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Important Formulae for Complex Numbers

1) z = x + iy where x = Real part of z and y = Imaginary part of z

2) c = a + ib where a = Real part of c and b = Imaginary part of c

3) z = re^iq = (sqrt(x^2 + y^2)) (cos q + i sin q)
where q = arctan (y / x), r = sqrt(x^2 + y^2) and "sqrt" means square root

4) z^n = r^n*e^inq = (sqrt(x^2 + y^2))^n (cos nq + i sin nq) ; r and q as above

5) sqrt(z) = (sqrt(r)sqrt(e^iq)) = (sqrt(sqrt(x^2 + y^2))) [cos (.5 arctan (y / x))
+ i sin (arctan (y / x))]

6) ln z = ln[sqrt(x^2 + y^2)] + i arctan (y / x)

7) e^z = e^x(cos y + i sin y)

8) sin z = sin x cosh y + i cos x sinh y = -i sinh iz = (e^iz - e^-iz) / 2i

9) cos z = cos x cosh y - i sin x sinh y = cosh iz = (e^iz + e^-iz) / 2

10) sinh z = - i sinh iz = (e^z - e^-z) / 2

11) cosh z = cos iz = (e^z + e^-z) / 2

12) tanh z = - i tan (iz) = (e^z - e^-z) / (e^z + e^-z)

13) sech z = sech (iz) = [cosh z] ^ -1

14) csch z = i csc (iz) = [sinh z] ^ -1

15) arcsinh z = ln(z + sqrt(z^2 + 1))

16) arccosh z = ln(z + sqrt(z^2 - 1)) , ln(z - sqrt(z^2 - 1))

17) arctanh z = .5 * ln[(1 + z) / (1 - z)]

18) arcsech z = ln[(1 + sqrt(z^2 + 1)) / z]

19) arccsch z = ln[(1 + sqrt(1 - z^2 )) / z] , ln[(1 - sqrt(1 - z^2 )) / z]

20) arccoth z = .5 * ln[(z + 1) / (z - 1)]

21) sin^2(z) + cos^2(z) = 1

22) cosh^2(z) - sinh^2(z) = 1

23) tan z = (sin 2x + i sinh 2y) / (cos 2x + cosh 2y)

24) cot z = (sin 2x - i sinh 2y) / (cosh 2y - cos 2x)

25) nth root of z = [nth root of (x^2 + y^2)](cos (q / n) + i sin (q / n))

26) Newton's Method z(n+1) = z(n) - [f(z(n)) / f '(z(n))]

27) Henon Attractor: (for z(n) = x(n) + iy(n)) , x(n+1) = ax(n) + y(n) and y(n+1)= bx(n)

28) Halley Map: z(n+1) = z(n) - L[(2f(z(n))f '(z(n))) / (2(f '(z(n)))^2 - f' '(z(n))f(z(n)))]

29) Lorenz Attractor: dx / dt = a(y - x) dy / dt = x(r - z) - y dz / dt = xy - bz

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Important Mathematical Constants

1) Pi --- The ratio of the circumference of a circle to its diameter, supposedly first discovered by Archimedes (287-212 BC). He surmised that pi was

3 10/17 < pi < 3 1/7

The first hundred digits of pi are given here though I understand that 1.24 trillion digits (!) have been calculated already (there is more about pi on my Math Fun page:

pi = 3. 1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 ...

Probably the most famous formula for determining pi is Leibnitz' formula:

pi = 4 - (4/3) + (4/5) - (4/7) + (4/9) - (4/11) =

Summation (from n=0 to infinity) of [(-1)^n][4/(2n+1)]

Another famous summation involving pi was discovered by Euler as:

(pi^2)/6 = 1/1 + 1/4 + 1/9 + 1/16 + 1/25 + 1/36 +.... + (1/n)^2

2) e --- The natural logarithm base, supposed named after the great mathmatician Leonhard Euler. The first hundred digits of e are given here as well:

e = 2.7182818284 5904523536 0287471352 6624977572 4709369995 9574966967 6277240766 3035354759 4571382178 5251664274 ...

For the first 500 digits of e, go HERE.

I offer my students two ways to remember how to calculate the value of e:

e = limit (as n -> infinity) of (1 + 1/n)^n

e = Summation (from n=0 to infinity) of simply (1/n!)

3) Feigenbaum's Number --- This number, first shown by Becker and Dorfler, was demonstrated by Mitchell Feigenbaum to be a fundamental constant of nature having to do with the ratio of intervals of growth rate versus the doubling of up and down cycles characteristic of that rate. Keith Briggs, a scientist from the University of Melbourne, Australia, has calculated the most precise Feigenbaum number to date:

F = 4. 6692016091 0299067185 3203820466 2016172581 8557747576 8632745651
3430041343 3021131473 7138689744 0239480138 17165984855 1898151344
0862714202 7932522312 4429888908 9085994493 5463236713 4115324817 1421994745 5644365823 7932020095 6105833057 5458617652 2220703854 1064674949 4284981453 3917262005 6875566595 2339875603 825637225

4) Square Root of two = 1. 41421 35623 73095 0488...

5) Square Root of three = 1. 73205 08075 68877 2935...

6) Square Root of five = 2. 23606 79774 99789 6964...

7) Square Root of pi = 1.77245 38509 05516 02729 8167... (also known as Gamma(.5))

8) Square Root of e = 1. 64872 12707 00128 1468...

9) The Golden Mean, phi = (1 + sqrt(5)) / 2 = 1.61803 39887 99894...

10) e ^ pi = 23. 14069 26327 79267 006...

11) pi ^ e = 22. 45915 77183 61045 47342 715...

12) e ^ e = 15. 15426

13) Euler's constant (usually given as lower case gamma) = .57721 56649 01532 86060 6512...

= limit (as n -> infinity) of (summation (from k=1 to n) of (1/k) - ln n) --- (thanks to N. Hobson)

14) 1 radian = the number of degrees that are subtended when the length of a radius is traced along the circumference of a circle.

1 radian = 180 / pi = 57. 29577 95130 8232...

15) ln 2 = .69315... = 1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + ...

= Summation (from n = 1 to infinity) of (-1)^(n+1) * (1/n)

16) ln 10 = 2.30259...

There is a library of more obscure mathematical constants HERE.

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Selected Fractal Book References List
(Listed alphabetically as TITLE, AUTHOR, PUBLISHER, DATE)

Please submit any important missing texts to me at fractali@lifesmith.com.

Advanced Fractal Programming in C and C++, Stevens, Henry Holt, New York, 1992

An Eye for Fractals: A Graphic & Photographic Essay, McGuire, Addison-Wesley, Redwood City, CA, 1991

An Introduction to Chaotic Dynamical Systems, Devaney, Addison-Wesley, New York, 1986

Bifurcation Theory and its Applications, Kaplan, Yorke,and Williams, New York Academy of Sciences, New York, 1979

Chaos: Making A New Science, Gleick, Viking Press, New York, 1987

Computers and the Imagination, Pickover, St. Martin's Press, New York, 1989

Computers, Pattern, Chaos, and Beauty: Graphics from an Unseen World, Pickover, St. Martin's Press, New York, 1990

Determistic Chaos, Schuster, Physik-Verlag, Weinheim, 1984

Does God Play Dice: The Mathematics of Chaos, Stewart, Basil Blackwell, Cambridge, 1990

Dynamical Systems and Evolution Equations, Walker, Plenum Press, New York, 1980

Dynamical Systems and Fractals, Becker and Dörfler, Cambridge Univ. Press, Cambridge, 1986

Exploring the Geometry of Nature: Computer Modeling of Chaos, Fractals, Cellular Automata, and Neural Networks, Rietman, Windcrest Books, 1989

Fractal Cosmos: The Art of Mathematical Design, Lifesmith, Amber Lotus, Oakland, 1994

Fractal Programming in C, Stevens, Henry Holt, New York, 1990

Fractal Programming in Turbo Pascal, Stevens, Henry Holt, New York, 1991

Fractals, Feder, Plenum Press, New York, 1988

Fractals, Lauwerier, Princeton univ. Press, Princeton, NJ 1992

Fractals and Multifractals, Mandelbrot, Springer-Verlag, New York, 1991

Fractals, Chaos, Power Laws, Schroeder, Freeman, New York, 1991

Fractals Everywhere, Barnsley, Academic Press, Boston, 1988

Fractals: Form, Chance, and Dimension, Mandelbrot, Freeman, San Francisco, 1977

Fractals, The Patterns of Chaos, Briggs, Touchstone/Simon & Schuster, New York, 1992

FractalVision: Put Fractals to Work for You, Oliver, Prentice-Hall, Carmel, IN, 1992

Fun with Fractals, Robbins, Sybex, Alameda, CA 1992

Handbook of Mathematical Functions, Abramowitz and Stegun, Dover, New York, 1968

Islands of Truth: A Mathematical Mystery Cruise, Peterson, Freeman, New York, 1990

Iterated Maps on the Interval as Dynamical Systems, Collet and Ekmann, Birkhauser, Boston, 1980

Mathematics and the Unexpected, Ekeland, Univ. of Chicago Press, 1988

Order Out of Chaos: Man's New Dialogue with Nature, Prigogine & Stengers, Bantam, New York, 1984

Symmetry in Chaos, Field & Golubitsky, Oxford Univ. Press, New York, 1992

The Beauty of Fractals, Peitgen and Richter, Springer-Verlag, Berlin, 1986

The Fractal Explorer, Garcia, Dynamic Press, Santa Cruz, CA, 1991

The Fractal Geometry of Nature, Mandelbrot, Freeman, New York, 1980

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Selected Papers

Please submit any important missing papers to me at fractali@lifesmith.com.

Dewdney, A. K., "Computer Recreations," Scientific American, Sept. 1986, pp. 140-145

Fatou, P. 1906. Sur les solutions uniformes de certains équations fonctionelles. Comptes rendus (Paris) 143, 546-548

Fatou, P. 1919-20. Sur les équations fonctionelles. Bull. Société Mathématique de France 47, 161-271; 48, 208-314

Feigenbaum, M. J., "Quantitative Universality for a Class of Nonlinear Transformations," Journal of Statistical Physics 19, 25-52 (1978)

Feigenbaum, M. J., "The Universal Metric Properties of Nonlinear Transformations," Journal of Statistical Physics 21, 669-706 (1979)

Hausdorff, F. 1919. "Dimension und äusseres Mass". Mathematische Annalen, 79, 157-79

Julia, G. 1918. Mémoire sur l'itération des fonctions rationnelles. J. de Mathématiques Pures et Appliqués 4, 47-245. Reprinted (with related texts) in Oeuvres de Gaston Julia, Paris, Gauthier-Villars. 1968, p. 121-319

Lorenz, E. 1963. "Deterministic Nonperiodic Flows", Journal of the Atmospheric Sciences, 20, 130-41

Mandelbrot, B. B. 1980. Fractal aspects of the iteration of z=lz(1-z) for complex l and z. Non Linear Dynamics, Ed. R.H.G. Helleman. Annals of New York Academy of Sciences, 357, 249-259

Verhulst, P. F., 1845. Récherches mathématiques sur la loi d'accroissement de la population. Nouv. Mém. de l'Acad. Roy. des Sciences et Belles-Lettres de Bruxelles XVIII. 8, 1-38

Voss, R. F. 1985. "Random Fractal Forgeries," Fundamental Algorithms for Computer Graph- ics, Springer-Verlag, Berlin

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Complex Equations Researched

Here are the equations that I have used during the past seventeen years to generate well over 400,000 Mandelbrot and Julia sets. I have over seven TERAbytes of fractal data! Feel free to continue to delve into them using whatever software (your own or canned) you have available. Because I wrote my own code in C language and a complex math library was not available when I first started, I had to resolve each of these equations into real, f(x), and imaginary, f(y), parts. Many, many long (but fun) hours doing just the basic algebra were spent in order to bring you the majestic beauty of these incredible forms.

1--F(Z) = Z^2 + C

2--F(Z) = Z^3 + C

3--F(Z) = (Z^2 + C) / (Z - C)

4--F(Z) = Z^2 - Z + C

5--F(Z) = Z^3 - Z^2 + Z + C

6--F(Z) = (1 + C)Z - CZ^2

7--F(Z) = Z^3 / (1 + CZ^2)

8--F(Z) = (Z - 1)(Z + .5)(Z^2 - 1) + C

9--F(Z) = (Z^2 + 1 + C) / (Z^2 - 1 - C)

10--F(Z) = Z^1.5 + C

11--F(Z) = exp(Z)-C

12--F(Z) = Z - 1 + Cexp(-Z)

13--F(Z) = CZ - 1 + Cexp(-Z)

14--F(Z) = (4Z^5 + C)/5Z^4

15--F(Z) = (6Z^7 + C)/7Z^6

16--F(Z) = Z^2 * exp(-Z) + C

17--F(Z) = Z^2 * Z^(-2) + C

18--F(Z) = Z * exp(-Z) + C

19--F(Z) = C * exp(-Z) + Z^2

20--F(Z) = Z^3 + Z + C

21--F(Z) = Z^4 + Z + C

22--F(Z) = Z^4 + CZ^2 + C

23--F(Z) = Z^2sin(Re Z) + CZcos(Im Z) + C

24--F(Z) = 2^Z * CZ^2

25--F(Z) = Z^5 - Z^3 + Z + C

26--F(Z) = (Z^2 + C)^2 + Z + C

27--F(Z) = (Z + sin(Z))^2 + C

28--F(Z) = Cexp(Z)

29--F(Z) = Z^2 + C^3

30--F(Z) = Cexp(CZ)

31--F(Z) = Z^2cos(ReZ)+CZsin(ImZ)+C

32--F(Z) = CZ^2 + ZC^2

33--F(Z) = exp(cos(CZ))

34--F(Z) =(1 + Jo(Re Z))^2 + (Jo(Im Z) + C)^2 (Here Jo represents the Bessel function)

35--F(Z) = C(sin Z + cos Z)

36--F(Z) = Z^(-.5) + C

37--F(Z) = CZ(1 - Z)

38--F(Z) = C^2Z(1 - Z)

39--F(Z) = ((Z^2+C)^2)/(Z-C)

40--F(Z) = (Z + sin Z)^2 + Z^-.5 + C

41--F(Z) = C*(sin Z + cos Z)*(Z^3+Z+C)

42--F(Z) = Cexp(Z) * exp(cosCZ)

43--F(Z) = (Z^3+Z+C)*C*(sinZ + cosZ)

44--F(Z) = ((1+C)Z-CZ^2)*((Z+sinZ)^2+C)

45--F(Z) = Z^2 + Z^1.5 + C

46--F(Z) = Z^2 + ZexpZ + C

47--F(Z) = (Z+sinZ)^2+Cexp(-Z)+Z^2+C

48--F(Z) = ((Z^3)/(1+CZ^2))+expZ-C

49--F(Z) = (Z^2*sin(ReZ) + CZ(ImZ) + (Z^2*cos(ReZ)+CZsin(ImZ)+C

50--F(Z) = (Z+sinZ)^2+Cexp(Z)+C

51-- F(Z) = Z^2 + 1/Z + C

52-- F(Z) = (Z^3 + C) / Z

53-- F(Z) = (Z^3 + C) / Z^2

54-- F(Z) = ((Z+1)^2 + C) / Z

55-- F(Z) = (Z + C)^2 + (Z + C)*

56-- F(Z) = (Z + C)^3 - (Z + C)^2

57-- F(Z) = (Z^3 - Z^2)^2 + C

58-- F(Z) = (Z^2 - Z)^2 + C

59-- F(Z) = (Z + ln Z)^2 + C

60-- F(Z) = (Z - sqrt(Z))^2 + C

61-- F(Z) = (Z + sqrt(Z))^2 + C

62-- F(Z) = Z^2exp(Z) - Zexp(Z) + C

63-- F(Z) = (exp(CZ) + C)^2

64-- F(Z) = Z * exp(Re Z/Im Z) + C

65-- F(Z) = exp(X^2*Y^2) + Im Z + C

66-- F(Z) = exp(Re Z)*(X-a) + exp(Im Z)*(Y-b)i

67-- F(Z) = X^2*exp(Y+b) + iaexp(Y+b)

68-- F(Z) = (a-X^2+Y^2)exp(b+X^2-Y^2) + i(b+X^2-Y^2)exp(a-X^2+Y^2)

69-- F(Z) = [(2X-Y^2+a)/(2X^2+Y-b)] + i[(2X^2+Y-a)/(2X-Y^2+b)]

70-- F(Z) = [(X^2+Y^2+a)/cos(X^2+Y^2)] + i[(X^2+Y^2+b)/sin(X^2+Y^2)]

71-- F(Z) = Z^5 + C

72-- F(Z) = Z^6 + C

73-- F(Z) = Z^7 + C

74-- F(Z) = (3Z^4 + C) / 4Z^3

75-- F(Z) = (2Z^3 + C) / 3Z^2

76-- F(Z) = Z^5 + CZ^3 + C

77-- F(Z) = Z^6 + CZ^4 + CZ^2 + C

78-- F(Z) = Z^8 + C

79-- F(Z) = Z^9 + C

80-- F(Z) = Z^8 + CZ^4 + CZ^2 + C

81-- F(Z) = Z^9 - CZ^6 + CZ^3 + C

82-- F(Z) = (Z^4 + C) / (Z - C)

83-- F(Z) = (Z^3 + Z + C) / (Z^2 - Z - C)

84-- F(Z) = (Z^3 + Z + C) / (Z - C)

85-- F(Z) = (Z^3 + Z + C) / Z

86-- X = X^2+XY+A ; Y = Y^2-XY+B

87-- X = X^3-(X^2)Y+XY^2-XY+A; Y = Y^3-XY^2+(X^2)Y+XY+B

88-- X = (X^2)sin Y + A ; Y = (Y^2)cos X + B

89-- X = X^4-3X^3+3X^2(Y^2)+A ; Y = Y^4+3XY^3-3X^2(Y^2)

90-- X = X^2(1+exp(-Y))+A ; Y = Y^2(1+exp(-X)+B

91-- F(Z) = C(Z^2 + 1)^2 / Z(Z^2 -1)

92-- F(Z) = CZ^2

93-- F(Z) = CZ^3

94-- F(Z) = CZ^4

95-- F(Z) = C*cos Z

96-- F(Z) = C*sin Z

97-- F(Z) = CZ*ln Z

98-- F(Z) = C*tan Z

99-- F(Z) = C*exp(CZ) / (exp(C) - 1)

100-- F(Z) = C*exp(Z)*sqrt(Z) /n

101 -- F(Z) = (Z^2(1+Z^2))/(Z+C)

102 -- F(Z) = Z(1+Z^2)/(Z+C)

103 -- F(Z) = (Z^5+C)/(Z^3+Z^2+Z+1)

104 -- F(Z) = (Z^3+C)/3Z^2

105 -- F(Z) = (Z^3+Z^2+Z+C)/(Z-C)

106 -- F(Z) = exp(Z^2+C)

107 -- F(Z) = Z^2*exp(Z^2)+C

108 -- F(Z) = exp(Z^2)/(Z+C)

109 -- F(Z) = (Z+exp(Z))^2+C

110 -- F(Z) = (Z^2+C)^2-exp(Z)+C

111 -- F(Z) = (1+iC)sin(Z)

112 -- F(Z) = (1+iC)cos(Z)

113 -- F(Z) = Z*tan(ln Z)+C

114 -- F(Z) = sqrt(Z^4+1)+C

115 -- F(Z) = sqrt(Z^4+C)

116 -- F(Z) = C^Z

117 -- F(Z) = C*arctan(Z)

118 -- F(Z) = (ZlnZ)/exp(C)

119 -- F(Z) = exp(Z)/lnZ+C

120 -- F(Z) = sqrt(Z^3+C)

121 -- F(Z) = sqrt(Z^3+1)+C

122 -- F(Z) = cubrt(Z^6+1)+C

123 -- F(Z) = (Z+exp(Z)+ln Z)^2+C

124 -- F(Z) = (Z^2+C+1)^2 / (2Z+C+2)^2

125 -- F(Z) = Z ^ 10 + C

126 -- F(Z) = Z ^ 11 + C

127 -- F(Z) = Z ^ 12 + C

128 -- F(Z) = Z^12 - Z^11 - Z^10 + C

129 -- F(Z) = Z ^ 13 + C

130 -- F(Z) = Z ^ 14 + C

131 -- F(Z) = Z ^ 15 + C

132 -- F(Z) = Z ^ 16 + C

133 -- F(Z) = Z ^ 17 + C

134 -- F(Z) = Z ^ 18 + C

135 -- F(Z) = Z ^ 19 + C

136 -- F(Z) = Z ^ 20 + C

137 -- F(Z) = Z ^ 21 + C

138 -- F(Z) = Z ^ 22 + C

139 -- F(Z) = Z ^ 23 + C

140 -- F(Z) = Z ^ 24 + C

141 -- F(Z) = Z ^ 25 + C

142 -- F(Z) = Z ^ 26 + C

143 -- F(Z) = Z ^ 27 + C

144 -- F(Z) = Z ^ 28 + C

145 -- F(Z) = Z ^ 29 + C

146 -- F(Z) = Z^30 + C

147 -- X=X^2+Y+A+X^2/Y ;Y=Y^2+X+B+Y^2/X

148 -- X=X^3+Y^2-X+A ;Y=Y^3-X^2+Y+B

149 -- X=X^2+2XY-Y+A ;Y=Y^2-2XY+X+B

150 -- X=X^3+AX^2+BY ;Y=Y^3+BY^2+AX

151 -- X=2X^2-3ABY+A ;Y=3Y^2+2ABX-B

152 -- X=X^4lnX+Y^2sinY+A; Y=Y^4lnY+X^2cosX+B

153 -- X=sqr(ln(X^2))+YsinX+A; Y=sqr(ln(Y^2))-XcosY+B

154 -- X=.5(X^2-Y^2)+.5(X+Y)+A; Y=.5(Y^2-X^2)-.5(X+Y)+B

155 -- X=sqr(X^3)+sqr(Y^3)+A; Y=sqr(Y^3)-sqr(X^3)+B

156 -- X=Y/sqrX+X/sqrY+A; Y=XsqrY+YsqrX+B

157 -- F(Z) = Z^30 - 30Z^29 - 870Z^28 + C

158 -- F(Z) = Z^27 - 27Z^26 - 702Z^25 + C

159 -- F(Z) = Z^24 - 24Z^23 - 552Z^22 + C

160 -- F(Z) = Z^21 - 21Z^20 - 420Z^19 + C

161 -- F(Z) = Z^18 - 18Z^17 - 306Z^16 + C

162 -- F(Z) = Z^15 - 15Z^14 - 210Z^13 + C

163 -- F(Z) = Z^30 - 30Z^29 - 870Z^28 + Z^27 - 27Z^26 - 702Z^25 + C

164 -- F(Z) = Z^27 - 27Z^26 - 702Z^25 + Z^24 - 24Z^23 - 552Z^22 + C

165 -- F(Z) = Z^24 - 24Z^23 - 552Z^22 + Z^21 - 21Z^20 - 420Z^19 + C

166 -- F(Z) = Z^21 - 21Z^20 - 420Z^19 + Z^18 - 18Z^17 - 306Z^16 + C

167 -- F(Z) = Z^18 - 18Z^17 - 306Z^16 + Z^15 - 15Z^14 - 210Z^13 + C

168 -- F(Z) = Z^30 - 30Z^29 - 870Z^28 + Z^27 - 27Z^26 - 702Z^25 + Z^24 - 24Z^23 -
552Z^22 + C

169 -- F(Z) = Z^27 - 27Z^26 - 702Z^25 + Z^24 - 24Z^23 - 552Z^22 + Z^21 - 21Z^20 -
420Z^19 + C

170 -- F(Z) = Z^24 - 24Z^23 - 552Z^22 + Z^21 - 21Z^20 - 420Z^19 + Z^18 - 18Z^17 -
306Z^16 + C

171 -- F(Z) = Z^21 - 21Z^20 - 420Z^19 + Z^18 - 18Z^17 - 306Z^16 + Z^15 - 15Z^14 -
210Z^13 + C

172 -- F(Z) = Z^30 - Z^29 + Z^28 - Z^27 + Z^26 - Z^25 + C

173 -- F(Z) = Z^24 - Z^23 + Z^22 - Z^21 + Z^20 - Z^19 + C

174 -- F(Z) = Z^18 - Z^17 + Z^16 - Z^15 + Z^14 - Z^13 + C

175 -- F(Z) = Z^15sinX - Z^14cosY - Z^13tanX + C

176 -- F(Z) = Z^12cosX - Z^11sinY - Z^10tanY + C

177 -- F(Z) = Z^15sinA - Z^14cosB - Z^13tanX - Z^12tanY + C

178 -- F(Z) = Z^12cosA - Z^11sinB - Z^10tanY - Z^9tanX + C

179 -- F(Z) = Z^30sinX - 30Z^29cosY + C

180 -- F(Z) = Z^28cosX - 28Z^27sinY + C

181 -- F(Z) = (Z^3+3Z(C-1)+(C-1)(C-2))^2

182 -- F(Z) = (3Z^2+3Z(C-2)+C^2-3C+3)^2

183 -- F(Z) = (Z^3+3Z(C-1)+(C-1)(C-2))^2 / (3Z^2+3Z(C-2)+C^2-3C+3)^2

184 -- F(Z) = Z ^ pi + C

185 -- F(Z) = pi ^ Z + C

186 -- F(Z) = Z ^ 4 + C

187 -- F(Z) = Z ^ pi + pi ^ C

188 -- F(Z) = C * Z ^ pi

189 -- F(Z) = Z ^ pi - Z ^ 3 + C

190 -- F(Z) = Z ^ pi - Z ^ 2 + C

191 -- F(Z) = Z ^ 2.5 + C

192 -- F(Z) = (5Z^6 + C)/6Z^5

193 -- F(Z) = Z ^ e + C

194 -- F(Z) = Z ^ (C * e)

195 -- F(Z) = (Z ^ e) ^ C

196 -- F(Z) = C * Z ^ e

197 -- F(Z) = Z ^ (pi * e)

198 -- F(Z) = Z * (C ^ e)

199 -- F(Z) = cbrt(Z ^ 7 + 1) + C

200 -- F(Z) = Z ^ 4.669 + C

201 -- F(Z) = (Z ^ 8 + 1) ^ 1/4 + C

202 -- F(Z) = (Z ^ 9 + 1) ^ 1/4 + C

203 -- F(Z) = ((Z ^ 2 * (ReZ - (ImZ)^2))/(1 - Z)) + C

204 -- F(Z) = (Z ^ 10 + C) ^ 1/4

205 -- F(Z) = (Z ^ 10 + 1) ^ 1/4 + C

206 -- F(Z) = (Z ^ 11 + C) ^ 1/4

207 -- F(Z) = (Z ^ 11 + 1) ^ 1/4 + C

208 -- F(Z) = (Z ^ 12 + C) ^ 1/4

209 -- F(Z) = (Z ^ 12 + 1) ^ 1/4 + C

210 -- F(Z) = YZ^2sinX - XZcosY + C

211 -- F(Z) = XZ^3cosY + YZ^2sinX + C

212 -- F(Z) = Z^4 - Z^2cosX + YsinY + C

213 -- F(Z) = XYZ^2 + C

214 -- F(Z) = Z^2 + X^2*Y^2 + C

215 -- F(Z) = Z^3 + X^2sinY + Y^2cosX + C

216 -- F(Z) = (Z ^ 13 + C) ^ 1/6

217 -- F(Z) = (Z ^ 5 + C) ^ 1/3

218 -- F(Z) = (Z ^ 4 + C) ^ 1/sin X

219 -- F(Z) = Z ^ 2 + iZ ^ 2 + C

220 -- F(Z) = Z ^ 3 + iZ ^ 3 + C

221 -- F(Z) = Z ^ 4 + iZ ^ 2 + C

222 -- F(Z) = (Z ^ 4 / Z + 1) + C

223 -- F(Z) = (Z ^ 6 / Z + 1) + C

224 -- F(Z) = (Z ^ 4 / Z + i) + C

225 -- F(Z) = (Z ^ 6 / Z + i) + C

226 -- F(Z) = (Z ^ 2 / (lnZ)^2) + C

227 -- F(Z) = (Z ^ 2 / (ln(Z^2)) + C

228 -- F(Z) = (Z ^ 3 / (lnZ)^3) + C

229 -- F(Z) = (Z ^ 3 / (ln(Z^3)) + C

230 -- F(Z) = (Z ^ 4 / (lnZ)^4) + C

231 -- F(Z) = (Z ^ 4 / (ln(Z^4)) + C

232 -- F(Z) = Z ^ 2 + Z / ln Z + C

233 -- F(Z) = Z ^ 2 + ln Z / Z + C

234 -- F(Z) = Z ^ 6 + Z ^ 4 + Z ^ 2 + C

235 -- F(Z) = Z ^ 6 - Z ^ 4 - Z ^ 2 + C

236 -- F(Z) = Z ^ (1/Z) + C

237 -- F(Z) = Z ^ 2 + sin Z / Z + C

238 -- F(Z) = Z ^ 2 + Z / sin Z + C

239 -- F(Z) = Z ^ iZ + C

240 -- F(Z) = Z ^ 2 * exp(X) + C

241 -- F(Z) = Z ^ 2 * exp(X ^ 2) + C

242 -- F(Z) = Z ^ 3 * exp(X) + Z ^ 2 * exp(Y) + C

243 -- F(Z) = exp(Z ^ Z) + C

244 -- F(Z) = (Z ^ 3) / (Z + 1) + C

245 -- F(Z) = Z ^ 2 / C

246 -- F(Z) = (Z ^ 4 + 1) / (Z + C)

247 -- F(Z) = (Z ^ 4 + C) / (Z ^ 2 + 1)

248 -- F(Z) = (Z ^ 4 + C) / (1 - Z ^ 2)

249 -- F(Z) = Z ^ 2 * exp(Z) / (Z + C)

250 -- F(Z) = Z ^ 2 - exp(Z) + sin(Z) + C

251 -- F(Z) = (Z ^ 4) / (Z ^ 2 + C)

252 -- F(Z) = Z ^ 2 + sqrt(Z ^ 2 + C)

253 -- F(X) = X^2 - Y^2 + XsinY + A; F(Y) = Y^2 - B

254 -- F(X) = X^2 + atan(Y/X) + A; F(Y) = Y^2 - A

255 -- F(X) = 1 - X - Y^2 + A; F(Y) = 1 - Y + X^2 + B

256 -- F(X) = exp(sqrt(X)) - exp(sqrt(Y)) + A; F(Y) = exp(XlnY) + B

257 -- F(Z) = C ^ 2 * ln(Z ^ 2)

258 -- F(Z) = Z ^ 2 ln(C)

259 -- F(Z) = Z ^ 2 ln(C) + C

260 -- F(Z) = Z ^ 2 ln(Z + C)

261 -- F(Z) = Z ^ -2 + C

262 -- F(Z) = ((X^2 + Y^2 + A) / (X^2 - Y^2)) + i[((X^2 - Y^2 - B) / (X^2 + Y^2))]

263 -- F(Z) = (X^3 - iY + C) / (X + Y + 1)

264 -- F(Z) = [(X^2 + A^2) / Y] + i[(Y^2 + B^2) / X]

265 -- F(Z) = [(X^3 + X^2 + X + A) / (Y^3 - Y^2 - Y - 1)] + i[(Y^3 + Y^2 + Y + B) /
(X^3 - X^2 - X - 1)]

266 -- F(Z) = [(X^4 - Y^2) / (X + Y + A)] + i[(X^2 + Y^4) / (X - Y - B)]

267 -- F(Z) = C ^ 3 / Z ^ 2

268 -- F(Z) = [Z^(1/2) / Z^(1/3)] + C

269 -- F(Z) = (Z ^ 2 + C) / (1 - C)

270 -- F(Z) = (exp(Z ^ 4)/ Z ^ 4) + C

271 -- F(Z) = (Z ^ 6 + 1) ^ (1/5) + C

272 -- F(Z) = Z ^ 2 + CZ + C * sin Y - Z * cos X + C

273 -- F(Z) = Z ^ 6 - Z ^ 5 - Z ^ 4 - Z ^ 3 - Z ^ 2 - Z + C

274 -- F(Z) = Z ^ 2 * (sin C / C)

275 -- F(Z) = exp(- Z ^ 2 / 2) + C

276 -- F(Z) = (Z ^ 3 + 3 * Z - 1) / (2 - Z)

277 -- F(Z) = Z ^ 3 * sin C + Z ^ 2 * cos C + XY + C

278 -- F(Z) = (Z ^ 2 + 1) ^ 2 / (Z + C) ^ 2

279 -- F(Z) = (Z ^ 2 + C + 1) ^ 2 / (Z - C - 1) ^ 2

280 -- F(Z) = Z ^ 4 * sin Y + Z ^ 2 * cos X + XY + C

281 -- F(Z) = Z * cos (XY) + C

282 -- F(Z) = Z ^ 2 * cos (X ^ 2 + Y ^ 2) + C

283 -- F(Z) = Z ^ (2 + ln C)

284 -- F(Z) = Z ^ (9/7) + C

285 -- F(Z) = Z ^ 5 * (1 - Z - (Z + C) ^ 2) + C

286 -- F(Z) = Z ^ 6 + Z ^ 5 + C

287 -- F(Z) = Z ^ 6 + Z ^ 4 + C

288 -- F(Z) = Z ^ 6 + Z ^ 3 + C

289 -- F(Z) = Z ^ 6 + Z ^ 2 + C

290 -- F(Z) = Z ^ 6 + Z + C

291 -- F(Z) = Z ^ 2 + cos Z + C

292 -- F(Z) = Z ^ 2 + cos 2Z + C

293 -- F(Z) = Z ^ 2 + cos 3Z + C

294 -- F(Z) = Z ^ 2 + cos 4Z + C

295 -- F(Z) = Z ^ 2 + cos 5Z + C

296 -- F(Z) = (Z ^ 7 + C) / Z ^ 5

297 -- F(Z) = (Z ^ 7 + C) / Z ^ 4

298 -- F(Z) = (Z ^ 7 + C) / Z ^ 3

299 -- F(Z) = (Z ^ 7 + C) / Z ^ 2

300 -- F(Z) = (Z ^ 7 + C) / Z

301 -- F(Z) = Z ^ 3 - Z ^ 2 - Z + C

302 -- F(Z) = Z ^ 4 - Z ^ 3 - Z ^ 2 + C

303 -- F(Z) = Z ^ 5 - Z ^ 4 - Z ^ 3 + C

304 -- F(Z) = Z ^ 6 - Z ^ 5 - Z ^ 4 + C

305 -- F(Z) = Z ^ 7 - Z ^ 6 - Z ^ 5 + C

306 -- F(Z) = Z ^ 2 * (cos(Z)) ^ 2 + C

307 -- F(Z) = Z ^ 2 * (cos(XY)) ^ 2 + C

308 -- F(Z) = Z ^ 4 * (sin(Z)) ^ 2 + C

309 -- F(Z) = Z ^ 3 * (sin(XY)) ^ 2 + C

310 -- F(Z) = Z ^ 3 * (cos(Z)*sin(Z)) + C

311 -- F(Z) = (Z ^ 2 / sin(Z)) + C

312 -- F(Z) = (Z ^ 4 / cos(Z)) + C

313 -- F(Z) = (Z ^ 6 + C) / (sin(Z) * cos(Z))

314 -- F(Z) = (Z ^ 3 + Z ^ 2 + Z + C) / (Z + cos(Z)

315 -- F(Z) = (Z ^ 2 * ln Z + Z + C) / (sin(Z)) ^ 2

316 -- F(Z) = Z ^ 4 + (cos X) ^ 2 + (sin Y) ^ 2 + C

317 -- F(Z) = Z ^ 3 + cos X * sin Y + C

318 -- F(Z) = Z ^ 4 + Z + cos C

319 -- F(Z) = Z ^ 2 + Z + tan C

320 -- F(Z) = Z ^ 3 + Z ^ 2 + exp(1 + sin X) + C

321 -- F(Z) = sqrt(Z ^ 4 + cos(theta) + C); theta = arctan (Im Z / Re Z)

322 -- F(Z) = sqrt(Z ^ 5 + Z ^ 3 + Z + C)

323 -- F(Z) = sqrt(Z ^ 4 + Z ^ 3 + Z ^ 2 + Z + C)

324 -- F(Z) = sqrt(Z ^ 6 - Z ^ 3 + C)

325 -- F(Z) = sqrt(ln (Z ^ 2) + Z ^ 2 * ln Z + C)

326 -- F(Z) = cos((Z ^ 2 + C) / XY)

327 -- F(Z) = cos((Z ^ 3 + C) / XY)

328 -- F(Z) = cos((Z ^ 4 + C) / XY)

329 -- F(Z) = ((Z ^ 4 + C) / XY) + cos((Z ^ 3 + C) / XY)

330 -- F(Z) = cos((Z ^ 4 + C) / XY) + cos((Z ^ 3 + C) / XY) + cos((Z ^ 2 + C) / XY)

331 -- F(Z) = Z ^ 3/2 + Z ^ 4/3 + C

332 -- F(Z) = Z ^ 4/3 + Z ^ 5/4 + C

333 -- F(Z) = Z ^ 5/4 + Z ^ 6/5 + C

334 -- F(Z) = Z ^ 5/2 + Z ^ 7/3 + C

335 -- F(Z) = Z ^ (pi/e) ^ 2 + C

336 -- F(Z) = Y * sin X * cos Y * exp(-X) + C

337 -- F(Z) = Z ^ 2 * cos X * cos Y * exp(-Y) + C

338 -- F(Z) = XYZ * sin X * sin Y * exp(Z) + C

339 -- F(Z) = Z ^ 3 + X ^ 2 * Y ^ 2 * cos X * sin Y + C

340 -- F(Z) = Z ^ 2 + X ^ 2 * sin Y + Y ^ 2 * cos X + C

341 -- F(Z) = 1 / Z + 1 / Z ^ 2 + C

342 -- F(Z) = Z ^ 2 / Z' + Z ^ 3 / Z' ^ 2 + C

343 -- F(Z) = Z ^ 3 / C' + Z ^ 2 + C

344 -- F(Z) = Z ^ 2 + Z' ^ 2 + C

345 -- F(Z) = Z ^ 3 + Z ^ 2 * Z' + Z * Z' ^ 2 + Z' ^ 3 + C

346 -- F(Z) = Z ^ 4 - Z ^ 3 * Z' + Z ^ 2 - Z' + C

347 -- F(Z) = Z ^ 5 - C * Z ^ 3 - C' * Z ^ 2 + Z' + C

348 -- F(Z) = Z ^ 4 * Z' ^ 2 - Z ^ 3 * Z' + C

349 -- F(Z) = Z ^ 6 + Z' ^ 5 + Z ^ 4 + Z' ^ 3 + Z ^ 2 + Z' + C

350 -- F(Z) = Z ^ 4 + Z ^ 2 / Z' + Z' ^ 3 / Z ^ 2 + C

351 -- F(Z) = arcsin(ln(Z)) + C

352 -- F(Z) = arctan(ln(Z)) + C

353 -- F(Z) = (arcsin(ln(Z))) ^ 2 + C

354 -- F(Z) = e ^ (1 + cos(ln(Z))) + C

355 -- F(Z) = e ^ (2 - e ^ cos(Z)) + C

356 -- F(Z) = XY * Z^2 - X^2 * YZ + X * Y ^ 2 * Z ^ 3 + C

357 -- F(Z) = X ^ 3 * Y ^ 4 + X ^ 2 * Z ^ 5 + C

358 -- F(Z) = XY^2Z^3 - X^3Y^2Z + X^2Y^2Z^2 + C

359 -- F(Z) = Z ^ 4 - X ^ 2 * cos(Y) + Y * sin(X) + C

360 -- F(Z) = Z ^ 3 - Y ^ 2 * cos(XY) - X ^ 2 * sin(X) - Y * cos(Y) + C

361 -- F(Z) = (Z ^ 3 + C) / (Z ^ 3 - C)

362 -- F(Z) = (Z ^ 3 + C ^ 2 + 1) / (Z ^ 3 - C ^ 2 - 1)

363 -- F(Z) = (Z ^ 3 + Z + C) / (Z ^ 3 - Z - 1)

364 -- F(Z) = (Z ^ 2 - Z ^ 3 + 1) / (Z ^ 4 + C)

365 -- F(Z) = (Z ^ 4 + C) / (4Z ^ 3 + 1)

366 -- F(Z) = (Z ^ C) / (Z + 1)

367 -- F(Z) = (Z ^ (1 + C)) / (1 + C)

368 -- F(Z) = (2 ^ Z) / C

369 -- F(Z) = 2 ^ Z + C

370 -- F(Z) = 2 ^ Z + (2 ^ Z) / C + C

371 -- F(Z) = XYZ ^ 2 - X ^ 2YZ + C

372 -- F(Z) = X ^ 4 * Y ^ 3 * Z ^ 2 + C

373 -- F(Z) = X^3*Y^3*Z^3 - X^2*Y^2*Z^2 + XYZ + C

374 -- F(Z) = XY^2Z + X^2YZ^4 + C

375 -- F(Z) = Z^2*sqrt(XY) + XY^2Z^4*sqrt(XY) + C

376 -- F(Z) = Z ^ (2XY) + C

377 -- F(Z) = X ^ (2YZ) + C

378 -- F(Z) = Y ^ (Z^2) + X + C

379 -- F(Z) = X ^ (2YZ) + Z ^ (2XY) + C

380 -- F(Z) = (XY) ^ (Z - C)

381 -- F(Z) = Z ^ 2C + C

382 -- F(Z) = Z ^ 2 + C ^ 2Z + C

383 -- F(Z) = X^Y + Z^X + Y^Z + C

384 -- F(Z) = Z ^ 2 + A ^ X + B ^ Y + C

385 -- F(Z) = Z ^ 3 + X ^ (AB) + Y ^ C

386 -- F(Z) = Z ^ 9 - Z ^ 8 - Z ^ 7 + C

387 -- F(Z) = Z^9 - 9Z^8 - 8Z^7 + C

388 -- F(Z) = Z^9 - 9Z^8 - 8Z^7 - 7Z^6 - 6Z^5 - 5Z^4 + C

389 -- F(Z) = Z^9 - 9Z^8 - 8Z^7 - 7Z^6 - 6Z^5 - 5Z^4 - 4Z^3 - 3Z^2 - 2Z +C

390 -- F(Z) = Z^9 + C^9

391 -- F(Z) = Z^3 + Z^2 + CsinX + C

392 -- F(Z) = Z^4 + X^2 - Y^2 - C^2*cosZ + C + A

393 -- F(Z) = Z^5 + Im(Z^4 + Z^3 + Z^2) + CZRe(Z^2 + C) + C

394 -- F(Z) = Z^3 + Z^2*cosY + ZsinX + C

395 -- F(Z) = Z^4 + (Z^2 / sinY) + (CZ^3 / cosX) + C

396 -- F(Z) = (Z + ln Z)^4 + C

397 -- F(Z) = Z + (ln Z)^4 + C

398 -- F(Z) = Z^2 + (ln Z)^3 + C

399 -- F(Z) = Z^3 + (ln Z)^2 + C

400 -- F(Z) = (ln Z)^2 + C^2 + C

401 -- F(Z) = Z^2 + C + A

402 -- F(Z) = Z^2 + C + iB

403 -- F(Z) = Z^2 + C + X

404 -- F(Z) = Z^2 + C + iY

405 -- F(Z) = Z^2 + C + iXY

406 -- F(Z) = X^3 + X^2Y - XY^2 + Y^3 + C

407 -- F(Z) = X^5 + iX^4 -iX^3Y + iX^2Y^2 - iXY^3 + iY^4 + C

408 -- F(Z) = Y^3 - Y^2 - Y - 1 - A + iX^3 + iX^2 + iX + i + iB

409 -- F(Z) = Y^4 + Y^2 + A - iX^4 - iX^2 - iB

410 -- F(Z) = X^2 + X + 1 + A + iY^2 - iY - i - iB

411 -- F(Z) = arcsin(Z) + C (3rd appr)

412 -- F(Z) = 1 - Z/X - Z^2/Y + C

413 -- F(Z) = arctan(Z) + C (5th appr)

414 -- F(Z) = X^3/Y^2 + Y^4/X^3 + A + i(Y^2/X) - i(X^3/Y) + iB

415 -- F(Z) = Z^4/(X+Y) + Z^3/(X-Y) + C

416 -- F(Z) = Z^(2-X) + C

417 -- F(Z) = Z^(2-X-Y) + C

418 -- F(Z) = Z^(3+C)

419 -- F(Z) = Z^(2X^2 - 3Y^2) + C

420 -- F(Z) = (X+Y)^2Z + C

421 -- F(Z) = Z^2 + sin(Z)*cos(Z) + C

422 -- F(Z) = Z^2 + sec(Z)*tan(Z) + C

423 -- F(Z) = Z^2 + sin(Z)*tan(Z) + C

424 -- F(Z) = Z^2 + cot(Z)*arcsin(Z) + C

425 -- F(Z) = Z^2 + tan(Z)*arccos(Z) + C

426 -- F(Z) = Z^(2-Z) + C

427 -- F(Z) = Z^(3-Z) + C

428 -- F(Z) = Z^(4-Z) + C

429 -- F(Z) = Z^(5-Z) + C

430 -- F(Z) = Z^(6-Z) + C

431 -- F(Z) = Z^(2-C) + C

432 -- F(Z) = Z^(3-C) + C

433 -- F(Z) = Z^(4-C) + C

434 -- F(Z) = Z^(5-C) + C

435 -- F(Z) = Z^(6-C) + C

436 -- F(Z) = Z^Z / Z^C

437 -- F(Z) = C^Z / Z^Z

438 -- F(Z) = (1-C)^2Z

439 -- F(Z) = (Z+C)^Z

440 -- F(Z) = (Z^2 + C^2)^2Z

441 -- F(Z) = Z^CZ

442 -- F(Z) = Z^(C^2) + C

443 -- F(Z) = C^(2^Z) + C

444 -- F(Z) = Z^(tan(Z) + C

445 -- F(Z) = Z^(sin(Z)cos(Z)) + C

446 -- F(Z) = 2CZ / (Z + C)

447 -- F(Z) = 2CZ / sqrt(Z + C)

448 -- F(Z) = 2 * Z^2 * C^2 / (Z + C)

449 -- F(Z) = 2CZ^3 / (Z + C)

450 -- F(Z) = sqrt(2CZ) / (Z + C)

451 -- F(Z) = Z ^ (Z/C)

452 -- F(Z) = Z^2 + Z ^ (Z/C)

453 -- F(Z) = Z ^ (Z/C) + Z + C

454 -- F(Z) = Z ^ (Z/C) Z ^ 2 + C

455 -- F(Z) = Z ^ (Z^2 / C)

456 -- F(Z) = Z ^ (C/Z)

457 -- F(Z) = Z ^ (C/Z) + Z

458 -- F(Z) = Z ^ (C/Z) + Z^2

459 -- F(Z) = Z ^ (C/Z) + C

460 -- F(Z) = Z ^ (C^2 / Z)

461 -- F(Z) = Z ^ ln(Z ^ Z/C)

462 -- F(Z) = Z ^ ln(Z ^ C/Z)

463 -- F(Z) = Z ^ Z/C + Z ^ C/Z

464 -- F(Z) = sqrt(X^2 + Y^2) + iarctan(Y/X) + C

465 -- F(Z) = Z^2 + sqrt(X^2 + Y^2) + iarctan(Y/X) + C

466 -- F(Z) = Z^3 + sqrt(X^2 + Y^2) + iarctan(Y/X) + C

467 -- F(Z) = Z^4 + sqrt(X^2 + Y^2) + iarctan(Y/X) + C

468 -- F(Z) = Z^2 + sqrt(A^2 + B^2) + iarctan(B/A)

469 -- F(Z) = Z^3 + sqrt(X^3 + Y^3) + C

470 -- F(Z) = Z^4 + sqrt(X^4 + Y^4) + C

471 -- F(Z) = Z^5 + sqrt(X^5 + Y^5) + C

472 -- F(Z) = Z^6 + sqrt(X^6 + Y^6) + C

473 -- F(Z) = Z^7 + sqrt(X^7 + Y^7) + C

474 -- F(Z) = Z^8 + sqrt(X^8 + Y^8) + C

475 -- F(Z) = Z^9 + sqrt(X^9 + Y^9) + C

476 -- F(Z) = sqrt(X^2 + Y^2) + isqrt(X^3 + Y^3)

477 -- F(Z) = sqrt(X^3 + Y^3) + isqrt(X^4 + Y^4)

478 -- F(Z) = sqrt(X^4 + Y^4) + isqrt(X^5 + Y^5)

479 -- F(Z) = sqrt(X^5 + Y^5) + isqrt(X^6 + Y^6)

480 -- F(Z) = sqrt(X^6 + Y^6) + isqrt(X^7 + Y^7)

481 -- F(Z) = Z^2 + exp(cot(Z))*cot(Z)

482 -- F(Z) = Z^2 + exp(tan(Z))*tan(Z)

483 -- F(Z) = Z^2 + exp(cos(Z))*cos(Z)

484 -- F(Z) = Z^2 + exp(sin(Z))*sin(Z)

485 -- F(Z) = Z^2 + (exp(cot(Z))/ Z) + C

486 -- F(Z) = Z * det|X A Y B| + i * det|A Y B X| + C

487 -- F(Z) = Z^2 + det|X A Y B| + C

488 -- F(Z) = Z^2 * C^2 * det|X A Y B| + C

489 -- F(Z) = Z^3 * C^3 * det|X A Y B| - C

490 -- F(Z) = 2XY - i(X^2 - Y^3) + C

491 -- F(Z) = 3*X^3*Y^2 - iZ^2 + C

492 -- F(Z) = ZC^3 - Z^2*C^2 - CZ^3 + C\n");

493 -- F(Z) = X^2*Y*Z^2 - Y^2*Z*sin(X) + X^2*cos(Y) + C

494 -- F(Z) = Y^3*Z^3 - ZX^2 - C^2 * sin(Y) + C

495 -- F(Z) = X^2*Y^2*Z^2*C^2 + CXYZ + iZ^3

496 -- F(Z) = X^3*Y^3*Z^3 + iCXZ^2

497 -- F(Z) = X^2*Y^8*Z^5 + iXYZC^4

498 -- F(Z) = X^2*Y^3*Z^4 + XY^2*Z^3 + Z^2 + C

499 -- F(Z) = XYZ^2*cos(X) + X^2 - Y^2 + BCZ + C

500 -- F(Z) = Z^2 + ABXY + C

The next fifty-two equations are exhibited in the
brand new 2006 "M" line of fractal imagery.

501 -- F(Z) = Z^C + SIN C

502 -- F(Z) = (SQRT Z + C) / C

503 -- F(Z) = (SIN(Z+1))^Z * COS(1/Z) + C

504 -- F(Z) = SIN(Z * COS Z + C) + C

505 -- F(Z) = COS(Z^2 / C) + SIN(C / Z^2) + C

506 -- F(Z) = Z^C + C^Z + C^(Z^2) + Z^(C^2)

507 -- F(Z) = ((Z + SIN(Z) + C)^2 + C) / Z

508 -- F(Z) = SIN(COS(Z^2 + C) + Z + C

509 -- F(Z) = SIN(Z^2) * LOG(C^2) * (Z * C^(SQRT(Z)) + C

510 -- F(Z) = Z^2 + Z * C^(SQRT(Z)) + C

511 -- F(Z) = SIN(Z^2 / C) + COS(Z / C) + Z^2 + C

512 -- F(Z) = ((Z^4 + C) / (Z^2 - C)) + Z^2 + C

513 -- F(Z) = ((Z^2 + C)^2 + Z - C^3) / 2(Z^2 + C)

514 -- F(Z) = e^(Z^2) + e^2Z - Z^2 - 1 + C

515 -- F(Z) = (2e^2Z / Z^3) + C

516 -- F(Z) = 3Z^2 * 3^(Z^3) - 3^Z + C

517 -- F(Z) = (1 + LOG(Z))^3 + C

518 -- F(Z) = (Z + 1/Z)^2.5 + C

519 -- F(Z) = (Z^(Z - 1) + 1)^2 + C

520 -- F(Z) = (1 / LOG(Z + C))

521 -- F(Z) = C * (Z + 1/Z) + Z^(-.5) + C

522 -- F(Z) = Z^2 + Z^1.75 + Z^1.5 + Z^1.25 + Z + C

523 -- F(Z) = ((Z^4 + Z^2 + C)^2 / (Z^5 + Z^3 + Z + C))

524 -- F(Z) = (1 / SINH(1 / Z^2)) + C

525 -- F(Z) = (Z ^ C + Z) / C

526 -- F(Z) = (Z + C^Z) / C

527 -- F(Z) = (Z^C + C) / Z

528 -- F(Z) = (C^Z + C) / Z

529 -- F(Z) = Z^1.75 + C

530 -- F(Z) = (Z^2.5 / LOG(Z)) + C

531 -- F(Z) = Z^2 * LOG(Z) - C*Z + Z + C

532 -- F(Z) = SQRT(Z^6 + Z + C) + C

533 -- F(Z) = (C * Z^2)^Y + C

534 -- F(Z) = SQRT(Z^4 + Z^3) + Z + C

535 -- F(Z) = (((Z^2 + X - 1)^2) / (2Z + X + 1)^2) + C

536 -- F(Z) = Z^2 + X + Y - X^2 * Y^2 + C

537 -- F(Z) = Z^(Z^2 + C) - Z + C

538 -- F(Z) = (Z + TAN(Z) + TAN(1/Z))^2 + C

539 -- F(Z) = Z^(C - 1) * (1 - Z - Z^2) + C

540 -- F(Z) = LOG(1 / (COS(Z^2 + C)))

541 -- F(Z) = Z^3 + (TAN(Z) + C * Z)^2

542 -- F(Z) = Z^2 * X - Z * Y + C

543 -- F(Z) = Z^2 * CSC(Z^2) + C

544 -- F(Z) = (Z + LOG(Z))^3 + C

545 -- F(Z) = Z^(LOG(Z^2)) - Z + C

546 -- F(Z) = (2Z^4 - Z + C) / (Z + C)

547 -- F(Z) = Z^5C - Z^3C + Z^C + C

548 -- F(Z) = Z^2 + ((Z^3) / (TAN(Z)^2))) + C

549 -- F(Z) = C^(Z^2 - Z + C)

550 -- F(Z) = Z^7 + ((Z^5) / (5 - Z)) + ((Z^3) / (3 - Z)) + Z / C

551 -- F(Z) = (TAN(C * Z))^2 + C

552 -- F(Z) = Z^3 + (1 + LOG(Z))^2 + C

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Lissajous Figures

A 3-D Lissajous figure is created using three parametric equations, one each for the x, y, and z coordinates, that is, each coordinate is a function of the independent parameter, time, t. These equations are sinusoidal functions (sines and cosines) so they are periodic, with the actual period depending on what values you enter. The values you input in these functions are the coefficients a and b, and the exponents i, j, and k. The value of t ranges from 0 to the number of spheres plotted minus one. Thanks to Aaron C. Caba for the info.

I used five different sets of equations. Here they are:

Set 1)

x(t) = r * (sin(a*t) * (cos(b*t)^i))

y(t) = r * (sin(a*t) * (sin(b*t)^j))

z(t) = r * (cos(a*t)^k)

Set 2)

x(t) = r * (sin(a*t) * (cos(b*t)^i))

y(t) = r * (cos(a*t) * (cos(b*t)^k))

z(t) = r * (sin(a*t)^k)

Set 3)

x(t) = r * (sin(a*t) * (sin(b*t)^i))

y(t) = r * (sin(a*t) * (cos(b*t)^j))

z(t) = r * (sin(a*t)^k)

Set 4)

x(t) = r/4 * (a * sin(2*(t-pi/13))^i)

y(t) = r/4 * (-b * cos(t)^j)

z(t) = r * (sin(a*t)^k)

Set 5)

x(t) = r * (sin(a*t) * (cos(a*t)^i))

y(t) = r * (sin(b*t) * (sin(b*t)^j))

z(t) = r * (sin(t)^k)

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Spherical Harmonics

Spherical harmonics are expressions in three-dimensional spherical coordinates which are primarily used to describe the theoretical hybrid electron orbital shapes in molecules. The three coordinates are r (for radius), theta (degrees in the traditional x-y plane), and phi (degrees in the y-z plane). You may also recognize this way of laying out spatial coordinates from Star Trek's "210 mark 45" designation for navigation as the degrees in theta and phi. As with the rectangular coordinates, x, y, and z, we can describe any point in three dimensional space using such a coordinate system. All types of scientists and engineers use spherical and cylindrical (rho, theta, and z) coordinate systems in addition to the familiar rectangular system to analyze various physical phenomena.

Here are a few of the examples we have used to produce our mathematical "flying saucers:"

r = (cos (theta))^2 + (cos(2 * theta))^4 + sin(4 * phi)

r = (cos(12 * theta))^5 + (cos(8 * theta))^3 + cos(6 * theta)

r = 2 * (cos(6 * theta))^6 - 4 * (cos(4 * theta))^4 - 2 * (cos(2 * theta))^2

rho = (sin(theta))^4 + (sin(2 * theta))^2 + e ^ (1 - sin(z))

rho = 4 * (cos(4 * theta))^4 - 2 * (cos(2 * theta))^2 + (1 + cos (z))^2

You can experiment with an infinite number of possibilities. You will soon discover what each coefficient, exponent, and function does to the overall shape of the object. Happy Hunting!

For a large and extended study of over 640 images, go HERE.

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Knots

Knots are a colloquial term for three dimensional figures very much akin to Lissajous figures. The program I used to generate these "knots" was created by Lloyd Burchill, a most clever programmer and mathematician who created this shareware gem. His program allows one to generate multiple parametrically calculated figures and their interaction produces some very surprising results. Additionally there are some special techniques and features available to produce other very tricky designs. You can get it HERE. You can e-mail Mr. Burchill at lloyd@kagi.com.

Also, there's the very clever and interesting KnotPlot software available from Robert Scharein HERE. I recommend this one, indeed. I have yet to delve into its mysteries.

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Polyhedra

Polyhedra, the plural of polyhedron, are three-dimensional solid figures with many geometrical faces to them. There are five commonly known regular polyhedra, regular meaning all faces are congruent and all angles are congruent. They are:

 Tetrahedron 4 faces equilateral triangle Hexahedron (Cube) 6 faces square Octahedron 8 faces equilateral triangle Dodecahedron 12 faces pentagon Icosahedron 20 faces equilateral triangle

There is information regarding formulas to find the volumes, surface areas, inscribed radii, and circumscribed radii of the above polyhedra HERE.

There are also the Archimedean solids, solid shapes whose faces are all regular polygons of two or more kinds, and whose vertices are all identical. There are 13 different kinds. Two (the snub cube and snub dodecahedron) come in paired mirror-image forms. Eleven of these solids can be formed by truncating (chopping the corners off) simpler solids. They have pleasingly symmetrical crystalline shapes, and are described below. These eleven are:

 Truncated Tetrahedron 8 faces (4 triangles, 4 hexagons) Truncated Cube 14 faces (8 triangles, 6 octagons) Truncated Octahedron 14 faces (6 squares, 8 hexagons) Cuboctahedron 14 faces (8 triangles, 6 squares) Truncated Dodecahedron 32 faces (20 triangles, 12 dodecagons) - soccer ball pattern Truncated Icosahedron 32 faces (12 pentagons, 20 hexagons) - soccer ball / fullerene shape

 Icosidodecahedron 32 faces (20 triangles, 12 pentagons) Small Rhombicuboctahedron 26 faces (8 triangles, 18 squares) Great Rhombicuboctahedron 26 faces (12 squares, 8 hexagons, 6 octagons) Small Rhombicosidodecahedron 62 faces (20 triangles, 30 squares, 12 pentagons) Great Rhombicosidodecahedron 62 faces (30 squares, 20 hexagons, 12 dodecagons)

Thanks to Grant Hutchison for the info.

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Stellated Polyhedra

Below are the stellated forms of the simple Platonic solids we use in our scenes.

Stellated Octahedron
Stellated Dodecahedron
Stellated Icosahedron

The cube and tetrahedron have no stellated forms.

The stellations come in several different forms, depending on the shape of their faces. The octahedron has a single stellation, the dodecahedron has three, and the icosahedron has fifty-nine different stellations, fifteen of which are given here.

The octahedron has only one stellation, the Stella Octangula.
The three stellations of the dodecahedron are:

Small stellated dodecahedron
Great dodecahedron
Great stellated dodecahedron

There are fifty-nine stellations of the icosahedron. A selection of the fifteen we use are:

Compound of five tetrahedra
Compound of ten tetrahedra
Compound of five octahedra
First stellation
Second stellation
Third stellation
Fourth stellation
Sixth stellation
Seventh stellation
Ninth stellation
Tenth stellation
Fourteenth stellation
Fifteenth stellation
Great icosahedron
Final stellation

Thanks to Grant Hutchison and Magnus Wenninger for the info.

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Non-Convex Uniform "Starface" Polyhedra

Uniform polyhedra are solids whose faces are all regular polygons, and whose vertices are identical. The convex uniform polyhedra (or Starfaces) are the five Platonic solids, the thirteen Archimedean solids, and an infinite series of rather mundane prisms and antiprisms.

Non-convex uniforms have faces which interpenetrate, making very complicated and pleasing solids. Four of these are the small stellated dodecahedron, the great stellated dodecahedron, the great dodecahedron and the great icosahedron.

There are at least another 53 forms, 21 of which are here. The 21 solids below can be grouped into seven families. They have been numbered from one to seven arbitrarily. Each member of a family shares some of its faces with other members.

The seven families are:

 Small Dodecahemicosahedron 12 pentagrams, 10 hexagons Great Dodecahemicosahedron 12 pentagons, 10 hexagons Dodecadodecahedron 12 pentagrams, 12 pentagons

 Small Ditrigonal Icosidodecahedron 12 pentagrams, 20 triangles Great Ditrigonal Icosidodecahedron 12 pentagons, 20 triangles Ditrigonal Dodecahedron 12 pentagrams, 12 pentagons

 Rhombidodecadodecahedron 12 pentagrams, 12 pentagons, 30 squares Rhombicosahedron 30 squares, 20 hexagons Icosidodecadodecahedron 12 pentagrams, 12 pentagons, 20 hexagons

 Great Ditrigonal Dodecicosidodecahedron 12 decagrams, 12 pentagons, 20 triangles Great Icosicosidodecahedron 12 pentagons, 20 triangles, 20 hexagons Great Dodecicosahedron 12 decagrams, 20 hexagons

 Small Ditrigonal Dodecicosidodecahedron 12 pentagrams, 12 decagons, 20 triangles Small Icosicosidodecahedron 12 pentagrams, 20 triangles, 20 hexagons Small Dodecicosahedron 12 decagons, 20 hexagons

 Great Dodecicosidodecahedron 12 pentagrams, 12 decagrams, 20 triangles Great Rhombidodecahedron 12 decagrams, 30 squares Quasirhombicosidodecahedron 12 pentagrams, 20 triangles, 30 squares

 Great Icosihemidodecahedron 6 decagrams, 20 triangles Great Icosidodecahedron 12 pentagrams, 20 triangles Great Dodecahemidodecahedron 6 decagrams, 12 pentgrams

Families one and two are similar - equalateral hexagons in one are replaced by pairs of triangles in two. Four and five are also related - pentagrams replace pentagons, decagons replace decagrams. Families six and seven have "small" versions that aren't listed here - they are dissected versions of the rhombicosidodecahedron and the icosidodecahedron, respectively, and they don't have any star-shaped faces. Thanks to Grant Hutchison and Magnus Wenninger for the info.

For a scholarly look at Polyhedra of all types, see George Hart's dynamite website HERE.

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Affine Transformations

(Due to the limitations of web publishing, our notation of matrices, symbols, subscripts etc. will be clumsily laid out...please bear with us...as our tools improve, so will our presentation.)

As given by Barnsley in "Fractals Everywhere," an affine transformation is a manipulation of a geometric set of points (here x1 and x2, or just x) using matrices and column vectors such that:

w(x1,x2) = (ax1 + bx2 + e , cx1 + dx2 + f)

A general affine two-dimensional transformation, is given by:

w(x) = Ax + t

where A is a 2 x 2 real matrix and t is the column vector:

 A = a b c d

 t = e f

In graphic terms, the A matrix transforms x by a linear transformation, which deforms space relative to the origin (involving rotation and rescaling), whereas the t vector merely translates (moves) the points once the deformation is complete.

The matrix A can always be written as:

 A = [ r1 cos g -r2 cos h ] r1 sin g r2 sin h

where r1 and r2 are scaling factors and g and h are rotation angles.

Barnsley continues in his book to describe Iterated Function Systems, a way of describing objects created by affine transformations. Using the letters a, b, c, d, e, and f as defined above, he offers a typical a typical fern designation in tidier "IFS code:"

 a b c d e f p 0 0 0 .16 0 0 .01 .85 .04 -.04 .85 0 1.6 .85 .2 -.26 .23 .22 0 1.6 .07 -.15 .28 .26 .24 0 .44 .07

Notice he provides a number p which corresponds to the probability that each of the four "w" transformations will be used given each point (x1,x2) that is to be manipulated. All of the p's must add up to one. Because of this probability factor, each time you generate a spleenwort fern, it will be a slightly different one, just like Nature. Thus we are not producing a "deterministic fractal," as are Mandelbrot and Julia sets (which are exactly reproducible), but more of a "random iteration" fractal. See the Barnsley textbook for more info, illustrations, IFS codes, etc.

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