Sierpinski Squared

https://docs.google.com/document/d/1lttzYs1g0NNCi8HtwlGNtZBsKK_jJQ0rj--extf77Io/edit#heading=h.bygdrodhii9f

Niloo,  Damanpreet , and Brendan

Theme

Sierpinski polygons are fractals based on iterations within each other’s shape.

It is a paradox:  they both can, and cannot, be identified as being finite or infinite.

What practicality might arise from this?

Specifications

The actual math here we’ll talk about:

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.[2] This process of recursively removing triangles is an example of a finite subdivision rule.

Shrinking and duplication[edit]

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 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.)
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."[3]

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 dA 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 dA ∪ dB ∪ dC.

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[edit]

Animated creation of a Sierpinski triangle using the chaos game

If one takes a point and applies each of the transformations dA, dB, and dC 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:[4]

Start by labeling p1, p2 and p3 as the corners of the Sierpinski triangle, and a random point v1. Set vn+1 = 1/2(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 vnis 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
7. 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.

                                                         Andrew Werth

http://www.andrewwerth.com/

In the painting Sierpinski Squared, I used several (four, I think) sets of Sierpinski Triangles, rotated and overlapped, to make the square painting. The Sierpinski Triangle is a fractal formed when you start with a triangle and then remove the middle triangular section (and then keep doing that to remaining triangles ad infinitum). The Wikipedia page has a nice explanation of that.  Here’s one of their illustrations:

This was all three of us were talking about in the class. We have made a drawing similar to what the artist had done, and also came up with some 3D shaped of the model as well.