The Sierpinski triangle (also with the original orthography Sierpiński), 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.^{[1]}
There are many different ways of constructing the Sierpinski triangle.
The Sierpinski triangle may be constructed from an equilateral triangle by repeated removal of triangular subsets:
Each removed triangle (a trema) is topologically an open set.^{[2]} This process of recursively removing triangles is an example of a finite subdivision rule.
The same sequence of shapes, converging to the Sierpinski triangle, can alternatively be generated by the following steps:
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."^{[3]}
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 d_{A} denote the dilation by a factor of 1/2 about a point A, then the Sierpinski triangle with corners A, B, and C is the fixed set of the transformation d_{A} ∪ d_{B} ∪ d_{C}.
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.
If one takes a point and applies each of the transformations d_{A}, d_{B}, and d_{C} 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:^{[4]}
Start by labeling p_{1}, p_{2} and p_{3} as the corners of the Sierpinski triangle, and a random point v_{1}. Set v_{n+1} = 1/2(v_{n} + p_{rn}), where r_{n} is a random number 1, 2 or 3. Draw the points v_{1} to v_{∞}. If the first point v_{1} was a point on the Sierpiński triangle, then all the points v_{n} lie on the Sierpinski triangle. If the first point v_{1} to lie within the perimeter of the triangle is not a point on the Sierpinski triangle, none of the points v_{n} will lie on the Sierpinski triangle, however they will converge on the triangle. If v_{1} is outside the triangle, the only way v_{n} will land on the actual triangle, is if v_{n} is on what would be part of the triangle, if the triangle was infinitely large.
Or more simply:
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 (if the starting point lies on the outline of the triangle, there are no 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. An interactive version of the chaos game can be found here.
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:
The resulting fractal curve is called the Sierpiński arrowhead curve, and its limiting shape is the Sierpinski triangle.^{[5]}
The Sierpinski triangle also appears in certain cellular automata (such as Rule 90), 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.^{[6]} A very long one cell thick line in standard life will create two mirrored Sierpinski triangles. The time-space diagram of a replicator pattern in a cellular automaton also often resembles a Sierpinski triangle, such as that of the common replicator in HighLife.^{[7]}
If one takes Pascal's triangle with 2^{n} 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 2^{n}-row Pascal triangle is the Sierpinski triangle.^{[8]}
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.^{[9]}
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 the Sierpinski triangle, doubling its side creates 3 copies of itself. Thus the Sierpinski triangle has Hausdorff dimension ^{log(3)}/_{log(2)} = log_{2} 3 ≈ 1.585, which follows from solving 2^{d} = 3 for d.^{[10]}
The area of a Sierpinski triangle is zero (in Lebesgue measure). 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.^{[11]}
The points of a Sierpinski triangle have a simple characterization in barycentric coordinates.^{[12]} If a point has coordinates (0.u_{1}u_{2}u_{3}…, 0.v_{1}v_{2}v_{3}…, 0.w_{1}w_{2}w_{3}…), expressed as binary numerals, then the point is in Sierpinski's triangle if and only if u_{i} + v_{i} + w_{i} = 1 for all i.
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 P^{n} 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 P^{2} 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.^{[13]}
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 (iteration-0) tetrahedron of side-length L is L^{2}√3. The next iteration consists of four copies with side length L/2, so the total area is 4(L/2)^{2}√3 = 4L^{2}√3/4 = L^{2}√3 again. Meanwhile the volume of the construction is halved at every step and therefore approaches zero. The limit of this process has neither volume nor surface but, like the Sierpinski gasket, is an intricately connected curve. Its Hausdorff dimension is log(4)/log(2) = 2. If all points are projected onto a plane that is parallel to two of the outer edges, they exactly fill a square of side length L/√2 without overlap.
A short code in the Mathematica internal language: the recursive procedure SiPyramid generates a 3D pyramid of arbitrary order n as the displayable graphic object Graphics3D:
vect[1] = {0, 0, 0};
vect[2] = {1, 0, 0};
vect[3] = {0.5, 3^0.5/2, 0};
vect[4] = {0.5, 1/3*3^0.5/2, ((3^0.5/2)^2 - (1/3*3^0.5/2)^2)^0.5};
Tetron[{i_, j_, k_}] :=
Tetrahedron[{vect[1] + {i, j, k}, vect[2] + {i, j, k}, vect[3] + {i, j, k}, vect[4] + {i, j, k}}];
SiPyramid[0, {i_, j_, k_}] := {Tetron[{i, j, k}]};
SiPyramid[n_, {i_, j_, k_}] :=
Module[{s = {}},
Do[s = Union[s,
SiPyramid[n - 1, 2^(n - 1)*vect[u] + {i, j, k}]], {u, 4}]; s];
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,^{[14]} and other places of central Italy, for carpets in many places such as the nave of the Roman Basilica of Santa Maria in Cosmedin,^{[15]} and for isolated triangles positioned in rotae in several churches and Basiliche.^{[1]} In the case of the isolated triangle, it is interesting to notice that the iteration is at least of three levels.
The Apollonian gasket was first described by Apollonius of Perga (3rd century BC) and further analyzed by Gottfried Leibniz (17th century), and is a curved precursor of the 20th-century Sierpiński triangle.^{[16]}
The usage of the word "gasket" to refer to the Sierpinski triangle refers to gaskets such as are found in motors, and which sometimes feature a series of holes of decreasing size, similar to the fractal; this usage was coined by Benoît Mandelbrot, who thought the fractal looked similar to "the part that prevents leaks in motors".^{[17]}
This article uses material from the Wikipedia article "Sierpinski triangle", which is released under the Creative Commons Attribution-Share-Alike License 3.0. There is a list of all authors in Wikipedia
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