1 #+TITLE: Creating a golden spiral in R
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2 #+AUTHOR: Frederik Vanrenterghem
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4 #+PROPERTY: session *R*
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5 #+PROPERTY: cache yes
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6 #+PROPERTY: results graphics
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7 #+PROPERTY: exports both
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8 #+PROPERTY: tangle yes
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10 #+date: <2019-09-16 22:03:03>
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13 After having read the first part of a Rcpp tutorial which compared native R vs C++ implementations of a Fibonacci sequence generator, I resorted to drawing the so-called Golden Spiral using R.
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16 Libraries used in this example are the following
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18 #+BEGIN_SRC R :session R
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34 In polar coordinates, this special instance of a logarithmic spiral's functional representation can be simplified to r(t) = e^(0.0635*t)
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35 For every quarter turn, the corresponding point on the spiral is a factor of phi further from the origin (r is this distance), with phi the golden ratio - the same one obtained from dividing any 2 sufficiently big successive numbers on a Fibonacci sequence, which is how the golden ratio, the golden spiral and Fibonacci sequences are linked concepts!
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36 #+BEGIN_SRC R :session R
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37 polar_golden_spiral <- function(theta) exp(0.30635*theta)
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42 Let's do 2 full circles. First, I create a sequence of angle values theta. Since 2 * PI is the equivalent of a circle in polar coordinates, we need to have distances from origin for values between 0 and 4 * PI.
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43 #+BEGIN_SRC R :session R
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44 seq_theta <- seq(0,4*pi,by=0.05)
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46 dist_from_origin <- sapply(seq_theta,polar_golden_spiral)
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51 | 1.01543541418402 |
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52 | 1.03110908037907 |
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53 | 1.04702467610363 |
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54 | 1.06318593564018 |
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55 | 1.07959665091141 |
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56 | 1.09626067236991 |
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57 | 1.11318190990159 |
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58 | 1.13036433374308 |
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59 | 1.14781197541325 |
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60 | 1.16552892865913 |
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61 | 1.18351935041644 |
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62 | 1.20178746178492 |
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63 | 1.22033754901874 |
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64 | 1.23917396453215 |
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65 | 1.25830112792076 |
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66 | 1.27772352699844 |
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67 | 1.29744571885033 |
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68 | 1.31747233090207 |
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69 | 1.33780806200553 |
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70 | 1.35845768354131 |
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71 | 1.37942604053823 |
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72 | 1.40071805281016 |
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73 | 1.42233871611032 |
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74 | 1.44429310330345 |
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75 | 1.46658636555607 |
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76 | 1.48922373354506 |
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78 | 1.53555211437434 |
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79 | 1.55925399726085 |
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80 | 1.58332172852666 |
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81 | 1.60776095519303 |
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82 | 1.63257741144533 |
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83 | 1.65777691997847 |
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84 | 1.68336539336305 |
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85 | 1.70934883543265 |
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86 | 1.73573334269253 |
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88 | 1.78973041076699 |
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89 | 1.81735564093491 |
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90 | 1.84540727797241 |
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91 | 1.87389190364612 |
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92 | 1.90281620131498 |
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93 | 1.93218695749834 |
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94 | 1.96201106346829 |
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95 | 1.99229551686655 |
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96 | 2.02304742334636 |
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97 | 2.05427399823962 |
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98 | 2.08598256824992 |
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100 | 2.15087556763495 |
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101 | 2.18407522287968 |
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102 | 2.21778732855389 |
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103 | 2.25201979454219 |
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104 | 2.28678065282156 |
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105 | 2.32207805934587 |
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106 | 2.3579202959595 |
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107 | 2.39431577234054 |
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108 | 2.43127302797395 |
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109 | 2.46880073415517 |
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110 | 2.50690769602466 |
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111 | 2.54560285463391 |
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112 | 2.58489528904321 |
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113 | 2.62479421845192 |
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114 | 2.66530900436155 |
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115 | 2.70644915277227 |
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116 | 2.7482243164133 |
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117 | 2.79064429700773 |
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118 | 2.83371904757232 |
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119 | 2.87745867475275 |
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120 | 2.92187344119496 |
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121 | 2.9669737679531 |
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122 | 3.01277023693458 |
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123 | 3.05927359338295 |
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124 | 3.10649474839905 |
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125 | 3.15444478150108 |
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126 | 3.20313494322417 |
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127 | 3.25257665776014 |
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128 | 3.30278152563795 |
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129 | 3.3537613264455 |
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130 | 3.40552802159354 |
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131 | 3.45809375712212 |
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132 | 3.51147086655048 |
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133 | 3.56567187377081 |
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134 | 3.62070949598677 |
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135 | 3.67659664669734 |
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136 | 3.7333464387267 |
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137 | 3.79097218730088 |
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138 | 3.84948741317197 |
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139 | 3.90890584579046 |
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140 | 3.96924142652657 |
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141 | 4.03050831194138 |
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142 | 4.09272087710833 |
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143 | 4.15589371898609 |
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144 | 4.22004165984341 |
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145 | 4.28517975073691 |
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146 | 4.35132327504251 |
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147 | 4.41848775204137 |
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148 | 4.48668894056115 |
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149 | 4.55594284267357 |
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150 | 4.62626570744896 |
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151 | 4.69767403476877 |
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152 | 4.77018457919694 |
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153 | 4.84381435391107 |
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154 | 4.91858063469419 |
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155 | 4.9945009639882 |
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156 | 5.07159315500985 |
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157 | 5.14987529593027 |
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158 | 5.22936575411901 |
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159 | 5.31008318045357 |
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160 | 5.39204651369547 |
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161 | 5.47527498493387 |
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162 | 5.55978812209773 |
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163 | 5.6456057545377 |
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164 | 5.73274801767868 |
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165 | 5.82123535774417 |
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166 | 5.91108853655362 |
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167 | 6.00232863639374 |
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168 | 6.09497706496509 |
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169 | 6.18905556040493 |
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170 | 6.28458619638769 |
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171 | 6.38159138730412 |
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172 | 6.48009389352033 |
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173 | 6.58011682671816 |
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174 | 6.6816836553178 |
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175 | 6.78481820998423 |
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176 | 6.88954468921862 |
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177 | 6.99588766503603 |
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178 | 7.10387208873074 |
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179 | 7.2135232967306 |
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180 | 7.32486701654172 |
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181 | 7.43792937278492 |
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182 | 7.55273689332534 |
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183 | 7.66931651549675 |
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184 | 7.78769559242179 |
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185 | 7.90790189942989 |
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186 | 8.0299636405742 |
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187 | 8.15390945524908 |
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188 | 8.27976842490986 |
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189 | 8.40757007989611 |
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190 | 8.53734440636049 |
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191 | 8.6691218533043 |
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192 | 8.80293333972179 |
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193 | 8.93881026185472 |
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194 | 9.07678450055882 |
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195 | 9.21688842878404 |
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196 | 9.35915491917023 |
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197 | 9.50361735176004 |
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198 | 9.6503096218309 |
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199 | 9.79926614784789 |
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200 | 9.95052187953938 |
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201 | 10.1041123060972 |
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202 | 10.2600734645037 |
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203 | 10.4184419479868 |
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204 | 10.5792549146061 |
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205 | 10.7425500959714 |
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206 | 10.9083658060953 |
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207 | 11.0767409503832 |
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208 | 11.2477150347615 |
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209 | 11.4213281749469 |
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210 | 11.5976211058588 |
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211 | 11.7766351911771 |
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212 | 11.958412433047 |
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213 | 12.1429954819344 |
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214 | 12.3304276466328 |
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215 | 12.5207529044246 |
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216 | 12.7140159114002 |
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217 | 12.9102620129349 |
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218 | 13.1095372543288 |
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219 | 13.3118883916102 |
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220 | 13.5173629025061 |
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221 | 13.726008997582 |
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222 | 13.9378756315533 |
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223 | 14.1530125147717 |
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224 | 14.3714701248888 |
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225 | 14.5932997186998 |
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226 | 14.8185533441694 |
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227 | 15.0472838526447 |
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228 | 15.2795449112548 |
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229 | 15.5153910155034 |
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230 | 15.7548775020547 |
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231 | 15.9980605617174 |
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232 | 16.2449972526286 |
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233 | 16.4957455136412 |
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234 | 16.7503641779184 |
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235 | 17.0089129867378 |
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236 | 17.271452603508 |
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237 | 17.5380446280028 |
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238 | 17.8087516108139 |
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239 | 18.0836370680272 |
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240 | 18.3627654961257 |
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241 | 18.6462023871224 |
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242 | 18.9340142439267 |
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243 | 19.2262685959479 |
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244 | 19.5230340149396 |
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245 | 19.8243801310889 |
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246 | 20.1303776493537 |
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247 | 20.4410983660522 |
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248 | 20.7566151857085 |
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249 | 21.0770021381583 |
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250 | 21.4023343959182 |
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251 | 21.7326882918241 |
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252 | 22.0681413369407 |
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253 | 22.4087722387478 |
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254 | 22.7546609196083 |
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255 | 23.1058885355194 |
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256 | 23.462537495155 |
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257 | 23.8246914792008 |
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258 | 24.1924354599887 |
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259 | 24.5658557214339 |
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260 | 24.9450398792791 |
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261 | 25.3300769016527 |
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262 | 25.7210571299428 |
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263 | 26.1180722999943 |
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264 | 26.5212155636329 |
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265 | 26.9305815105213 |
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266 | 27.3462661903527 |
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267 | 27.7683671353873 |
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268 | 28.1969833833359 |
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269 | 28.6322155005976 |
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270 | 29.0741656058555 |
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271 | 29.5229373940367 |
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272 | 29.9786361606425 |
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273 | 30.4413688264541 |
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274 | 30.911243962619 |
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275 | 31.3883718161253 |
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276 | 31.8728643356692 |
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277 | 32.3648351979214 |
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278 | 32.8643998341988 |
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279 | 33.371675457549 |
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280 | 33.8867810902509 |
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281 | 34.4098375917422 |
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282 | 34.9409676869756 |
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283 | 35.4802959952146 |
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284 | 36.0279490592724 |
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285 | 36.584055375203 |
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286 | 37.1487454224504 |
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287 | 37.7221516944627 |
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288 | 38.3044087297792 |
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289 | 38.8956531435973 |
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290 | 39.4960236598267 |
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291 | 40.105661143638 |
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292 | 40.7247086345141 |
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293 | 41.3533113798114 |
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294 | 41.9916168688395 |
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295 | 42.6397748674667 |
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296 | 43.2979374532595 |
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297 | 43.9662590511644 |
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298 | 44.644896469741 |
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299 | 45.3340089379542 |
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300 | 46.0337581425336 |
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301 | 46.7443082659106 |
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303 Plotting the function using coord_polar in ggplot2 does not work as intended. Unexpectedly, the x axis keeps extending instead of circling back once a full circle is reached. Turns out coord_polar might not really be intended to plot elements in polar vector format.
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304 #+BEGIN_SRC R :session R :results output graphics :file ../pics/golden_spiral-coord_polar-fail.png :exports both
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305 ggplot(data.frame(x = seq_theta, y = dist_from_origin), aes(x,y)) +
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307 coord_polar(theta="x")
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311 [[file:../pics/golden_spiral-coord_polar-fail.png]]
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313 To ensure what I was trying to do is possible, I employ a specialised plotfunction instead
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314 #+BEGIN_SRC R :session R :results output graphics :file ../pics/golden_spiral-plotrix.png :exports both
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315 plotrix::radial.plot(dist_from_origin, seq_theta,rp.type="s", point.col = "blue")
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318 With that established and the original objective of the exercise achieved, it still would be nice to be able to accomplish this using ggplot2. To do so, the created sequence above needs to be converted to cartesian coordinates.
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319 The rectangular function equivalent of the golden spiral function r(t) defined above is a(t) = (r(t) cos(t), r(t) sin(t))
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320 It's not too hard to come up with a hack to convert one to the other.
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321 #+BEGIN_SRC R :session R
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322 cartesian_golden_spiral <- function(theta) {
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323 a <- polar_golden_spiral(theta)*cos(theta)
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324 b <- polar_golden_spiral(theta)*sin(theta)
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331 Applying that function to the same series of angles from above and stitching the resulting coordinates in a data frame. Note I'm enclosing the first expression in brackets, which prints it immediately, which is useful when the script is run interactively.
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332 #+BEGIN_SRC R :session R :exports code
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333 (serie <- sapply(seq_theta,cartesian_golden_spiral))
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334 df <- data.frame(t(serie))
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341 With everything now ready in the right coordinate system, it's now only a matter of setting some options to make the output look acceptable.
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342 #+BEGIN_SRC R :session R :results output graphics :file ../pics/golden_spiral-ggplot-coord-fixed.png :width 800 :height 800 :exports both
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343 ggplot(df, aes(x=X1,y=X2)) +
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344 geom_path(color="blue") +
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345 theme(panel.grid.minor = element_blank(),
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346 axis.text.x = element_blank(),
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347 axis.text.y = element_blank()) +
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348 scale_y_continuous(breaks = seq(-20,20,by=10)) +
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349 scale_x_continuous(breaks = seq(-20,50,by=10)) +
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351 labs(title = "Golden spiral",
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352 subtitle = "Another view on the Fibonacci sequence",
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353 caption = "Maths from https://www.intmath.com/blog/mathematics/golden-spiral-6512\nCode errors mine.",
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358 * Note on how this post was written.
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359 After a long hiatus, I set about using emacs, org-mode and ESS together to create this post. All code is part of an .org file, and gets exported to markdown using the orgmode conversion - C-c C-e m m.
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