The Sierpià Âski triangle, also called the Sierpià Âski gasket or Sierpià Âski sieve, is a fractal with the overall shape of an equilateral triangle, subdivided recursively into smaller equilateral triangles. Originally constructed as a Sierpià Âski curve, this is one of the basic examples of self-similar setsâÂÂthat is, it is a mathematically generated pattern reproducible at any magnification or reduction. It is named after the Polish mathematician Wacà Âaw Sierpià Âski but appeared as a decorative pattern many centuries before the work of Sierpià Âski.
There are many different ways of constructing the Sierpià Âski triangle.
The Sierpià Âski triangle may be constructed from an equilateral triangle by repeated removal of triangular subsets:
Each removed triangle (a trema) is topologically an open set. This process of recursively removing triangles is an example of a finite subdivision rule.
The same sequence of shapes, converging to the Sierpià Âski triangle, can alternatively be generated by the following steps:
This infinite process is not dependent upon the starting shape being a triangleâÂÂit is just clearer that way. The first few steps starting, for example, from a square also tend towards a Sierpià Âski triangle (as illustrated below), and 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 d<sub>A</sub> denote the dilation by a factor of about a point A, then the Sierpià Âski triangle with corners A, B, and C is the fixed set of the transformation .
This is an attractive fixed set, so that when the operation is applied to any other set repeatedly, the images converge on the Sierpià Âski 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<sub>A</sub>, d<sub>B</sub>, and d<sub>C</sub> to it randomly, the resulting points will be dense in the Sierpià Âski triangle, so the following algorithm will again generate arbitrarily close approximations to it:
Start by labeling p<sub>1</sub>, p<sub>2</sub> and p<sub>3</sub> as the corners of the Sierpià Âski triangle, and a random point v<sub>1</sub>. Set , where r<sub>n</sub> is a random number 1, 2 or 3. Draw the points v<sub>1</sub> to v<sub>âÂÂ</sub>. If the first point v<sub>1</sub> was a point on the Sierpià Âski triangle, then all the points v<sub>n</sub> lie on the Sierpià Âski triangle. If the first point v<sub>1</sub> to lie within the perimeter of the triangle is not a point on the Sierpià Âski triangle, none of the points v<sub>n</sub> will lie on the Sierpià Âski triangle, however they will converge on the triangle. If v<sub>1</sub> is outside the triangle, the only way v<sub>n</sub> will land on the actual triangle, is if v<sub>n</sub> is on what would be part of the triangle, if the triangle was infinitely large.
Or more simply:
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 Sierpià Âski Gasket with a few leftover points (if the starting point lies on the outline of the triangle, there are no leftover points). With pencil and paper, a brief outline is formed after placing approximately one hundred points, and detail begins to appear after a few hundred.
Another construction for the Sierpià Âski gasket 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:
At every iteration, this construction gives a continuous curve. In the limit, these approach a curve that traces out the Sierpià Âski triangle by a single continuous directed (infinitely wiggly) path, which is called the Sierpià Âski arrowhead. In fact, the aim of Sierpià Âski's original article in 1915 was to show an example of a curve (a Cantorian curve), as the title of the article itself declares.
The Sierpià Âski 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 Sierpià Âski triangle. A very long, one cellâÂÂthick line in standard life will create two mirrored Sierpià Âski triangles. The time-space diagram of a replicator pattern in a cellular automaton also often resembles a Sierpià Âski triangle, such as that of the common replicator in HighLife. The Sierpià Âski triangle can also be found in the Ulam-Warburton automaton and the Hex-Ulam-Warburton automaton.
If one takes Pascal's triangle with rows and colors the even numbers white, and the odd numbers black, the result is an approximation to the Sierpià Âski triangle. More precisely, the limit as approaches infinity of this parity-colored -row Pascal triangle is the Sierpià Âski triangle.
As the proportion of black numbers tends to zero with increasing n, a corollary is that the proportion of odd binomial coefficients tends to zero as n tends to infinity.
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 -disk puzzle, and the allowable moves from one state to another, form an undirected graph, the Hanoi graph, that can be represented geometrically as the intersection graph of the set of triangles remaining after the th step in the construction of the Sierpià Âski triangle. Thus, in the limit as goes to infinity, this sequence of graphs can be interpreted as a discrete analogue of the Sierpià Âski triangle.
For objects of integer dimension , scaling a figure by a factor of 2 (that is, applying a homothetic transformation of ratio 2) produces congruent copies that fit inside the enlarged figure: for instance, doubling a line segment (1-dimensional) yields 2 copies, a square (2-dimensional) yields 4, and a cube (3-dimensional) yields 8.
For the Sierpià Âski triangle, when the figure is scaled by a factor of 2, the enlarged version can be exactly partitioned into 3 scaled copies of itself, each of scale ratio . Therefore, the relationship holds, and solving for gives the Hausdorff dimension
At each iterative stage of its construction, the area that remains equals of that from the previous stage. Consequently, after iterations, the total area is of the original. In the limit as , the total area tends to zero (in the sense of Lebesgue measure).
The points of a Sierpià Âski triangle have a simple characterization in barycentric coordinates. If a point has barycentric coordinates , expressed in binary numeral form, then the point lies in the Sierpià Âski triangle if and only if for
A generalization of the Sierpià Âski triangle can also be generated using Pascal's triangle if a different modulus is used. Iteration can be generated by taking a Pascal's triangle with rows and coloring numbers by their value modulo . As approaches infinity, a fractal is generated.
The same fractal can be achieved by dividing a triangle into a tessellation of 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 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.
The Sierpià Âski tetrahedron or tetrix is the three-dimensional analogue of the Sierpià Âski 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.
The Sierpià Âski tetrahedron can also be formed by starting with a single tetrahedron, removing octahedra from it or recursively combining quadruples of tetrahedra into larger tetrahedra.
A tetrix constructed from an initial tetrahedron of side-length 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 is . The next iteration consists of four copies with side length , so the total area is again. Subsequent iterations again quadruple the number of copies and halve the side length, preserving the overall area. 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 Sierpià Âski gasket, is an intricately connected curve. Its Hausdorff dimension is ; here "log" denotes the natural logarithm, the numerator is the logarithm of the number of copies of the shape formed from each copy of the previous iteration, and the denominator is the logarithm of the factor by which these copies are scaled down from the previous iteration. 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 without overlap.
Wacà Âaw Sierpià Âski described the Sierpià Âski triangle in 1915. However, similar patterns appear already as a common motif of 13th-century Cosmatesque inlay stonework.
The Apollonian gasket, named for Apollonius of Perga (3rd century BC), was first described by Gottfried Leibniz (17th century) and is a curved precursor of the 20th-century Sierpià Âski triangle.
The usage of the word "gasket" to refer to the Sierpià Âski 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 Benoit Mandelbrot, who thought the fractal looked similar to "the part that prevents leaks in motors".