SPACE-FILLING CURVE - ANTICIPATING THE HILBERT CURVE

PEANO, GIUSEPPE.

Sur une courbe, qui remplit toute une aire plane.

Leipzig, B. G. Teubner, 1890. 8vo. Bound in recent full black cloth with gilt lettering to spine. In "Mathematische Annalen", Volume 36., 1890. Entire volume offered. Library label pasted on to pasted down front free end-paper. Small library stamp to lower part of title title page and verso of title page. Very fine and clean. Pp. 157160. [Entire volume: IV, 602 pp.].

First printing of Peano's seminal paper in which he for the very first time discovered a space-filling curve in a 2-dimensional plane, it is often referred to as Peano curves.

Peano's ground-breaking paper contained no illustrations of his construction, which is defined in terms of ternary expansions and a mirroring operator. But the graphical construction was perfectly clear to him-he made an ornamental tiling showing a picture of the curve in his home in Turin. Peano's paper also ends by observing that the technique can be obviously extended to other odd bases besides base 3. His choice to avoid any appeal to graphical visualization was no doubt motivated by a desire for a well-founded, completely rigorous proof owing nothing to pictures. At that time (the beginning of the foundation of general topology), graphical arguments were still included in proofs, yet were becoming a hindrance to understanding often counter-intuitive results.

Peano' purpose was to construct a continuous mapping from the unit interval onto the unit square. Peano was inspired by Georg Cantor's earlier counterintuitive result that the infinite number of points in a unit interval is the same cardinality as the infinite number of points in any finite-dimensional manifold, such as the unit square. The problem Peano solved was whether such a mapping could be continuous; i.e., a curve that fills a space.

Order-nr.: 47189