Sierpiński triangle

Originally constructed as a curve, this is one of the basic examples of self-similar sets—that is, it is a mathematically generated pattern reproducible at any magnification or reduction.

[1] This process of recursively removing triangles is an example of a finite subdivision rule.

The same sequence of shapes, converging to the Sierpiński triangle, can alternatively be generated by the following steps: 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 Sierpiński triangle.

Michael Barnsley used an image of a fish to illustrate this in his paper "V-variable fractals and superfractals.

"[2][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 dA denote the dilation by a factor of ⁠1/2⁠ about a point A, then the Sierpiński triangle with corners A, B, and C is the fixed set of the transformation ⁠

If one takes a point and applies each of the transformations dA, dB, and dC to it randomly, the resulting points will be dense in the Sierpiński triangle, so the following algorithm will again generate arbitrarily close approximations to it:[4] Start by labeling p1, p2 and p3 as the corners of the Sierpiński triangle, and a random point v1.

Set vn+1 = ⁠1/2⁠(vn + prn), where rn is a random number 1, 2 or 3.

Or more simply: This method is also called the chaos game, and is an example of an iterated function system.

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 Sierpiński triangle by a single continuous directed (infinitely wiggly) path, which is called the Sierpiński arrowhead.

[6][7] The Sierpiński 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 Sierpiński triangle.

[8] A very long, one cell–thick line in standard life will create two mirrored Sierpiński triangles.

The states of an n-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 nth step in the construction of the Sierpiński triangle.

Thus, in the limit as n goes to infinity, this sequence of graphs can be interpreted as a discrete analogue of the Sierpiński triangle.

For the Sierpiński triangle, doubling its side creates 3 copies of itself.

[15] The points of a Sierpiński triangle have a simple characterization in barycentric coordinates.

, expressed as binary numerals, then the point is in Sierpiński's triangle if and only if

Conversely, the fractal can also be generated by beginning with a triangle and duplicating it and arranging

[17] The Sierpiński tetrahedron or tetrix is the three-dimensional analogue of the Sierpiński 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.

has the property that the total surface area remains constant with each iteration.

Subsequent iterations again quadruple the number of copies and halve the side length, preserving the overall area.

The limit of this process has neither volume nor surface but, like the Sierpiński gasket, is an intricately connected curve.

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

However, similar patterns appear already as a common motif of 13th-century Cosmatesque inlay stonework.

[19] 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 Sierpiński triangle.

[20][21][22] The usage of the word "gasket" to refer to the Sierpiński 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".