Sierpiński triangle

File:Sierpinski Triangle.svg

Sierpiński triangle

The Sierpiński triangle, also called the Sierpiński gasket or the Sierpiński Sieve, is a fractal named after Wacław Sierpiński who described it in 1915. 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.

Comparing the Sierpinski Triangle or the Sierprinski Carpet to equivalent repetitive tiling arrangements, it is evident that similar structures can be built into any rep-tile arrangements.

Construction

"Zooming in"—it never ends.

Animated construction ^(Enlarge)

An algorithm for obtaining arbitrarily close approximations to the Sierpiński triangle is as follows:

Note: each removed triangle (a trema) is topologically an open set.

1. Start with any triangle in a plane (any closed, bounded region in the plane will actually work). The canonical Sierpiński triangle uses an equilateral triangle with a base parallel to the horizontal axis (first image).
2. 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 (image 2). 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. (Holes are an important feature of Sierpiński's Triangle.)
3. Repeat step 2 with each of the smaller triangles (image 3 and so on).

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 Sierpiński gasket. 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 $d_{a}$ note the dilation by a factor of ½ about a point a, then the Sierpiński triangle with corners a, b, and c is the fixed set of the transformation $d_{a}$ U $d_{b}$ U $d_{c}$.

This is an attractive fixed set, so that when the operation is applied to any other set repeatedly, the images converge on the Sierpiński 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 Sierpiński triangle, so the following algorithm will again generate arbitrarily close approximations to it:

Start by labelling p₁, p₂ and p₃ as the corners of the Sierpiński triangle, and a random point v₁. Set v_n+1 = ½ ( v_n + p_rn ), where r_n 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 v_n lie on the Sierpiński triangle. If the first point v₁ to lie within the perimeter of the triangle is not a point on the Sierpiński triangle, none of the points v_n will lie on the Sierpiński triangle, however they will converge on the triangle. If v₁ 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.

Animated creation of a Sierpiński triangle using the chaos game

Or more simply:

1. Take 3 points in a plane, and form a triangle
2. Randomly select any point inside the triangle and move half the distance from that point to any of the 3 vertex points. Plot the current position.
3. Repeat from step 2.

Note: This method is also called the Chaos game. You can start from any point outside or inside the triangle, and it would eventually form the Sierpiński 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.

Or using an Iterated function system

An alternative way of computing the Sierpiński triangle uses an Iterated function system and starts by a point in the origin (x₀ = 0, 'y₀ = 0) and then the new points are iteratively computed by randomly applying (with equal probability) one of the following three coordinate transformations (using the so called chaos game):

Sierpiński triangle using IFS

x_n+1 = 0.5 x_n
y_n+1 = 0.5 y_n; a half-size copy
when this coordinate transformation is used, the point is drawn in yellow in the figure

x_n+1 = 0.5 x_n + 0.5
y_n+1 = 0.5 y_n + 0.5; a half-size copy shifted right and up
when this coordinate transformation is used, the point is drawn in red

x_n+1 = 0.5 x_n + 1
y_n+1 = 0.5 y_n; a half-size copy doubled shifted to the right
when this coordinate transformation is used, the point is drawn in blue

Or using an L-system — The Sierpiński triangle drawn using an L-system.

Other means — The Sierpiński triangle also appears in certain cellular automata (such as Rule 90), including those relating to Conway's Game of Life. The automaton "12/1" when applied to a single cell will generate four approximations of the Sierpiński triangle.

Properties

The Sierpiński triangle has Hausdorff dimension log(3)/log(2) ≈ 1.585, which follows from the fact that it is a union of three copies of itself, each scaled by a factor of 1/2.

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 Sierpiński triangle. More precisely, the limit as n approaches infinity of this parity-colored 2ⁿ-row Pascal triangle is the Sierpiński triangle.

The area of a Sierpiński triangle is zero (in Lebesgue measure). This can be seen from the infinite iteration, where we remove 25% of the area left at the previous iteration. Therefore the proportion of even numbers in Pascal's triangle must tend to 1 as the number of rows of the triangle tends to infinity.

Analogs in higher dimension

A Sierpiński square-based pyramid and its 'inverse'

A Sierpiński triangle-based pyramid as seen from above (5 main sections highlighted).

The tetrix is the three-dimensional analog 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. This can also be done with a 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 $L^{2}{\sqrt {3}}$. At the next iteration, the side-length is halved

$L\rightarrow {L \over 2}$

and there are 4 such smaller tetrahedra. Therefore, the total surface area after the first iteration is:

$4\left(\left({L \over 2}\right)^{2}{\sqrt {3}}\right)=4{{L^{2}} \over 4}{\sqrt {3}}=L^{2}{\sqrt {3}}.$

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 (factor of 0.5) 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 ${\frac {\ln 4}{\ln 2}}=2$ which agrees with the finite area of the figure. (A Hausdorff dimension between 2 and 3 would indicate 0 volume and infinite area.)

See also

• List of fractals by Hausdorff dimension
• Sierpiński carpet
• Pascal's triangle

External links

• Weisstein, Eric W. "Sierpinski Sieve". MathWorld.
• Sierpinski Triangle C++ code
• 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
• Animated 3D model loop of Sierpinski's Triangle and Article on similarities with Sacred Geometry
• Article explaining Sierpinski's Triangle created with a bitwise XOR (example program in Macromedia Flash ActionScript)
• Article explaining Sierpinski's Triangle created with the Chaos Game (example program in Macromedia Flash ActionScript)
• Sierpinski Triangle and Machu Picchu. Fractal illustration with animation and sound.
• VisualBots - Freeware multi-agent simulator in Microsoft Excel. Sample programs include Sierpinski Triangle.
• IFS Fractal fern and Sierpinski triangle - JAVA applet
• Contains a section where the Sierpinski triangle can be seen step by step -- Shockwave
• The artist Richard Marquis has created murrine Sierpinski triangles which can be viewed on his website here, here, and elsewhere on his recent work page. See also the recent book by Barry Behrstock: The Way of the Artist: Reflections on Creativity and the Life, Home, Art,and Collections of Richard Marquis.
• Another reference, removed triangles are open sets
• Online Sierpinski Triangle Generator
• Sierpinski Triangle in Chaotic Scattering Problem