The
Sierpinski triangle (also with the original orthography Sierpiński), also called the
Sierpinski gasket or the
Sierpinski Sieve, is a
fractalA fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity...
and attractive fixed set named after the
PolishPoland , officially the Republic of Poland , is a country in Central Europe bordered by Germany to the west; the Czech Republic and Slovakia to the south; Ukraine, Belarus and Lithuania to the east; and the Baltic Sea and Kaliningrad Oblast, a Russian exclave, to the north...
mathematicianA mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
Wacław Sierpiński who described it in 1915. However, similar patterns appear already in the 13th-century
CosmatiThe Cosmati were a Roman family, seven members of which, for four generations, were skilful architects, sculptors and workers in decorative geometric mosaic, mostly for church floors...
mosaicMosaic is the art of creating images with an assemblage of small pieces of colored glass, stone, or other materials. It may be a technique of decorative art, an aspect of interior decoration, or of cultural and spiritual significance as in a cathedral...
s in the cathedral of
AnagniAnagni is an ancient town and comune in Latium, central Italy, in the hills east-southeast of Rome. It is a historical center in Ciociaria.-Geography:...
,
ItalyItaly , officially the Italian Republic languages]] under the European Charter for Regional or Minority Languages. In each of these, Italy's official name is as follows:;;;;;;;;), is a unitary parliamentary republic in South-Central Europe. To the north it borders France, Switzerland, Austria and...
. and other places, such as
in the nave of the roman Basilica of
Santa Maria in CosmedinThe Basilica of Saint Mary in Cosmedin is a minor basilica church in Rome, Italy. It is located in the rione of Ripa.- History :The church was built in the 8th century during the Byzantine Papacy over the remains of the Templum Herculis Pompeiani in the Forum Boarium and of the Statio annonae, one...
.
Originally constructed as a curve, this is one of the basic examples of
self-similarIn mathematics, a self-similar object is exactly or approximately similar to a part of itself . Many objects in the real world, such as coastlines, are statistically self-similar: parts of them show the same statistical properties at many scales...
sets, i.e. it is a mathematically generated pattern that can be reproducible at any magnification or reduction.
Comparing the Sierpinski triangle or the
Sierpinski carpetThe Sierpinski carpet is a plane fractal first described by Wacław Sierpiński in 1916. The carpet is a generalization of the Cantor set to two dimensions . Sierpiński demonstrated that this fractal is a universal curve, in that any possible one-dimensional graph, projected onto the two-dimensional...
to equivalent repetitive tiling arrangements, it is evident that similar structures can be built into any
rep-tileIn the geometry of tessellations, a shape that can be dissected into smaller copies of the same shape is called a reptile or rep-tile. The shape is labelled as rep-n if the dissection uses n copies...
arrangements.
Construction
An algorithm for obtaining arbitrarily close approximations to the Sierpinski triangle is as follows:
Note: each removed triangle (a trema) is
topologicallyTopology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
an
open setThe concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
.
- Start with any triangle in a plane (any closed, bounded region in the plane will actually work). The canonical Sierpinski triangle uses an equilateral triangle with a base parallel to the horizontal axis (first image).
- 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 Sierpinski's triangle.)
- 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 Sierpinski triangle.
Michael BarnsleyMichael Fielding Barnsley is a British mathematician, researcher and an entrepreneur who has worked on fractal compression; he holds several patents on the technology. He received his Ph.D in Theoretical Chemistry from University of Wisconsin–Madison in 1972...
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.
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 (
vn +
prn ), 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
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 gameIn mathematics, the term chaos game, as coined by Michael Barnsley, originally referred to a method of creating a fractal, using a polygon and an initial point selected at random inside it...
. 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.
Or using an Iterated function system
An alternative way of computing the Sierpinski triangle uses an
Iterated function systemIn mathematics, iterated function systems or IFSs are a method of constructing fractals; the resulting constructions are always self-similar....
and starts by a point at the origin (x
0 = 0, y
0 = 0). The new points are iteratively computed by randomly applying (with equal probability) one of the following three coordinate transformations (using the so called
chaos gameIn mathematics, the term chaos game, as coined by Michael Barnsley, originally referred to a method of creating a fractal, using a polygon and an initial point selected at random inside it...
):
x
n+1 = 0.5 x
n
y
n+1 = 0.5 y
n; a half-size copy
This coordinate transformation is drawn in yellow in the figure.
x
n+1 = 0.5 x
n + 0.25
y
n+1 = 0.5 y
n + 0.5
; a half-size copy shifted right and up
This coordinate transformation is drawn using red color in the figure.
x
n+1 = 0.5 x
n + 0.5
y
n+1 = 0.5 y
n; a half-size copy doubled shifted to the right
When this coordinate transformation is used, the triangle is drawn in blue.
Or using an L-system — The Sierpinski triangle drawn using an L-system.
bitwise AND - The 2D AND function, z=AND(x,y) can also produce a white on black right angled Sierpinski triangle if all pixels of which z=0 are white, and all other values of z are black.
bitwise XOR - The values of the discrete, 2D XOR function, z=XOR(x,y) also exhibit structures related to the Sierpinski triangle. For example, one could generate the Sierpinski triangle by setting up a 2 dimensional matrix, [rows][columns] placing the uppermost point on [1][n/2], then cycling through the remaining cells row by row the value of the cell being XOR([i-1][j-1],[i-1][j+1])
Other means — The Sierpinski triangle also appears in certain cellular automata (such as
Rule 90Rule 90 is an elementary cellular automaton based on the exclusive or function. It consists of a one-dimensional array of cells, each of which can hold either a 0 or a 1 value; in each time step all values are simultaneously replaced by the exclusive or of the two neighboring values...
), including those relating to
Conway's Game of LifeThe Game of Life, also known simply as Life, is a cellular automaton devised by the British mathematician John Horton Conway in 1970....
. The automaton "12/1" when applied to a single cell will generate four approximations of the Sierpinski triangle.
Another method is create a triangle of summations and shade all odd numbers. That is, the first row is "1", the second row is "1 1", the third being "1 2 1", and all subsequent rows being "1" followed by summations of the pairs of numbers directly above it in a triangle, followed by an ending one. The fourth row is thus "1 (1+2) (2+1) 1". If only odd numbers are shaded, this represents a Sierpinski Triangle to any discrete resolution.
Properties
The Sierpinski triangle has
Hausdorff dimensionthumb|450px|Estimating the Hausdorff dimension of the coast of Great BritainIn mathematics, the Hausdorff dimension is an extended non-negative real number associated with any metric space. The Hausdorff dimension generalizes the notion of the dimension of a real vector space...
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 triangleIn mathematics, Pascal's triangle is a triangular array of the binomial coefficients in a triangle. It is named after the French mathematician, Blaise Pascal...
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
limitThe limit of a sequence is, intuitively, the unique number or point L such that the terms of the sequence become arbitrarily close to L for "large" values of n...
as n approaches infinity of this parity-colored 2
n-row Pascal triangle is the Sierpinski triangle.
The area of a Sierpinski triangle is zero (in
Lebesgue measureIn measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space. For n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called...
). 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. Intuitively one can see this applies to any geometrical construction with an infinite number of iterations, each of which decreases the size by an amount proportional to a previous iteration.
Analogues in higher dimensions
The tetrix is the three-dimensional analogue of the Sierpinski triangle, formed by repeatedly shrinking a regular
tetrahedronIn geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...
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
pyramidIn geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle. It is a conic solid with polygonal base....
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
. 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 (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 dimensionthumb|450px|Estimating the Hausdorff dimension of the coast of Great BritainIn mathematics, the Hausdorff dimension is an extended non-negative real number associated with any metric space. The Hausdorff dimension generalizes the notion of the dimension of a real vector space...
of such a construction is
which agrees with the finite area of the figure. (A Hausdorff dimension strictly between 2 and 3 would indicate 0 volume and infinite area.)
See also
- Apollonian gasket
In mathematics, an Apollonian gasket or Apollonian net is a fractal generated from triples of circles, where each circle is tangent to the other two. It is named after Greek mathematician Apollonius of Perga.-Construction:...
- Chaos game
In mathematics, the term chaos game, as coined by Michael Barnsley, originally referred to a method of creating a fractal, using a polygon and an initial point selected at random inside it...
- List of fractals by Hausdorff dimension
- Pascal's triangle
In mathematics, Pascal's triangle is a triangular array of the binomial coefficients in a triangle. It is named after the French mathematician, Blaise Pascal...
- Rule 90
Rule 90 is an elementary cellular automaton based on the exclusive or function. It consists of a one-dimensional array of cells, each of which can hold either a 0 or a 1 value; in each time step all values are simultaneously replaced by the exclusive or of the two neighboring values...
- Sierpinski carpet
The Sierpinski carpet is a plane fractal first described by Wacław Sierpiński in 1916. The carpet is a generalization of the Cantor set to two dimensions . Sierpiński demonstrated that this fractal is a universal curve, in that any possible one-dimensional graph, projected onto the two-dimensional...
- Sierpiński arrowhead curve
The Sierpiński arrowhead curve is a fractal curve similar in appearance and identical in limit to the Sierpiński triangle.-Representation as Lindenmayer system:The Sierpiński arrowhead curve can be expressed by a rewrite system .-Literature:...
External links
- 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
Cut-the-knot is a free, advertisement-funded educational website maintained by Alexander Bogomolny and devoted to popular exposition of many topics in mathematics. The site has won more than 20 awards from scientific and educational publications, including a Scientific American Web Award in 2003,...
- Sierpinski Gasket and Tower of Hanoi at cut-the-knot
Cut-the-knot is a free, advertisement-funded educational website maintained by Alexander Bogomolny and devoted to popular exposition of many topics in mathematics. The site has won more than 20 awards from scientific and educational publications, including a Scientific American Web Award in 2003,...
- 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)
- 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
Richard Marquis is an American studio glass artist who was born September 17, 1945 in Bumblebee, Arizona. He studied both ceramics and glass at the University of California, Berkeley, where he received a BA in 1969 and an MA in 1972...
has created murrineMurrine is an Italian term for colored patterns or images made in a glass cane that are revealed when cut in cross-sections. Murrine can be made in infinite designs—some styles are more familiar, such as millefiore...
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 set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
s
- Online Sierpinski Triangle Generator
- Sierpinski Triangle in Chaotic Scattering Problem
- 3D printed Stage 5 Sierpinski Tetrahedron