The document discusses the Minkowski curve, a fractal curve first introduced by Hermann Minkowski. It has the property of self-similarity, where portions of the curve exactly replicate the whole curve at different scales. The construction of the Minkowski curve is based on a recursive procedure where at each step an 8-sided generator is applied to line segments, increasing the complexity. As the number of iterations increases, the length of the curve tends towards infinity, and its fractal dimension is calculated to be 1.5, demonstrating its self-similarity. Variations include starting with geometric shapes other than a straight line.

1. The Minkowski Curve
Name : laxmi chandolia
Cental University of Rajasthan
Enrollment no. : 2013IMSCS009

2. Introduction
The euclidean geometry uses objects that have integer
topological dimensions. A line or a curve is an object that
have a topological dimension of one while a surface is
described as an object with two topological dimensions and a
cuve as an object with three dimensions. This geometry
adequately describes the regular objects but failed to be
applicable when it comes to consider natural irregular shapes.
Benoit B. Mandelbrot introduced a new concepts, that he
called fractals, that are useful to describe natural shapes as
islands, clouds, landscapes or other fragmented structures.
Fractals curves exhibit a very interesting property known as
self-similarity. If you observe precisely the details of a fractal
curve, it appears that a portion of the curve replicates exactly
the whole curve but on a different scale.

3. Author Biography
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. 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.
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.

4. 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.

5. Construction
As almost all fractals curves, the construction of the Minkowski
curve is based on a recursive procedure.
At each recursion, a 8-sides generator is applied to each line
segment of the curve. As the first step starst with a straigth line, it
gives:
there are 8 differents segments
The same generator is applied to the 8 segments formed at
the first iteration to produce a somewhat more complex curve:

6. he 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.

7. Propreties
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 by 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 Hausdorff-Besicovitch equation gives:
D = log(8) / log(4) = 1.5
Self-Similarity
Two successive iterations of the drawing process provides
graphical evidence that this property is also shared by this
curve.

8. Variations
Basic Geometric Figure
Instead of starting with a straight line, the drawing can start
from a triangle or a square, leading to interesting curves.