCE)
concept of the Good
Aristotle (384 – 322 BCE)
actual vs. potential
A Historiographic Look at Infinity
on to the Middle Ages
General attitudes
St. Augustine (354 - 430)
A Historiographic Look at Infinity
in the Modern Age
Galileo Galilei (1564 – 1642)
Georg Cantor (1845 – 1918)
Kurt Gödel (1906 – 1978)
“These difficulties are real … but let us remember
that we are dealing with infinites and indivisibles,
both of which transcend our finite understanding.
This I think is wrong, for we cannot speak of
infinite quantities as being the one greater or less
than or equal to another.”
Galileo Galilei
The Background of the Symbol
Lemniscate - 1700s – conic sections
[(x-a)2 + y2][(x+a)2 + y2] = a4
“The appropriateness of the symbol lies in the
fact that one can travel endlessly around the
curve … demolition style, if you will”
(R. Rucker)
The Background of the Symbol
The symbol caught on popularly and
was used on the Tarot card of the
Magician (Juggler).
Qualities: The Magician strides the
webs of time, the bridges of our years;
He sees the past and future, knows our
hopes and fears.
Source: http://www.tarot-card-meaning.co.uk
Types of Infinity
• Temporal Infinities (music)
• Spatial Infinities (art)
of the large
in the small
• Infinity of the Mindscape (Rucker, 1995)
How we first perceive infinity
• Successive Theory (Lakoff)
• Unconsciousness Theory (Baldwin)
• Cause and Effect Theory
(refutation: Hume)
Questions Raised by M. C. Escher in
his Approaches to Infinity
How can a composer succeed in evoking
a suggestion of something that does not
end? Music is not there either before it
begins or after it ends.
How can an artist represent something
that continues forever when he or she is
bounded by a canvas or piece of paper
that has limits?
Gödel, Escher, and Bach: An Eternal
Golden Braid
Douglas Hofstadter (1979)
Strange Loops
• artists can exploit our perceptions
• the artist will make the medium
appear to be increasing
• but, eventually the audience
member will find her/himself in the
same place that she/he began
Strange Loops: Visual Art
M. C. Escher
Waterfall
(lithograph, 1961)
6 Steps
Water is continually falling
throughout the entire piece
Strange Loops: Visual Art II
M. C. Escher
Ascending and
Descending
(lithograph, 1960)
Appears that monks are
either continually walking
down the stairs or
continually walking up
the stairs depending on
the perspective.
Strange Loops: Visual Art III
M. C. Escher
Metamorphosis II
(woodcut, 1938-1940)
Begins and ends with the
same design after going
through many different
transformations.
Strange Loops: Music
J. S. Bach
“Canon per Tonas,” The Musical Offering
Canon appears to be continually rising in steps, but “we
unexpectedly find ourselves where we started again”
From C to C
Can go as high or as low as the physical constraints of the ear
Strange Loops: Math
Epimenides (Liar) Paradox
“All Cretans are liars.”
or more modernly,
“This statement is false.”
If you think the statement is true …
If you think the statement is false …
Gödel’s Incompleteness Theorem hinges upon the writing of a
self-referential mathematical statement, in the same way as the
Epimenides paradox is a self-referential paradox of language.
Types of Impossible Figures
The Necker (Rib) Cube
The Tribar
The Möbius Strip
Necker Cube
In 1832, a Swiss crystallographer
named Necker published pictures of
an unusual cube that appeared to
assume different orientations as one
continued to look at it.
The effect works because the
drawing of the cube carefully
eliminates all depth cues. In
attempting to fit the expected model
of a cube to the picture, our brain
must resolve the ambiguity as to
which corner of the cube is closer.
Cuboid (Necker Cube)
Unlike the previous figure, the
cuboid is not ambiguous.
Instead it appears to be clearly
wrong. Our depth cues tells us
that the arrangement of the
bars is impossible because
some bars impossibly occlude
others.
Our brain thinks locally to create a global picture.
Cuboid (Necker Cube) II
M. C. Escher
Belvedere
(lithograph, 1958)
Boy with cuboid who is
starring at the necker
cube on the sheet of
paper below.
Cuboid (Necker Cube) III
M. C. Escher
Belvedere
(lithograph, 1958)
Six of the eight columns
are impossible. The
position of the two people
on the ladder and hence,
likewise impossible.
Tribar
The tribar was discovered in 1958 by
R. Penrose.
Try and rotate the figure around in
your head (it can’t be done).
The direction of any two bars is
conflicting with their connection
elsewhere. One bar approaches while
on bar recedes.
Even though you know the
relationship cannot exist, you still see
it as a 3-D object.
Perception v. Conception (R. Gregory)
Brain hardwiring not designed for this
and gravitates to the simplest
solution.
More local to global thinking !
Tribar II
Breda Pieter Brueghel
Magpie on the Gallows
(oil, 1568)
The relationship between
the 3D cues in the picture
are contradictory.
Tribar II
Oscar Reutersvärd
(stamp collection, 1980)
The arrangement of the
blocks again is
impossible.
Tribar III
Escher: The right side shows a reproduction of a Penrose drawing. I
have indicated with red shades a level that out to be horizontal, which
explains the trick.
The Möbius Strip
The Möbius Strip (1958) is named
for August Möbius, who was a
mathematician and astronomer. It
resulted out of his investigations
about the nature of polyhedra.
The Möbius strip is amazing
because it has only one continuous
edge. A ladybug could go from any
point on the ribbon to another point
without every crossing an edge.
The Möbius strip has been
commercially in conveyor belts and
continuous loop recording devices.
The Möbius Strip II
The Möbius Strip II
M. C. Escher
(lithograph, 1958)
An endless ring-shaped band
usually has two distinct surfaces,
one inside and one outside.
Yet on this strip nine red ants
crawl after each other and travel
the front side as well as the
reverse side. Therefore the strip
has only one surface.
The Möbius Strip III
The recycling symbol designed to
be released on Earth Day 1970
was designed by a graduate
architectural student from
University of Southern California
named Gary Anderson.
He used the frame of the Möbius
Strip because in his words, "a
finite object, but its one surface is
infinite in a way."
http://home.att.net/~DyerConsequences/recycling_symbol.html
Bibliography and Further Reading
Berbaum, K., Tharp, D., & Mroczek, K. (1983). Depth perception of surfaces in pictures: Looking for
conventions of depiction in Pandora's box. Perception, 12, 5-20.
Biederman, I. (1987). Scene Perception. Scientific American.
Bloomer, Carolyn M. (1976). Principles of Visual Perception. New York: Litton Educational Publishing,
Inc.
Carterette, Edward C. (1975). Handbook of Perception. Volume V. New York: Academic Press, Inc.
Coren, Stanley. (1978). Seeing is Deceiving: The Psychology of Visual Illusions. Hillsdale, New Jersey:
Lawrence Erlbaum Associates,
Escher, M. C. (1986). Escher on Escher. New York: Harry N. Abrams, Inc.
Escher, M. C. (1967). The Graphic Work of M. C. Escher. New York: Meredith Press.
Escher, M. C. (1971). The Work of M. C. Escher. New York: Harry N. Abrams, Inc.
Fineman, Mark B. (1981). The Inquisitive Eye. Oxford: Oxford University Press.
Gregory, R. L. (1973). Illusion in Nature and Art. London: Gerald Duckworth & Company, Limited.
Bibliography and Further Reading
Koffka, K. (1935). Principles of Gestalt Psychology. New York: Harcourt Brace.
Matlin, Margaret W. (1988). Sensation and Perception. Boston: Allyn and Bacon, Inc.
Peterson, Ivars (2001). Fragments of Infinity: A kaleidoscope of Math and Art. New York: John Wiley &
Sons.
Rainey, Patricia Ann. (1973). Illusions: A Journey into Perception. Connecticut: The Shoe String Press,
Inc.
Ramachadran, V. S. (1973). Utilitarian theory of perception. Washington, D.C.: American Psychological
Association.
Rucker, Rudy (1995). Infinity and the Mind. Princeton: Princeton University Press.
Schattscneider, Doris (1990). Visions of Symmetry: Notebooks, Periodic Drawings,
and Related Work of M. C. Escher. New York: W. H. Freeman and Company.
Wade, Nicholas. (1980). Visual Allusions: Pictures of Perception. Hove, UK: Lawrence Erlbaum
Associates Ltd.
Visual Images
http://noordnet.net/optical_illusion/triangle.html
http://www.worldofescher.com/gallery
http://www.aspecialplace.net/illusions/necker_cube.htm
http://www.public.asu.edu/~sbroder/Escher/figure4.html
http://ccat.sas.upenn.edu/jod/augustine.html