The Sierpinski triangle, also called the Sierpinski gasket or the Sierpinski
Sieve, is a fractal and attractive fixed set with the overall shape of an
equilateral triangle, subdivided recursively into smaller equilateral
triangles. Originally constructed as a curve, this is one of the basic examples
of self-similar sets, i.e., it is a mathematically generated pattern that
can be reproducible at any magnification or reduction. It is named after the
Polish mathematician Wacław Sierpiński but appeared as a decorative pattern
many centuries prior to the work of Sierpiński.
Constructions There are many different ways of
constructing the Sierpinski triangle. = Removing triangles=
The Sierpinski triangle may be constructed from an equilateral triangle
by repeated removal of triangular subsets:
Start with an equilateral triangle. Subdivide it into four smaller congruent
equilateral triangles and remove the central one.
Repeat step 2 with each of the remaining smaller triangles
Each removed triangle is topologically an open set. This process of recursively
removing triangles is an example of a finite subdivision rule.
= Shrinking and duplication= The same sequence of shapes, converging
to the Sierpinski triangle, can alternatively be generated by the
following steps: Start with any triangle in a plane. The
canonical Sierpinski triangle uses an equilateral triangle with a base
parallel to the horizontal axis. Shrink the triangle to ½ height and ½
width, make three copies, and position the three shrunken triangles so that
each triangle touches the two other triangles at a corner. Note the
emergence of the central hole - because the three shrunken triangles can between
them cover only 3/4 of the area of the original.
Repeat step 2 with each of the smaller triangles.
Note that this infinite process is not dependent upon the starting shape being
a triangle—it is just clearer that way. The first few steps starting, for
example, from a square also tend towards a Sierpinski triangle. Michael Barnsley
used an image of a fish to illustrate this in his paper "V-variable fractals
and superfractals." The actual fractal is what would be
obtained after an infinite number of iterations. More formally, one describes
it in terms of functions on closed sets of points. If we let note the dilation
by a factor of ½ about a point a, then the Sierpinski triangle with corners a,
b, and c is the fixed set of the transformation U U .
This is an attractive fixed set, so that when the operation is applied to any
other set repeatedly, the images converge on the Sierpinski triangle.
This is what is happening with the triangle above, but any other set would
suffice. = Chaos game=
If one takes a point and applies each of the transformations , , and to it
randomly, the resulting points will be dense in the Sierpinski triangle, so the
following algorithm will again generate arbitrarily close approximations to it:
Start by labeling p1, p2 and p3 as the corners of the Sierpinski triangle, and
a random point v1. Set vn+1 = ½, where rn is a random number 1, 2 or 3. Draw
the points v1 to v∞. If the first point v1 was a point on the Sierpiński
triangle, then all the points vn lie on the Sierpinski triangle. If the first
point v1 to lie within the perimeter of the triangle is not a point on the
Sierpinski triangle, none of the points vn will lie on the Sierpinski triangle,
however they will converge on the triangle. If v1 is outside the triangle,
the only way vn will land on the actual triangle, is if vn is on what would be
part of the triangle, if the triangle was infinitely large.
Or more simply: Take 3 points in a plane to form a
triangle, you need not draw it. Randomly select any point inside the
triangle and consider that your current position.
Randomly select any one of the 3 vertex points.
Move half the distance from your current position to the selected vertex.
Plot the current position. Repeat from step 3.
Note: This method is also called the chaos game, and is an example of an
iterated function system. You can start from any point outside or inside the
triangle, and it would eventually form the Sierpinski Gasket with a few
leftover points. It is interesting to do this with pencil and paper. A brief
outline is formed after placing approximately one hundred points, and
detail begins to appear after a few hundred.
= Arrowhead curve= Another construction for the Sierpinski
triangle shows that it can be constructed as a curve in the plane. It
is formed by a process of repeated modification of simpler curves,
analogous to the construction of the Koch snowflake:
Start with a single line segment in the plane
Repeatedly replace each line segment of the curve with three shorter segments,
forming 120° angles at each junction between two consecutive segments, with
the first and last segments of the curve either parallel to the original line
segment or forming a 60° angle with it. The resulting fractal curve is called
the Sierpiński arrowhead curve, and its limiting shape is the Sierpinski
triangle. = Cellular automata=
The Sierpinski triangle also appears in certain cellular automata, including
those relating to Conway's Game of Life. For instance, the life-like cellular
automaton B1/S12 when applied to a single cell will generate four
approximations of the Sierpinski triangle. The time-space diagram of a
replicator pattern in a cellular automaton also often resembles a
Sierpinski triangle. = Pascal's triangle=
If one takes Pascal's triangle with 2n rows and colors the even numbers white,
and the odd numbers black, the result is an approximation to the Sierpinski
triangle. More precisely, the limit as n approaches infinity of this
parity-colored 2n-row Pascal triangle is the Sierpinski triangle.
= Towers of Hanoi= The Towers of Hanoi puzzle involves
moving disks of different sizes between three pegs, maintaining the property
that no disk is ever placed on top of a smaller disk. The states of an n-disk
puzzle, and the allowable moves from one state to another, form an undirected
graph that can be represented geometrically as the intersection graph
of the set of triangles remaining after the nth step in the construction of the
Sierpinski triangle. Thus, in the limit as n goes to infinity, this sequence of
graphs can be interpreted as a discrete analogue of the Sierpinski triangle.
Properties For integer number of dimensions d, when
doubling a side of an object, 2 d copies of it are created, i.e. 2 copies for
1-dimensional object, 4 copies for 2-dimensional object and 8 copies for
3-dimensional object. For Sierpinski triangle doubling its side creates 3
copies of itself. Thus Sierpinski triangle has Hausdorff dimension
log(3)/log(2) ≈ 1.585, which follows from solving 2 d = 3 for d.
The area of a Sierpinski triangle is zero. The area remaining after each
iteration is clearly 3/4 of the area from the previous iteration, and an
infinite number of iterations results in zero.
The points of a Sierpinski triangle have a simple characterization in Barycentric
coordinates. If a point has coordinates, expressed as Binary numbers, then the
point is in Sierpinski's triangle if and only if ui+vi+wi=1 for all i.
Generalization to other Moduli A generalization of the Sierpinski
triangle can also be generated using Pascal's Triangle if a different Modulo
is used. Iteration n can be generated by taking a Pascal's triangle with Pn rows
and coloring numbers by their value for x mod P. As n approaches infinity, a
fractal is generated. The same fractal can be achieved by
dividing a triangle into a tessellation of P2 similar triangles and removing the
triangles that are upside-down from the original, then iterating this step with
each smaller triangle. Conversely, the fractal can also be
generated by beginning with a triangle and duplicating it and arranging
n(n+1)/2 of the new figures in the same orientation into a larger similar
triangle with the vertices of the previous figures touching, then
iterating that step. Analogues in higher dimensions
The Sierpinski tetrahedron or tetrix is the three-dimensional analogue of the
Sierpinski triangle, formed by repeatedly shrinking a regular
tetrahedron to one half its original height, putting together four copies of
this tetrahedron with corners touching, and then repeating the process. This can
also be done with a square pyramid and five copies instead. A tetrix
constructed from an initial tetrahedron of side-length L has the property that
the total surface area remains constant with each iteration.
The initial surface area of the tetrahedron of side-length L is . At the
next iteration, the side-length is halved
and there are 4 such smaller tetrahedra. Therefore, the total surface area after
the first iteration is: This remains the case after each
iteration. Though the surface area of each subsequent tetrahedron is 1/4 that
of the tetrahedron in the previous iteration, there are 4 times as
many—thus maintaining a constant total surface area.
The total enclosed volume, however, is geometrically decreasing with each
iteration and asymptotically approaches 0 as the number of iterations increases.
In fact, it can be shown that, while having fixed area, it has no
3-dimensional character. The Hausdorff dimension of such a construction is
which agrees with the finite area of the figure.
History Wacław Sierpiński described the
Sierpinski triangle in 1915. However, similar patterns appear already in the
13th-century Cosmati mosaics in the cathedral of Anagni, Italy, and other
places of central Italy, for carpets in many places such as the nave of the
Roman Basilica of Santa Maria in Cosmedin, and for isolated triangles
positioned in rotae in several churches and Basiliche. In the case of the
isolated triangle, it is interesting to notice that the iteration is at least of
three levels. See also
Apollonian gasket, a set of mutually tangent circles with the same
combinatorial structure as the Sierpinski triangle
List of fractals by Hausdorff dimension Sierpinski carpet, another fractal named
after Sierpinski and formed by repeatedly removing squares from a
larger square References
^ "Sierpinski Gasket by Trema Removal" ^ Michael Barnsley; et al., V-variable
fractals and superfractals CS1 maint: Explicit use of et al.
^ Feldman, David P., "17.4 The chaos game", Chaos and Fractals: An Elementary
Introduction, Oxford University Press, pp. 178–180, ISBN 9780199566440 .
^ Prusinkiewicz, P., "Graphical applications of L−systems", Proceedings
of Graphics Interface '86 / Vision Interface '86, pp. 247–253 .
^ Rumpf, Thomas, "Conway's Game of Life accelerated with OpenCL", Proceedings of
the Eleventh International Conference on Membrane Computing, pp. 459–462 .
^ Bilotta, Eleonora; Pantano, Pietro, "Emergent patterning phenomena in 2D
cellular automata", Artificial Life 11: 339–362, doi:10.1162/1064546054407167 .
^ Stewart, Ian, How to Cut a Cake: And other mathematical conundrums, Oxford
University Press, p. 145, ISBN 9780191500718 .
^ Romik, Dan, "Shortest paths in the Tower of Hanoi graph and finite
automata", SIAM Journal on Discrete Mathematics 20: 610–62,
arXiv:math.CO/0310109, doi:10.1137/050628660, MR 2272218 .
^ Falconer, Kenneth. Fractal geometry: mathematical foundations and
applications. Chichester: John Wiley. p. 120. ISBN 0-471-92287-0. Zbl 0689.28003.
^ Helmberg, Gilbert, Getting Acquainted with Fractals, Walter de Gruyter, p. 41,
ISBN 9783110190922 . ttp:www.cut-the-knot.orgSierpinski.shtml
^ Shannon & Bardzell, Kathleen & Michael, "Patterns in Pascal's Triangle
- with a Twist - First Twist: What is It?", maa.org, retrieved 29 March 2015
^ Wolfram, Stephen, A New Kind of Science, Wolfram Media, pp. 43, 873
^ "Geometric floor mosaic, nave of Santa Maria in Cosmedin, Forum Boarium, Rome",
5 September 2011, Flickr ^ Conversano, Elisa; Tedeschini-Lalli,
Laura, "Sierpinski Triangles in Stone on Medieval Floors in Rome"", APLIMAT
Journal of Applied Mathematics 4: 114, 122
External links Hazewinkel, Michiel, ed., "Sierpinski
gasket", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
Weisstein, Eric W., "Sierpinski Sieve", MathWorld.
Paul W. K. Rothemund, Nick Papadakis, and Erik Winfree, Algorithmic
Self-Assembly of DNA Sierpinski Triangles, PLoS Biology, volume 2, issue
12, 2004. Sierpinski Gasket by Trema Removal at
cut-the-knot Sierpinski Gasket and Tower of Hanoi at
cut-the-knot 3D printed Stage 5 Sierpinski
Tetrahedron