Fraktale geometry
Fraktale geometry (of lat. fractus = broken) marks a subsection of mathematics, which concerns itself with the geometrical characteristics of objects, which exhibit a high self similarity. While in Euclidean geometry only integral dimensions occur, can occur with such bodies or quantities also real dimensions.
In the Fraktalen geometry therefore with generalized terms of the dimension ( the Hausdorff dimension or Minkowski dimension) one works, the also real valuesto assume can.
Such an object with broken dimension and more highly Self similarity is called Fraktal. To the most important personalities in connection with the Fraktalen geometry the mathematicians Benoît B. belong. Almond bread, Waclaw Sierpinski, gas clay/tone Julia.
Fraktaler dimension term
the fraktale dimension term covers itself in thatCase of Euclidean geometry with the classical dimension term, is however expanded on fraktale quantities. Thus the fraktale dimension term corresponds to a Verallgemeinerung of the dimension term. One defines z. B. the Minkowski dimension by one given to one ball of the radius r, thosesmallpossible number determines from balls, which cover the regarded fraktale quantity. Then one computes the logaritmierte relationship from number to radius. Then one must regard the Limes, if the radius goes against zero. To the Hausdorff dimension one looks best simplyunder the left after.
mentioned
, the Minkowski dimension has examples the flake curve , or also Koch's curve: dim [Mink] = log (4) /log (3) ~ 1,2618595… , since one with ever 4 balls of the radius 1/3, which curve of first stage covers. Those CAN gate quantity has the Minkowski dimension: dim [Mink] = log (2) /log (3)
