Sierpinski Gasket

Sierpinski

Start with a solid (filled) equilateral triangle S(0). Divide this into four smaller equilateral triangles using the midpoints of the three sides of the original triangle as the new vertices. Remove the interior of the middle triangle (that is, do not remove the boundary) to get S(1). Now repeat this procedure on each of the three remaining solid equilateral triangles to obtain S(2).

Continue to repeat the construction to obtain a decreasing sequence of sets

The Sierpinski gasket, also known as the Sierpinski triangle, is the intersection of all the sets in this sequence, that is, the set of points that remain after this construction is repeated infinitely often.

The set S(1) can also be obtained by scaling three copies of S(0) each by a factor of r=1/2 and then translating two of the smaller triangles to form the desired arrangement. If we imagine the bottom side of S(0) to lie along the x-axis with the vertices at the origin and at the point (1,0), then two of the scaled triangles would have to be translated so that the lower left vertices are at (1/2, 0) and (1/4, sqrt(3)/4) respectively. This yields the following iterated function system

IFS
Animation
(18Kb)
3 different
initial sets

Watch
Sierpinski
converge to
his own
triangle!

The Sierpinski gasket is the unique invariant set for this IFS.

Because of the rotational symmetry of an equilateral triangle, there is second iterated function system that has the Sierpinski gasket as its invariant set. In addition to scaling by a factor of r=1/2, rotate two of the triangles by 120°and -120°, then translate.

This yields the IFS given by

The axiom we have listed forms the boundary of the original equilateral triangle. Suppose we started with just the axiom F, that is, just one side of the triangle. The first iteration generated by the rule given above would be the following

We can see how the rule generates the boundary of the removed inside triangle. The axiom F+F+F would trace this inside triangle three times.

L-system
Animation
(11Kb)

Another L-system that generates Sierpinski's gasket is given by

Angle 60
Axiom FX
F --> Z
X --> +FX-FX-FY+
Y --> -FX+FY+FX-

This a bit different in that each curve never intersects with itself as with the first L-system. As such, this construction illustrates that the Sierpinski gasket is actually a plane curve. It corresponds to the second of the iterated function systems given above.

The following figure shows six Sierpinski gaskets put together in a hexagonal shape. This "curve" also has some interesting topological properties that Sierpinski discussed in his 1915 and 1916 papers. Also see reference [3].

We remove "all" of the area of the initial triangle in constructing the Sierpinski gasket. But of course there are many points still left in the gasket.

• Area calculations
• What points are in the Sierpinski gasket? An exercise in binary numbers.

What is the average distance between two points in a unit Sierpinski gasket? The answer is 466 / 885 = 0.52655. Ian Stewart [15] credits this result to Andreas Hinz and describes how the use of graph theory to analyze the n-disk Tower of Hanoi puzzle can be used to calculate this average distance.

There are many posssible variations on the construction of the Sierpinski gasket, both from the geometric point of view of "removing" sections and the IFS point of view. An interesting connection also exists between Pascal's Triangle and the Sierpinski Gasket. There's even some applications to electric circuits!

• Sierpinski Carpet
   
• Sierpinski Pentagon
   
• Pascal's Triangle (mod 2)
   
• Overcoming Resistance with Fractals

Could the Sierpinski gasket have captured the imagination of neolithic people in Ireland 5000 years ago? Look at these images and decide for yourself.

1. Bannon, Thomas. "Fractals and Transformations," Mathematics Teacher, March 1991.
2. Barnsley, Michael F. Fractals Everywhere, 2nd Edition, Academic Press Professional, 1993.
3. Ciesielski, Krzysztof and Zdzislaw Pogoda. "The Beginning of Polish Topology," The Mathematical Intelligencer, 18 (3), 1996, 32-39.
4. Devaney, Robert. A First Course in Chaotic Dynamical Systems, Addison-Wesley Publishing Co., 1992.
5. Edgar, Gerald A. Measure, Topology, and Fractal Geometry, Springer-Verlag, 1990.
6. Jones, Juw. "Fractals Before Mandelbrot-A Selective History," in Fractals and Chaos, Crilly, Earnshaw, and Jones, Editors, Springer-Verlag 1991.
7. Lauwerier, Hans. Fractals, Endlessly Repeated Geometrical Figures, translated by Sophia Gill-Hoffstadt, Princeton University Press, 1991.
8. Mandelbrot, Benoit. The Fractal Geometry of Nature, W.H. Freeman and Co. 1983.
9. Peitgen, Heinz-Otto, Hartmut Jurgens and Dietmar Saupe. Fractals for the Classroom, Part One: Introduction to Fractals and Chaos, Springer-Verlag New York, Inc. 1990.
10. Prusinkiewicz, Przemyslaw and James Hanan. Lindenmayer Systems, Fractals, and Plants, Lecture Notes in Biomathematics #79, Springer-Verlag 1989.
11. Prusinkiewicz, Przemyslaw and Aristid Lindenmayer. The Algorithmic Beauty of Plants, Springer-Verlag, 1990.
12. Sierpinski, Warclaw. "Sur une courbe dont tout point est un point de ramification," Compt. Rendus Acad. Sci. Paris 160 (1915), 302-305.
13. Sierpinski, Warclaw. "On a curve every point of which is a point of ramification," Prace Mat. Fiz. 27 (1916), 77-86 [Polish].
14. Sokolov, I.M. "A ride on Sierpinski's carpet," Quantum, May/June 1992, 7-11.
15. Stewart, Ian. "Four Encounters with Sierpinski's Gasket," The Mathematical Intelligencer, 17, No. 1 (1995), 52-64.