1 { ======================================================================== } 2 { File originally distributed with FRAC'Cetera Vol 2 Iss 7 } 3 {========================================================================== 4 = Compiled by Jon Horner for FRAC'Cetera from several sources - Jul 93 = 5 = F'names, where present, represent FRAC'Cetera created variations or = 6 = derivatives based, often quite loosely, on the author's originals. = 7 ==========================================================================} 8 9{ IKENAGA - Formula originally discovered by Bruce Ikenaga, at Western Reserve 10 University, Indiana. Documented in Dewdney's `Armchair Universe". 11 12 The Ikenaga set is: Z(n+1) =Z(n)^3 + (C-1) * Z(n) - C 13 where: Z(n) = x + yi and C = a + bi 14 15 In Basic (with Mandelbrot for comparison): 16 17 Mandelbrot: Ikenaga: 18 19 newX = x*x - y*y + a newX = x*x*x - 3*x*y*y + a*x - b*y - x - a 20 newY = 2*x*y + b newY = 3*x*x*y - y*y*y + a*y + b*x - y - b 21 22 Ikenaga is also in R.F.J. Stewart's MANDPLOT; Larry Cobb's DRAGONS4; 23 Mike Curnow's !Fractal for the Archimedes, and Tim Harris's CAL. 24 25 ==========================================================================} 26 27!_Press_F2_! {; There is text in this formula file. Shell to DOS with 28 ; the <d> key and use a text reader to browse the file. 29 } 30 31Ikenaga(XAXIS) { ; this version correct per Roderick Stewart - 32 ; announcement in Fractal Report 25 p2. 33 ; same as letter, Joyce Haslam Mar 1993 34 ; z=z*z*z+ (c-1)*z-c produces same results. 35 z = (0,0), c = pixel : 36 z = z * z * z + z * (c-1) - c , 37 |z|<=4 38 } 39 40IkenagaJ.1(XAXIS) { ; this version correct per Roderick Stewart - 41 ; announcement in Fractal Report 25 p2. 42 ; same as letter, Joyce Haslam Mar 1993 43 ; z=z*z*z+ (c-1)*z-c produces same results. 44 z = (0,0), c = pixel : 45 z = z * (z * z + (c-1)) - c , 46 |z|<=4 47 } 48 49IkenagaJUL { ; formula from a letter from Joyce Haslam Mar 1993. 50 ; Asymmetric. try p1 = (0.56667, 0.36) 51 ; Next line, from Haslam article Fractal Report 24 p5 52 ; z=z*z*z+ (c-1)*z-c produces same results. 53 ; Same as Julike in REB001.FRM 54 z = pixel, c = p1 : 55 z = z * z * z + z * (c-1) - c , 56 |z| <= 4 57 } 58 59IkenagaJULJ.1 { ; formula from a letter from Joyce Haslam Mar 1993. 60 ; Asymmetric. try p1 = (0.56667, 0.36) 61 ; Next line, from Haslam article Fractal Report 24 p5 62 ; z=z*z*z+ (c-1)*z-c produces same results. 63 ; Same as Julike in REB001.FRM 64 z = pixel, c = p1 : 65 z = z * (z * z + (c-1)) - c , 66 |z| <= 4 67 } 68 69{ The rest are all variations on the two basic formulas } 70 71IkenagaPwr(XAXIS) { ; from Jon Horner 72 z = (0,0), c = pixel : 73 z = z * z * z + z * (c-1) - c ^ p1 , 74 |z| <=4 75} 76 77IkenagaPwrJul { ; from Jon Horner - asymmetric 78 ; try p1 = (0.035, -0.35), p2 = 2 79 z = pixel, c = p1 : 80 z = z * z * z + z * (c-1) - c ^ p2 , 81 |z| <=4 82} 83 84Ikenaga4(XAXIS) { ; CAL v3.8 calls it Ikenaga-4Set 85 ; Per Haslam, Fractal Report 25 p2 (IKE4) 86 ; z^4 is used instead of z^3 and multiplication 87 ; is used in place of addition. 88 z = (0,0), c = pixel: 89 z = z * z * z * z * (c-1) - c , 90 |z|<= 4 91 } 92 93Ikenaga4JUL(ORIGIN) { ; Jon Horner, from IKE4 - Fractal Report 25 p2 94 ; try p1 = (0.3874, 0.85) 95 z = pixel, c = p1 : 96 z = z * z * z * z * (c-1) - c, 97 |z| <= 4 98 } 99 100IkenagaABS(XAXIS) { ; Jon Horner 101 ; Ikenaga with alternative bailout 102 z = (0,0), c = pixel : 103 z = z * z * z + z * (c-1) - c , 104 abs(z)<=4 105 } 106 107IkenagaABSJUL { ; Jon Horner 108 ; IkenagaJul with alternative bailout 109 z = pixel, c = p1 : 110 z = z * z * z + z * (c-1) - c , 111 abs(z) <= 4 112 } 113 114Ikenaga4BIO(XAXIS) { ; Ikenaga4 variation - Jon Horner 115 ; float=y 116 z = (0,0), c = pixel : 117 z = z * z * z * z * (c-1) - c , 118 |real(z)|<=4 || |imag(z)|<=4 119 } 120 121Ikenaga4BIOJUL(ORIGIN){ ; Jon Horner, from IKE4 - FR 25, p2 122 ; try p1 = (0.3874, 0.85) float=y 123 z = pixel, c = p1 : 124 z = z * z * z * z * (c-1) - c, 125 |real(z)|<=4 || |imag(z)|<=4 126 } 127 128IkenagaFN(XAXIS) {; Jon Horner 129 ; derived from Ikenaga 130 z = (0,0), c = fn1(pixel) : 131 z = z * z * z + z * (c-1) - c , 132 |z| <=4 } 133 134IkenagaFNJUL { ; Jon Horner 135 ; derived from IkenagaJul 136 ; asymmetric: try p1 = (0.56667, 0.36) 137 z = fn1(pixel), c = p1 : 138 z = z * z * z + z * (c-1) - c, 139 |z| <= 4 140 } 141 142IkenagaJUL1+(ORIGIN){ ; formula from an article by Joyce Haslam 143 ; in Fractal Report 24 (w/+pixel stead of +c) 144 ; symmetric 145 z = pixel, c = p1 : 146 z = z * z * z + (c-1) * z + pixel , 147 |z|<=4 148 } 149 150Ikenaga2(XAXIS) { ; from Joyce Haslam article, Fractal Report Iss 24 p5. 151 ; Uses pixel instead of c !!!! 152 ; CAL v4.0 calls it UnamiSet. FR 25 calls it Unknown. 153 ; CAL v4.0 IkenagaSet(XAXIS) = 154 ; {z=0:z=z*z*z+z*(pixel-1)-pixel,|z|<=|2| 155 z = pixel: 156 z = z * z * z + (pixel-1) * z - pixel, 157 |z|<=4 158 } 159 160IkenagaMap(XAXIS) {; from REB001.FRM - by Ron Barnett 70153,1233 161 ; The initial starting point allows the function to provide 162 ; a "map" for the corresponding Julia (IkenagaJul) 163 z = ((1-pixel)/3)^0.5: 164 z = z*z*z + (pixel-1)*z - pixel, 165 |z| <= 4 166 } 167 168IkeNewtMand { ; from REB001.FRM - by Ron Barnett 70153,1233 169 ; p1 > 0, < 2, float=yes 170 z = c = pixel: 171 zf = z*z*z + (c-1)*z - c; 172 zd = 3*z*z + c-1; 173 z = z - p1*zf/zd, 174 0.001 <= |zf| 175 } 176 177IkeNewtJul { ; from REB001.FRM - by Ron Barnett 70153,1233 178 ; p1 > 0, < 2, float=yes 179 z = pixel: 180 zf = z*z*z + (p2-1)*z - p2; 181 zd = 3*z*z + p2-1; 182 z = z - p1*zf/zd, 183 0.001 <= |zf| 184 } 185 186RecipIke { ; from REB001.FRM - by Ron Barnett 70153,1233 187 z = pixel: 188 z = 1/(z*z*z + (p1-1)*z - p1), 189 |z| <= 4 190 } 191 192F'FunctionIke { ; generalized by Jon Horner 193 ; from RecipIke in REB001.FRM 194 : - by Ron Barnett 70153,1233 195 z = pixel: 196 z = fn1(z*z*z + (p1-1) * z - p1), 197 |z| <= 4 198 } 199 200IkeGenM {; from REB002.FRM - by Ron Barnett 70153,1233 201 z = ((1-pixel)/(3*p1))^0.5: 202 z =p1*z*z*z + (pixel-1)*z - pixel, 203 |z| <= 100 204 } 205 206IkeGenJ {; from REB002.FRM - by Ron Barnett 70153,1233 207 z = pixel: 208 z =p1*z*z*z + (p2-1)*z - p2, 209 |z| <= 100 210 } 211 212IkeFrRbtGenM {; from REB002.FRM - by Ron Barnett 70153,1233 213 z = 2*(1-pixel)/(3*p1): 214 z = p1*z*z*z + (pixel-1)*z*z - pixel, 215 |z| <= 100 216 } 217 218IkeFrRbtGenJ {; from REB002.FRM - by Ron Barnett 70153,1233 219 z = pixel: 220 z = p1*z*z*z + (p2-1)*z*z - p2, 221 |z| <= 100 222 } 223 224