/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 29 Refer to the chaos game as descr... [FREE SOLUTION] | 91影视

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Chapter 12: Problem 29

Refer to the chaos game as described in Section 12.2. You should use graph paper for these exercises. Start with an isosceles right triangle \(A B C\) with \(A B=A C=32,\) as shown in Fig. \(12-40 .\) Choose vertex \(A\) for a roll of 1 or \(2,\) vertex \(B\) for a roll of 3 or \(4,\) and vertex \(C\) for \(a\) roll of 5 or \(6 .\) Using a rectangular coordinate system with \(A\) at \((0,0), B\) at \((32,0),\) and \(C\) at \((0,32),\) complete Table \(12-21\) $$ \begin{array}{c|c|c} \text { Roll } & \text { Point } & \text { Coordinates } \\ \hline 3 & P_{1} & (32,0) \\ \hline 1 & P_{2} & (16,0) \\ \hline 2 & P_{3} & \\ \hline 3 & P_{4} & \\ \hline 5 & P_{5} & \\ \hline 5 & P_{6} & \\ \hline \end{array} $$

Short Answer

Expert verified
The coordinates for the points are \( P3 = (8,0) \), \( P4 = (20,0) \), \( P5 = (10,16) \), and \( P6 = (5,24) \)

Step by step solution

01

Compute Point P3 Coordinates

Given that the die roll was a 2, we move halfway between the current point P2 (16,0) and the vertex A(0,0). The formula for the midpoint between two points (x1, y1) and (x2, y2) is ((x1+x2)/2, (y1+y2)/2). Therefore, P3 is ((16+0)/2, (0+0)/2) which gives \( P3 = (8,0) \).
02

Compute Point P4 Coordinates

The roll for P4 was a 3, therefore we calculate the midpoint between the current point P3 (8,0) and vertex B(32,0). Using the same midpoint formula, we get \( P4 = ((8+32)/2, (0+0)/2) \) which gives \( P4 = (20,0) \).
03

Compute Point P5 Coordinates

The next roll was a 5, so the new point will be the midpoint between P4 (20,0) and vertex C(0,32). Applying the midpoint formula, we find \( P5 = ( (20+0)/2, (0+32)/2) \) which gives \( P5 = (10,16) \).
04

Compute Point P6 Coordinates

Finally, the die roll was again a 5, so we compute the midpoint between the current point P5 (10,16) and vertex C(0,32). Using the midpoint formula, we compute \( P6 = ( (10+0)/2, (16+32)/2) \) which gives \( P6 = (5,24) \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fractals
Fractals are complex geometric shapes that display self-similarity across different scales. This means that regardless of how much you zoom in or out, the pattern remains intricate and similar to the original shape. Fractals are formed by repeating a simple process over and over again in an ongoing feedback loop.
They are often used to model seemingly chaotic patterns in nature, such as snowflakes, mountain ranges, lightning, and coastlines. By following a set of rules or mathematical algorithms, complex patterns can emerge from the repetition of basic shapes.
In the context of the chaos game, fractals are created by plotting points iteratively according to specific rules. These plotted points often reveal complex patterns like the famous Sierpinski triangle or other unique fractal designs. The simplicity of the process, paired with the complexity of the results, makes fractals a fascinating object of study in mathematics and art.
Midpoint Calculation
Midpoint calculation is an essential mathematical concept frequently used in geometry and other disciplines to determine the point that lies exactly halfway between two specified points. It is Calculated using the formula:
\[ \text{Midpoint} \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two endpoints. By averaging the x-coordinates and the y-coordinates separately, the formula outputs the coordinates of the midpoint.
In exercises such as the chaos game, midpoint calculations help determine the successive positions of a point being iteratively placed according to a roll of dice or a set of random decisions. This process not only creates a sequence of points but also the striking fractal patterns seen within the chaos game. Understanding how to calculate midpoints enables students to manually develop these patterns and appreciate their intricate beauty.
Rectangular Coordinate System
The rectangular coordinate system, also known as the Cartesian coordinate system, is a two-dimensional plane defined by an x-axis (horizontal) and a y-axis (vertical). Each point in this plane is expressed as an ordered pair \((x, y)\) indicating its precise location relative to the two axes.
This system allows for easy plotting of points, shapes, and mathematical functions. It's especially useful in geometry and algebra for visualizing and solving problems that involve distance, location, and spatial relationships.
In the context of the chaos game, using a rectangular coordinate system allows for the clear plotting of points and vertices like in the isosceles right triangle \(ABC\). By assigning specific coordinates to each vertex, such as \(A(0,0)\), \(B(32,0)\), and \(C(0,32)\), students can accurately apply processes like midpoint calculation and track the formation of fractal patterns more effectively, enhancing both comprehension and interest in these mathematical concepts.

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Most popular questions from this chapter

Think of the construction of the Sierpinski carpet as a process where you start with a solid blue square, punch a square "hole" in it in Step 1 , and continue punching smaller square "holes" at each step of the construction. Using this interpretation, (a) find the number of square "holes" at Step 3 . (b) find the number of square "holes" at Step 5 . (c) give a general formula (in terms of \(N\) ) for the number of square "holes" at Step \(N\). [Hint: Here you will need to use the geometric sum formula (see p.275).]Suppose that the seed square (Step 0 ) of the Sierpinski carpet has sides of length 1 . (a) Find the length of the boundary of the figures at Steps \(1,2,\) and \(3 .\) (b) Find the length of the boundary of the figure at Step 5 . (c) Find the length of the boundary of the figure at Step 6 .

Refer to the Sierpinski ternary gasket, \(a\) variation of the Sierpinski gasket defined by the following recursive replacement rule. Assume that the seed triangle of the Sierpinski ternary gasket has perimeter \(P\). Let \(U\) denote the number of shaded triangles at a particular step, \(V\) the perimeter of each shaded triangle, and \(W\) the length of the boundary of the "gasket" obtained at a particular step of the construction. Complete the missing entries in Table \(12-20\). $$ \begin{array}{l|l|l|l} & U & V & W \\ \hline \text { Start } & 1 & \mathrm{P} & \mathrm{P} \\ \hline \text { Step 1 } & 6 & \frac{P}{3} & 2 P \\ \hline \text { Step 2 } & & & \\ \hline \text { Step 3 } & & & \\ \hline \text { Step 4 } & & & \\ \hline \text { Step N } & & & \end{array} $$

Consider the Mandelbrot sequence with seed \(s=-0.25\). (a) Using a calculator find \(s_{1}\) through \(s_{10}\), rounded to six decimal places. (b) Suppose you are given \(s_{N}=-0.207107\). Using a calculator find \(s_{N+1}\), rounded to six decimal places. (c) Is this Mandelbrot sequence escaping, periodic, or attracted? Explain.

Refer to a variation of the Koch snowflake called the Koch antisnowflake. The Koch antisnowflake is much like the Koch snowflake, but it is based on a recursive rule that removes equilateral triangles. The recursive replacement rule for the Koch antisnowflake is as follows: Assume that the seed triangle of the Koch antisnowflake has area \(A=81\). Let \(R\) denote the number of triangles subtracted at a particular step, \(S\) the area of each subtracted triangle, \(T\) the total area subtracted, and \(Q\) the area of the shape obtained at a particular step of the construction. Complete the missing entries in Table \(12-11\). $$ \begin{array}{l|c|c|c|c} & R & S & T & Q \\ \hline \text { Start } & 0 & 0 & 0 & 81 \\ \hline \text { Step 1 } & 3 & 9 & 27 & 54 \\ \hline \text { Step 2 } & 12 & 1 & 12 & 42 \\ \hline \text { Step 3 } & & & & \\ \hline \text { Step 4 } & & & & \\ \hline \text { Step 5 } & & & & \end{array} $$

Assume that the seed triangle of the Sierpinski gasket has perimeter \(P=20 .\) Let \(U\) denote the number of solid triangles at a particular step, \(V\) the perimeter of each solid triangle, and \(W\) the length of the boundary of the "gasket" obtained at a particular step of the construction. Complete the missing entries in Table \(12-16\). $$ \begin{array}{l|l|l|l} & U & V & W \\ \hline \text { Start } & 1 & 20 & 20 \\ \hline \text { Step 1 } & 3 & 10 & 30 \\ \hline \text { Step 2 } & & & \\ \hline \text { Step 3 } & & & \\ \hline \text { Step 4 } & & & \\ \hline \text { Step 5 } & & & \end{array} $$
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