1. Fractal Glass Pictures
2. Fractal Gallery

Introduction to Fractal Analysis

• Find fractals stock images in HD and millions of other royalty-free stock photos, illustrations and vectors in the Shutterstock collection. Thousands of new, high-quality pictures added every day.
• The Fractal Gallery, An online catalog of the fractal art modern abstract oil paintings and lifestyle art by Eric R Doolin, eco luxe designer and artist creating green, ecologically conscious green living furniture which has been elevated to functional sculpture for bringing the outdoor inside. Doolin has been dedicated to developing artwork, particularly oil painting, and expanding.

NICO'S FRACTAL MACHINE. The shape you see is the combined output of the controls below. Mouse over them to see what they do. If the page gets too slow, turn some of the parameters down. Press H or to hide the controls. Find out more in this blog post.

How would you characterize the images on this page? Describing them using traditional features such as 'size' and overall 'shape' wouldn't really say much, although it could be very informative for some other forms, such as when characterizing 'a 10 cm, round orange' or 'a 300 cm x 100 cm oblong watermelon', for instance. But using such descriptors for the images here would oversimplify the detail in their patterns. Describing these patterns using the terms of fractal analysis with FracLac, however, can convey some of the complexity inherent in their design.

These images show diffusion limited aggregation, which is a type of fractal growth that can be analyzed with FracLac. The images were generated with the DLA Generator for ImageJ.

What is Fractal Analysis?

Fractal analysis is a contemporary method of applying nontraditional mathematics to patterns that defy understanding with traditional Euclidean concepts.

In essence, it measures complexity using the fractal dimension.

Fractal Glass Pictures

The field was developed to describe computer-generated fractals such as the diffusion limited aggregates shown on this page, but fractals are not necessarily computer-generated images. Rather, whereas we see Euclidean geometry in familiar shapes like oranges and watermelons, we see fractal geometry in familiar forms like meandering coastlines, growing crystals, and swirling galaxies.

Fractal Patterns

Fractals are not necessarily physical forms - they can be spatial or temporal patterns, as well. In general, fractals can be any type of infinitely scaled and repeated pattern. In this regard, it is important to be aware that theoretical fractals are abstractions, but the subjects of fractal analysis, such as digital images limited by screen resolution, are generally not true fractals in the strictest sense. Similarly, the so-called fractals typically found in nature are not infinitely scaled, thus, like finite computer generated patterns, are generally only approximations to fractals in the strictest sense.

Imagine a real fractal.

The computer generated pattern shown here was constructed to emulate a 32-segment quadric fractal.

It's essential pattern is scaled and repeated to make the whole figure, but as illustrated, the scaling is not infinite - as highlighted in the upper box, it ends when the pattern becomes 1 pixel wide, a limit imposed by pixels on a computer screen.

Click on the image to see it in higher resolution or click here to read about digital images of fractals.

Fractal Analysis in Practice

Notwithstanding the fact that what we actually investigate by fractal analysis merely approximates fractals, this sort of analysis is very useful. Investigators use different types of fractal analysis to study a host of otherwise intractable phenomena including the complex geometries of many types of biological cells (Kam et al. 2009) and complex patterns such as tree growth, river paths, tumour growth (Cross 1997), heart rates (Huikiri & Stein 2012), diabetic retinopathy (Karperien et al. 2008), gene expression (Aldrich et al 2010), forest fire progression (Turcotte et al. 2002), economic trends, and cellular differentiation in space and time (Waliszewski & Konarski 2002) (search PubMed or see FracLac Citations for more information). The next page discusses fractal analysis in more detail.

Fractal of the Day

Every day at a few minutes past midnight (local Wisconsin time), a new fractal is automatically posted using a variation of the program included with the book Strange Attractors: Creating Patterns in Chaos by Julien C. Sprott. The figure above is today's fractal. Click on it or on any of the cases below to see them at higher (640 x 480) resolution with a code that identifies them according to a scheme described in the book. Older Fractals of the Day are saved in an archive. If your browser supports Java, you might enjoy the applet that creates a new fractal image every five seconds or so. If you would like to place the Fractal of the Day on your Web page, you may do so provided you mention that it is from Sprott's Fractal Gallery and provide a link back to this page. If you want to make your own fractals, I recommend the Chaoscope freeware.

• fracday0.gif Today's fractal of the day
• fracday1.gif Yesterday's fractal of the day
• fracday2.gif Fractal of the day from 2 days ago
• fracday3.gif Fractal of the day from 3 days ago
• fracday4.gif Fractal of the day from 4 days ago
• fracday5.gif Fractal of the day from 5 days ago
• fracday6.gif Fractal of the day from 6 days ago
• fracday7.gif Fractal of the day from 7 days ago
• fracday8.gif Fractal of the day from 8 days ago
• fracday9.gif Fractal of the day from 9 days ago
• fracdaya.gif Fractal of the day from 10 days ago
• fracdayb.gif Fractal of the day from 11 days ago
• fracdayc.gif Fractal of the day from 12 days ago
• fracdayd.gif Fractal of the day from 13 days ago
• fracdaye.gif Fractal of the day from 14 days ago
• fracdayf.gif Fractal of the day from 15 days ago

Chaos Demonstrations

The following rather standard fractals are low-resolution sample screen captures from the Chaos Demonstrations program by J. C. Sprott and G. Rowlands. You may also want to view an index of these and many other screen captures from the program.

• Catalog#0 (34,311 bytes) - catalog of the following
• anaglyph (15,045 bytes) - Torus (requires red/blue glasses)
• automata (28,216 bytes) - One-D cellular automaton
• chirikov (12,627 bytes) - Chirikov (standard) map
• cml (57,803 bytes) - Coupled-logistic-map lattice
• curtains (13,130 bytes) - Cantor curtains
• diff (14,848 bytes) - Diffusion (random walk of 16 particles)
• dla (11,178 bytes) - Diffusion-limited aggregation
• fern (16,353 bytes) - Fractal fern
• henon (10,764 bytes) - Henon map with basin of attraction
• juliaset (18,088 bytes) - Julia set for c = 0 + 1.0 i
• life (5,546 bytes) - Game of life equilibrium
• logistic (15,037 bytes) - Bifurcation diagram of logistic equation
• lorenz (15,596 bytes) - Lorenz attractor
• manbrot (21,408 bytes) - Mandelbrot set
• predator (36,669 bytes) - Predator-prey attractor
• rossler (26,376 bytes) - Rossler attractor

Strange Attractors

The following fractals are low-resolution sample color plates from the book Strange Attractors: Creating Patterns in Chaos by Julien C. Sprott. You may also want to view an index of all 371 figures from the book.

• Catalog#1 (43,570 bytes) - catalog of the following
• plate03 (26,778 bytes) - Three-dimensional quadratic map
• plate05 (21,047 bytes) - Three-dimensional quadratic map
• plate08 (38,420 bytes) - Three-dimensional cubic map projected onto a sphere
• plate09 (19,806 bytes) - Three-dimensional anaglyph (requires red/blue glasses)
• plate16 (25,192 bytes) - Three-dimensional anaglyph (requires red/blue glasses)
• plate17 (44,259 bytes) - Four-dimensional quadratic map with colored bands
• plate18 (21,914 bytes) - Four-dimensional analglyph (requires red/blue glasses)
• plate19 (27,960 bytes) - Four-dimensional quadratic map with shadowed colors
• plate20 (25,098 bytes) - Four-dimensional quadratic map with banded colors
• plate21 (15,513 bytes) - Stereo pair of four-dimensional quadratic map
• plate23 (19,746 bytes) - Three-dimensional quartic ODE
• plate24 (13,172 bytes) - Three-dimensional anaglyph (requires red/blue glasses)
• plate27 (26,408 bytes) - Four-dimensional quadratic ODE with shadowed colors
• plate29 (30,009 bytes) - Stereo pair of four-dimensional quadratic ODE
• plate31 (37,470 bytes) - Stochastic web map
• plate32 (55,509 bytes) - Stochastic web map

More Strange Attractors

The following 3-dimensional strange attractors are mostly from the book Strange Attractors: Creating Patterns in Chaos by Julien C. Sprott but are here rendered in higher (800 x 600) resolution with the third dimension mapped to a palette of 256 colors. Additional such cases can be produced automatically by the programsa256.exe that searches for chaotic solutions of a general system of quadratic maps with 30 coefficients.

• Catalog#2 (43,625 bytes) - catalog of the following
• IIPPSGTM (76,254 bytes)
• IJKRADSX (52,872 bytes)
• IJMYRKHS (138,682 bytes)
• IKUELCPY (37,751 bytes)
• ILRRHAEY (63,118 bytes)
• ILURCEGO (38,544 bytes)
• IMNGCLHT (54,101 bytes)
• IMTISVBK (165,607 bytes)
• INKRCPBN (79,877 bytes)
• INRRXLCE (56,469 bytes)
• IOGBGSHO (116,278 bytes)
• IOHGWFIH (87,623 bytes)
• IOLORGSF (40,700 bytes)
• IQNBDVIS (45,941 bytes)
• ISYINLLU (32,149 bytes)
• IWUBBBVG (58,085 bytes)

Julia Sets

The following fractals are standard Julia sets of the function z^2 + c. They were produced automatically by the programjulia256.exe by J. C. Sprott that searches the complex-c plane for interesting cases. The names consist of two four-digit hexadecimal numbers p and q such that c is given by c = -2 + p / 21845 + i q / 43691. The plots cover the range z = (-0.02, 0.02) + (-0.02, 0.02) i.

• Catalog#3 (103,694 bytes) - catalog of the following
• 3a54 0b31 (228,457 bytes)
• 484d 3017 (134,352 bytes)
• 4a02 2466 (151,915 bytes)
• 578c 2a9b (184,666 bytes)
• 6ac3 0a04 (141,401 bytes)
• 6ac5 0878 (97,487 bytes)
• 6ad6 0e5d (156,110 bytes)
• 6b4c 1311 (162,886 bytes)
• 7268 3a16 (214,382 bytes)
• 94e2 6edc (85,619 bytes)
• 9d56 6ee3 (189,298 bytes)
• a111 984a (235,732 bytes)
• a813 7ff2 (126,131 bytes)
• af20 6d1a (82,870 bytes)
• b3ed 6b3c (195,514 bytes)
• c340 5247 (183,601 bytes)

Quadratic Map Basins

The following fractals are generalized Julia sets of 2-D quadratic maps with twelve coefficients. They were produced automatically by the programgenjulia.exe by J. C. Sprott that searches the 12-dimensional space of coefficients for interesting cases. The technique is described in a paper 'Automatic Generation of General Quadratic Map Basins' by J. C. Sprott and C. A. Pickover. The coefficients are coded into the names according to a scheme described in the book Strange Attractors: Creating Patterns in Chaos. The plots cover the range -1 < x < 1 and -1 < y < 1. A few additional images of this type produced with the program Fractal eXtreme by Cygnus Software are available.

• Catalog#4 (103,178 bytes) - catalog of the following
• EHVPROJHKKLVO (151,780 bytes)
• EHVUXMENMGVIS (146,999 bytes)
• EICHEJYQKROCP (127,330 bytes)
• EJBILKQIONDBL (204,771 bytes)
• EKESXTWLQRVQH (188,735 bytes)
• ELEWECYNQOQGC (153,424 bytes)
• ELLRHCHOQOQLK (185,975 bytes)
• EMDIOJQKPLEDN (230,593 bytes)
• EMMOICKNWORMK (193,550 bytes)
• EOXKKVYIESBID (154,804 bytes)
• EPARXHHPRGSAS (106,710 bytes)
• EQSFMFWSYHFLJ (205,375 bytes)
• ERBIQHQRRKEBO (100,696 bytes)
• ESDNLHLRRMPDM (282,668 bytes)
• EWTGLOEOIDNNN (154,120 bytes)
• EXKMPTMRFKNKO (249,058 bytes)

Iterated Function Systems

The following fractals are iterated function systems generated by the random iteration algorithm using two linear affine transformations. Color is introduced according to the number of successive applications of each transform. They were produced automatically by the program ifs256.exe by J. C. Sprott that searches the 12-dimensional space of coefficients for interesting cases. The coefficients are coded into the names according to a scheme described in the paper 'Automatic Generation of Iterated Function Systems'.

• Catalog#5 (33,130 bytes) - catalog of the following
• aGNGUVETSNWK (52,756 bytes)
• aGOHRHRNFLNO (20,106 bytes)
• aIFEROWOIRQI (35,432 bytes)
• aIIDQIKFHUSK (65,121 bytes)
• aIPUDJIHEIQY (24,045 bytes)
• aKDOWQMMEMWD (27,602 bytes)
• aKJPSVUHVJNR (50,394 bytes)
• aKWMRVRUOWWB (63,774 bytes)
• aLGMRYBSSSQR (43,610 bytes)
• aMCGMGGJORXH (20,305 bytes)
• aNYDEQPKOOKY (90,193 bytes)
• aRFUSIENJVCN (31,732 bytes)
• aRIGIEDMBOWR (27,593 bytes)
• aTLPVPLMEIFC (52,941 bytes)
• aTTGUJNMLWBR (68,894 bytes)
• aUIPUDROPLPM (48,380 bytes)

Strange Attractor Symmetric Icons

The following icons are produced from 3-D strange attractors by mapping the x-coordinate to radius and the y-coordinate to angle and replicating the pattern with different orientations. The z-coordinate is represented by one of 256 colors, and a shadow is added to enhance the illusion of depth. The technique is described in a paper 'Strange Attractor Symmetric Icons' by J. C. Sprott. The equations used are coded into the name according to a scheme described in the book Strange Attractors: Creating Patterns in Chaos by Julien C. Sprott. Additional such cases can be produced automatically by the programicon256.exe. If you like these, you can view an index of 100 additional such examples, or an index of cyclic symmetric attractor anaglyphs produced by a different method. You can also view an index of fractal tilings useful as Windows wallpaper or HTML backgrounds (as on this page).

• Catalog#6 (97,546 bytes) - catalog of the following
• ^AJVLNLDT (131,381 bytes)
• ^DIWJDMES (221,318 bytes)
• ^FDTNMVCD (217,328 bytes)
• ^HARPVCIW (169,994 bytes)
• HHTLOWIB (60,153 bytes)
• ]IGDFKI (169,892 bytes)
• QIRVHHLOA (63,399 bytes)
• ZJEKCJCOA (70,035 bytes)
• ]LEXDJE (186,388 bytes)
• QOFNACAWL (87,498 bytes)
• ]PUBLNE (128,327 bytes)
• IQOGCBNMV (146,279 bytes)
• QSANAXXGB (67,789 bytes)
• ^XSCJSNCO (179,586 bytes)
• ^YXBQKEGY (195,678 bytes)
• ^YXGSGGCK (132,148 bytes)

Newsgroup Collection

You might also like to look at an index of many thousands of fractals I've collected off the net, mostly from the newsgroup alt.binaries.pictures.fractals and from the World Wide Web. In most cases, I don't know the original source, and so I apologize to anyone whose copyright may have been violated. Many of the nicest of these images are the work of Paul Carlson whose Fractal Gallery you may wish to visit. Here's a few cases I've selected as the best of the best:

• Catalog#7 (112,366 bytes) - catalog of the following
• 1044 (121,529 bytes)
• blumetal (146,746 bytes)
• bubbls04 (129,295 bytes)
• byphx85 (130,680 bytes)
• candy3 (181,562 bytes)
• cmphx09 (107,663 bytes)
• cymgpk02 (145,052 bytes)
• grapes (192,639 bytes)
• intrcolr (128,706 bytes)
• lyap001 (147,468 bytes)
• medusa (123,194 bytes)
• myczc201 (192,466 bytes)
• quatern5 (73,151 bytes)
• spky2j10 (126,698 bytes)
• stolives (74,355 bytes)
• trivet (162,987 bytes)

Animated GIF Attractors

These images are from the iterated mapping xnew = a + bx + cx² + dy + ez + f sin(pi t/8), ynew = x, znew = y, where a through f are constants coded into the file name as described above. If your browser supports animated GIFs, you will see 16 looping frames (t mod 16). These images illustrate the stretching that causes chaos and the folding that produces the fractal microstructure of strange attractors. The DOS programs that were used to produce them are available. The individual frames were assembled using the GIF Construction Set by Alchemy Mindworks Inc. You might also like to look at an index of other fractal GIF animations, which includes 19 of the simplest known chaotic flows.

• Catalog#8 (44,617 bytes) - catalog of the following
• AROJTP (112,292 bytes)
• CYOJTP (110,221 bytes)
• FGKJUK (82,867 bytes)
• HWOJUJ (103,654 bytes)
• KTXEUN (75,704 bytes)
• LNQFVO (137,633 bytes)
• NHRISK (94,787 bytes)
• NPJCTB (110,422 bytes)
• OMHGTR (79,341 bytes)
• PSDJTL (91,503 bytes)
• RKLVRD (78,413 bytes)
• SHNKUE (113,776 bytes)
• TILSHF (97,695 bytes)
• TOKHUU (89,525 bytes)
• VCLUTQ (121,428 bytes)
• WLJFQE (85,678 bytes)

Fractal Gallery

Natural Fractals

These images are real-world fractals photographed by J. C. Sprott. More images of this type can be found in the index of natural fractals.

• Catalog#9 (44,617 bytes) - catalog of the following
• Ashes (103,736 bytes)
• Bark (114,511 bytes)
• Branches (92,423 bytes)
• Brick (148,456 bytes)
• Broccoli (48,229 bytes)
• Bubbles (95,751 bytes)
• Carpet (86,844 bytes)
• Fabric (149,813 bytes)
• Ice (80,564 bytes)
• Ice Crystals (74,388 bytes)
• Moss (110,881 bytes)
• Pavement (103,656 bytes)
• Rocks (84,558 bytes)
• Shrub (89,055 bytes)
• Snow Tree (98,996 bytes)
• Trees (149,156 bytes)

Publication Quality Attractors

These very high resolution (4400 x 3200) strange attractors (c) by J. C. Sprott are included for anyone who would like to make publication quality prints or display art. You may publish or display them without further permission provided you acknowledge their source. Thousands more like these are available upon request. See also the slideshow and high-resolution images from the CD-ROM that accompanies the book 'Images of a Complex World: The Art and Poetry of Chaos' and the strange attractor prints offered for sale.

• Catalog#10 (61,512 bytes) - catalog of the following
• IGFVGXKP (1,571,565 bytes)
• JQDKIBIH (781,094 bytes)
• JVKNGSNN (1,587,411 bytes)
• KALKRQMM (992,973 bytes)
• MFJDOTHB (1,095,259 bytes)
• NCHXNTNI (1,655,068 bytes)
• NXRGJVMB (650,899 bytes)
• OIORIOLL (1,070,168 bytes)
• OYLORAIX (1,475,032 bytes)
• PMJKNQHP (3,980,653 bytes)
• QKLJHHMD (830,421 bytes)
• RIRFVAMM (840,584 bytes)
• RMCEPOJG (3,336,763 bytes)
• RYEOSQKP (2,157,019 bytes)
• SKEANAMO (1,339,740 bytes)
• SMKBNXQA (822,502 bytes)

HELP | NEXT 5 | RANDOM | SKIP 1Statistics

I hope you'll visit my home page and leave a comment.

Total accesses: