FRACTALS
1
What is a fractal? A fractal is a never ending pattern that repeats itself at different scales. This property is called ‘self similarity’. There are many sources of information on fractals to be found online. See for instance:https://thefractalforge.com/what-is-a-fractal/ https://fractalfoundation.org https://mathigon.org
Where can we find fractals? Natural fractals include branching patterns like trees, ferns, river networks, broccoli and blood vessels; also spiral patterns like tornados and spiral galaxies. Mathematical fractals are produced using repeated calculations based on simple equations. A well known example is the ‘Mandelbrot Set’, which has been described as ‘The Thumbprint of God’ owing to its association with concepts of eternity and infinity.
2
Spiral galaxy (Image courtesy of NASA)
Winter tree in Mist
Fractal image from Apophysis
Fractal image from the Mandelbrot Set
3
How can we produce fractals for viewing on a computer or for high quality printing? There are many programmes on the web which can produce fractal images. Most of these programmes are free to download though a few require a fee.
Windows Programmes Tierazon ($35) Ultrafractal ($29 - $99) Xenodream ($119) Jux (free but $24 for high res Xaos Apophysis Fractal Science Kit Visions of Chaos ($45) Mind Boggling Fractals LE Chaoscope Mandelbulber
Incendia Fractint SterlingFractice Fractal Forge Mandala Explorer Kaos Rhei Fractal Extreme Gimp )plug-in) Terry Gintz ($45 upwards) Mark Eggleston J Wildfire
Mac programmes Ultrafractal Mandelbulb 3D Mandelbulber Fractal Architect Fractal Domaines FractalASM Oxidiser Gnofract Aros Fractals Fractastic FractalWorks Spangfract
Programmes I have tried include Mandelbrot Explorer, Apophysis and Malin Christersson’s Image Fractals,. Some details of these programmes and examples of fractals which can be produced by them are shown in the next few pages.
4
Malin Christersson’s Image Fractals Web link http://www.malinc.se/m/ImageFractals.php Although this is one of the simpler programmes to use it is capable of producing fascinating and complex fractals including some.which can be based on an imported image. It is an online programme; that is, the programme is not downloaded and the calculations are carried out online. There are no tutorials but the process is clear from instructions on the site.There are many variants. When a fractal of sufficient interest has been generated it is possible for certain variants to be downloaded at a resolution of 2500x2500 px as jpegs or pngs suitable for display or printing. Where this is not possible the images can be saved at lower resolutions by a screen shot or the command ‘save image’. Colours can be changed by subsequent editing in Photoshop. Initial calculations are done at low resolution and take only a few seconds. Calculations for a high resolution image can take a few minutes. Downloading time is dependent on Internet speeds but is generally less than 1 minute. The image shown on page 6 is a ‘Julia 5’ based on a red flower photographed against white blinds. The blue background has been added in Photoshop. On page 7 is a fractal produced from a Mandelbrot variant based on another flower, this time an Early Purple Orchid (with a background from Photoshop). On page 8 is another Mandelbrot variant, this one based on an image of shadows against conservatory blinds.
5
Julia 5 Red Flower against White Blinds
6
Mandelbrot variant Early Purple Orchid
7
Mandelbrot variant Shadows against conservatory blinds
8
Malin Christersson’s Image Fractals (cont.) Other fractal variants which can be explored on this site include:Orbit Trap Fractals Animated Julia Sets Mobius Transformations Rolling Hypocycloids and Epicycloids Focal Points, Ellipses and Ovals Hyperbolic Tilings of Images The Mandelbrot Set Paint Circle - Inverted Mondrian Damped Lissajous Curves Most of these have to be be captured by a screen shot or ‘save image’ but one which can be downloaded is a ‘Hyperbolic Tiling’. This fractal variant again uses an imported image.. An example, based on an image of bluebells in a wood, is shown on page 10. An example of an ‘Orbit Trap Mandelbrot Set’ is shown on page 11 On page 12 is an ‘Orbit Trap Julia Set’.whilst on page 13 is a Mandelbrot fractal similar to those which can be produced from ‘Mandelbrot Explorer’ (see later). For each of these fractal variants there are a vast number of variations of shape and colour which can be set in the programmes. The three examples above have all been produced initally by the online programmes, saved as images and then edited further in Photoshop.
Mandelbrot variant Early purple orchid 9
Hyperbolic Tiling Bluebells in a Wood 10
Orbit Trap Mandelbrot Set 11
Orbit Trap Julia Set
12
A deep part of the Mandelbrot Set
13
Mandelbrot Explorer Web link https://www.mandel.org.uk/ This is a simple programme developed to allow exploration of the Mandelbrot Set, a fractal discovered by Prof. Benoit Mandelbrot in 1980 using IBM computers to draw the first examples. It is easy to use and allows high resolution images to be rendered and saved for printing or display. It is free to download. On page 15 is a screen shot of the main window after maximising and choosing ‘view>fit to screen’. To explore the details, choose a colour gradient and then the + symbol. Double clicking anywhere in the image will then enlarge that part and this can be continued until an interesting area is found. Clicking on the - symbol allows the process to be reversed. The default resolution is 300x300 px. To produce a high resolution fractal go to the drop down menu top right and choose ‘next area to draw’ and then ‘edit this value’. Within this box choose ‘resolution > edit’. A box appears where you can insert a new resolution. Just remember that the higher the resolution the longer it takes to render. Click ‘apply change’ and then ‘draw next’ to produce the new image. Save as a jpeg. If desired, open in Photoshop or other editing software to change the colours.
Some fractal images produced in this way are shown on pages 16-19.
14
15
Mandelbrot Set 1
16
Mandelbrot Set 2
17
Mandelbrot Set 3
18
Mandelbrot Set 4
19
Apophysis Apophysis is one of the most powerful and versatile fractal programmes available and is frequenly used by dedicated exponents to make striking ‘Fractal Art’ images. It is based on ‘iterative function systems’ which allow a vast number of quite different fractals to be produced of high quality and resolution. There are several versions of this programme available. The most recent appears to be Apophysis 7X which can be found and downloaded freely using the link below:-. https://sourceforge.net/projects/apophysis7x/ It is not a very intuitive programme but fortunately there are some useful tutorials which describe the basics of the programme and some of these include step by step instructions for producing particular fractals - see for instance:https://design.tutsplus.com/articles/fractal-art-an-introduction-to-apophysis--cms-22248 https://design.tutsplus.com/tutorials/fractal-art-create-a-3d-julian-fractal-in-apophysis--cms-23669 http://librarianchick.com/sdid/wp-content/uploads/2007/03/ApophysisTutorial_for_Beginners1.pdf This booklet is not intended to be a detailed tutorial on using Apophysis - better to use those on the web sites listed above - but the main features are briefly described on the next few pages. Opposite (page 21) is a screen shot of the main window. On the left is a list of random fractals (flames) generated by the programme each time it is opened whilst at the top is the main tool bar. The large image is the first random flame (highlighted). .Other random flames can be selected and viewed and these can be the starting point for further modification. The number of random flames generated can be preselected and these flames can also be restricted to particular variants (tools>settings>random). The most important items on the tool bar are ‘flame’ whch provides for the final result to be rendered as a high quality image, and ‘tools’ which gives access to settings (options). Below these are various buttons of which the most important are fx which opens the editor (see pages 22/23) and the colour gradient selector.. The colour gradient determines the colour range whilst the editor is the main window which determines the final flame generated. There is a box where the quality can be set. Use low values for the latter - 10 or 15 - when working, otherwise calculation times become very long. The quality for the final rendered flame can be set separately.
20
Screen shot of Apophysis 7X - main window
21
Apophysis - cont On page 23 opposite is a screenshot of the editor. This shows triangles representing ‘transforms’ associated with the current fractal/flame and a box top right where a low resolution image of the fractal is displayed before the full calculations have been completed. Below the latter is a box showing pop-up menus which allow changes to be made to a selected transform. A flame may contain several transforms. The type of transform and its position on the grid determine the flame to be generated The most important factors determining transforms are:Variations: there are over 100 of these in the 7X version of Apophysis and several of these can be allocated for each transform. Variables: these are additional factors which can be allocated for certain Variations Transforms can be added using the toolbar at the top of the editor and all transforms can be assigned a weight (determining the relative effect on the flame), and also a colour which will be part of the original colour gradient. The triangles representing each transform can be moved around the grid to change the fractal/flame. This is done either by dragging the corners or by using the pop-up triangle menu. Hence, a work flow might be:Create new flame (top right button in editor).This will produce a first transform. Assign one or more variations to this transform and a weight. Create a second transform and repeat. Move the triangle representing the second transform around the grid watching the effect in the preview window. Create a third transform and repeat. Assign colours to each of the transforms Continue until a pleasing result is obtained. Open the adjustment window and zoom/centralise. Open flame menu, select size and final quality required (1,000 to 10,000). Higher qualities like 10.000 may need an hour or more of computing time to render. Save as a png to allow editing in Photoshop or other. This provides for a background colour to be selected and other colours/tones changed if required.
22
23
Apophysis - cont Trying to make interesting fractal images can be quite frustrating for the first time user. There are so many variables to choose from that it is difficult to know where to start. Often no image at all appears. I have found the best approach is to follow in detail one or more of the on-line tutorials to get a basic feel for what works and the type of fractal produced by some of the Variables. Such images can be quite satisfying in their own right but essentially are the result of someone else’s efforts. However, once you have found an interesting combination it is not too difficult to to modify it to make your own unique image. Just to get an appreciation of the immense variety of images possible, note that with 100 Variations and (say) two applied to each of 3 transforms gives the number of possible fractals as around 1012 (1,000,000,000,000). In addition there are variables such as the weight and colour assigned to each transform and other factors which can be associated with many of the Variations. The scope is enormous!
On the following pages are a few fractal flames produced along the lines outlined. All have had a final edit in Photoshop. The main parameters for each one are>Page 25 - 4 linear transforms plus one epicyclic Page 26 to 31 - various combinations of Radial Blur, Pre-blur and Waves 2 Page 32 and 33 - Pre-blur and Loonie In all cases the triangles associated with each transform (see page 23) were manipulated to produce the final result. A quality of around 4000 was used which required render times of 5 to 30 minutes
24
25
26
27
28
29
30
31
32
33
Combining Photographic Images with Fractals Whilst fractal images are often of sufficient interest to be appreciated in their own right they would not be acceptable in any purely photographic competition or exhibition (although many years ago I did see a ‘Mandelbrot’ fractal accepted as an image in MidPhot!). However, there appears no reason why fractal images should not be combined with photographs as montages where the shapes and colours of the fractal complements and enhances the visual experience. This would be on a par with use of complex manipulations and extensive use of filters to produce images generally described as ‘Creative’. A few examples of such combinations are shown in the next few pages. All fractals were produced in Apophysis Page 35 - Silhouette of dancer with gnarl fractal, Page 36 - Damselfly and spiral fractal. Page 37 - Images of dancer in ‘Loonie’ bubbles. Page 38 and 39 - Butterflies in crystal bubbles. Page 40 - High speed flash image of water drops in gnarl fractal. Page 41 - Fern with gnarl fractal.
34
35
36
37
38
39
40
41
In Conclusion It is possible to spend many happy (wasted?) hours producing an infinite variety of fractal images and perhaps combining some of them with original photographs. Here are a few ideas on presenting such work to a wider audience.
Make and frame prints to hang on your wall.
Arrange related prints into a creative panel.
Collect images together and make a book (hard copy or on-line via ISSUU)
Make cards for birthday or Christmas
Make an Audio-Visual presentation. There is a lot of scope here for exciting visual effects and choice of music.
Enter art exhibitions or creative photographic exhibitions - either selected or artists choice.
If you get really good at fractal art you might join those artists who are selling their fractal compositions. In addition, you may be interested to know that fractal work is used in films such as Star Trek and Star Wars and in the growing field of computer games.
42
David Eaves August 2020