Flaesketorvet 68
1711 Koebenhavn V
Denmark
Contact Christian Westergaard
Nils Lundvang
Telephone +45 2762 8014
+45 2075 1233
Fax +45 6991 8469
Specialists Mathematics, Physics, Chemistry, Medicine, Anatomy, Astronomy, General Science.
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Flaesketorvet 68
1711 Koebenhavn V
Denmark
Contact Christian Westergaard
Nils Lundvang
Telephone +45 2762 8014
+45 2075 1233
Fax +45 6991 8469
Specialists Mathematics, Physics, Chemistry, Medicine, Anatomy, Astronomy, General Science.
Sophia Rare Books
Stand

The first geometric fractals

KOCH, Helge von

Sur une courbe continue sans tangente, obtenue par une construction geometrique elementaire, Stockholm: P.A. Norstedt & Soner 1904

First edition, extremely rare offprint issue, inscribed by the author to the great Swedish mathematician Ivar Fredholm (the founder of functional analysis). Koch’s paper contains the first examples of geometric fractals, the famous ‘Koch curve’ and ‘Koch snowflake’. The term ‘fractal’ was coined much later, by Benoit Mandelbrot in his 1975 book Les objets fractals, forme, hasard et dimension. Today, fractals have found a bewildering variety of applications in both the arts and sciences (fractal patterns have been found in the works of Jackson Pollock).

The first example of a fractal was the ‘Cantor set,’ introduced by Georg Cantor in 1883, but this was not a genuine curve. Modifying Cantor’s construction, Koch started with a straight line segment, erected an equilateral triangle with base the middle third of the original segment, and then erased that middle third; this produces a shape with four line segments; the same construction is now applied to each of these four segments; the Koch curve is the result of iterating this process indefinitely. It has the properties essential to fractals: it is self-similar, in the sense that any part of it, when magnified, looks the same as part of the original curve; and it has ‘fractional’ dimension – log 4/log 3 = 1.26186 – between a curve (dimension 1) and a plane (dimension 2).
£2,681