Sierpinski triangle

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Sierpinski triangle with recursion depth 7

The Sierpinski triangle is a fractal described by Wacław Sierpiński in 1915 - sometimes also called the Sierpinski surface or seal , which is a self-similar subset of a triangle (mostly shown on equilateral). If the triangle is divided into four congruent triangles that are similar to the starting triangle, the corner points of which are the side centers of the starting triangle, then the subsets of the fractal in the three outer triangles are scaled copies of the entire fractal, while the middle partial triangle does not belong to the fractal. This division of the fractal into scaled copies can be continued recursively in the outer partial triangles . The fractal dimension of the Sierpinski triangle is

construction

algorithm

To represent the Sierpinski triangle, an equilateral triangle is usually chosen as the starting triangle ; this is not mandatory, a Sierpinski triangle can be generated from every triangle.

Step-by-step construction of the Sierpinski triangle

The "classic" algorithm used to graphically demonstrate the concept of fractal is as follows:

  1. Draw a triangle ("initiator")
  2. Connect the midpoints of the sides ("generator") (this will split the original triangle into four congruent partial triangles)
  3. Remove the middle of the four sub-triangles (the other three sub-triangles remain)
  4. Apply steps 2 and 3 to the three remaining sub-triangles, and so on.

This algorithm makes the connection clear. With each iteration step, three triangles (similar to the initiator) with half the side and a quarter of the area are created at the corners, which are colored. The fourth “inner” small triangle that arises can be imagined as being “cut out” of the triangular area of ​​the previous step.

The actual Sierpinski triangle in the strictly mathematical sense is that point set that remains as a “boundary object” after an infinite number of iteration steps. To put it clearly, the Sierpinski triangle consists of an infinite number of "corner points". An iteration or recursion depth of at most ten is sufficient for the display, which is usually implemented with recursive computer programs and displayed or printed out on a screen as required . Due to the resolution of the medium ( monitor , printer, etc.) and the human eye, these structures can no longer be distinguished from the border object. In classic planimetric area measurement, the area tends to zero with increasing iteration depth.

Lindenmayer system

The Sierpinski triangle can be described by a Lindenmayer system with the following properties:

  • Angle: 120 °
  • Start string:
  • Derivation rules:

Representation in Wolfram's one-dimensional universe

In Stephen Wolfram's one-dimensional cellular automaton , a single living cell usually creates a Sierpinski triangle.

Mathematical relationships

The Sierpinski triangle is self-similar

As a classic fractal, the Sierpinski triangle is a prime example of exact self-similarity : The outer sub-triangles created in each step contain scaled-down, exact copies of the entire fractal. A suitable scaling of any triangular part of the fractal appears like the entire object itself. It is therefore scale-invariant .

In terms of the underlying mathematical principles, the Sierpinski triangle is closely related to the Cantor set . Its dimension is its reciprocal , namely . This results from the fact that exactly new partial triangles with the side length are created with each step . The fractal arises as a border object when it approaches infinity, more precisely by forming the average of all intermediate steps of the construction and it can therefore be understood as a "geometric analogue" to a limit value . From the formation regulation it can also be calculated which points of the original area belong to the boundary object.

The following formula can be used to calculate the number of triangles after k iterations:

For

topology

In topology one considers the Sierpinski triangle subspace of the Euclidean metric provided . It represents a nowhere dense , locally coherent , metric continuum and, together with the Sierpinski carpet , is considered a particularly remarkable topological space .

Representation using the Hutchinson operator

A Sierpinski triangle can also be represented as an attractor of a dynamic feedback process, a deterministically iterated function system with suitable parameters from almost any geometric figure. Multiple transformations of the initial object are carried out repeatedly , these images are appropriately arranged with a mapping rule, the Hutchinson operator , and this procedure is applied again to the resulting overall image, etc. -Triangle, which in this case is the attractor of the functional system.

In this sense, the Sierpinski triangle is characterized as that compact subset of the plane that is identical to the union of its three images under the three similarity maps, which map the entire triangle to the three half-size triangles.

1st stage 2nd stage 3rd stage 4th stage 5th stage 6th stage 7th stage 8th stage
step 1 2 3 4th 5 6th 7th 8th

Sierpinski tetrahedron

A Sierpinski tetrahedron

A representation of the Sierpinski triangle is, similar to the Menger sponge , also possible in the third dimension: The starting figure is a tetrahedron . In each iteration step, an octahedron with half the edge length is cut out of its center . What remains are four tetrahedra, from each of which an octahedron is cut out, etc. The dimension for this structure is although it is a figure in three-dimensional space. With an increasing number of iteration steps, the volume of the figure tends to zero; however, the surface remains constant.

Chaos game

Apart from the recursive representation, there is also a random point algorithm for the approximate construction of the Sierpinski triangle: The Chaos Game .

An equilateral triangle with corners A, B, C is recorded and a random point is chosen inside the triangle (it can also be outside without significantly changing the result). Now a corner is selected randomly for each step (the probabilities for the corners are always the same) and the point is mentally connected to the drawn corner. The middle of this route now marks the point for the next lap. If this is repeated very often, the points form an approximation of the Sierpinski triangle. If you also color the points differently depending on the selected corner, e.g. B. A = green, B = red and C = blue, then you get three differently colored Sierpinski triangles in the Sierpinski triangle.

From the iterative structure of the Sierpinski triangle, one can prove that a point obtained using this algorithm belongs to the Sierpinski triangle if and only if the starting point is also part of the Sierpinski triangle. For example, if you choose a point on the segment AB as the starting point, you will have constructed a Sierpinski triangle after an infinite number of iterations.

Relation to Pascal's triangle

The Pascal triangle is related to the Sierpinski triangle . The even numbers in the Pascal triangle correspond to the gaps in the Sierpinski triangle. Both triangles have a simple iteration rule, from which a geometric similarity always emerges: If every initiator triangle is replaced in one step in the Sierpinski triangle according to the rule already described above, the Pascal triangle merely doubles the number of lines (odd numbers with more than a decimal place are shown below as # for the sake of clarity).

Initiator:

           1
          1 1

Step 1

           1
          1 1
         1   1
        1 3 3 1

2nd step

           1
          1 1
         1   1
        1 3 3 1
       1       1
      1 5     5 1
     1   #   #   1
    1 7 # # # # 7 1

3. ...

This similarity is given for an infinite Pascal triangle as well as a Sierpinski triangle after an infinite number of iteration steps.

The creation of this similarity can also be viewed from a different point of view: The Pascal triangle itself is always infinite as an idea and thus as a geometrical blueprint, we just cannot write it down completely. An iteratively generated Sierpinski triangle is always limited by its outer shell. Thus, by continuous iteration, not the Pascal triangle is aligned with a Sierpinski triangle, but the Sierpinski triangle with the infinite geometric construction plan and thus with the infinite Pascal triangle.

nature

This pattern occurs naturally on the shell of the snail species Cymbiola innexa .

literature

  • PS Alexandroff : Introduction to set theory and general topology (=  university books for mathematics . Volume 85 ). VEB Deutscher Verlag der Wissenschaften, Berlin 1984.

See also

Individual evidence

  1. ^ W. Sierpinski, Sur une courbe dont tout point est un point de ramification. // Comptes rendus hebdomadaires des séances de l'Académie des sciences. - Paris. - Tome 160, Janvier - Juin 1915. - Pp. 302 - 305. - [1]
  2. ^ PS Alexandroff: Introduction to set theory and general topology. 1984, pp. 191-192.
  3. http://www.3d-meier.de/tut10/Seite20.html
  4. Max Planck Institute for Developmental Biology: Shell Shell: Explaining Patterns with Formulas. In: Spiegel online. October 12, 2009. Retrieved October 12, 2009 .
  5. Bjørn Jamtveit, Paul Meakin (ed.): Growth, Dissolution and Pattern Formation in Geosystems . Kluwer Academic Publishers, Dordrecht 1999, p. 234 ( digitized from Google Books ).

Web links

Commons : Sierpinski triangle  - album with pictures, videos and audio files
This version was added to the list of articles worth reading on August 19, 2005 .