Why do you think the game chaos produce a Sierpinski triangle?
In 10-20 steps, the point off is practically indistinguishable from a similar point on . Finally, and this is subtle, we’ll show that randomness ensures the chaos game (eventually) get arbitrarily close to every point on . This guarantees that the chaos game generates the whole Sierpinski triangle.
What is the chaos game and how do you win?
The goal of the chaos game is to roll the die many hundreds of times and predict what the resulting pattern of points will be. Most students who are unfamiliar with the game guess that the resulting image will be a random smear of points. Others predict that the points will eventually fill the entire triangle.
What is the formula for Sierpinski triangle?
We can break up the Sierpinski triangle into 3 self similar pieces (n=3) then each can be magnified by a factor m=2 to give the entire triangle. The formula for dimension d is n = m^d where n is the number of self similar pieces and m is the magnification factor.
Who invented the chaos game?
The term “Chaos Game” was coined by Michael Barnsley , who developed this ingenious technique for generating mathematical objects called fractals. We have discussed a particular fractal set on this blog: see Cantor’s Ternary Set.
How many triangles are in Sierpinski’s triangle?
This leaves us with three triangles, each of which has dimensions exactly one-half the dimensions of the original triangle, and area exactly one-fourth of the original area.
What dimension is the Sierpinski triangle?
For the Sierpinski triangle, doubling its side creates 3 copies of itself. Thus the Sierpinski triangle has Hausdorff dimension log(3)log(2) = log2 3 ≈ 1.585, which follows from solving 2d = 3 for d. The area of a Sierpinski triangle is zero (in Lebesgue measure).
What is chaos tag?
If you and the other person tag each other at the same time, then you do “Rock-Paper-Scissors”, and whoever wins, kneels in place, and watches until the person who “beat” you is tagged or beaten, at which point you can get up and resume playing the game. The game is won when only one person remains.
What is the name of the two similar triangles?
The concept of similar triangles and congruent triangles are two different terms that are closely related. Similar triangles are two or more triangles with the same shape, equal pair of corresponding angles, and the same ratio of the corresponding sides.
Are Fractals two-dimensional?
No small piece of it is line-like, but rather it is composed of an infinite number of segments joined at different angles. The fractal dimension of a curve can be explained intuitively thinking of a fractal line as an object too detailed to be one-dimensional, but too simple to be two-dimensional.
What is a tag infinity?
How to make a Sierpinski triangle in chaos?
The steps we use in the chaos game to create a Sierpinski Triangle are as follows: Start with an equilateral triangle. Choose one of the triangles vertices and a random point inside the triangle. Plot the point that is midway between the vertex and random point.
What kind of polygon is the Sierpinski triangle?
Using an equilateral triangle as the polygon, and a fraction of 1/2 will result in the Sierpinski Triangle. Huh, no wonder it’s called the chaos game! That sounds a bit confusing! Let’s step it out using an equilateral triangle and 1/2 as our fraction to better our understanding of the construction process and the chaos game itself.
How does the chaos game work in math?
Let’s explore! In mathematics, the chaos game is a manipulation of a polygon in such a way that a fractal is often, but not always, created. It involves starting with an initial random point within the polygon, and plotting points within the polygon that are a given fraction between vertices of the polygon and those points.