vastuniversity.blogg.se - Construction of triangle tessellation

It is formed by a process of repeated modification of simpler curves, analogous to the construction of the Koch snowflake:

Īnother construction for the Sierpinski triangle shows that it can be constructed as a curve in the plane. An interactive version of the chaos game can be found here.Ĭonstruction of the Sierpiński arrowhead curve. With pencil and paper, a brief outline is formed after placing approximately one hundred points, and detail begins to appear after a few hundred. 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). This method is also called the chaos game, and is an example of an iterated function system. Move half the distance from your current position to the selected vertex.Randomly select any one of the three vertex points.Randomly select any point inside the triangle and consider that your current position.Take three points in a plane to form a triangle, you need not draw it.If the angle is reduced, the triangle can be continuously transformed into a fractal resembling a tree. 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.Ī Sierpinski Triangle is outlined by a fractal tree with three branches forming an angle of 60° between each other. If the first point v 1 to lie within the perimeter of the triangle is not a point on the Sierpinski triangle, none of the points v n will lie on the Sierpinski triangle, however they will converge on the triangle. If the first point v 1 was a point on the Sierpiński triangle, then all the points v n lie on the Sierpinski triangle. Set v n+1 = 1 / 2( v n + p r n), where r n is a random number 1, 2 or 3. Start by labeling p 1, p 2 and p 3 as the corners of the Sierpinski triangle, and a random point v 1. 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: This is what is happening with the triangle above, but any other set would suffice. 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. 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. More formally, one describes it in terms of functions on closed sets of points. The actual fractal is what would be obtained after an infinite number of iterations. Michael Barnsley used an image of a fish to illustrate this in his paper "V-variable fractals and superfractals." The first few steps starting, for example, from a square also tend towards a Sierpinski triangle. Note that this infinite process is not dependent upon the starting shape being a triangle-it is just clearer that way.

Repeat step 2 with each of the smaller triangles (image 3 and so on).

(Holes are an important feature of Sierpinski's triangle.) 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.

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).

The canonical Sierpinski triangle uses an equilateral triangle with a base parallel to the horizontal axis (first image).

Start with any triangle in a plane (any closed, bounded region in the plane will actually work).

The same sequence of shapes, converging to the Sierpinski triangle, can alternatively be generated by the following steps: This process of recursively removing triangles is an example of a finite subdivision rule.

Repeat step 2 with each of the remaining smaller triangles forever.Įach removed triangle (a trema) is topologically an open set.

Subdivide it into four smaller congruent equilateral triangles and remove the central triangle.

The Sierpinski triangle may be constructed from an equilateral triangle by repeated removal of triangular subsets: The evolution of the Sierpinski triangle.