 12.1: Consider the construction of a Koch snowflake starting with a seed ...
 12.2: Consider the construction of a Koch snowflake starting with a seed ...
 12.3: Consider the construction of a Koch snowflake starting with a seed ...
 12.4: Consider the construction of a Koch snowflake starting with a seed ...
 12.5: Assume that the seed square of the quadratic Koch fractal has sides...
 12.6: Assume that the seed square of the quadratic Koch fractal has sides...
 12.7: Assume that the seed square of the quadratic Koch fractal has area ...
 12.8: Assume that the seed square of the quadratic Koch fractal has area ...
 12.9: Exercises 9 through 12 refer to a variation of the Koch snowflake c...
 12.10: Exercises 9 through 12 refer to a variation of the Koch snowflake c...
 12.11: Exercises 9 through 12 refer to a variation of the Koch snowflake c...
 12.12: Exercises 9 through 12 refer to a variation of the Koch snowflake c...
 12.13: Exercises 13 through 16 refer to the construction of the quadratic ...
 12.14: Exercises 13 through 16 refer to the construction of the quadratic ...
 12.15: Exercises 13 through 16 refer to the construction of the quadratic ...
 12.16: Exercises 13 through 16 refer to the construction of the quadratic ...
 12.17: Consider the construction of a Sierpinski gasket starting with a se...
 12.18: Consider the construction of a Sierpinski gasket starting with a se...
 12.19: Assume that the seed triangle of the Sierpinski gasket has perimete...
 12.20: Assume that the seed triangle of the Sierpinski gasket has perimete...
 12.21: Let A denote the area of the seed triangle of the Sierpinski gasket...
 12.22: Let P denote the perimeter of the seed triangle of the Sierpinski g...
 12.23: Exercises 23 through 26 refer to a square version of the Sierpinski...
 12.24: Exercises 23 through 26 refer to a square version of the Sierpinski...
 12.25: Exercises 23 through 26 refer to a square version of the Sierpinski...
 12.26: Exercises 23 through 26 refer to a square version of the Sierpinski...
 12.27: Exercises 27 through 30 refer to the Sierpinski ternary gasket, a v...
 12.28: Exercises 27 through 30 refer to the Sierpinski ternary gasket, a v...
 12.29: Exercises 27 through 30 refer to the Sierpinski ternary gasket, a v...
 12.30: Exercises 27 through 30 refer to the Sierpinski ternary gasket, a v...
 12.31: Assume that that the seed square for the box fractal has area A = 1...
 12.32: Assume that the seed square for the box fractal has sides of length...
 12.33: Exercises 33 through 36 refer to the chaos game as described in Sec...
 12.34: Exercises 33 through 36 refer to the chaos game as described in Sec...
 12.35: Exercises 33 through 36 refer to the chaos game as described in Sec...
 12.36: Exercises 33 through 36 refer to the chaos game as described in Sec...
 12.37: Exercises 37 through 40 refer to a variation of the chaos game. In ...
 12.38: Exercises 37 through 40 refer to a variation of the chaos game. In ...
 12.39: Exercises 37 through 40 refer to a variation of the chaos game. In ...
 12.40: Exercises 37 through 40 refer to a variation of the chaos game. In ...
 12.41: Exercises 41 through 46 are a review of complex number arithmetic. ...
 12.42: Exercises 41 through 46 are a review of complex number arithmetic. ...
 12.43: Exercises 41 through 46 are a review of complex number arithmetic. ...
 12.44: Exercises 41 through 46 are a review of complex number arithmetic. ...
 12.45: Exercises 41 through 46 are a review of complex number arithmetic. ...
 12.46: Exercises 41 through 46 are a review of complex number arithmetic. ...
 12.47: Consider the Mandelbrot sequence with seed s = 2. (a) Find s1, s2,...
 12.48: Consider the Mandelbrot sequence with seed s = 2. (a) Find s1, s2, ...
 12.49: Consider the Mandelbrot sequence with seed s = 0.5. (a) Using a ca...
 12.50: Consider the Mandelbrot sequence with seed s = 0.25. (a) Using a c...
 12.51: Consider the Mandelbrot sequence with seed s = i. (a) Find s1 thro...
 12.52: Consider the Mandelbrot sequence with seed s = 1 + i. Find s1, s2, ...
 12.53: Let H denote the total number of holes (i.e., white triangles) in t...
 12.54: Exercises 54 and 55 refer to the Menger sponge, a threedimensional ...
 12.55: Exercises 54 and 55 refer to the Menger sponge, a threedimensional ...
 12.56: Exercises 56 and 57 refer to reflection and rotation symmetries and...
 12.57: Exercises 56 and 57 refer to reflection and rotation symmetries and...
 12.58: Consider the Mandelbrot sequence with seed s = 0.75. Show that thi...
 12.59: Consider the Mandelbrot sequence with seed s = 0.25. Is this Mandel...
 12.60: Consider the Mandelbrot sequence with seed s = 1.25. Is this Mande...
 12.61: Consider the Mandelbrot sequence with seed s = 22. Is this Mandelbr...
 12.62: Suppose that we play the chaos game using triangle ABC and that M1,...
 12.63: Consider the following variation of the chaos game. The game is pla...
 12.64: (a) Show that the complex number s = 0.25 + 0.25i is in the Mandel...
 12.65: Show that the Mandelbrot set has a reflection symmetry. (Hint: Comp...
 12.66: Exercises 66 through 68 refer to the concept of fractal dimension. ...
 12.67: Exercises 66 through 68 refer to the concept of fractal dimension. ...
 12.68: Exercises 66 through 68 refer to the concept of fractal dimension. ...
Solutions for Chapter 12: Fractal Geometry
Full solutions for Excursions in Modern Mathematics  8th Edition
ISBN: 9781292022048
Solutions for Chapter 12: Fractal Geometry
Get Full SolutionsExcursions in Modern Mathematics was written by and is associated to the ISBN: 9781292022048. This expansive textbook survival guide covers the following chapters and their solutions. Since 68 problems in chapter 12: Fractal Geometry have been answered, more than 11873 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Excursions in Modern Mathematics, edition: 8. Chapter 12: Fractal Geometry includes 68 full stepbystep solutions.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.