Acheron 2.0 Menu
Fast Track What's New in Acheron 2.0 Introduction to Fractals Overview of Acheron 2.0 Fractals Curves in Acheron 2.0 Von Koch Curve Mandelbrot Curve Hilbert Curve Cesaro Curve Heighway Curve Minkowski Curve Construction Properties Variations Author Biography Peano Curve Square Curve Sierpinski Curve Sierpinski Objects
Feedback about Acheron 2.0 Download Counters of Acheron 2.0 Support of Acheron 2.0 Safe Use of Acheron 2.0
Visitors Counter
17972 visitors (since Jan 2010)





All pictures are from Acheron 2.0,
a free explorer of geometrical fractals. You can download Acheron 2.0
here 
Acheron 2.0 Screen Overview

The Minkowski Curve is also called the Minkowski sausage. According to
Mandelbrot, the origin of the curve is uncertain and was dated back at least to Hermann Minkowski.

Construction Back to Top
As almost all fractals curves, the construction of the Minkowski curve is
based on a recursive procedure.
At each recursion, a 8sides generator is applied to each line segment of
the curve. As the first step starst with a straigth line, it gives:
Note that there are 8 differents segments (and not 7, as it can be thought at
first sight ..)
The same generator is applied to the 8 segments formed at the first
iteration to produce a somewhat more complex curve:
The third iteration already gives a nice picture:
The first stages of the procedure modify heavily the appearance of the
curve. However, quite soon, the curve remains roughly the same whatever the
recursion level, only the time required to drawn the curve increases.
Propreties Back to Top
 Curve Length
The length of the Minkowski curve increases at each iteration.
On each iteration, the length of the segments is divided by four and
the number of segments is multiplied by eight, hence the total curve
length is multiplied by 2 with each iteration.
Obviously, the length of the curve tends to infinity as the iteration
number increases.
 Fractal Dimension
The fractal dimension is computed using the
HausdorffBesicovitch equation:
D = log (N) / log ( r)
Replacing r by four ( as each segment is divided by four on each
iteration) and N by eight ( as the drawing process yields 8 segments) in
the HausdorffBesicovitch equation gives:
D = log(8) / log(4) = 1.5
 SelfSimilarity
Looking at two successive iterations of the drawing process provides graphical
evidence that this property is also shared by this curve.
Variations Back to Top
All Variations described are available using Acheron 2.0
 Iteration Level
Five recursion levels are available. Above this iteration number, the overall
aspect of the curve remains essentially unaffected.
 Basic Geometric Figure
Instead of starting with a straight line, the drawing can start from a triangle
or a square, leading to interesting curves.
Author Biography Back to Top
Born: 22 June 1864 in Alexotas, Russian Empire (now Kaunas, Lithuania)
Died: 12 Jan 1909 in Gottingen, Germany
Hermann Minkowski studied at the Universities of Berlin and Konigsberg.
He received his doctorate in 1885 from Konigsberg. He taught at several
universities, Bonn, Konigsberg and Zurich. In Zurich, Einstein was a student
in several of the courses he gave.
Minkowski accepted a chair in 1902 at the University of Gottingen, where he
stayed for the rest of his life. At Gottingen he learnt mathematical physics
from Hilbert and his associates.
By 1907 Minkowski realised that the work of Lorentz and Einstein could be
best understood in a noneduclidean space. He considered space and time, which
were formerly thought to be independent, to be coupled together in a
fourdimensional 'spacetime continuum'. This spacetime continuum provided a
framework for all later mathematical work in relativity.
Minkowski was mainly interested in pure mathematics and spent much of his
time investigating quadratic forms and continued fractions. His
most original achievement, however, was his 'geometry of numbers'.
At the young age of 44, Minkowski died suddenly from a ruptured appendix.
Biography From
School of Mathematics and Statistics  University of StAndrews, Scotland




