Explanation A tree is a network that has no circuits. In a tree there is only one path connecting any two vertices. Given: Consider the construction of a Koch snowflake starting with a seed triangle having area A = 81 . Let R denote the number of triangles added at a particular step, S the area of each added triangle, T the total new area added, and Q the area of the “snowflake” obtained at a particular step of the construction. The given table is, R S T Q Start 0 0 0 81 Step 1 3 9 27 108 Step 2 12 1 12 120 Step 3 Step 4 Step 5 Calculation: Firstly a shaded equilateral triangle is obtained that is also called seed of the Koch snowflake, with area as 81 cm. Figure (1) Construct an equilateral triangle with side equal to 1 3 of the original side to middle third of each of the sides of the triangle. The result obtained is a 12 − sided snowflake. Figure (2) The area of an equilateral triangle of side s is calculated by A = 3 4 s 2 . The area of each of the triangle added in step 1 is 1 9 of the original triangle. Area of each triangle is, Area = 1 9 × 81 = 9 Area of three added triangles is, Area = 3 × 9 = 27 Calculate the total area, Total area = 27 + 81 = 108 Add an equilateral triangle with side equal to one-third of that side to the middle third of each of the 12 sides of the snowflake. The result obtained is a 48 − sided snowflake. Area of each of the 12 added triangles is, Area = 1 9 × 9 = 1 Total area of the triangles is, Total area = 12 × 1 = 12 Total area of the snowflake is, Total area = 12 + 108 = 120 Figure (3) Continue in the same way. The result will be a snowflake having 192 sides. Area of each of the 12 added triangles is, Area = 1 9 × 9 = 1 And total area of the triangles in step 2 is,

Excursions in Modern Mathematics (9th Edition)

9th Edition

ISBN: 9780134468372

Author: Peter Tannenbaum

Publisher: PEARSON

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Chapter 12, Problem 3E

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Excursions in Modern Mathematics (9th Edition)

Ch. 12 - Consider the construction of a Koch snowflake...Ch. 12 - Consider the construction of a Koch snowflake...Ch. 12 - Prob. 3E Ch. 12 - Prob. 4E Ch. 12 - Prob. 5E Ch. 12 - Prob. 6E Ch. 12 - Prob. 7E Ch. 12 - Prob. 8E Ch. 12 - Prob. 9E Ch. 12 - Exercises 9 through 12 refer to a variation of the...

Ch. 12 - Exercises 9 through 12 refer to a variation of the...Ch. 12 - Exercises 9 through 12 refer to a variation of the...Ch. 12 - Prob. 13E Ch. 12 - Prob. 14E Ch. 12 - Exercises 13 through 16 refer to the construction...Ch. 12 - Prob. 16E Ch. 12 - Prob. 17E Ch. 12 - Prob. 18E Ch. 12 - Prob. 19E Ch. 12 - Prob. 20E Ch. 12 - Prob. 21E Ch. 12 - Assume that the seed triangle of the Sierpinski...Ch. 12 - Prob. 23E Ch. 12 - Prob. 24E Ch. 12 - Prob. 25E Ch. 12 - Prob. 26E Ch. 12 - Prob. 27E Ch. 12 - Prob. 28E Ch. 12 - Prob. 29E Ch. 12 - Prob. 30E Ch. 12 - Prob. 31E Ch. 12 - Exercises 31 through 34 refer to a variation of...Ch. 12 - Prob. 33E Ch. 12 - Prob. 34E Ch. 12 - Prob. 35E Ch. 12 - Prob. 36E Ch. 12 - Prob. 37E Ch. 12 - Prob. 38E Ch. 12 - Exercises 35 through 40 are a review of complex...Ch. 12 - Prob. 40E Ch. 12 - Prob. 41E Ch. 12 - Prob. 42E Ch. 12 - Prob. 43E Ch. 12 - Prob. 44E Ch. 12 - Prob. 45E Ch. 12 - Prob. 46E Ch. 12 - Prob. 47E Ch. 12 - Prob. 48E Ch. 12 - Prob. 49E Ch. 12 - Exercises 49 and 50 refer to the Menger sponge, a...Ch. 12 - Prob. 51E Ch. 12 - Prob. 52E Ch. 12 - Consider the Mandelbrot sequence with seed s=1.25....Ch. 12 - Consider the Mandelbrot sequence with seed s=2. Is...Ch. 12 - Prob. 55E Ch. 12 - Prob. 56E Ch. 12 - Prob. 57E Ch. 12 - Prob. 58E Ch. 12 - Prob. 59E Ch. 12 - Prob. 60E

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The missing entries in the given table.

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