## Sierpiński triangle |

The **Sierpinski triangle** (also with the original *Sierpiński*), also called the **Sierpinski gasket** or **Sierpinski sieve**, is a ^{[1]}^{[2]}

- constructions
- properties
- generalization to other moduli
- analogues in higher dimensions
- history
- etymology
- see also
- references
- external links

There are many different ways of constructing the Sierpinski triangle.

The Sierpinski triangle may be constructed from an

- 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 ^{[3]}
This process of recursively removing triangles is an example of a

The same sequence of shapes, converging to the Sierpinski 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 Sierpinski triangle uses an
equilateral triangle with a base parallel to the horizontal axis (first image). - Shrink the triangle to 1/2 height and 1/2 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. ^{[4]}^{[5]}

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 1/2 about a point A, then the Sierpinski triangle with corners A, B, and C is the fixed set of the transformation *d*_{A} ∪ *d*_{B} ∪ *d*_{C}.

This is an

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 Sierpinski triangle, so the following algorithm will again generate arbitrarily close approximations to it:^{[6]}

Start by labeling **p**_{1}, **p**_{2} and **p**_{3} as the corners of the Sierpinski triangle, and a random point **v**_{1}. Set **v**_{n+1} = 1/2(**v**_{n} + **p**_{rn}), where *r _{n}* is a random number 1, 2 or 3. Draw the points

Or more simply:

- Take three 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 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

Another construction for the Sierpinski gasket shows that it can be constructed as a

- 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 Sierpenski triangle by a single continuous directed (infinitely wiggly) path, which is called the ^{[7]} 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.^{[8]}^{[2]}

The Sierpinski triangle also appears in certain ^{[9]} A very long one cell thick line in standard life will create two mirrored Sierpinski triangles. The time-space diagram of a replicator pattern in a cellular automaton also often resembles a Sierpinski triangle, such as that of the common replicator in HighLife.^{[10]} The Sierpinski triangle can also be found in the ^{[11]}

If one takes ^{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 *n* approaches infinity of this ^{n}-row Pascal triangle is the Sierpinski triangle.^{[12]}

The *n*-disk puzzle, and the allowable moves from one state to another, form an *n*th step in the construction of the Sierpinski triangle. Thus, in the limit as *n* goes to infinity, this sequence of graphs can be interpreted as a discrete analogue of the Sierpinski triangle.^{[13]}

Other Languages

العربية: مثلث سيربنسكي

български: Триъгълник на Серпински

català: Triangle de Sierpiński

čeština: Sierpińského trojúhelník

dansk: Sierpinski-trekant

Deutsch: Sierpinski-Dreieck

Ελληνικά: Τρίγωνο Σιερπίνσκι

español: Triángulo de Sierpinski

Esperanto: Triangulo de Sierpinski

français: Triangle de Sierpiński

galego: Triángulo de Sierpinski

한국어: 시에르핀스키 삼각형

hrvatski: Trokut Sierpińskog

italiano: Triangolo di Sierpiński

עברית: משולש שרפינסקי

magyar: Sierpiński-háromszög

Nederlands: Driehoek van Sierpiński

日本語: シェルピンスキーのギャスケット

polski: Trójkąt Sierpińskiego

português: Triângulo de Sierpinski

русский: Треугольник Серпинского

Simple English: Sierpinski triangle

کوردی: سێگۆشەی سیرپینسکی

српски / srpski: Троугао Сјерпињског

srpskohrvatski / српскохрватски: Trokut Sierpińskog

suomi: Sierpińskin kolmio

svenska: Sierpinskitriangel

українська: Трикутник Серпінського

中文: 謝爾賓斯基三角形