Sierpinski triangle

Sierpinski triangle
Generated using a random algorithm
Sierpinski triangle in logic: The first 16 conjunctions of lexicographically ordered arguments
The columns interpreted as binary numbers give 1, 3, 5, 15, 17, 51... (sequence A001317 in OEIS)

The Sierpinski triangle (also with the original orthography Sierpiński), also called the Sierpinski gasket or the Sierpinski Sieve, is a fractal and attractive fixed set with the overall shape of an equilateral triangle, subdivided recursively into smaller equilateral triangles. 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. It is named after the Polish mathematician Wacław Sierpiński but appeared as a decorative pattern many centuries prior to 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 by repeated removal of triangular subsets:

  1. Start with an equilateral triangle.
  2. Subdivide it into four smaller congruent equilateral triangles and remove the central one.
  3. Repeat step 2 with each of the remaining smaller triangles

Each removed triangle (a trema) is topologically an open set.[1] This process of recursively removing triangles is an example of a finite subdivision rule.

Shrinking and duplication

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

  1. 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).
  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 Sierpinski'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 Sierpinski triangle. Michael Barnsley used an image of a fish to illustrate this in his paper "V-variable fractals and superfractals."[2]

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 Sierpinski 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 Sierpinski triangle. This is what is happening with the triangle above, but any other set would suffice.

Chaos game

Animated creation of a Sierpinski triangle using the 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 Sierpinski triangle, so the following algorithm will again generate arbitrarily close approximations to it:[3]

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 rn 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.


Animated construction of a Sierpinski triangle

Or more simply:

  1. Take 3 points in a plane to form a triangle, you need not draw it.
  2. Randomly select any point inside the triangle and consider that your current position.
  3. Randomly select any one of the 3 vertex points.
  4. Move half the distance from your current position to the selected vertex.
  5. Plot the current position.
  6. Repeat from step 3.

Note: This method is also called the chaos game, and is an example of an iterated function system. You can start from any point outside or inside the triangle, and it would eventually form the Sierpinski Gasket with a few leftover points (if the starting point lies on the outline of the triangle, there are no 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.

Sierpinski triangle using an iterated function system

Arrowhead curve

Construction of the Sierpiński arrowhead curve

Another construction for the Sierpinski triangle shows that it can be constructed as a curve in the plane. It is formed by a process of repeated modification of simpler curves, analogous to the construction of the Koch snowflake:

  1. Start with a single line segment in the plane
  2. 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.

The resulting fractal curve is called the Sierpiński arrowhead curve, and its limiting shape is the Sierpinski triangle.[4]

Cellular automata

The Sierpinski 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 Sierpinski triangle.[5] The time-space diagram of a replicator pattern in a cellular automaton also often resembles a Sierpinski triangle.[6]

Pascal's triangle

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 Sierpinski triangle. More precisely, the limit as n approaches infinity of this parity-colored 2n-row Pascal triangle is the Sierpinski triangle.[7]

Towers of Hanoi

The Towers of Hanoi 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 n-disk puzzle, and the allowable moves from one state to another, form an undirected 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 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.[8]

Properties

For integer number of dimensions d, when doubling a side of an object, 2 d 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 Sierpinski triangle doubling its side creates 3 copies of itself. Thus Sierpinski triangle has Hausdorff dimension log(3)/log(2) ≈ 1.585, which follows from solving 2 d = 3 for d.[9]

The area of a Sierpinski triangle is zero (in Lebesgue measure). 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.[10]

The points of a Sierpinski triangle have a simple characterization in Barycentric coordinates.[11] If a point has coordinates (0.u1u2u3…,0.v1v2v3…,0.w1w2w3…), expressed as Binary numbers, then the point is in Sierpinski's triangle if and only if ui+vi+wi=1 for all i.

Generalization to other Moduli

A generalization of the Sierpinski triangle can also be generated using Pascal's Triangle if a different Modulo is used. Iteration n can be generated by taking a Pascal's triangle with Pn rows and coloring numbers by their value for x mod P. As n approaches infinity, a fractal is generated.

The same fractal can be achieved by dividing a triangle into a tessellation of P2 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 n(n+1)/2 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. [12]

Analogues in higher dimensions

A Sierpinski square-based pyramid and its 'inverse'
Sierpinski pyramid recursion progression (7 steps)
A Sierpiński triangle-based pyramid as seen from above (4 main sections highlighted). Note the self-similarity in this 2-dimensional projected view, so that the resulting triangle could be a 2D fractal in itself.

The Sierpinski tetrahedron or tetrix is the three-dimensional analogue of the Sierpinski 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 square 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 \textstyle\frac{\ln 4}{\ln 2}=2 which agrees with the finite area of the figure. (A Hausdorff dimension strictly between 2 and 3 would indicate 0 volume and infinite area.)

Numerical generation

Short code in the Mathematica internal language: The recursive procedure SiPyramid generates pyramid (3D) of the arbitrary order n as the displayable graphic object Graphics3D:

vect[1] = {0, 0, 0};
vect[2] = {1, 0, 0};
vect[3] = {0.5, 3^0.5/2, 0};
vect[4] = {0.5, 1/3*3^0.5/2, ((3^0.5/2)^2 - (1/3*3^0.5/2)^2)^0.5};
Tetron[{i_, j_, k_}] := 
 Tetrahedron[{vect[1] + {i, j, k}, vect[2] + {i, j, k}, vect[3] + {i, j, k}, vect[4] + {i, j, k}}];
SiPyramid[0, {i_, j_, k_}] := {Tetron[{i, j, k}]};
SiPyramid[n_, {i_, j_, k_}] := 
  Module[{s = {}}, 
   Do[s = Union[s, 
      SiPyramid[n - 1, 2^(n - 1)*vect[u] + {i, j, k}]], {u, 4}]; s];


History

Wacław Sierpiński described the Sierpinski triangle in 1915. However, similar patterns appear already in the 13th-century Cosmati mosaics in the cathedral of Anagni, Italy,[13] and other places of central Italy, for carpets in many places such as the nave of the Roman Basilica of Santa Maria in Cosmedin,[14] and for isolated triangles positioned in rotae in several churches and Basiliche.[15] In the case of the isolated triangle, it is interesting to notice that the iteration is at least of three levels.

Etymology

The usage of the word "gasket" to refer to the Sierpinksi 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 Mandlebrot, who thought the fractal looked similar to "the part that prevents leaks in motors".[16]

See also

References

  1. ↑ "Sierpinski Gasket by Trema Removal"
  2. ↑ Michael Barnsley; et al., V-variable fractals and superfractals (PDF)
  3. ↑ Feldman, David P. (2012), "17.4 The chaos game", Chaos and Fractals: An Elementary Introduction, Oxford University Press, pp. 178–180, ISBN 9780199566440.
  4. ↑ Prusinkiewicz, P. (1986), "Graphical applications of L−systems" (PDF), Proceedings of Graphics Interface '86 / Vision Interface '86, pp. 247–253.
  5. ↑ Rumpf, Thomas (2010), "Conway's Game of Life accelerated with OpenCL" (PDF), Proceedings of the Eleventh International Conference on Membrane Computing (CMC 11), pp. 459–462.
  6. ↑ Bilotta, Eleonora; Pantano, Pietro (Summer 2005), "Emergent patterning phenomena in 2D cellular automata", Artificial Life 11 (3): 339–362, doi:10.1162/1064546054407167.
  7. ↑ Stewart, Ian (2006), How to Cut a Cake: And other mathematical conundrums, Oxford University Press, p. 145, ISBN 9780191500718.
  8. ↑ Romik, Dan (2006), "Shortest paths in the Tower of Hanoi graph and finite automata", SIAM Journal on Discrete Mathematics 20 (3): 610–62, arXiv:math.CO/0310109, doi:10.1137/050628660, MR 2272218.
  9. ↑ Falconer, Kenneth (1990). Fractal geometry: mathematical foundations and applications. Chichester: John Wiley. p. 120. ISBN 0-471-92287-0. Zbl 0689.28003.
  10. ↑ Helmberg, Gilbert (2007), Getting Acquainted with Fractals, Walter de Gruyter, p. 41, ISBN 9783110190922.
  11. ↑ http://www.cut-the-knot.org/ctk/Sierpinski.shtml
  12. ↑ Shannon & Bardzell, Kathleen & Michael, "Patterns in Pascal's Triangle - with a Twist - First Twist: What is It?", maa.org (Mathematical association of America), retrieved 29 March 2015
  13. ↑ Wolfram, Stephen (2002), A New Kind of Science, Wolfram Media, pp. 43, 873
  14. ↑ "Geometric floor mosaic (Sierpinski triangles), nave of Santa Maria in Cosmedin, Forum Boarium, Rome", 5 September 2011, Flickr
  15. ↑ Conversano, Elisa; Tedeschini-Lalli, Laura (2011), "Sierpinski Triangles in Stone on Medieval Floors in Rome"" (PDF), APLIMAT Journal of Applied Mathematics 4: 114, 122
  16. ↑ Benedetto, John; Wojciech, Czaja. Integration and Modern Analysis. p. 408.

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