--- /home/fdroid/fdroiddata/tmp/uk.ac.ed.inf.mandelbrotmaps_2.apk +++ /home/fdroid/fdroiddata/tmp/sigcp_uk.ac.ed.inf.mandelbrotmaps_2.apk │┄ Command `apktool d -f -k -m -o «TEMP»/diffoscope_0qdj4lum_tmp/tmpx4lu50zn_apk /home/fdroid/fdroiddata/tmp/uk.ac.ed.inf.mandelbrotmaps_2.apk` failed with exit code 1. (No output) │┄ Archive contents identical but files differ, possibly due to different compression levels. 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-00001b60: 5765 6c63 6f6d 6520 746f 203c 7370 616e Welcome to Mande -00001b90: 6c62 726f 7420 4d61 7073 3c2f 7370 616e lbrot Maps!

This -00001bb0: 6170 7020 6973 2064 6573 6967 6e65 6420 app is designed -00001bc0: 746f 2073 686f 7763 6173 6520 7468 6520 to showcase the -00001bd0: 7265 6c61 7469 6f6e 7368 6970 2062 6574 relationship bet -00001be0: 7765 656e 2074 6865 204d 616e 6465 6c62 ween the Mandelb -00001bf0: 726f 7420 7365 7420 616e 6420 4a75 6c69 rot set and Juli -00001c00: 6120 7365 7420 6672 6163 7461 6c73 2e20 a set fractals. -00001c10: 3c70 3e4d 6f76 6520 6172 6f75 6e64 2074

Move around t -00001c20: 6865 2066 7261 6374 616c 7320 7769 7468 he fractals with -00001c30: 206f 6e65 2066 696e 6765 7220 616e 6420 one finger and -00001c40: 7069 6e63 6820 746f 207a 6f6f 6d2e 2057 pinch to zoom. W -00001c50: 6865 6e20 796f 7520 7a6f 6f6d 2c20 7468 hen you zoom, th -00001c60: 6520 6672 6163 7461 6c20 7769 6c6c 2072 e fractal will r -00001c70: 652d 7265 6e64 6572 2077 6974 6820 6d6f e-render with mo -00001c80: 7265 2064 6574 6169 6c2e 203c 703e 486f re detail.

Ho -00001c90: 6c64 2079 6f75 7220 6669 6e67 6572 206f ld your finger o -00001ca0: 6e20 6120 706f 696e 7420 696e 2074 6865 n a point in the -00001cb0: 204d 616e 6465 6c62 726f 7420 6f72 2073 Mandelbrot or s -00001cc0: 656c 6563 7420 3c62 3e41 6464 204a 756c elect Add Jul -00001cd0: 6961 3c2f 623e 2066 726f 6d20 7468 6520 ia from the -00001ce0: 6d65 6e75 2074 6f20 6469 7370 6c61 7920 menu to display -00001cf0: 6120 4a75 6c69 6120 6672 6163 7461 6c2e a Julia fractal. -00001d00: 2054 6865 7265 2773 2061 2075 6e69 7175 There's a uniqu -00001d10: 6520 4a75 6c69 6120 6672 6163 7461 6c20 e Julia fractal -00001d20: 666f 7220 6561 6368 2070 6f69 6e74 2069 for each point i -00001d30: 6e20 7468 6520 4d61 6e64 656c 6272 6f74 n the Mandelbrot -00001d40: 2073 6574 202d 2063 6861 6e67 6520 7468 set - change th -00001d50: 6520 706f 696e 7420 6465 6669 6e69 6e67 e point defining -00001d60: 2069 7420 6279 2064 7261 6767 696e 6720 it by dragging -00001d70: 7468 6520 636f 6c6f 7572 6564 2070 696e the coloured pin -00001d80: 2e20 5461 7020 6974 2074 6f20 7365 6520 . Tap it to see -00001d90: 6974 2069 6e20 6d6f 7265 2064 6574 6169 it in more detai -00001da0: 6c2e 203c 703e 3c62 3e4d 616e 6465 6c62 l.

Mandelb -00001db0: 726f 7420 4d61 7073 3c2f 623e 2069 7320 rot Maps is -00001dc0: 6265 696e 6720 6465 7665 6c6f 7065 6420 being developed -00001dd0: 6173 206d 7920 756e 6465 7267 7261 6475 as my undergradu -00001de0: 6174 6520 6469 7373 6572 7461 7469 6f6e ate dissertation -00001df0: 2070 726f 6a65 6374 2c20 736f 2061 6e79 project, so any -00001e00: 2066 6565 6462 6163 6b20 796f 7520 6c65 feedback you le -00001e10: 6176 6520 6f6e 2074 6865 2047 6f6f 676c ave on the Googl -00001e20: 6520 506c 6179 2053 746f 7265 2070 6167 e Play Store pag -00001e30: 6520 6973 2067 7265 6174 6c79 2061 7070 e is greatly app -00001e40: 7265 6369 6174 6564 2e20 3c70 3e3c 623e reciated.

-00001e50: 4d61 6e64 656c 6272 6f74 204d 6170 733c Mandelbrot Maps< -00001e60: 2f62 3e20 6973 206f 7065 6e2d 736f 7572 /b> is open-sour -00001e70: 6365 2073 6f66 7477 6172 652c 2072 656c ce software, rel -00001e80: 6173 6564 2075 6e64 6572 2074 6865 2047 ased under the G -00001e90: 4e55 2047 656e 6572 616c 2050 7562 6c69 NU General Publi -00001ea0: 6320 4c69 6365 6e73 6520 2876 3329 2e20 c License (v3). -00001eb0: 5468 6520 736f 7572 6365 2063 6f64 6520 The source code -00001ec0: 6361 6e20 6265 2066 6f75 6e64 203c 6120 can be found here......

Abou -00001f20: 7420 6672 6163 7461 6c73 3c2f 6831 3e20 t fractals

-00001f30: 3c70 3e41 2066 7261 6374 616c 2069 7320

A fractal is -00001f40: 6120 2273 656c 662d 7369 6d69 6c61 7222 a "self-similar" -00001f50: 2073 6861 7065 2c20 6d65 616e 696e 6720 shape, meaning -00001f60: 7468 6174 2074 6865 2073 616d 6520 6465 that the same de -00001f70: 7461 696c 6564 2070 6174 7465 726e 2072 tailed pattern r -00001f80: 6570 6561 7473 206f 7665 7220 616e 6420 epeats over and -00001f90: 6f76 6572 2061 6761 696e 2061 7320 796f over again as yo -00001fa0: 7520 7a6f 6f6d 2069 6e2e 2049 6e20 7468 u zoom in. In th -00001fb0: 6520 4d61 6e64 656c 6272 6f74 2073 6574 e Mandelbrot set -00001fc0: 2066 7261 6374 616c 2c20 666f 7220 6578 fractal, for ex -00001fd0: 616d 706c 652c 2074 686f 7365 2064 6973 ample, those dis -00001fe0: 7469 6e63 7469 7665 2068 6561 7274 2d73 tinctive heart-s -00001ff0: 6861 7065 6420 6275 6c62 7320 7769 6c6c haped bulbs will -00002000: 206b 6565 7020 6170 7065 6172 696e 672e keep appearing. -00002010: 203c 703e 4672 6163 7461 6c73 2061 7265

Fractals are -00002020: 2067 656e 6572 6174 6564 2062 7920 6974 generated by it -00002030: 6572 6174 696e 6720 6120 6d61 7468 656d erating a mathem -00002040: 6174 6963 616c 2066 756e 6374 696f 6e20 atical function -00002050: 6f6e 2061 2073 6574 206f 6620 706f 696e on a set of poin -00002060: 7473 2e20 446f 696e 6720 6d6f 7265 2069 ts. Doing more i -00002070: 7465 7261 7469 6f6e 7320 7461 6b65 7320 terations takes -00002080: 6120 6c6f 6e67 6572 2074 696d 6520 6275 a longer time bu -00002090: 7420 7072 6f64 7563 6573 206d 6f72 6520 t produces more -000020a0: 6465 7461 696c 6564 2073 6861 7065 732e detailed shapes. -000020b0: 204d 6f72 6520 6974 6572 6174 696f 6e73 More iterations -000020c0: 2068 6176 6520 746f 2062 6520 646f 6e65 have to be done -000020d0: 2077 6865 6e20 7a6f 6f6d 696e 6720 696e when zooming in -000020e0: 2c20 746f 206b 6565 7020 7468 6520 7361 , to keep the sa -000020f0: 6d65 206c 6576 656c 206f 6620 6170 7061 me level of appa -00002100: 7265 6e74 2064 6574 6169 6c2e 203c 6831 rent detail.

About the Mande -00002120: 6c62 726f 7420 616e 6420 4a75 6c69 6120 lbrot and Julia -00002130: 7365 7473 3c2f 6831 3e20 3c70 3e50 726f sets

Pro -00002140: 6261 626c 7920 7468 6520 6d6f 7374 2072 bably the most r -00002150: 6563 676f 6e69 7361 626c 6520 6672 6163 ecgonisable frac -00002160: 7461 6c20 6f66 2074 6865 6d20 616c 6c2c tal of them all, -00002170: 2074 6865 204d 616e 6465 6c62 726f 7420 the Mandelbrot -00002180: 7365 7420 7761 7320 6469 7363 6f76 6572 set was discover -00002190: 6564 2061 6e64 2070 6f70 756c 6172 6973 ed and popularis -000021a0: 6564 2062 7920 6d61 7468 656d 6174 6963 ed by mathematic -000021b0: 6961 6e20 4265 6e6f 6974 204d 616e 6465 ian Benoit Mande -000021c0: 6c62 726f 742c 2077 686f 2061 6c73 6f20 lbrot, who also -000021d0: 696e 7665 6e74 6564 2074 6865 2074 6572 invented the ter -000021e0: 6d20 2266 7261 6374 616c 2220 746f 2064 m "fractal" to d -000021f0: 6573 6372 6962 6520 7468 6520 7368 6170 escribe the shap -00002200: 6573 2068 6520 7374 7564 6965 642e 2049 es he studied. I -00002210: 7420 7761 7320 6f6e 6520 6f66 2074 6865 t was one of the -00002220: 2066 6972 7374 2066 7261 6374 616c 7320 first fractals -00002230: 746f 2062 6520 7669 7375 616c 6973 6564 to be visualised -00002240: 206f 6e20 6120 636f 6d70 7574 6572 2c20 on a computer, -00002250: 6f6e 204d 6172 6368 2031 7374 2c20 3139 on March 1st, 19 -00002260: 3830 2c20 696e 2049 424d 2773 2054 686f 80, in IBM's Tho -00002270: 6d61 7320 4a2e 2057 6174 736f 6e20 5265 mas J. Watson Re -00002280: 7365 6172 6368 2043 656e 7472 652e 203c search Centre. < -00002290: 703e 4a75 6c69 6120 7365 7473 2061 7265 p>Julia sets are -000022a0: 2061 6e20 6f6c 6465 7220 6964 6561 2c20 an older idea, -000022b0: 6669 7273 7420 7265 7365 6172 6368 6564 first researched -000022c0: 2069 6e20 7468 6520 6561 726c 7920 3230 in the early 20 -000022d0: 7468 2063 656e 7475 7279 2062 7920 4761 th century by Ga -000022e0: 7374 6f6e 204a 756c 6961 2e20 5768 656e ston Julia. When -000022f0: 2061 204a 756c 6961 2073 6574 2069 7320 a Julia set is -00002300: 6465 6669 6e65 6420 7573 696e 6720 7468 defined using th -00002310: 6520 7361 6d65 2066 756e 6374 696f 6e20 e same function -00002320: 7468 6174 2064 6566 696e 6573 2074 6865 that defines the -00002330: 204d 616e 6465 6c62 726f 7420 7365 742c Mandelbrot set, -00002340: 2073 6f6d 6520 7665 7279 2069 6e74 6572 some very inter -00002350: 6573 7469 6e67 2062 6568 6176 696f 7572 esting behaviour -00002360: 2061 7070 6561 7273 2e20 4966 2074 6865 appears. If the -00002370: 204a 756c 6961 2073 6574 2069 7320 7061 Julia set is pa -00002380: 7261 6d65 7469 7269 7365 6420 6279 2061 rametirised by a -00002390: 2070 6f69 6e74 2069 6e73 6964 6520 7468 point inside th -000023a0: 6520 4d61 6e64 656c 6272 6f74 2073 6574 e Mandelbrot set -000023b0: 2c20 6974 2077 696c 6c20 6265 2063 6f6e , it will be con -000023c0: 6e65 6374 6564 2e20 4d6f 7669 6e67 206f nected. Moving o -000023d0: 7574 206f 6620 7468 6520 4d61 6e64 656c ut of the Mandel -000023e0: 6272 6f74 2073 6574 2063 6175 7365 7320 brot set causes -000023f0: 7468 6520 4a75 6c69 6120 6672 6163 7461 the Julia fracta -00002400: 6c20 746f 2062 7265 616b 2061 7061 7274 l to break apart -00002410: 2069 6e74 6f20 756e 636f 6e6e 6563 7465 into unconnecte -00002420: 6420 7069 6563 6573 2e20 3c70 3e54 6865 d pieces.

The -00002430: 204d 616e 6465 6c62 726f 7420 616e 6420 Mandelbrot and -00002440: 4a75 6c69 6120 7365 7473 2061 6c73 6f20 Julia sets also -00002450: 6c6f 6f6b 2073 696d 696c 6172 2069 6e20 look similar in -00002460: 6365 7274 6169 6e20 7265 6769 6f6e 732c certain regions, -00002470: 2070 6172 7469 6375 6c61 726c 7920 6f6e particularly on -00002480: 2074 6865 2062 6f72 6465 7273 206f 6620 the borders of -00002490: 7468 6520 4d61 6e64 656c 6272 6f74 2e20 the Mandelbrot. -000024a0: 4974 2069 7320 7468 6573 6520 7265 6c61 It is these rela -000024b0: 7469 6f6e 7368 6970 7320 7468 6174 203c tionships that < -000024c0: 623e 4d61 6e64 656c 6272 6f74 204d 6170 b>Mandelbrot Map -000024d0: 733c 2f62 3e20 7761 7320 6465 7665 6c6f s was develo -000024e0: 7065 6420 746f 2064 656d 6f6e 7374 7261 ped to demonstra -000024f0: 7465 2e20 3c68 313e 436f 6e74 726f 6c73 te.

Controls -00002500: 3c2f 6831 3e20 3c70 3e44 7261 6720 7769

Drag wi -00002510: 7468 2061 2073 696e 676c 6520 6669 6e67 th a single fing -00002520: 6572 2074 6f20 6d6f 7665 2061 726f 756e er to move aroun -00002530: 6420 7468 6520 6672 6163 7461 6c73 2e20 d the fractals. -00002540: 416e 7920 6e65 7720 6172 6561 7320 7769 Any new areas wi -00002550: 6c6c 2066 696c 6c20 7468 656d 7365 6c76 ll fill themselv -00002560: 6573 2069 6e20 6f6e 6365 2079 6f75 2073 es in once you s -00002570: 746f 702e 203c 703e 5573 6520 7477 6f20 top.

Use two -00002580: 6669 6e67 6572 7320 746f 207a 6f6f 6d20 fingers to zoom -00002590: 696e 2061 6e64 206f 7574 2e20 4167 6169 in and out. Agai -000025a0: 6e2c 2074 6865 2066 7261 6374 616c 2077 n, the fractal w -000025b0: 696c 6c20 7265 2d72 656e 6465 7220 6f6e ill re-render on -000025c0: 6365 2079 6f75 2073 746f 702e 203c 703e ce you stop.

-000025d0: 546f 2063 6f6d 7061 7265 2074 6865 204d To compare the M -000025e0: 616e 6465 6c62 726f 7420 616e 6420 4a75 andelbrot and Ju -000025f0: 6c69 6120 7365 7473 2c20 6c6f 6e67 2063 lia sets, long c -00002600: 6c69 636b 2061 2070 6f69 6e74 2069 6e20 lick a point in -00002610: 7468 6520 4d61 6e64 656c 6272 6f74 2073 the Mandelbrot s -00002620: 6574 206f 7220 7461 7020 3c62 3e41 6464 et or tap Add -00002630: 204a 756c 6961 3c2f 623e 206f 6e20 7468 Julia on th -00002640: 6520 6d65 6e75 2e20 4472 6167 2074 6865 e menu. Drag the -00002650: 2063 6f6c 6f75 7265 6420 6369 7263 6c65 coloured circle -00002660: 2061 726f 756e 6420 746f 2063 6861 6e67 around to chang -00002670: 6520 7468 6520 706f 696e 7420 7061 7261 e the point para -00002680: 6d65 7469 7269 7369 6e67 2074 6865 204a metirising the J -00002690: 756c 6961 2073 6574 2e20 546f 2073 6565 ulia set. To see -000026a0: 2074 6865 2066 7261 6374 616c 2069 6e20 the fractal in -000026b0: 6d6f 7265 2064 6574 6169 6c2c 2074 6170 more detail, tap -000026c0: 2069 742e 2054 6865 207a 6f6f 6d20 6c65 it. The zoom le -000026d0: 7665 6c20 616e 6420 7265 6769 6f6e 2064 vel and region d -000026e0: 6973 706c 6179 6564 2077 696c 6c20 7374 isplayed will st -000026f0: 6179 2063 6f6e 7374 616e 7420 6173 2079 ay constant as y -00002700: 6f75 2073 7761 7020 6265 7477 6565 6e20 ou swap between -00002710: 7468 6520 7477 6f20 6672 6163 7461 6c73 the two fractals -00002720: 2e20 5461 7020 3c62 3e52 6573 6574 3c2f . Tap Reset on the menu t -00002740: 6f20 7265 7475 726e 2074 6f20 7468 6520 o return to the -00002750: 6465 6661 756c 7420 6172 6561 2e20 4469 default area. Di -00002760: 736d 6973 7320 7468 6520 736d 616c 6c20 smiss the small -00002770: 6672 6163 7461 6c20 6279 2074 6170 7069 fractal by tappi -00002780: 6e67 203c 623e 5265 6d6f 7665 204a 756c ng Remove Jul -00002790: 6961 2f4d 616e 6465 6c62 726f 743c 2f62 ia/Mandelbrot on the menu. I -000027b0: 6620 796f 7520 6c6f 7365 2074 7261 636b f you lose track -000027c0: 206f 6620 7468 6520 4a75 6c69 6120 7069 of the Julia pi -000027d0: 6e2c 206c 6f6e 6720 636c 6963 6b20 6167 n, long click ag -000027e0: 6169 6e20 6f72 2061 6464 2061 6e64 2072 ain or add and r -000027f0: 656d 6f76 6520 7468 6520 4a75 6c69 6120 emove the Julia -00002800: 6672 6f6d 2074 6865 206d 656e 7520 746f from the menu to -00002810: 2062 7269 6e67 2069 7420 746f 2079 6f75 bring it to you -00002820: 2e20 3c68 313e 4f74 6865 7220 4d65 6e75 .

Other Menu -00002830: 204f 7074 696f 6e73 3c2f 6831 3e20 3c70 Options

Under Prefer -00002850: 656e 6365 733c 2f62 3e2c 2079 6f75 2063 ences, you c -00002860: 616e 2063 686f 6f73 6520 7468 6520 636f an choose the co -00002870: 6c6f 7572 696e 6720 7363 6865 6d65 2066 louring scheme f -00002880: 6f72 2065 6163 6820 6672 6163 7461 6c2c or each fractal, -00002890: 2074 6865 2063 6f6c 6f75 7220 6f66 2074 the colour of t -000028a0: 6865 204a 756c 6961 2070 696e 2c20 7768 he Julia pin, wh -000028b0: 6574 6865 7220 6f72 206e 6f74 2074 6f20 ether or not to -000028c0: 646f 2061 6e20 696e 6974 6961 6c20 6372 do an initial cr -000028d0: 7564 6520 7265 6e64 6572 696e 6720 2861 ude rendering (a -000028e0: 2062 6c6f 636b 6965 7220 6669 7273 7420 blockier first -000028f0: 7061 7373 2074 6861 7420 6769 7665 7320 pass that gives -00002900: 6120 6661 7374 6572 2066 6972 7374 2069 a faster first i -00002910: 6d70 7265 7373 696f 6e20 6275 7420 7461 mpression but ta -00002920: 6b65 7320 736c 6967 6874 6c79 206c 6f6e kes slightly lon -00002930: 6765 7220 6f76 6572 616c 6c29 2c20 616e ger overall), an -00002940: 6420 7768 6574 6865 7220 6f72 206e 6f74 d whether or not -00002950: 2074 6f20 7368 6f77 2074 6865 2074 696d to show the tim -00002960: 6520 666f 7220 6561 6368 2072 656e 6465 e for each rende -00002970: 7220 6166 7465 7220 6974 2773 2063 6f6d r after it's com -00002980: 706c 6574 652e 203c 703e 5461 7020 3c62 plete.

Tap Detail level to control the -000029b0: 206c 6576 656c 206f 6620 6465 7461 696c level of detail -000029c0: 206f 6e20 6561 6368 2066 7261 6374 616c on each fractal -000029d0: 2e20 4869 6768 6572 2064 6574 6169 6c20 . Higher detail -000029e0: 6d65 616e 7320 6d6f 7265 2069 7465 7261 means more itera -000029f0: 7469 6f6e 7320 7769 6c6c 2062 6520 646f tions will be do -00002a00: 6e65 2077 6865 6e20 7265 6e64 6572 696e ne when renderin -00002a10: 672c 2074 616b 696e 6720 6c6f 6e67 6572 g, taking longer -00002a20: 2062 7574 2067 6976 696e 6720 6120 636c but giving a cl -00002a30: 6561 7265 7220 7069 6374 7572 652e 203c earer picture. < -00002a40: 703e 3c62 3e53 6176 652f 5368 6172 653c p>Save/Share< -00002a50: 2f62 3e20 6c65 7473 2079 6f75 2073 6176 /b> lets you sav -00002a60: 6520 616e 2069 6d61 6765 206f 6620 7468 e an image of th -00002a70: 6520 6375 7272 656e 7420 6672 6163 7461 e current fracta -00002a80: 6c20 696e 746f 2079 6f75 7220 5069 6374 l into your Pict -00002a90: 7572 6573 2066 6f6c 6465 7220 6f72 2073 ures folder or s -00002aa0: 6861 7265 2069 7420 7573 696e 6720 656d hare it using em -00002ab0: 6169 6c2c 2046 6163 6562 6f6f 6b2c 2054 ail, Facebook, T -00002ac0: 7769 7474 6572 2c20 6574 632e 2028 6966 witter, etc. (if -00002ad0: 2079 6f75 2068 6176 6520 7468 6f73 6520 you have those -00002ae0: 6170 7073 2069 6e73 7461 6c6c 6564 292e apps installed). -00002af0: 203c 6831 3e41 626f 7574 204d 616e 6465

About Mande -00002b00: 6c62 726f 7420 4d61 7073 3c2f 6831 3e20 lbrot Maps

-00002b10: 3c70 3e3c 623e 4d61 6e64 656c 6272 6f74

Mandelbrot -00002b20: 204d 6170 733c 2f62 3e20 6973 2062 6569 Maps is bei -00002b30: 6e67 2064 6576 656c 6f70 6564 2061 7320 ng developed as -00002b40: 6d79 2075 6e64 6572 6772 6164 7561 7465 my undergraduate -00002b50: 2064 6973 7365 7274 6174 696f 6e20 7072 dissertation pr -00002b60: 6f6a 6563 742c 2073 6f20 616e 7920 6665 oject, so any fe -00002b70: 6564 6261 636b 2079 6f75 206c 6561 7665 edback you leave -00002b80: 206f 6e20 7468 6520 476f 6f67 6c65 2050 on the Google P -00002b90: 6c61 7920 5374 6f72 6520 7061 6765 2069 lay Store page i -00002ba0: 7320 6772 6561 746c 7920 6170 7072 6563 s greatly apprec -00002bb0: 6961 7465 642e 2049 6620 796f 7520 6669 iated. If you fi -00002bc0: 6e64 2061 6e79 2062 7567 732c 2070 6c65 nd any bugs, ple -00002bd0: 6173 6520 696e 636c 7564 6520 6173 206d ase include as m -00002be0: 7563 6820 6465 7461 696c 2061 7320 706f uch detail as po -00002bf0: 7373 6962 6c65 2c20 7375 6368 2061 7320 ssible, such as -00002c00: 7468 6520 6465 7669 6365 2079 6f75 2772 the device you'r -00002c10: 6520 7573 696e 672c 2077 6861 7420 796f e using, what yo -00002c20: 7520 7765 7265 2064 6f69 6e67 2077 6865 u were doing whe -00002c30: 6e20 7468 6520 6572 726f 7220 6f63 6375 n the error occu -00002c40: 7272 6564 2c20 616e 6420 7768 6174 2076 rred, and what v -00002c50: 6572 7369 6f6e 206f 6620 416e 6472 6f69 ersion of Androi -00002c60: 6420 796f 7527 7265 2072 756e 6e69 6e67 d you're running -00002c70: 2e20 3c70 3e3c 623e 4d61 6e64 656c 6272 .

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Welcome t +00004810: 6f20 3c73 7061 6e20 7374 796c 653d 666f o Mandelbrot Map +00004840: 733c 2f73 7061 6e3e 213c 2f68 323e 203c s!

< +00004850: 703e 5468 6973 2061 7070 2069 7320 6465 p>This app is de +00004860: 7369 676e 6564 2074 6f20 7368 6f77 6361 signed to showca +00004870: 7365 2074 6865 2072 656c 6174 696f 6e73 se the relations +00004880: 6869 7020 6265 7477 6565 6e20 7468 6520 hip between the +00004890: 4d61 6e64 656c 6272 6f74 2073 6574 2061 Mandelbrot set a +000048a0: 6e64 204a 756c 6961 2073 6574 2066 7261 nd Julia set fra +000048b0: 6374 616c 732e 203c 703e 4d6f 7665 2061 ctals.

Move a +000048c0: 726f 756e 6420 7468 6520 6672 6163 7461 round the fracta +000048d0: 6c73 2077 6974 6820 6f6e 6520 6669 6e67 ls with one fing +000048e0: 6572 2061 6e64 2070 696e 6368 2074 6f20 er and pinch to +000048f0: 7a6f 6f6d 2e20 5768 656e 2079 6f75 207a zoom. When you z +00004900: 6f6f 6d2c 2074 6865 2066 7261 6374 616c oom, the fractal +00004910: 2077 696c 6c20 7265 2d72 656e 6465 7220 will re-render +00004920: 7769 7468 206d 6f72 6520 6465 7461 696c with more detail +00004930: 2e20 3c70 3e48 6f6c 6420 796f 7572 2066 .

Hold your f +00004940: 696e 6765 7220 6f6e 2061 2070 6f69 6e74 inger on a point +00004950: 2069 6e20 7468 6520 4d61 6e64 656c 6272 in the Mandelbr +00004960: 6f74 206f 7220 7365 6c65 6374 203c 623e ot or select +00004970: 4164 6420 4a75 6c69 613c 2f62 3e20 6672 Add Julia fr +00004980: 6f6d 2074 6865 206d 656e 7520 746f 2064 om the menu to d +00004990: 6973 706c 6179 2061 204a 756c 6961 2066 isplay a Julia f +000049a0: 7261 6374 616c 2e20 5468 6572 6527 7320 ractal. There's +000049b0: 6120 756e 6971 7565 204a 756c 6961 2066 a unique Julia f +000049c0: 7261 6374 616c 2066 6f72 2065 6163 6820 ractal for each +000049d0: 706f 696e 7420 696e 2074 6865 204d 616e point in the Man +000049e0: 6465 6c62 726f 7420 7365 7420 2d20 6368 delbrot set - ch +000049f0: 616e 6765 2074 6865 2070 6f69 6e74 2064 ange the point d +00004a00: 6566 696e 696e 6720 6974 2062 7920 6472 efining it by dr +00004a10: 6167 6769 6e67 2074 6865 2063 6f6c 6f75 agging the colou +00004a20: 7265 6420 7069 6e2e 2054 6170 2069 7420 red pin. Tap it +00004a30: 746f 2073 6565 2069 7420 696e 206d 6f72 to see it in mor +00004a40: 6520 6465 7461 696c 2e20 3c70 3e3c 623e e detail.

+00004a50: 4d61 6e64 656c 6272 6f74 204d 6170 733c Mandelbrot Maps< +00004a60: 2f62 3e20 6973 2062 6569 6e67 2064 6576 /b> is being dev +00004a70: 656c 6f70 6564 2061 7320 6d79 2075 6e64 eloped as my und +00004a80: 6572 6772 6164 7561 7465 2064 6973 7365 ergraduate disse +00004a90: 7274 6174 696f 6e20 7072 6f6a 6563 742c rtation project, +00004aa0: 2073 6f20 616e 7920 6665 6564 6261 636b so any feedback +00004ab0: 2079 6f75 206c 6561 7665 206f 6e20 7468 you leave on th +00004ac0: 6520 476f 6f67 6c65 2050 6c61 7920 5374 e Google Play St +00004ad0: 6f72 6520 7061 6765 2069 7320 6772 6561 ore page is grea +00004ae0: 746c 7920 6170 7072 6563 6961 7465 642e tly appreciated. +00004af0: 203c 703e 3c62 3e4d 616e 6465 6c62 726f

Mandelbro +00004b00: 7420 4d61 7073 3c2f 623e 2069 7320 6f70 t Maps is op +00004b10: 656e 2d73 6f75 7263 6520 736f 6674 7761 en-source softwa +00004b20: 7265 2c20 7265 6c61 7365 6420 756e 6465 re, relased unde +00004b30: 7220 7468 6520 474e 5520 4765 6e65 7261 r the GNU Genera +00004b40: 6c20 5075 626c 6963 204c 6963 656e 7365 l Public License +00004b50: 2028 7633 292e 2054 6865 2073 6f75 7263 (v3). The sourc +00004b60: 6520 636f 6465 2063 616e 2062 6520 666f e code can be fo +00004b70: 756e 6420 3c61 2068 7265 663d 6874 7470 und here......< +00004bc0: 6831 3e41 626f 7574 2066 7261 6374 616c h1>About fractal +00004bd0: 733c 2f68 313e 203c 703e 4120 6672 6163 s

A frac +00004be0: 7461 6c20 6973 2061 2022 7365 6c66 2d73 tal is a "self-s +00004bf0: 696d 696c 6172 2220 7368 6170 652c 206d imilar" shape, m +00004c00: 6561 6e69 6e67 2074 6861 7420 7468 6520 eaning that the +00004c10: 7361 6d65 2064 6574 6169 6c65 6420 7061 same detailed pa +00004c20: 7474 6572 6e20 7265 7065 6174 7320 6f76 ttern repeats ov +00004c30: 6572 2061 6e64 206f 7665 7220 6167 6169 er and over agai +00004c40: 6e20 6173 2079 6f75 207a 6f6f 6d20 696e n as you zoom in +00004c50: 2e20 496e 2074 6865 204d 616e 6465 6c62 . In the Mandelb +00004c60: 726f 7420 7365 7420 6672 6163 7461 6c2c rot set fractal, +00004c70: 2066 6f72 2065 7861 6d70 6c65 2c20 7468 for example, th +00004c80: 6f73 6520 6469 7374 696e 6374 6976 6520 ose distinctive +00004c90: 6865 6172 742d 7368 6170 6564 2062 756c heart-shaped bul +00004ca0: 6273 2077 696c 6c20 6b65 6570 2061 7070 bs will keep app +00004cb0: 6561 7269 6e67 2e20 3c70 3e46 7261 6374 earing.

Fract +00004cc0: 616c 7320 6172 6520 6765 6e65 7261 7465 als are generate +00004cd0: 6420 6279 2069 7465 7261 7469 6e67 2061 d by iterating a +00004ce0: 206d 6174 6865 6d61 7469 6361 6c20 6675 mathematical fu +00004cf0: 6e63 7469 6f6e 206f 6e20 6120 7365 7420 nction on a set +00004d00: 6f66 2070 6f69 6e74 732e 2044 6f69 6e67 of points. Doing +00004d10: 206d 6f72 6520 6974 6572 6174 696f 6e73 more iterations +00004d20: 2074 616b 6573 2061 206c 6f6e 6765 7220 takes a longer +00004d30: 7469 6d65 2062 7574 2070 726f 6475 6365 time but produce +00004d40: 7320 6d6f 7265 2064 6574 6169 6c65 6420 s more detailed +00004d50: 7368 6170 6573 2e20 4d6f 7265 2069 7465 shapes. More ite +00004d60: 7261 7469 6f6e 7320 6861 7665 2074 6f20 rations have to +00004d70: 6265 2064 6f6e 6520 7768 656e 207a 6f6f be done when zoo +00004d80: 6d69 6e67 2069 6e2c 2074 6f20 6b65 6570 ming in, to keep +00004d90: 2074 6865 2073 616d 6520 6c65 7665 6c20 the same level +00004da0: 6f66 2061 7070 6172 656e 7420 6465 7461 of apparent deta +00004db0: 696c 2e20 3c68 313e 4162 6f75 7420 7468 il.

About th +00004dc0: 6520 4d61 6e64 656c 6272 6f74 2061 6e64 e Mandelbrot and +00004dd0: 204a 756c 6961 2073 6574 733c 2f68 313e Julia sets

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Probably the +00004df0: 206d 6f73 7420 7265 6367 6f6e 6973 6162 most recgonisab +00004e00: 6c65 2066 7261 6374 616c 206f 6620 7468 le fractal of th +00004e10: 656d 2061 6c6c 2c20 7468 6520 4d61 6e64 em all, the Mand +00004e20: 656c 6272 6f74 2073 6574 2077 6173 2064 elbrot set was d +00004e30: 6973 636f 7665 7265 6420 616e 6420 706f iscovered and po +00004e40: 7075 6c61 7269 7365 6420 6279 206d 6174 pularised by mat +00004e50: 6865 6d61 7469 6369 616e 2042 656e 6f69 hematician Benoi +00004e60: 7420 4d61 6e64 656c 6272 6f74 2c20 7768 t Mandelbrot, wh +00004e70: 6f20 616c 736f 2069 6e76 656e 7465 6420 o also invented +00004e80: 7468 6520 7465 726d 2022 6672 6163 7461 the term "fracta +00004e90: 6c22 2074 6f20 6465 7363 7269 6265 2074 l" to describe t +00004ea0: 6865 2073 6861 7065 7320 6865 2073 7475 he shapes he stu +00004eb0: 6469 6564 2e20 4974 2077 6173 206f 6e65 died. It was one +00004ec0: 206f 6620 7468 6520 6669 7273 7420 6672 of the first fr +00004ed0: 6163 7461 6c73 2074 6f20 6265 2076 6973 actals to be vis +00004ee0: 7561 6c69 7365 6420 6f6e 2061 2063 6f6d ualised on a com +00004ef0: 7075 7465 722c 206f 6e20 4d61 7263 6820 puter, on March +00004f00: 3173 742c 2031 3938 302c 2069 6e20 4942 1st, 1980, in IB +00004f10: 4d27 7320 5468 6f6d 6173 204a 2e20 5761 M's Thomas J. Wa +00004f20: 7473 6f6e 2052 6573 6561 7263 6820 4365 tson Research Ce +00004f30: 6e74 7265 2e20 3c70 3e4a 756c 6961 2073 ntre.

Julia s +00004f40: 6574 7320 6172 6520 616e 206f 6c64 6572 ets are an older +00004f50: 2069 6465 612c 2066 6972 7374 2072 6573 idea, first res +00004f60: 6561 7263 6865 6420 696e 2074 6865 2065 earched in the e +00004f70: 6172 6c79 2032 3074 6820 6365 6e74 7572 arly 20th centur +00004f80: 7920 6279 2047 6173 746f 6e20 4a75 6c69 y by Gaston Juli +00004f90: 612e 2057 6865 6e20 6120 4a75 6c69 6120 a. When a Julia +00004fa0: 7365 7420 6973 2064 6566 696e 6564 2075 set is defined u +00004fb0: 7369 6e67 2074 6865 2073 616d 6520 6675 sing the same fu +00004fc0: 6e63 7469 6f6e 2074 6861 7420 6465 6669 nction that defi +00004fd0: 6e65 7320 7468 6520 4d61 6e64 656c 6272 nes the Mandelbr +00004fe0: 6f74 2073 6574 2c20 736f 6d65 2076 6572 ot set, some ver +00004ff0: 7920 696e 7465 7265 7374 696e 6720 6265 y interesting be +00005000: 6861 7669 6f75 7220 6170 7065 6172 732e haviour appears. +00005010: 2049 6620 7468 6520 4a75 6c69 6120 7365 If the Julia se +00005020: 7420 6973 2070 6172 616d 6574 6972 6973 t is parametiris +00005030: 6564 2062 7920 6120 706f 696e 7420 696e ed by a point in +00005040: 7369 6465 2074 6865 204d 616e 6465 6c62 side the Mandelb +00005050: 726f 7420 7365 742c 2069 7420 7769 6c6c rot set, it will +00005060: 2062 6520 636f 6e6e 6563 7465 642e 204d be connected. M +00005070: 6f76 696e 6720 6f75 7420 6f66 2074 6865 oving out of the +00005080: 204d 616e 6465 6c62 726f 7420 7365 7420 Mandelbrot set +00005090: 6361 7573 6573 2074 6865 204a 756c 6961 causes the Julia +000050a0: 2066 7261 6374 616c 2074 6f20 6272 6561 fractal to brea +000050b0: 6b20 6170 6172 7420 696e 746f 2075 6e63 k apart into unc +000050c0: 6f6e 6e65 6374 6564 2070 6965 6365 732e onnected pieces. +000050d0: 203c 703e 5468 6520 4d61 6e64 656c 6272

The Mandelbr +000050e0: 6f74 2061 6e64 204a 756c 6961 2073 6574 ot and Julia set +000050f0: 7320 616c 736f 206c 6f6f 6b20 7369 6d69 s also look simi +00005100: 6c61 7220 696e 2063 6572 7461 696e 2072 lar in certain r +00005110: 6567 696f 6e73 2c20 7061 7274 6963 756c egions, particul +00005120: 6172 6c79 206f 6e20 7468 6520 626f 7264 arly on the bord +00005130: 6572 7320 6f66 2074 6865 204d 616e 6465 ers of the Mande +00005140: 6c62 726f 742e 2049 7420 6973 2074 6865 lbrot. It is the +00005150: 7365 2072 656c 6174 696f 6e73 6869 7073 se relationships +00005160: 2074 6861 7420 3c62 3e4d 616e 6465 6c62 that Mandelb +00005170: 726f 7420 4d61 7073 3c2f 623e 2077 6173 rot Maps was +00005180: 2064 6576 656c 6f70 6564 2074 6f20 6465 developed to de +00005190: 6d6f 6e73 7472 6174 652e 203c 6831 3e43 monstrate.

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7474 696e brot Maps settin │ -00001ae0: 6773 0001 0173 0006 0644 6574 6169 6c00 gs...s...Detail. │ -00001af0: 6565 2041 6e20 416e 6472 6f69 6420 706f ee An Android po │ -00001b00: 7274 206f 6620 4d61 6e64 656c 6272 6f74 rt of Mandelbrot │ -00001b10: 204d 6170 732c 2063 7572 7265 6e74 6c79 Maps, currently │ -00001b20: 2063 6170 6162 6c65 206f 6620 6469 7370 capable of disp │ -00001b30: 6c61 7969 6e67 2061 6e64 206d 6f76 696e laying and movin │ -00001b40: 6720 6120 4d61 6e64 656c 6272 6f74 2066 g a Mandelbrot f │ -00001b50: 7261 6374 616c 2e00 83b7 83b7 3c68 323e ractal......

│ -00001b60: 5765 6c63 6f6d 6520 746f 203c 7370 616e Welcome to Mande │ -00001b90: 6c62 726f 7420 4d61 7073 3c2f 7370 616e lbrot Maps!

This │ -00001bb0: 6170 7020 6973 2064 6573 6967 6e65 6420 app is designed │ -00001bc0: 746f 2073 686f 7763 6173 6520 7468 6520 to showcase the │ -00001bd0: 7265 6c61 7469 6f6e 7368 6970 2062 6574 relationship bet │ -00001be0: 7765 656e 2074 6865 204d 616e 6465 6c62 ween the Mandelb │ -00001bf0: 726f 7420 7365 7420 616e 6420 4a75 6c69 rot set and Juli │ -00001c00: 6120 7365 7420 6672 6163 7461 6c73 2e20 a set fractals. │ -00001c10: 3c70 3e4d 6f76 6520 6172 6f75 6e64 2074

Move around t │ -00001c20: 6865 2066 7261 6374 616c 7320 7769 7468 he fractals with │ -00001c30: 206f 6e65 2066 696e 6765 7220 616e 6420 one finger and │ -00001c40: 7069 6e63 6820 746f 207a 6f6f 6d2e 2057 pinch to zoom. W │ -00001c50: 6865 6e20 796f 7520 7a6f 6f6d 2c20 7468 hen you zoom, th │ -00001c60: 6520 6672 6163 7461 6c20 7769 6c6c 2072 e fractal will r │ -00001c70: 652d 7265 6e64 6572 2077 6974 6820 6d6f e-render with mo │ -00001c80: 7265 2064 6574 6169 6c2e 203c 703e 486f re detail.

Ho │ -00001c90: 6c64 2079 6f75 7220 6669 6e67 6572 206f ld your finger o │ -00001ca0: 6e20 6120 706f 696e 7420 696e 2074 6865 n a point in the │ -00001cb0: 204d 616e 6465 6c62 726f 7420 6f72 2073 Mandelbrot or s │ -00001cc0: 656c 6563 7420 3c62 3e41 6464 204a 756c elect Add Jul │ -00001cd0: 6961 3c2f 623e 2066 726f 6d20 7468 6520 ia from the │ -00001ce0: 6d65 6e75 2074 6f20 6469 7370 6c61 7920 menu to display │ -00001cf0: 6120 4a75 6c69 6120 6672 6163 7461 6c2e a Julia fractal. │ -00001d00: 2054 6865 7265 2773 2061 2075 6e69 7175 There's a uniqu │ -00001d10: 6520 4a75 6c69 6120 6672 6163 7461 6c20 e Julia fractal │ -00001d20: 666f 7220 6561 6368 2070 6f69 6e74 2069 for each point i │ -00001d30: 6e20 7468 6520 4d61 6e64 656c 6272 6f74 n the Mandelbrot │ -00001d40: 2073 6574 202d 2063 6861 6e67 6520 7468 set - change th │ -00001d50: 6520 706f 696e 7420 6465 6669 6e69 6e67 e point defining │ -00001d60: 2069 7420 6279 2064 7261 6767 696e 6720 it by dragging │ -00001d70: 7468 6520 636f 6c6f 7572 6564 2070 696e the coloured pin │ -00001d80: 2e20 5461 7020 6974 2074 6f20 7365 6520 . Tap it to see │ -00001d90: 6974 2069 6e20 6d6f 7265 2064 6574 6169 it in more detai │ -00001da0: 6c2e 203c 703e 3c62 3e4d 616e 6465 6c62 l.

Mandelb │ -00001db0: 726f 7420 4d61 7073 3c2f 623e 2069 7320 rot Maps is │ -00001dc0: 6265 696e 6720 6465 7665 6c6f 7065 6420 being developed │ -00001dd0: 6173 206d 7920 756e 6465 7267 7261 6475 as my undergradu │ -00001de0: 6174 6520 6469 7373 6572 7461 7469 6f6e ate dissertation │ -00001df0: 2070 726f 6a65 6374 2c20 736f 2061 6e79 project, so any │ -00001e00: 2066 6565 6462 6163 6b20 796f 7520 6c65 feedback you le │ -00001e10: 6176 6520 6f6e 2074 6865 2047 6f6f 676c ave on the Googl │ -00001e20: 6520 506c 6179 2053 746f 7265 2070 6167 e Play Store pag │ -00001e30: 6520 6973 2067 7265 6174 6c79 2061 7070 e is greatly app │ -00001e40: 7265 6369 6174 6564 2e20 3c70 3e3c 623e reciated.

│ -00001e50: 4d61 6e64 656c 6272 6f74 204d 6170 733c Mandelbrot Maps< │ -00001e60: 2f62 3e20 6973 206f 7065 6e2d 736f 7572 /b> is open-sour │ -00001e70: 6365 2073 6f66 7477 6172 652c 2072 656c ce software, rel │ -00001e80: 6173 6564 2075 6e64 6572 2074 6865 2047 ased under the G │ -00001e90: 4e55 2047 656e 6572 616c 2050 7562 6c69 NU General Publi │ -00001ea0: 6320 4c69 6365 6e73 6520 2876 3329 2e20 c License (v3). │ -00001eb0: 5468 6520 736f 7572 6365 2063 6f64 6520 The source code │ -00001ec0: 6361 6e20 6265 2066 6f75 6e64 203c 6120 can be found here......

Abou │ -00001f20: 7420 6672 6163 7461 6c73 3c2f 6831 3e20 t fractals

│ -00001f30: 3c70 3e41 2066 7261 6374 616c 2069 7320

A fractal is │ -00001f40: 6120 2273 656c 662d 7369 6d69 6c61 7222 a "self-similar" │ -00001f50: 2073 6861 7065 2c20 6d65 616e 696e 6720 shape, meaning │ -00001f60: 7468 6174 2074 6865 2073 616d 6520 6465 that the same de │ -00001f70: 7461 696c 6564 2070 6174 7465 726e 2072 tailed pattern r │ -00001f80: 6570 6561 7473 206f 7665 7220 616e 6420 epeats over and │ -00001f90: 6f76 6572 2061 6761 696e 2061 7320 796f over again as yo │ -00001fa0: 7520 7a6f 6f6d 2069 6e2e 2049 6e20 7468 u zoom in. In th │ -00001fb0: 6520 4d61 6e64 656c 6272 6f74 2073 6574 e Mandelbrot set │ -00001fc0: 2066 7261 6374 616c 2c20 666f 7220 6578 fractal, for ex │ -00001fd0: 616d 706c 652c 2074 686f 7365 2064 6973 ample, those dis │ -00001fe0: 7469 6e63 7469 7665 2068 6561 7274 2d73 tinctive heart-s │ -00001ff0: 6861 7065 6420 6275 6c62 7320 7769 6c6c haped bulbs will │ -00002000: 206b 6565 7020 6170 7065 6172 696e 672e keep appearing. │ -00002010: 203c 703e 4672 6163 7461 6c73 2061 7265

Fractals are │ -00002020: 2067 656e 6572 6174 6564 2062 7920 6974 generated by it │ -00002030: 6572 6174 696e 6720 6120 6d61 7468 656d erating a mathem │ -00002040: 6174 6963 616c 2066 756e 6374 696f 6e20 atical function │ -00002050: 6f6e 2061 2073 6574 206f 6620 706f 696e on a set of poin │ -00002060: 7473 2e20 446f 696e 6720 6d6f 7265 2069 ts. Doing more i │ -00002070: 7465 7261 7469 6f6e 7320 7461 6b65 7320 terations takes │ -00002080: 6120 6c6f 6e67 6572 2074 696d 6520 6275 a longer time bu │ -00002090: 7420 7072 6f64 7563 6573 206d 6f72 6520 t produces more │ -000020a0: 6465 7461 696c 6564 2073 6861 7065 732e detailed shapes. │ -000020b0: 204d 6f72 6520 6974 6572 6174 696f 6e73 More iterations │ -000020c0: 2068 6176 6520 746f 2062 6520 646f 6e65 have to be done │ -000020d0: 2077 6865 6e20 7a6f 6f6d 696e 6720 696e when zooming in │ -000020e0: 2c20 746f 206b 6565 7020 7468 6520 7361 , to keep the sa │ -000020f0: 6d65 206c 6576 656c 206f 6620 6170 7061 me level of appa │ -00002100: 7265 6e74 2064 6574 6169 6c2e 203c 6831 rent detail.

About the Mande │ -00002120: 6c62 726f 7420 616e 6420 4a75 6c69 6120 lbrot and Julia │ -00002130: 7365 7473 3c2f 6831 3e20 3c70 3e50 726f sets

Pro │ -00002140: 6261 626c 7920 7468 6520 6d6f 7374 2072 bably the most r │ -00002150: 6563 676f 6e69 7361 626c 6520 6672 6163 ecgonisable frac │ -00002160: 7461 6c20 6f66 2074 6865 6d20 616c 6c2c tal of them all, │ -00002170: 2074 6865 204d 616e 6465 6c62 726f 7420 the Mandelbrot │ -00002180: 7365 7420 7761 7320 6469 7363 6f76 6572 set was discover │ -00002190: 6564 2061 6e64 2070 6f70 756c 6172 6973 ed and popularis │ -000021a0: 6564 2062 7920 6d61 7468 656d 6174 6963 ed by mathematic │ -000021b0: 6961 6e20 4265 6e6f 6974 204d 616e 6465 ian Benoit Mande │ -000021c0: 6c62 726f 742c 2077 686f 2061 6c73 6f20 lbrot, who also │ -000021d0: 696e 7665 6e74 6564 2074 6865 2074 6572 invented the ter │ -000021e0: 6d20 2266 7261 6374 616c 2220 746f 2064 m "fractal" to d │ -000021f0: 6573 6372 6962 6520 7468 6520 7368 6170 escribe the shap │ -00002200: 6573 2068 6520 7374 7564 6965 642e 2049 es he studied. I │ -00002210: 7420 7761 7320 6f6e 6520 6f66 2074 6865 t was one of the │ -00002220: 2066 6972 7374 2066 7261 6374 616c 7320 first fractals │ -00002230: 746f 2062 6520 7669 7375 616c 6973 6564 to be visualised │ -00002240: 206f 6e20 6120 636f 6d70 7574 6572 2c20 on a computer, │ -00002250: 6f6e 204d 6172 6368 2031 7374 2c20 3139 on March 1st, 19 │ -00002260: 3830 2c20 696e 2049 424d 2773 2054 686f 80, in IBM's Tho │ -00002270: 6d61 7320 4a2e 2057 6174 736f 6e20 5265 mas J. Watson Re │ -00002280: 7365 6172 6368 2043 656e 7472 652e 203c search Centre. < │ -00002290: 703e 4a75 6c69 6120 7365 7473 2061 7265 p>Julia sets are │ -000022a0: 2061 6e20 6f6c 6465 7220 6964 6561 2c20 an older idea, │ -000022b0: 6669 7273 7420 7265 7365 6172 6368 6564 first researched │ -000022c0: 2069 6e20 7468 6520 6561 726c 7920 3230 in the early 20 │ -000022d0: 7468 2063 656e 7475 7279 2062 7920 4761 th century by Ga │ -000022e0: 7374 6f6e 204a 756c 6961 2e20 5768 656e ston Julia. When │ -000022f0: 2061 204a 756c 6961 2073 6574 2069 7320 a Julia set is │ -00002300: 6465 6669 6e65 6420 7573 696e 6720 7468 defined using th │ -00002310: 6520 7361 6d65 2066 756e 6374 696f 6e20 e same function │ -00002320: 7468 6174 2064 6566 696e 6573 2074 6865 that defines the │ -00002330: 204d 616e 6465 6c62 726f 7420 7365 742c Mandelbrot set, │ -00002340: 2073 6f6d 6520 7665 7279 2069 6e74 6572 some very inter │ -00002350: 6573 7469 6e67 2062 6568 6176 696f 7572 esting behaviour │ -00002360: 2061 7070 6561 7273 2e20 4966 2074 6865 appears. If the │ -00002370: 204a 756c 6961 2073 6574 2069 7320 7061 Julia set is pa │ -00002380: 7261 6d65 7469 7269 7365 6420 6279 2061 rametirised by a │ -00002390: 2070 6f69 6e74 2069 6e73 6964 6520 7468 point inside th │ -000023a0: 6520 4d61 6e64 656c 6272 6f74 2073 6574 e Mandelbrot set │ -000023b0: 2c20 6974 2077 696c 6c20 6265 2063 6f6e , it will be con │ -000023c0: 6e65 6374 6564 2e20 4d6f 7669 6e67 206f nected. Moving o │ -000023d0: 7574 206f 6620 7468 6520 4d61 6e64 656c ut of the Mandel │ -000023e0: 6272 6f74 2073 6574 2063 6175 7365 7320 brot set causes │ -000023f0: 7468 6520 4a75 6c69 6120 6672 6163 7461 the Julia fracta │ -00002400: 6c20 746f 2062 7265 616b 2061 7061 7274 l to break apart │ -00002410: 2069 6e74 6f20 756e 636f 6e6e 6563 7465 into unconnecte │ -00002420: 6420 7069 6563 6573 2e20 3c70 3e54 6865 d pieces.

The │ -00002430: 204d 616e 6465 6c62 726f 7420 616e 6420 Mandelbrot and │ -00002440: 4a75 6c69 6120 7365 7473 2061 6c73 6f20 Julia sets also │ -00002450: 6c6f 6f6b 2073 696d 696c 6172 2069 6e20 look similar in │ -00002460: 6365 7274 6169 6e20 7265 6769 6f6e 732c certain regions, │ -00002470: 2070 6172 7469 6375 6c61 726c 7920 6f6e particularly on │ -00002480: 2074 6865 2062 6f72 6465 7273 206f 6620 the borders of │ -00002490: 7468 6520 4d61 6e64 656c 6272 6f74 2e20 the Mandelbrot. │ -000024a0: 4974 2069 7320 7468 6573 6520 7265 6c61 It is these rela │ -000024b0: 7469 6f6e 7368 6970 7320 7468 6174 203c tionships that < │ -000024c0: 623e 4d61 6e64 656c 6272 6f74 204d 6170 b>Mandelbrot Map │ -000024d0: 733c 2f62 3e20 7761 7320 6465 7665 6c6f s was develo │ -000024e0: 7065 6420 746f 2064 656d 6f6e 7374 7261 ped to demonstra │ -000024f0: 7465 2e20 3c68 313e 436f 6e74 726f 6c73 te.

Controls │ -00002500: 3c2f 6831 3e20 3c70 3e44 7261 6720 7769

Drag wi │ -00002510: 7468 2061 2073 696e 676c 6520 6669 6e67 th a single fing │ -00002520: 6572 2074 6f20 6d6f 7665 2061 726f 756e er to move aroun │ -00002530: 6420 7468 6520 6672 6163 7461 6c73 2e20 d the fractals. │ -00002540: 416e 7920 6e65 7720 6172 6561 7320 7769 Any new areas wi │ -00002550: 6c6c 2066 696c 6c20 7468 656d 7365 6c76 ll fill themselv │ -00002560: 6573 2069 6e20 6f6e 6365 2079 6f75 2073 es in once you s │ -00002570: 746f 702e 203c 703e 5573 6520 7477 6f20 top.

Use two │ -00002580: 6669 6e67 6572 7320 746f 207a 6f6f 6d20 fingers to zoom │ -00002590: 696e 2061 6e64 206f 7574 2e20 4167 6169 in and out. Agai │ -000025a0: 6e2c 2074 6865 2066 7261 6374 616c 2077 n, the fractal w │ -000025b0: 696c 6c20 7265 2d72 656e 6465 7220 6f6e ill re-render on │ -000025c0: 6365 2079 6f75 2073 746f 702e 203c 703e ce you stop.

│ -000025d0: 546f 2063 6f6d 7061 7265 2074 6865 204d To compare the M │ -000025e0: 616e 6465 6c62 726f 7420 616e 6420 4a75 andelbrot and Ju │ -000025f0: 6c69 6120 7365 7473 2c20 6c6f 6e67 2063 lia sets, long c │ -00002600: 6c69 636b 2061 2070 6f69 6e74 2069 6e20 lick a point in │ -00002610: 7468 6520 4d61 6e64 656c 6272 6f74 2073 the Mandelbrot s │ -00002620: 6574 206f 7220 7461 7020 3c62 3e41 6464 et or tap Add │ -00002630: 204a 756c 6961 3c2f 623e 206f 6e20 7468 Julia on th │ -00002640: 6520 6d65 6e75 2e20 4472 6167 2074 6865 e menu. Drag the │ -00002650: 2063 6f6c 6f75 7265 6420 6369 7263 6c65 coloured circle │ -00002660: 2061 726f 756e 6420 746f 2063 6861 6e67 around to chang │ -00002670: 6520 7468 6520 706f 696e 7420 7061 7261 e the point para │ -00002680: 6d65 7469 7269 7369 6e67 2074 6865 204a metirising the J │ -00002690: 756c 6961 2073 6574 2e20 546f 2073 6565 ulia set. To see │ -000026a0: 2074 6865 2066 7261 6374 616c 2069 6e20 the fractal in │ -000026b0: 6d6f 7265 2064 6574 6169 6c2c 2074 6170 more detail, tap │ -000026c0: 2069 742e 2054 6865 207a 6f6f 6d20 6c65 it. The zoom le │ -000026d0: 7665 6c20 616e 6420 7265 6769 6f6e 2064 vel and region d │ -000026e0: 6973 706c 6179 6564 2077 696c 6c20 7374 isplayed will st │ -000026f0: 6179 2063 6f6e 7374 616e 7420 6173 2079 ay constant as y │ -00002700: 6f75 2073 7761 7020 6265 7477 6565 6e20 ou swap between │ -00002710: 7468 6520 7477 6f20 6672 6163 7461 6c73 the two fractals │ -00002720: 2e20 5461 7020 3c62 3e52 6573 6574 3c2f . Tap Reset on the menu t │ -00002740: 6f20 7265 7475 726e 2074 6f20 7468 6520 o return to the │ -00002750: 6465 6661 756c 7420 6172 6561 2e20 4469 default area. Di │ -00002760: 736d 6973 7320 7468 6520 736d 616c 6c20 smiss the small │ -00002770: 6672 6163 7461 6c20 6279 2074 6170 7069 fractal by tappi │ -00002780: 6e67 203c 623e 5265 6d6f 7665 204a 756c ng Remove Jul │ -00002790: 6961 2f4d 616e 6465 6c62 726f 743c 2f62 ia/Mandelbrot on the menu. I │ -000027b0: 6620 796f 7520 6c6f 7365 2074 7261 636b f you lose track │ -000027c0: 206f 6620 7468 6520 4a75 6c69 6120 7069 of the Julia pi │ -000027d0: 6e2c 206c 6f6e 6720 636c 6963 6b20 6167 n, long click ag │ -000027e0: 6169 6e20 6f72 2061 6464 2061 6e64 2072 ain or add and r │ -000027f0: 656d 6f76 6520 7468 6520 4a75 6c69 6120 emove the Julia │ -00002800: 6672 6f6d 2074 6865 206d 656e 7520 746f from the menu to │ -00002810: 2062 7269 6e67 2069 7420 746f 2079 6f75 bring it to you │ -00002820: 2e20 3c68 313e 4f74 6865 7220 4d65 6e75 .

Other Menu │ -00002830: 204f 7074 696f 6e73 3c2f 6831 3e20 3c70 Options

Under Prefer │ -00002850: 656e 6365 733c 2f62 3e2c 2079 6f75 2063 ences, you c │ -00002860: 616e 2063 686f 6f73 6520 7468 6520 636f an choose the co │ -00002870: 6c6f 7572 696e 6720 7363 6865 6d65 2066 louring scheme f │ -00002880: 6f72 2065 6163 6820 6672 6163 7461 6c2c or each fractal, │ -00002890: 2074 6865 2063 6f6c 6f75 7220 6f66 2074 the colour of t │ -000028a0: 6865 204a 756c 6961 2070 696e 2c20 7768 he Julia pin, wh │ -000028b0: 6574 6865 7220 6f72 206e 6f74 2074 6f20 ether or not to │ -000028c0: 646f 2061 6e20 696e 6974 6961 6c20 6372 do an initial cr │ -000028d0: 7564 6520 7265 6e64 6572 696e 6720 2861 ude rendering (a │ -000028e0: 2062 6c6f 636b 6965 7220 6669 7273 7420 blockier first │ -000028f0: 7061 7373 2074 6861 7420 6769 7665 7320 pass that gives │ -00002900: 6120 6661 7374 6572 2066 6972 7374 2069 a faster first i │ -00002910: 6d70 7265 7373 696f 6e20 6275 7420 7461 mpression but ta │ -00002920: 6b65 7320 736c 6967 6874 6c79 206c 6f6e kes slightly lon │ -00002930: 6765 7220 6f76 6572 616c 6c29 2c20 616e ger overall), an │ -00002940: 6420 7768 6574 6865 7220 6f72 206e 6f74 d whether or not │ -00002950: 2074 6f20 7368 6f77 2074 6865 2074 696d to show the tim │ -00002960: 6520 666f 7220 6561 6368 2072 656e 6465 e for each rende │ -00002970: 7220 6166 7465 7220 6974 2773 2063 6f6d r after it's com │ -00002980: 706c 6574 652e 203c 703e 5461 7020 3c62 plete.

Tap Detail level to control the │ -000029b0: 206c 6576 656c 206f 6620 6465 7461 696c level of detail │ -000029c0: 206f 6e20 6561 6368 2066 7261 6374 616c on each fractal │ -000029d0: 2e20 4869 6768 6572 2064 6574 6169 6c20 . Higher detail │ -000029e0: 6d65 616e 7320 6d6f 7265 2069 7465 7261 means more itera │ -000029f0: 7469 6f6e 7320 7769 6c6c 2062 6520 646f tions will be do │ -00002a00: 6e65 2077 6865 6e20 7265 6e64 6572 696e ne when renderin │ -00002a10: 672c 2074 616b 696e 6720 6c6f 6e67 6572 g, taking longer │ -00002a20: 2062 7574 2067 6976 696e 6720 6120 636c but giving a cl │ -00002a30: 6561 7265 7220 7069 6374 7572 652e 203c earer picture. < │ -00002a40: 703e 3c62 3e53 6176 652f 5368 6172 653c p>Save/Share< │ -00002a50: 2f62 3e20 6c65 7473 2079 6f75 2073 6176 /b> lets you sav │ -00002a60: 6520 616e 2069 6d61 6765 206f 6620 7468 e an image of th │ -00002a70: 6520 6375 7272 656e 7420 6672 6163 7461 e current fracta │ -00002a80: 6c20 696e 746f 2079 6f75 7220 5069 6374 l into your Pict │ -00002a90: 7572 6573 2066 6f6c 6465 7220 6f72 2073 ures folder or s │ -00002aa0: 6861 7265 2069 7420 7573 696e 6720 656d hare it using em │ -00002ab0: 6169 6c2c 2046 6163 6562 6f6f 6b2c 2054 ail, Facebook, T │ -00002ac0: 7769 7474 6572 2c20 6574 632e 2028 6966 witter, etc. (if │ -00002ad0: 2079 6f75 2068 6176 6520 7468 6f73 6520 you have those │ -00002ae0: 6170 7073 2069 6e73 7461 6c6c 6564 292e apps installed). │ -00002af0: 203c 6831 3e41 626f 7574 204d 616e 6465

About Mande │ -00002b00: 6c62 726f 7420 4d61 7073 3c2f 6831 3e20 lbrot Maps

│ -00002b10: 3c70 3e3c 623e 4d61 6e64 656c 6272 6f74

Mandelbrot │ -00002b20: 204d 6170 733c 2f62 3e20 6973 2062 6569 Maps is bei │ -00002b30: 6e67 2064 6576 656c 6f70 6564 2061 7320 ng developed as │ -00002b40: 6d79 2075 6e64 6572 6772 6164 7561 7465 my undergraduate │ -00002b50: 2064 6973 7365 7274 6174 696f 6e20 7072 dissertation pr │ -00002b60: 6f6a 6563 742c 2073 6f20 616e 7920 6665 oject, so any fe │ -00002b70: 6564 6261 636b 2079 6f75 206c 6561 7665 edback you leave │ -00002b80: 206f 6e20 7468 6520 476f 6f67 6c65 2050 on the Google P │ -00002b90: 6c61 7920 5374 6f72 6520 7061 6765 2069 lay Store page i │ -00002ba0: 7320 6772 6561 746c 7920 6170 7072 6563 s greatly apprec │ -00002bb0: 6961 7465 642e 2049 6620 796f 7520 6669 iated. If you fi │ -00002bc0: 6e64 2061 6e79 2062 7567 732c 2070 6c65 nd any bugs, ple │ -00002bd0: 6173 6520 696e 636c 7564 6520 6173 206d ase include as m │ -00002be0: 7563 6820 6465 7461 696c 2061 7320 706f uch detail as po │ -00002bf0: 7373 6962 6c65 2c20 7375 6368 2061 7320 ssible, such as │ -00002c00: 7468 6520 6465 7669 6365 2079 6f75 2772 the device you'r │ -00002c10: 6520 7573 696e 672c 2077 6861 7420 796f e using, what yo │ -00002c20: 7520 7765 7265 2064 6f69 6e67 2077 6865 u were doing whe │ -00002c30: 6e20 7468 6520 6572 726f 7220 6f63 6375 n the error occu │ -00002c40: 7272 6564 2c20 616e 6420 7768 6174 2076 rred, and what v │ -00002c50: 6572 7369 6f6e 206f 6620 416e 6472 6f69 ersion of Androi │ -00002c60: 6420 796f 7527 7265 2072 756e 6e69 6e67 d you're running │ -00002c70: 2e20 3c70 3e3c 623e 4d61 6e64 656c 6272 .

Mandelbr │ -00002c80: 6f74 204d 6170 733c 2f62 3e20 6973 206f ot Maps is o │ -00002c90: 7065 6e2d 736f 7572 6365 2073 6f66 7477 pen-source softw │ -00002ca0: 6172 652c 2072 656c 6173 6564 2075 6e64 are, relased und │ -00002cb0: 6572 2074 6865 2047 4e55 2047 656e 6572 er the GNU Gener │ -00002cc0: 616c 2050 7562 6c69 6320 4c69 6365 6e73 al Public Licens │ -00002cd0: 6520 2876 3329 2e20 5468 6520 736f 7572 e (v3). 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an │ +000047e0: 6420 6d6f 7669 6e67 2061 204d 616e 6465 d moving a Mande │ +000047f0: 6c62 726f 7420 6672 6163 7461 6c2e 0083 lbrot fractal... │ +00004800: b783 b73c 6832 3e57 656c 636f 6d65 2074 ...

Welcome t │ +00004810: 6f20 3c73 7061 6e20 7374 796c 653d 666f o Mandelbrot Map │ +00004840: 733c 2f73 7061 6e3e 213c 2f68 323e 203c s!

< │ +00004850: 703e 5468 6973 2061 7070 2069 7320 6465 p>This app is de │ +00004860: 7369 676e 6564 2074 6f20 7368 6f77 6361 signed to showca │ +00004870: 7365 2074 6865 2072 656c 6174 696f 6e73 se the relations │ +00004880: 6869 7020 6265 7477 6565 6e20 7468 6520 hip between the │ +00004890: 4d61 6e64 656c 6272 6f74 2073 6574 2061 Mandelbrot set a │ +000048a0: 6e64 204a 756c 6961 2073 6574 2066 7261 nd Julia set fra │ +000048b0: 6374 616c 732e 203c 703e 4d6f 7665 2061 ctals.

Move a │ +000048c0: 726f 756e 6420 7468 6520 6672 6163 7461 round the fracta │ +000048d0: 6c73 2077 6974 6820 6f6e 6520 6669 6e67 ls with one fing │ +000048e0: 6572 2061 6e64 2070 696e 6368 2074 6f20 er and pinch to │ +000048f0: 7a6f 6f6d 2e20 5768 656e 2079 6f75 207a zoom. When you z │ +00004900: 6f6f 6d2c 2074 6865 2066 7261 6374 616c oom, the fractal │ +00004910: 2077 696c 6c20 7265 2d72 656e 6465 7220 will re-render │ +00004920: 7769 7468 206d 6f72 6520 6465 7461 696c with more detail │ +00004930: 2e20 3c70 3e48 6f6c 6420 796f 7572 2066 .

Hold your f │ +00004940: 696e 6765 7220 6f6e 2061 2070 6f69 6e74 inger on a point │ +00004950: 2069 6e20 7468 6520 4d61 6e64 656c 6272 in the Mandelbr │ +00004960: 6f74 206f 7220 7365 6c65 6374 203c 623e ot or select │ +00004970: 4164 6420 4a75 6c69 613c 2f62 3e20 6672 Add Julia fr │ +00004980: 6f6d 2074 6865 206d 656e 7520 746f 2064 om the menu to d │ +00004990: 6973 706c 6179 2061 204a 756c 6961 2066 isplay a Julia f │ +000049a0: 7261 6374 616c 2e20 5468 6572 6527 7320 ractal. There's │ +000049b0: 6120 756e 6971 7565 204a 756c 6961 2066 a unique Julia f │ +000049c0: 7261 6374 616c 2066 6f72 2065 6163 6820 ractal for each │ +000049d0: 706f 696e 7420 696e 2074 6865 204d 616e point in the Man │ +000049e0: 6465 6c62 726f 7420 7365 7420 2d20 6368 delbrot set - ch │ +000049f0: 616e 6765 2074 6865 2070 6f69 6e74 2064 ange the point d │ +00004a00: 6566 696e 696e 6720 6974 2062 7920 6472 efining it by dr │ +00004a10: 6167 6769 6e67 2074 6865 2063 6f6c 6f75 agging the colou │ +00004a20: 7265 6420 7069 6e2e 2054 6170 2069 7420 red pin. Tap it │ +00004a30: 746f 2073 6565 2069 7420 696e 206d 6f72 to see it in mor │ +00004a40: 6520 6465 7461 696c 2e20 3c70 3e3c 623e e detail.

│ +00004a50: 4d61 6e64 656c 6272 6f74 204d 6170 733c Mandelbrot Maps< │ +00004a60: 2f62 3e20 6973 2062 6569 6e67 2064 6576 /b> is being dev │ +00004a70: 656c 6f70 6564 2061 7320 6d79 2075 6e64 eloped as my und │ +00004a80: 6572 6772 6164 7561 7465 2064 6973 7365 ergraduate disse │ +00004a90: 7274 6174 696f 6e20 7072 6f6a 6563 742c rtation project, │ +00004aa0: 2073 6f20 616e 7920 6665 6564 6261 636b so any feedback │ +00004ab0: 2079 6f75 206c 6561 7665 206f 6e20 7468 you leave on th │ +00004ac0: 6520 476f 6f67 6c65 2050 6c61 7920 5374 e Google Play St │ +00004ad0: 6f72 6520 7061 6765 2069 7320 6772 6561 ore page is grea │ +00004ae0: 746c 7920 6170 7072 6563 6961 7465 642e tly appreciated. │ +00004af0: 203c 703e 3c62 3e4d 616e 6465 6c62 726f

Mandelbro │ +00004b00: 7420 4d61 7073 3c2f 623e 2069 7320 6f70 t Maps is op │ +00004b10: 656e 2d73 6f75 7263 6520 736f 6674 7761 en-source softwa │ +00004b20: 7265 2c20 7265 6c61 7365 6420 756e 6465 re, relased unde │ +00004b30: 7220 7468 6520 474e 5520 4765 6e65 7261 r the GNU Genera │ +00004b40: 6c20 5075 626c 6963 204c 6963 656e 7365 l Public License │ +00004b50: 2028 7633 292e 2054 6865 2073 6f75 7263 (v3). The sourc │ +00004b60: 6520 636f 6465 2063 616e 2062 6520 666f e code can be fo │ +00004b70: 756e 6420 3c61 2068 7265 663d 6874 7470 und here......< │ +00004bc0: 6831 3e41 626f 7574 2066 7261 6374 616c h1>About fractal │ +00004bd0: 733c 2f68 313e 203c 703e 4120 6672 6163 s

A frac │ +00004be0: 7461 6c20 6973 2061 2022 7365 6c66 2d73 tal is a "self-s │ +00004bf0: 696d 696c 6172 2220 7368 6170 652c 206d imilar" shape, m │ +00004c00: 6561 6e69 6e67 2074 6861 7420 7468 6520 eaning that the │ +00004c10: 7361 6d65 2064 6574 6169 6c65 6420 7061 same detailed pa │ +00004c20: 7474 6572 6e20 7265 7065 6174 7320 6f76 ttern repeats ov │ +00004c30: 6572 2061 6e64 206f 7665 7220 6167 6169 er and over agai │ +00004c40: 6e20 6173 2079 6f75 207a 6f6f 6d20 696e n as you zoom in │ +00004c50: 2e20 496e 2074 6865 204d 616e 6465 6c62 . In the Mandelb │ +00004c60: 726f 7420 7365 7420 6672 6163 7461 6c2c rot set fractal, │ +00004c70: 2066 6f72 2065 7861 6d70 6c65 2c20 7468 for example, th │ +00004c80: 6f73 6520 6469 7374 696e 6374 6976 6520 ose distinctive │ +00004c90: 6865 6172 742d 7368 6170 6564 2062 756c heart-shaped bul │ +00004ca0: 6273 2077 696c 6c20 6b65 6570 2061 7070 bs will keep app │ +00004cb0: 6561 7269 6e67 2e20 3c70 3e46 7261 6374 earing.

Fract │ +00004cc0: 616c 7320 6172 6520 6765 6e65 7261 7465 als are generate │ +00004cd0: 6420 6279 2069 7465 7261 7469 6e67 2061 d by iterating a │ +00004ce0: 206d 6174 6865 6d61 7469 6361 6c20 6675 mathematical fu │ +00004cf0: 6e63 7469 6f6e 206f 6e20 6120 7365 7420 nction on a set │ +00004d00: 6f66 2070 6f69 6e74 732e 2044 6f69 6e67 of points. Doing │ +00004d10: 206d 6f72 6520 6974 6572 6174 696f 6e73 more iterations │ +00004d20: 2074 616b 6573 2061 206c 6f6e 6765 7220 takes a longer │ +00004d30: 7469 6d65 2062 7574 2070 726f 6475 6365 time but produce │ +00004d40: 7320 6d6f 7265 2064 6574 6169 6c65 6420 s more detailed │ +00004d50: 7368 6170 6573 2e20 4d6f 7265 2069 7465 shapes. More ite │ +00004d60: 7261 7469 6f6e 7320 6861 7665 2074 6f20 rations have to │ +00004d70: 6265 2064 6f6e 6520 7768 656e 207a 6f6f be done when zoo │ +00004d80: 6d69 6e67 2069 6e2c 2074 6f20 6b65 6570 ming in, to keep │ +00004d90: 2074 6865 2073 616d 6520 6c65 7665 6c20 the same level │ +00004da0: 6f66 2061 7070 6172 656e 7420 6465 7461 of apparent deta │ +00004db0: 696c 2e20 3c68 313e 4162 6f75 7420 7468 il.

About th │ +00004dc0: 6520 4d61 6e64 656c 6272 6f74 2061 6e64 e Mandelbrot and │ +00004dd0: 204a 756c 6961 2073 6574 733c 2f68 313e Julia sets

│ +00004de0: 203c 703e 5072 6f62 6162 6c79 2074 6865

Probably the │ +00004df0: 206d 6f73 7420 7265 6367 6f6e 6973 6162 most recgonisab │ +00004e00: 6c65 2066 7261 6374 616c 206f 6620 7468 le fractal of th │ +00004e10: 656d 2061 6c6c 2c20 7468 6520 4d61 6e64 em all, the Mand │ +00004e20: 656c 6272 6f74 2073 6574 2077 6173 2064 elbrot set was d │ +00004e30: 6973 636f 7665 7265 6420 616e 6420 706f iscovered and po │ +00004e40: 7075 6c61 7269 7365 6420 6279 206d 6174 pularised by mat │ +00004e50: 6865 6d61 7469 6369 616e 2042 656e 6f69 hematician Benoi │ +00004e60: 7420 4d61 6e64 656c 6272 6f74 2c20 7768 t Mandelbrot, wh │ +00004e70: 6f20 616c 736f 2069 6e76 656e 7465 6420 o also invented │ +00004e80: 7468 6520 7465 726d 2022 6672 6163 7461 the term "fracta │ +00004e90: 6c22 2074 6f20 6465 7363 7269 6265 2074 l" to describe t │ +00004ea0: 6865 2073 6861 7065 7320 6865 2073 7475 he shapes he stu │ +00004eb0: 6469 6564 2e20 4974 2077 6173 206f 6e65 died. It was one │ +00004ec0: 206f 6620 7468 6520 6669 7273 7420 6672 of the first fr │ +00004ed0: 6163 7461 6c73 2074 6f20 6265 2076 6973 actals to be vis │ +00004ee0: 7561 6c69 7365 6420 6f6e 2061 2063 6f6d ualised on a com │ +00004ef0: 7075 7465 722c 206f 6e20 4d61 7263 6820 puter, on March │ +00004f00: 3173 742c 2031 3938 302c 2069 6e20 4942 1st, 1980, in IB │ +00004f10: 4d27 7320 5468 6f6d 6173 204a 2e20 5761 M's Thomas J. Wa │ +00004f20: 7473 6f6e 2052 6573 6561 7263 6820 4365 tson Research Ce │ +00004f30: 6e74 7265 2e20 3c70 3e4a 756c 6961 2073 ntre.

Julia s │ +00004f40: 6574 7320 6172 6520 616e 206f 6c64 6572 ets are an older │ +00004f50: 2069 6465 612c 2066 6972 7374 2072 6573 idea, first res │ +00004f60: 6561 7263 6865 6420 696e 2074 6865 2065 earched in the e │ +00004f70: 6172 6c79 2032 3074 6820 6365 6e74 7572 arly 20th centur │ +00004f80: 7920 6279 2047 6173 746f 6e20 4a75 6c69 y by Gaston Juli │ +00004f90: 612e 2057 6865 6e20 6120 4a75 6c69 6120 a. When a Julia │ +00004fa0: 7365 7420 6973 2064 6566 696e 6564 2075 set is defined u │ +00004fb0: 7369 6e67 2074 6865 2073 616d 6520 6675 sing the same fu │ +00004fc0: 6e63 7469 6f6e 2074 6861 7420 6465 6669 nction that defi │ +00004fd0: 6e65 7320 7468 6520 4d61 6e64 656c 6272 nes the Mandelbr │ +00004fe0: 6f74 2073 6574 2c20 736f 6d65 2076 6572 ot set, some ver │ +00004ff0: 7920 696e 7465 7265 7374 696e 6720 6265 y interesting be │ +00005000: 6861 7669 6f75 7220 6170 7065 6172 732e haviour appears. │ +00005010: 2049 6620 7468 6520 4a75 6c69 6120 7365 If the Julia se │ +00005020: 7420 6973 2070 6172 616d 6574 6972 6973 t is parametiris │ +00005030: 6564 2062 7920 6120 706f 696e 7420 696e ed by a point in │ +00005040: 7369 6465 2074 6865 204d 616e 6465 6c62 side the Mandelb │ +00005050: 726f 7420 7365 742c 2069 7420 7769 6c6c rot set, it will │ +00005060: 2062 6520 636f 6e6e 6563 7465 642e 204d be connected. M │ +00005070: 6f76 696e 6720 6f75 7420 6f66 2074 6865 oving out of the │ +00005080: 204d 616e 6465 6c62 726f 7420 7365 7420 Mandelbrot set │ +00005090: 6361 7573 6573 2074 6865 204a 756c 6961 causes the Julia │ +000050a0: 2066 7261 6374 616c 2074 6f20 6272 6561 fractal to brea │ +000050b0: 6b20 6170 6172 7420 696e 746f 2075 6e63 k apart into unc │ +000050c0: 6f6e 6e65 6374 6564 2070 6965 6365 732e onnected pieces. │ +000050d0: 203c 703e 5468 6520 4d61 6e64 656c 6272

The Mandelbr │ +000050e0: 6f74 2061 6e64 204a 756c 6961 2073 6574 ot and Julia set │ +000050f0: 7320 616c 736f 206c 6f6f 6b20 7369 6d69 s also look simi │ +00005100: 6c61 7220 696e 2063 6572 7461 696e 2072 lar in certain r │ +00005110: 6567 696f 6e73 2c20 7061 7274 6963 756c egions, particul │ +00005120: 6172 6c79 206f 6e20 7468 6520 626f 7264 arly on the bord │ +00005130: 6572 7320 6f66 2074 6865 204d 616e 6465 ers of the Mande │ +00005140: 6c62 726f 742e 2049 7420 6973 2074 6865 lbrot. It is the │ +00005150: 7365 2072 656c 6174 696f 6e73 6869 7073 se relationships │ +00005160: 2074 6861 7420 3c62 3e4d 616e 6465 6c62 that Mandelb │ +00005170: 726f 7420 4d61 7073 3c2f 623e 2077 6173 rot Maps was │ +00005180: 2064 6576 656c 6f70 6564 2074 6f20 6465 developed to de │ +00005190: 6d6f 6e73 7472 6174 652e 203c 6831 3e43 monstrate.

C │ +000051a0: 6f6e 7472 6f6c 733c 2f68 313e 203c 703e ontrols

│ +000051b0: 4472 6167 2077 6974 6820 6120 7369 6e67 Drag with a sing │ +000051c0: 6c65 2066 696e 6765 7220 746f 206d 6f76 le finger to mov │ +000051d0: 6520 6172 6f75 6e64 2074 6865 2066 7261 e around the fra │ +000051e0: 6374 616c 732e 2041 6e79 206e 6577 2061 ctals. Any new a │ +000051f0: 7265 6173 2077 696c 6c20 6669 6c6c 2074 reas will fill t │ +00005200: 6865 6d73 656c 7665 7320 696e 206f 6e63 hemselves in onc │ +00005210: 6520 796f 7520 7374 6f70 2e20 3c70 3e55 e you stop.

U │ +00005220: 7365 2074 776f 2066 696e 6765 7273 2074 se two fingers t │ +00005230: 6f20 7a6f 6f6d 2069 6e20 616e 6420 6f75 o zoom in and ou │ +00005240: 742e 2041 6761 696e 2c20 7468 6520 6672 t. Again, the fr │ +00005250: 6163 7461 6c20 7769 6c6c 2072 652d 7265 actal will re-re │ +00005260: 6e64 6572 206f 6e63 6520 796f 7520 7374 nder once you st │ +00005270: 6f70 2e20 3c70 3e54 6f20 636f 6d70 6172 op.

To compar │ +00005280: 6520 7468 6520 4d61 6e64 656c 6272 6f74 e the Mandelbrot │ +00005290: 2061 6e64 204a 756c 6961 2073 6574 732c and Julia sets, │ +000052a0: 206c 6f6e 6720 636c 6963 6b20 6120 706f long click a po │ +000052b0: 696e 7420 696e 2074 6865 204d 616e 6465 int in the Mande │ +000052c0: 6c62 726f 7420 7365 7420 6f72 2074 6170 lbrot set or tap │ +000052d0: 203c 623e 4164 6420 4a75 6c69 613c 2f62 Add Julia on the menu. D │ +000052f0: 7261 6720 7468 6520 636f 6c6f 7572 6564 rag the coloured │ +00005300: 2063 6972 636c 6520 6172 6f75 6e64 2074 circle around t │ +00005310: 6f20 6368 616e 6765 2074 6865 2070 6f69 o change the poi │ +00005320: 6e74 2070 6172 616d 6574 6972 6973 696e nt parametirisin │ +00005330: 6720 7468 6520 4a75 6c69 6120 7365 742e g the Julia set. │ +00005340: 2054 6f20 7365 6520 7468 6520 6672 6163 To see the frac │ +00005350: 7461 6c20 696e 206d 6f72 6520 6465 7461 tal in more deta │ +00005360: 696c 2c20 7461 7020 6974 2e20 5468 6520 il, tap it. The │ +00005370: 7a6f 6f6d 206c 6576 656c 2061 6e64 2072 zoom level and r │ +00005380: 6567 696f 6e20 6469 7370 6c61 7965 6420 egion displayed │ +00005390: 7769 6c6c 2073 7461 7920 636f 6e73 7461 will stay consta │ +000053a0: 6e74 2061 7320 796f 7520 7377 6170 2062 nt as you swap b │ +000053b0: 6574 7765 656e 2074 6865 2074 776f 2066 etween the two f │ +000053c0: 7261 6374 616c 732e 2054 6170 203c 623e ractals. Tap │ +000053d0: 5265 7365 743c 2f62 3e20 6f6e 2074 6865 Reset on the │ +000053e0: 206d 656e 7520 746f 2072 6574 7572 6e20 menu to return │ +000053f0: 746f 2074 6865 2064 6566 6175 6c74 2061 to the default a │ +00005400: 7265 612e 2044 6973 6d69 7373 2074 6865 rea. Dismiss the │ +00005410: 2073 6d61 6c6c 2066 7261 6374 616c 2062 small fractal b │ +00005420: 7920 7461 7070 696e 6720 3c62 3e52 656d y tapping Rem │ +00005430: 6f76 6520 4a75 6c69 612f 4d61 6e64 656c ove Julia/Mandel │ +00005440: 6272 6f74 3c2f 623e 206f 6e20 7468 6520 brot on the │ +00005450: 6d65 6e75 2e20 4966 2079 6f75 206c 6f73 menu. If you los │ +00005460: 6520 7472 6163 6b20 6f66 2074 6865 204a e track of the J │ +00005470: 756c 6961 2070 696e 2c20 6c6f 6e67 2063 ulia pin, long c │ +00005480: 6c69 636b 2061 6761 696e 206f 7220 6164 lick again or ad │ +00005490: 6420 616e 6420 7265 6d6f 7665 2074 6865 d and remove the │ +000054a0: 204a 756c 6961 2066 726f 6d20 7468 6520 Julia from the │ +000054b0: 6d65 6e75 2074 6f20 6272 696e 6720 6974 menu to bring it │ +000054c0: 2074 6f20 796f 752e 203c 6831 3e4f 7468 to you.

Oth │ +000054d0: 6572 204d 656e 7520 4f70 7469 6f6e 733c er Menu Options< │ +000054e0: 2f68 313e 203c 703e 556e 6465 7220 3c62 /h1>

Under Preferences │ +00005500: 2c20 796f 7520 6361 6e20 6368 6f6f 7365 , you can choose │ +00005510: 2074 6865 2063 6f6c 6f75 7269 6e67 2073 the colouring s │ +00005520: 6368 656d 6520 666f 7220 6561 6368 2066 cheme for each f │ +00005530: 7261 6374 616c 2c20 7468 6520 636f 6c6f ractal, the colo │ +00005540: 7572 206f 6620 7468 6520 4a75 6c69 6120 ur of the Julia │ +00005550: 7069 6e2c 2077 6865 7468 6572 206f 7220 pin, whether or │ +00005560: 6e6f 7420 746f 2064 6f20 616e 2069 6e69 not to do an ini │ +00005570: 7469 616c 2063 7275 6465 2072 656e 6465 tial crude rende │ +00005580: 7269 6e67 2028 6120 626c 6f63 6b69 6572 ring (a blockier │ +00005590: 2066 6972 7374 2070 6173 7320 7468 6174 first pass that │ +000055a0: 2067 6976 6573 2061 2066 6173 7465 7220 gives a faster │ +000055b0: 6669 7273 7420 696d 7072 6573 7369 6f6e first impression │ +000055c0: 2062 7574 2074 616b 6573 2073 6c69 6768 but takes sligh │ +000055d0: 746c 7920 6c6f 6e67 6572 206f 7665 7261 tly longer overa │ +000055e0: 6c6c 292c 2061 6e64 2077 6865 7468 6572 ll), and whether │ +000055f0: 206f 7220 6e6f 7420 746f 2073 686f 7720 or not to show │ +00005600: 7468 6520 7469 6d65 2066 6f72 2065 6163 the time for eac │ +00005610: 6820 7265 6e64 6572 2061 6674 6572 2069 h render after i │ +00005620: 7427 7320 636f 6d70 6c65 7465 2e20 3c70 t's complete.

Tap Detail l │ +00005640: 6576 656c 3c2f 623e 2074 6f20 636f 6e74 evel to cont │ +00005650: 726f 6c20 7468 6520 6c65 7665 6c20 6f66 rol the level of │ +00005660: 2064 6574 6169 6c20 6f6e 2065 6163 6820 detail on each │ +00005670: 6672 6163 7461 6c2e 2048 6967 6865 7220 fractal. Higher │ +00005680: 6465 7461 696c 206d 6561 6e73 206d 6f72 detail means mor │ +00005690: 6520 6974 6572 6174 696f 6e73 2077 696c e iterations wil │ +000056a0: 6c20 6265 2064 6f6e 6520 7768 656e 2072 l be done when r │ +000056b0: 656e 6465 7269 6e67 2c20 7461 6b69 6e67 endering, taking │ +000056c0: 206c 6f6e 6765 7220 6275 7420 6769 7669 longer but givi │ +000056d0: 6e67 2061 2063 6c65 6172 6572 2070 6963 ng a clearer pic │ +000056e0: 7475 7265 2e20 3c70 3e3c 623e 5361 7665 ture.

Save │ +000056f0: 2f53 6861 7265 3c2f 623e 206c 6574 7320 /Share lets │ +00005700: 796f 7520 7361 7665 2061 6e20 696d 6167 you save an imag │ +00005710: 6520 6f66 2074 6865 2063 7572 7265 6e74 e of the current │ +00005720: 2066 7261 6374 616c 2069 6e74 6f20 796f fractal into yo │ +00005730: 7572 2050 6963 7475 7265 7320 666f 6c64 ur Pictures fold │ +00005740: 6572 206f 7220 7368 6172 6520 6974 2075 er or share it u │ +00005750: 7369 6e67 2065 6d61 696c 2c20 4661 6365 sing email, Face │ +00005760: 626f 6f6b 2c20 5477 6974 7465 722c 2065 book, Twitter, e │ +00005770: 7463 2e20 2869 6620 796f 7520 6861 7665 tc. (if you have │ +00005780: 2074 686f 7365 2061 7070 7320 696e 7374 those apps inst │ +00005790: 616c 6c65 6429 2e20 3c68 313e 4162 6f75 alled).

Abou │ +000057a0: 7420 4d61 6e64 656c 6272 6f74 204d 6170 t Mandelbrot Map │ +000057b0: 733c 2f68 313e 203c 703e 3c62 3e4d 616e s

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M │ +00005920: 616e 6465 6c62 726f 7420 4d61 7073 3c2f andelbrot Maps is open-sourc │ +00005940: 6520 736f 6674 7761 7265 2c20 7265 6c61 e software, rela │ +00005950: 7365 6420 756e 6465 7220 7468 6520 474e sed under the GN │ +00005960: 5520 4765 6e65 7261 6c20 5075 626c 6963 U General Public │ +00005970: 204c 6963 656e 7365 2028 7633 292e 2054 License (v3). T │ +00005980: 6865 2073 6f75 7263 6520 636f 6465 2069 he source code i │ +00005990: 7320 6c69 6e6b 6564 2066 726f 6d20 7468 s linked from th │ +000059a0: 6520 476f 6f67 6c65 2050 6c61 7920 5374 e Google Play St │ +000059b0: 6f72 6520 7061 6765 2e00 0505 4162 6f75 ore page....Abou │ +000059c0: 7400 1818 4269 746d 6170 2028 6e6f 7420 t...Bitmap (not │ +000059d0: 696d 706c 656d 656e 7465 6429 0015 1553 implemented)...S │ +000059e0: 6964 652d 6279 2d73 6964 6520 6672 6163 ide-by-side frac │ +000059f0: 7461 6c73 0003 0372 6564 0000 0002 2001 tals...red.... . │ +00005a00: e00e 0000 7f00 0000 7500 6b00 2e00 6100 ........u.k...a. │ +00005a10: 6300 2e00 6500 6400 2e00 6900 6e00 6600 c...e.d...i.n.f. │ +00005a20: 2e00 6d00 6100 6e00 6400 6500 6c00 6200 ..m.a.n.d.e.l.b. │ +00005a30: 7200 6f00 7400 6d00 6100 7000 7300 0000 r.o.t.m.a.p.s... │ +00005a40: 0000 0000 0000 0000 0000 0000 0000 0000 ................ │ +00005a50: 0000 0000 0000 0000 0000 0000 0000 0000 ................ │ +00005a60: 0000 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