Page 220 - J. C. Turner - History and Science of Knots

210 History and Science of Knots

special type of knots which project into such a picture alternating knots.
The Celts made extensive use of such pictures (Fig. 4), for decorative and
presumably religious purposes. We surmize that their features had to sym-
bolize a number of things. The line would represent time, or possibly life. In
the case of a closed curve, the knot's periodicity might relate to the regularity
of seasonal changes, with the alternating aspect symbolizing night and day.
They seem to have been aware of the non-trivial fact that an alternating knot
could be made to correspond with any simple closed planar curve. Their de-
sire to draw such knots posed geometrical problems. This contributed to the
process of mathematization of their worldviews, because they had to discover
how to geometrically create the truly knotted curves and zoomorphics which
they employed to adorn surfaces [12], [24].

Fig. 4. Celtic Knotwork
In a sense the foregoing examples of diagrammatic representations of knots
and their uses are like an overture. They witness of relatively primitive math-
ematical thought, and were described in order to illustrate the transition from
intuitive knot theory, lacking any apparant formalism, to a vestigial form of
the subject. The introduction of a kind of planar geometry was doubtless
not directly an attempt to understand knots. The geometry of the Celts is
of an essentially different kind from Euclidean, but nevertheless it involves
elusive properties like transformation and symmetry. The awareness of such
problems posed refined demands, requiring the development of new ideas in
mathematics.

3. The Birth of Knot Theory

The subject's next steps were related to spirals and closed intertwined curves,
and were mainly a German affair. As far back as 1679 Leibniz, in his Char-
acteristica Geometrica, tried to formulate basic (geometric) properties of ge-
ometrical figures by using special symbols to represent them, and to combine
these properties under operations so as to produce other properties. He called