"The Hands of Cantor" -a 44 page descriptive volume on the making of this sculpture.

This sculpture is a compendium of strange ideas from the history of science and mathematics - namely, Klein Bottles, Atomic Knot Theory, and the Continuum. It is an exploded wonder cabinet , floating overhead. The sculpture is a touchstone on ideas about the infinite, and some of the various attempts to both quanitfy and qualify what this concept means.

ALTERNATING COMPOUND KNOT

ATOMIC KNOT TABLE

FOOTNOTES

KLEIN BOTTLES

KLEIN BOTTLES TESSELATING ACROSS SPACE IN AN ATTEMPT TO BECOME A PATTERN

  CORNER COUNTERWEIGHTS ALSO SHOWN

MOBIUS STRIP SML KLEIN BOTTLES ELLIPSES CANTOR'S CONTINUUM EQUATION HANDS OF CANTOR OCTAVE

  4 VS. 3

  AMPERSAND INFINITY 1 > INFINITY 2 3 VS. 2

THE FOOTNOTES FOR THE SCULPTURE DEPICT VARIOUS SYMBOLS FOR INFINITY

The Hands of Cantor  is made up of a series of three floating screens surrounding a giant alternating compound knot.

(note: use text scroll-buttons in lower left corner)

The left-hand screen is the realization of Lord Kelvin’s and Peter Tait’s Atomic Knot Table, out to seven crossings. Kelvin had proposed that atoms were actually ‘knotted vortices floating in ether’ and his colleague Peter Tait went to the trouble to distinguish what made one knot mathematically different from another, thus creating a forerunner to the Periodic Table of the Elements. Inasmuch as Kelvin was technically wrong about how atoms behave, the research into knots that his conjecture inspired has proved useful in contemporary studies of the knotting and folding of DNA and proteins.

On the opposite screen are a series of tiled, or tessellated, Klein Bottles. After the Mobius Strip, the Klein Bottle may be the next most recognizable symbol of infinity. This curiosity of topology was invented by the great Felix Klein, who inspired an industry of strange surfaces to be produced from Munich for Departments of Mathematics around the globe. As a vessel that drinks itself, the Klein Bottle is a sculptural version of a tautology. It also represents a departure point: the moment when mathematics went from geometry to topology, and therefore out of the realm of that which is describable in three dimensions.

On the rear screen are the footnotes for the sculpture. The footnotes comprise many symbols for infinity, such as the eight lying on its side, the ampersand (which is really just a broken eight), the ellipses that symbolize ”and so on”, and Cantor’s famous continuum equation. The beauty of Georg Cantor's legacy are his proofs that there are many types of infinity, and that some are bigger than others.

This particular equation states that the infinity of all the irrational numbers (numbers like .569258…, that simply don’t have an end), when placed side by side between two whole numbers, such as 1 and 2, is a larger infinity than the infinity you get when counting 1,2,3,4 etc…etc…out to infinity. It sounds confusing at first because we’re taught to think that infinity is just counting forever. But, as life goes, it quickly becomes more complicated. Just imagine that whole numbers are the stars, and irrationals are the space in between, and you start to get the picture.

Joining the footnotes are some corollary figures that translate both issues of basic counting and issues of infinity into the realm of Music. A couple of glyphs depict the phenomenon of how the number three relates to the number four, and the issue of ”space” versus ”thing”. For example, if you have four things, you are going to have three spaces in between those things, and then you will have all the garbage space outside that is not in between those things. In music, this is called phase harmonics or beat harmonics. It would be wrong not to therefore include the Octave, which is derived directly from the harmonics produced from bowing a string of given length. The wave shape produced by the octave is reminiscent of the conventional symbol for infinity. It makes plain the infinite combinations present in the notes on the musical scale.

In the center is the Giant Knot. This particular knot is the first order of ”alternating compound knot”— a ”trefoil” knot and a ”figure eight” knot, which are the two simplest knots joined together. This knot is also a one sided surface; it is a Mobius strip that would be almost 200 feet in length if its edges were unzipped and extended. It is shown here as a foil to the conventional infinity symbol, possibly a stand-in waiting in the wings.

This sculpture not only chronicles a few interesting, if obscure, mathematical ideas, it also addresses the process of invention itself. The concept of infinity may have remained in the realm of difficult mathematical thinking and religion, but the popular impression of it hasn’t really changed since the days of Aristotle. This is an attempt to put some of its other guises on the table. The sculpture describes the enthusiasm of comprehending strange things for the first time and the excitement of discovery. Discovery and invention are an exercise in expanding, rather than explaining, an idea. We have a collective, vested interest in expansiveness.