The Sierpiński triangle (sometimes spelled ''Sierpinski''), also called the Sierpiński gasket or Sierpiński sieve, is a
fractal
In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illu ...
attractive fixed set with the overall shape of an
equilateral triangle
In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each othe ...
, subdivided
recursively
Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematics ...
into smaller equilateral triangles. Originally constructed as a curve, this is one of the basic examples of
self-similar
__NOTOC__
In mathematics, a self-similar object is exactly or approximately similar to a part of itself (i.e., the whole has the same shape as one or more of the parts). Many objects in the real world, such as coastlines, are statistically se ...
sets—that is, it is a mathematically generated pattern that is reproducible at any magnification or reduction. It is named after the
Polish
Polish may refer to:
* Anything from or related to Poland, a country in Europe
* Polish language
* Poles
Poles,, ; singular masculine: ''Polak'', singular feminine: ''Polka'' or Polish people, are a West Slavic nation and ethnic group, w ...
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change.
History
On ...
Wacław Sierpiński
Wacław Franciszek Sierpiński (; 14 March 1882 – 21 October 1969) was a Polish mathematician. He was known for contributions to set theory (research on the axiom of choice and the continuum hypothesis), number theory, theory of functions, and t ...
, but appeared as a decorative pattern many centuries before the work of Sierpiński.
Constructions
There are many different ways of constructing the Sierpinski triangle.
Removing triangles
The Sierpinski triangle may be constructed from an
equilateral triangle
In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each othe ...
by repeated removal of triangular subsets:
# Start with an equilateral triangle.
# Subdivide it into four smaller congruent equilateral triangles and remove the central triangle.
# Repeat step 2 with each of the remaining smaller triangles infinitely.
Each removed triangle (a ''trema'') is
topologically an
open set
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suf ...
.
This process of recursively removing triangles is an example of a
finite subdivision rule
In mathematics, a finite subdivision rule is a recursive way of dividing a polygon or other two-dimensional shape into smaller and smaller pieces. Subdivision rules in a sense are generalizations of regular geometric fractals. Instead of repeati ...
.
Shrinking and duplication
The same sequence of shapes, converging to the Sierpiński triangle, can alternatively be generated by the following steps:
#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
In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each othe ...
with a base parallel to the horizontal axis (first image).
#Shrink the triangle to height and 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 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 Barnsley
Michael Fielding Barnsley (born 1946) 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 Wi ...
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 denote the dilation by a factor of about a point A, then the Sierpiński 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 Sierpiński triangle. This is what is happening with the triangle above, but any other set would suffice.
Chaos game
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 labeling p
1, p
2 and p
3 as the corners of the Sierpinski triangle, and a random point v
1. Set , 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 Sierpiński triangle. If the first point v
1 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 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:
# Take three points in a plane to form a triangle.
# Randomly select any point inside the triangle and consider that your current position.
# Randomly select any one of the three vertex points.
# Move half the distance from your current position to the selected vertex.
# Plot the current position.
# Repeat from step 3.
This method is also called the
chaos game
In mathematics, the term chaos game originally referred to a method of creating a fractal, using a polygon and an initial point selected at random inside it. The fractal is created by iteratively creating a sequence of points, starting with the ...
, and is an example of an
iterated function system
In mathematics, iterated function systems (IFSs) are a method of constructing fractals; the resulting fractals are often self-similar. IFS fractals are more related to set theory than fractal geometry. They were introduced in 1981.
IFS fractals, ...
. You can start from any point outside or inside the triangle, and it would eventually form the Sierpiński 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.
Arrowhead construction of Sierpiński gasket
Another construction for the Sierpinski gasket shows that it can be constructed as a
curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight.
Intuitively, a curve may be thought of as the trace left by a moving point (ge ...
in the plane. It is formed by a process of repeated modification of simpler curves, analogous to the construction of the
Koch snowflake
The Koch snowflake (also known as the Koch curve, Koch star, or Koch island) is a fractal curve and one of the earliest fractals to have been described. It is based on the Koch curve, which appeared in a 1904 paper titled "On a Continuous Curv ...
:
# Start with a single line segment in the plane
# Repeatedly replace each line segment of the curve with three shorter segments, forming 120° angles at each junction between two consecutive segments, with the first and last segments of the curve either parallel to the original line segment or forming a 60° angle with it.
At every iteration, this construction gives a continuous curve. In the limit, these approach a curve that traces out the Sierpinski triangle by a single continuous directed (infinitely wiggly) path, which is called the
Sierpinski arrowhead. In fact, the aim of the original article by Sierpinski of 1915, was to show an example of a curve (a Cantorian curve), as the title of the article itself declares.
Cellular automata
The Sierpinski triangle also appears in certain
cellular automata
A cellular automaton (pl. cellular automata, abbrev. CA) is a discrete model of computation studied in automata theory. Cellular automata are also called cellular spaces, tessellation automata, homogeneous structures, cellular structures, tessel ...
(such as
Rule 90
In the mathematics, mathematical study of cellular automaton, cellular automata, 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 ...
), including those relating to
Conway's Game of Life
The Game of Life, also known simply as Life, is a cellular automaton devised by the British mathematician John Horton Conway in 1970. It is a zero-player game, meaning that its evolution is determined by its initial state, requiring no further ...
. For instance, the
Life-like cellular automaton
Life-Like was a manufacturer of model trains and accessories. In 1960, the company purchased the assets of the defunct Varney Scale Models and began manufacturing model trains and accessories under the name Life-Like in 1970. In 2005 the parent co ...
B1/S12 when applied to a single cell will generate four approximations of the Sierpinski triangle. A very long one cell thick line in standard life will create two mirrored Sierpiński triangles. The time-space diagram of a replicator pattern in a cellular automaton also often resembles a Sierpiński triangle, such as that of the common replicator in HighLife. The Sierpinski triangle can also be found in the
Ulam-Warburton automaton and the Hex-Ulam-Warburton automaton.
Pascal's triangle
If one takes
Pascal's triangle
In mathematics, Pascal's triangle is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, although ot ...
with
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
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
as
approaches infinity of this
parity-colored
-row Pascal triangle is the Sierpinski triangle.
Towers of Hanoi
The
Towers of Hanoi
The Tower of Hanoi (also called The problem of Benares Temple or Tower of Brahma or Lucas' Tower and sometimes pluralized as Towers, or simply pyramid puzzle) is a mathematical game or puzzle consisting of three rods and a number of disks of va ...
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
In discrete mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called '' v ...
, the
Hanoi graph, that can be represented geometrically as the
intersection graph
In graph theory, an intersection graph is a graph that represents the pattern of intersections of a family of sets. Any graph can be represented as an intersection graph, but some important special classes of graphs can be defined by the types o ...
of the set of triangles remaining after the
th step in the construction of the Sierpinski triangle. Thus, in the limit as
goes to infinity, this sequence of graphs can be interpreted as a discrete analogue of the Sierpinski triangle.
Properties
For integer number of dimensions
, when doubling a side of an object,
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 Sierpiński triangle, doubling its side creates 3 copies of itself. Thus the Sierpiński triangle has
Hausdorff dimension
In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of a ...
, which follows from solving
for
.
The area of a Sierpiński triangle is zero (in
Lebesgue measure
In measure theory, a branch of mathematics, 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 wit ...
). The area remaining after each iteration is
of the area from the previous iteration, and an infinite number of iterations results in an area approaching zero.
The points of a Sierpinski triangle have a simple characterization in
barycentric coordinates
In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related ...
. If a point has barycentric coordinates
, expressed as
binary numeral
A binary number is a number expressed in the base-2 numeral system or binary numeral system, a method of mathematical expression which uses only two symbols: typically "0" (zero) and "1" ( one).
The base-2 numeral system is a positional notatio ...
s, then the point is in Sierpiński's triangle if and only if
for
Generalization to other moduli
A generalization of the Sierpiński triangle can also be generated using
Pascal's triangle
In mathematics, Pascal's triangle is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, although ot ...
if a different modulus
is used. Iteration
can be generated by taking a
Pascal's triangle
In mathematics, Pascal's triangle is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, although ot ...
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.
Analogues in higher dimensions
The Sierpinski tetrahedron or tetrix is the three-dimensional analogue of the Sierpiński triangle, formed by repeatedly shrinking a regular
tetrahedron
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...
to one half its original height, putting together four copies of this tetrahedron with corners touching, and then repeating the process.
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 Sierpinski gasket, is an intricately connected curve. Its
Hausdorff dimension
In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of a ...
is
; here "log" denotes the
natural logarithm
The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
, 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.
History
Wacław Sierpiński
Wacław Franciszek Sierpiński (; 14 March 1882 – 21 October 1969) was a Polish mathematician. He was known for contributions to set theory (research on the axiom of choice and the continuum hypothesis), number theory, theory of functions, and t ...
described the Sierpiński triangle in 1915. However, similar patterns appear already as a common motif of 13th-century
Cosmatesque
Cosmatesque, or Cosmati, is a style of geometric decorative inlay stonework typical of the architecture of Medieval Italy, and especially of Rome and its surroundings. It was used most extensively for the decoration of church floors, but was also u ...
inlay stonework.
The
Apollonian gasket
In mathematics, an Apollonian gasket or Apollonian net is a fractal generated by starting with a triple of circles, each tangent to the other two, and successively filling in more circles, each tangent to another three. It is named after Greek mat ...
was first described by
Apollonius of Perga
Apollonius of Perga ( grc-gre, Ἀπολλώνιος ὁ Περγαῖος, Apollṓnios ho Pergaîos; la, Apollonius Pergaeus; ) was an Ancient Greek geometer and astronomer known for his work on conic sections. Beginning from the contribution ...
(3rd century BC) and further analyzed by
Gottfried Leibniz
Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathem ...
(17th century), and is a curved precursor of the 20th-century Sierpiński triangle.
Etymology
The usage of the word "gasket" to refer to the Sierpiński triangle refers to
gasket
Some seals and gaskets
A gasket is a mechanical seal which fills the space between two or more mating surfaces, generally to prevent leakage from or into the joined objects while under compression. It is a deformable material that is used to c ...
s such as are found in
motor
An engine or motor is a machine designed to convert one or more forms of energy into mechanical energy.
Available energy sources include potential energy (e.g. energy of the Earth's gravitational field as exploited in hydroelectric power gen ...
s, and which sometimes feature a series of holes of decreasing size, similar to the fractal; this usage was coined by
Benoit Mandelbrot
Benoit B. Mandelbrot (20 November 1924 – 14 October 2010) was a Polish-born French-American mathematician and polymath with broad interests in the practical sciences, especially regarding what he labeled as "the art of roughness" of phy ...
, who thought the fractal looked similar to "the part that prevents leaks in motors".
See also
*
Apollonian gasket
In mathematics, an Apollonian gasket or Apollonian net is a fractal generated by starting with a triple of circles, each tangent to the other two, and successively filling in more circles, each tangent to another three. It is named after Greek mat ...
, a set of mutually tangent circles with the same combinatorial structure as the Sierpinski triangle
*
List of fractals by Hausdorff dimension
According to Benoit Mandelbrot, "A fractal is by definition a set for which the Hausdorff-Besicovitch dimension strictly exceeds the topological dimension."
Presented here is a list of fractals, ordered by increasing Hausdorff dimension, to illus ...
*
Sierpiński carpet
The Sierpiński carpet is a plane fractal first described by Wacław Sierpiński in 1916. The carpet is a generalization of the Cantor set to two dimensions; another is Cantor dust.
The technique of subdividing a shape into smaller copies of ...
, another fractal named after Sierpiński and formed by repeatedly removing squares from a larger square
*
Triforce
The is a fictional artifact and icon of Nintendo's ''The Legend of Zelda'' series of video games. It first appeared in the original 1986 action-adventure game ''The Legend of Zelda'' and is a focus of subsequent games in the series, including ...
, a relic in the ''
Legend of Zelda
''The Legend of Zelda'' is an action-adventure game franchise created by the Japanese game designers Shigeru Miyamoto and Takashi Tezuka. It is primarily developed and published by Nintendo, although some portable installments and re-rele ...
'' series
References
External links
*
*
*
Sierpinski Gasket by Trema Removalat
cut-the-knot
Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet-born Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and Math ...
Sierpinski Gasket and Tower of Hanoiat
cut-the-knot
Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet-born Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and Math ...
Real-time GPU generated Sierpinski Triangle in 3DPythagorean triangles Waclaw Sierpinski, Courier Corporation, 2003
A067771 Number of vertices in Sierpiński triangle of order n.''at''
OEIS
The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching at AT&T Labs. He transferred the intellectual property and hosting of the OEIS to the ...
Interactive version of the chaos game
{{Authority control
Factorial and binomial topics
Curves
Topological spaces
Types of triangles
Cellular automaton patterns
Science and technology in Poland
L-systems