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HTTP/1.1 200 OKContent-Type: text/htmlContent-Length: 51743Connection: keep-alivex-amz-meta-sha256: c2e0e4102713fa566347ea6c85db63671c99a793fc63a4f32f2a6a50ebaf1f05x-amz-meta-s3b-last-modified: 20250713T231934ZLast-Modified: Sun, 13 Jul 2025 23:29:45 GMTServer: AmazonS3Date: Fri, 05 Dec 2025 02:21:01 GMTETag: dd28ab09eca30ece189875f6c201a082X-Cache: Hit from cloudfrontVia: 1.1 e765e4220a4552156eaeff06c1304e0c.cloudfront.net (CloudFront)X-Amz-Cf-Pop: HIO52-P4Alt-Svc: h3:443; ma86400X-Amz-Cf-Id: zKAbhfKf0CmIUIIifrWIavbM5n6CJsyY0iH-PnGMxg5SpRGrtzGlEQAge: 51036 !doctype html>html langen>head> title>Fractal Art and Design by Gregory B. Searle/title> meta namedescription contentFractal art by Gregory B. Searle /> meta namekeywords contentfractal,mandelbrot,art,artist,mandelbrot set,Gregory B. Searle,multibrot,negabrot,burning ship,mandelbar /> meta nameX-UA-Compatible contentIEedge /> meta nameviewport contentwidthdevice-width,initial-scale1,user-scalableyes /> meta http-equivContent-Type contenttext/html; charsetutf-8 /> script srcjs/scripts.js?update201612051005 typetext/javascript>/script> link relicon typeimage/png hrefimages/mandelbrot-icon.png /> link relstylesheet hrefcss/styles.css?update201703161520 typetext/css /> link relstylesheet hreffonts/proza-libre.css typetext/css /> link relpayment hrefbitcoin:bc1qfyghxp8qflp6ufwqju32zay6txxf45c8q5833s titleBitcoin cryptocurrency payments accepted /> link relpayment hrefethereum:0x4CD6570228BE61E163bb3d97dfc744e9c8c63A34 titleEthereum cryptocurrency payments accepted /> noscript> style> div.content { display: inherit !important; } #svg_loading { display: none !important; } /style> /noscript>/head>body> script typetext/javascript> !-- // Catch DNS subdomain redirection failures. if (document.location.href.indexOf(fractal.fractalartdesign.com) ! -1) document.location.href document.location.href.replace(fractal.fractalartdesign, www.fractaleverywhere); //--> /script> a hrefindex.html>img classlogo srcimages/logo.jpg width160 height160 border0 alt />/a> h1>Fractal Art and Design/h1> h2>Gregory B. Searle/h2> ul classmenu> li>a href#welcome>Welcome/a>/li> li>a href#about>About Fractals/a>/li> li>a href#math>Math & Art/a>/li> li>a hrefgallery.html#gallery>Gallery/a>/li> !-- li>a hrefprints.html>Prints/a>/li> --> !-- li>a hrefgifts.html>Gifts/a>/li> --> li>a href#generator>Fractal Everywhere/a>/li> !--li>a href#app>Android App/a>/li>--> li>a href#resources>Resources/a>/li> li>a hrefwallpaper.html#wallpaper>Wallpaper/a>/li> li>a href#contact>Contact/a>/li> /ul> a namewelcome>/a> svg width300 height300 idsvg_loading classdrawing onclickstyle.displaynone>LOADING/svg> noscript> h3>Note/h3> p>This web site works best with scripting enabled in your web browser./p> /noscript> div classcontent idwelcome> h3>About the Artist/h3> p> Gregory B. Searle is a digital computer artist with a Bachelor in Fine Arts from the University of Lowell (now U-Mass at Lowell) and a computer programming background. He combines these seemingly opposing skill-sets to create unique computer-generated “fractal” imagery using his own custom computer code. This allows him to explore a whole world of mathematically-generated imagery, carefully crafting the limitless parameters to produce one-of-a-kind, high-quality fractal prints. /p> p> At this time he is exploring trigonometric variations of the em>Mandelbrot/em>, em>Tower of Powers/em>, and em>Newton/em> fractals at various real exponents. The combination of formulas, trig functions, fractional exponents, and different rendering techniques provides a limitless world of form and texture to explore. The expressive forms that emerge are enhanced by low-saturation or purely grayscale color schemes, highlighting shading, texture, and motion. /p> h3>About this Site/h3> p> This page is intended as a space to explore computer-generated em>fractal/em> imagery as an art form. See the em>About Fractals/em> page for more information. For frequent updates, works-in-progress, and other interesting items, see the a hrefhttps://www.facebook.com/fractalartanddesign>Fractal Art and Design/a> page on Facebook. /p> h3>Site Index/h3> ul> li>em>About Fractals/em> provides more information on this medium./li> li>em>Gallery/em> shows finished works that I have printed for sale./li> !-- li>em>Prints/em> lists prints that are available through Fine Art America./li> --> !-- li>em>Gifts/em> highlights some other items available./li> --> li>em>Fractal Everywhere/em> is a link to the tool I built to explore and create./li> li>em>Math & Art/em> goes into more detail on various subjects./li> li>em>Resources/em> lists other pages of interest on this subject./li> li>em>Wallpaper/em> is available for downloading for your phone or tablet./li> li>em>Contact/em> me through the links on this page./li> /ul> h3>Social Media/h3> p> You can now find me on Facebook as a hrefhttps://www.facebook.com/fractalartanddesign>img srcimages/facebook.png width16 height16 border0 altFacebook />FractalArtAndDesign/a> and on Instagram under a hrefhttps://www.instagram.com/fractalartdesign/>img srcimages/instagram.png width16 height16 border0 altInstagram />FractalArtDesign/a>. Ill be posting my latest finished works here, as well as following my favorite fractal artists. /p> /div> a nameabout>/a> div classcontent idabout> h3>About Fractals/h3> p> The word em>fractal/em> roughly means “fractional dimensions,” referring to the mathematical tendency of a fractal shape to form somewhere “between” classical geometric dimensions. Fractals are created by iterating a mathmatical formula, geometric progression, or another repeating process to generate an increasingly complex, often beautiful result. /p> svg height228 width200 idsvg_koch classright drawing> polygon points100,0,0,173,200,173,100,0 /> polygon points100,228,0,55,200,55,100,228 /> polygon points100,75,67,19,133,19,100,75 /> polygon points100,153,67,209,133,209,100,153 /> polygon points33,38,0,94,68,94,33,38 /> polygon points165,38,132,94,200,94,165,38 /> polygon points33,190,0,134,68,134,33,190 /> polygon points165,190,132,134,200,134,165,190 /> /svg> p> A classic example of a simple, geometric fractal is a em>Koch Curve/em>. This fractal shape is formed by overlaying an isosceles triangle on top of an inverted isosceles triangle of the same size, creating a six-point star. This forms six more triangles. Repeat for each of these triangles, and again and again... This ultimately creates a sort of fuzzy-star shape, somewhat organic-looking. The figure shows an example of the first two iterations of this process. /p> h3>Mandelbrot Set/h3> p> img classleft srcimages/base_set.jpg width240 height240 border0 altMandelbrot Set /> The em>Mandelbrot Set/em> was discovered by the mathmetician Benoit Mandelbrot at IBM in 1979. He discovered that repeated application of a deceptively simple formula, zz²+c produced an increasingly complicated result. The value of em>z/em> starts at zero, and em>c/em> is a complex (imaginary) number. The “set” is those values of em>c/em> where the formula result never exceeds 2, (or “escapes”) no matter how many times the formula is iterated. /p> p> The colorful graphics that can be generated by a computer are created not by the points that exist in the set, but by those outside that escape the formula. Each escape point took a certain number of iterations of the formula to escape. Different colors can be assigned to the different iteration counts, such as a color gradient or an interference pattern (sine waves). Since complex numbers consist of two axes, real and imaginary, this can be plotted on a two-dimensional graph to create a fractal image. /p> h3>Multibrot Space/h3> p> The Mandelbrot set can be extended into three dimensions by adding another variable, the exponent. This exponent is traditionally fixed at the value em>two/em> (or squared) in the Mandelbrot formula, zz²+c. Since em>c/em> is a complex number, the results are mapped onto a two-dimensional plane, with the real component mapped to the horizontal axis and the imaginary component mapped to the vertical axis. /p> p> Altering the formula to zzsup>n/sup>+c, we now can vary the exponent, em>n/em>. A em>Multibrot/em> simply changes the exponent to another fixed value and draws the resulting set. However, the exponent can be mapped to a third axis, creating a three-dimensional “Multibrot space” for further exploration. /p> p> img classright srcimages/vertical_multibrot.jpg width200 height320 border0 altMultibrot with real horizontal and exponent vertical /> How do we explore such as space? One possibility is to define a plane, or a “slice” through this space and calculate the resulting set on this plane. This is the approach I have taken, allowing a plane of arbitrary angle and position to be requested. See the “Multibrot Slice” formula option in my fractal generator. The image at the left is rendered with the real value on the horizontal axis, the imaginary value fixed at zero, and the exponent on the vertical axis. /p> p> Please see the em>Math/em> and em>Fractal Everywhere/em> pages for more details. /p> /div> a namegenerator>/a> div classcontent idgenerator> h3>Fractal Everywhere/h3> p> This is a fractal generator application that will run em>everywhere/em>, in any modern web browser. Browser technology has advanced far enough to allow efficient, intense number-crunching of the calculations required to create fractals right in the browser. The application currently renders the Mandelbrot set and variations. /p> p> This is a web-based application, so em>the browser matters!/em> It currently works best in a hrefhttps://www.google.com/chrome/>Google Chrome/a>. !--It currently works best in a hrefhttp://www.firefox.com/>Mozilla Firefox/a>.--> Though this application is standards-compliant, other browsers tend to have various issues: /p> ul> li>Chrome will throttle performance, especially in the background (though this has improved)./li> li>The old Internet Explorer 11 and the old Edge compiler is not as optimized and will swamp your CPU./li> li>Mobile WebKit browsers may not render the fractal (blank screen)./li> li>Mac Safari may display artifacts as the rendered fractal slices are assembled./li> li>Firefox will now also throttle performance. See em>Performance Notes/em>, below./li> /ul> p> The link below will bring you to the full version, the same version that I use to create the pieces in the gallery. A painter uses paints; I use a CPU. I am giving you access to my paints! Please be aware that this is always a work in progress. It will change at a whim. /p> noscript> p>em>strong>Note: scripting must be enabled in your web browser!/strong>/em>/p> /noscript> p aligncenter> a hrefhttp://www.fractaleverywhere.com/> img srcfractal/mandelbrot-icon.png width32 height32 border0 alticon alignabsmiddle hspace8 />Fractal Everywhere /a> /p> p stylefont-size:smaller> em> By using this application you agree to the following terms. This application is provided “as-is” and “at your own risk” without warranty as to suitability of use. I am sharing access without asking for compensation. No support is provided. I retain complete ownership and copyright on the code. The application does not track usage in any way above the normal web-host statistics typical of any and all other web sites. Your clicks and results within the application are not recorded. Any images produced are the sole property of the user under international copyright law. It is the users sole responsibility to store and preserve any results. This CPU-intensive application will place a high demand on your battery. If your tablet overheats from CPU load, contact the manufacturer, not me. /em> /p> h3>Performance Notes/h3> p> Browsers are taking steps to reduce the impact of CPU-intensive web pages on your web browsing experience. If you plan on performing some serious rendering in this application, you should consider disabling the CPU throttling in your browser. /p> ul> li> strong>Firefox/strong>: Enter “about:config” into the address bar and accept the warning that comes up. Search for em>dom.ipc.processPriorityManager.enabled/em>. If you dont see it, you can right-click on the list and create it. Set it to strong>false/strong>. /li> li> strong>Chrome/strong> no longer allows control of this feature. However, its task management has improved greatly. When rendering, keep the rendering tab frontmost and active for top performance. Other windows and tabs will be de-prioritized, but they will render. Chrome handles multiple tasks very nicely now. /li> /ul> p> To draw the most performance from your CPU, enable the em>Performance/em> item under the em>Options/em> menu. This is off by default. /p> h3>Fractal Variations/h3> p stylemin-height:90px> a hreffractal?l3&c17&resettrue> img classleft srcimages/icon_mandelbrot.png width80 height80 border1 altMandelbrot /> /a> There are multiple interesting variations of the strong>Mandelbrot/strong> set that arise through modifications to the underlying formula. You may switch the em>formula/em> in use under the em>Control/em> pane. /p> p stylemin-height:90px> a hreffractal?f7&c17&l3&resettrue> img classleft srcimages/icon_mandelbar.png width80 height80 border1 altMandelbar /> /a> strong>Mandelbar/strong> creates a three-lobed figure by using the em>complex conjugate/em> of the traditional formula, which calculates the real portion em>minus/em> the imaginary portion. /p> p stylemin-height:90px> a hreffractal?f6&c17&l3&resettrue> img classleft srcimages/icon_burning_ship.png width80 height80 border1 altBurning Ship /> /a> strong>Burning Ship/strong> calculates the absolute value (positive only) of the formula. The result looks like a ship at sea, on fire. Magnifying behind the “ship” reveals some tall ships, also seemingly on fire. This pattern appears to the left of the Multibrot set rendered with the y-axis mapped to the exponent instead of to the imaginary component. (See em>Multibrot Slice/em>, below.) /p> p stylemin-height:90px> a hreffractal?f3&c17&l3&resettrue> img classleft srcimages/icon_cubed.png width80 height80 border1 altCubed /> /a> strong>Cubed/strong> raises the power of the formula to three, instead of two. This creates a mirrored-image of the set, with some interesting differences. /p> p stylemin-height:90px> a hreffractal?f9&s3&c17&l3&16&resettrue> img classleft srcimages/icon_multibrot.png width80 height80 border1 altMultibrot /> /a> strong>Multibrot/strong>* allows you to set the exponent of the formula to something other than the traditional second power. Larger numbers create a fringed circle effect, while non-integer values add some interesting complications. Negative values change the behavior altogether, and utilizes em>period mapping/em> instead of the traditional em>escape/em> method. /p> p stylemin-height:90px> a hreffractal?f10&c17&l3&resettrue> img classleft srcimages/icon_multislice.png width80 height80 border1 altMultibrot Slice /> /a> strong>Multibrot Slice/strong>* takes a cross-section, or “slice,” of the multibrot set rendered in three-dimensions. The real and imaginary components are still mapped to the x- and y-axes, and the exponent is added for the z-axis. To take a “slice” of this form, extra parameters are available to define the plane of a cross-section. /p> p> em> This is a superset of almost all variations presented on this page. There is so much to explore, even without changing the parameters! /em> /p> p> First, an em>Offset/em> defines the distance of the center of the plane from the origin (0,0,0). em>Angle/em> determines the angle of the plane from the z-axis (the exponent). em>Rotation/em> specifies the rotation of the plane around the z-axis. Angles are in degrees, (0,0) facing “down” at a traditional multibrot rendering, in which case the offset is equivalent to the exponent. /p> p> It is very easy to get lost and end up with a blank screen! If this happens, zoom out, or reduce your offset to single digits, or em>Reset/em> the parameters to start over. The active set is a narrow, vertical column. Keep in mind that you are rotating your thin render plane around the column, and not all solutions intersect. At this time, the parameters rotate around the origin (0,0,0). /p> p stylemin-height:90px> a hreffractal?f8&c17&l3&rm0&resettrue> img classleft srcimages/icon_negabrot_escape.png width80 height80 border1 altNegabrot (Escape) /> /a> strong>Negabrot/strong> shows what occurs when the exponent is changed to -2. This implementation utilizes the same em>escape/em> method as the Mandelbrot (though technically incorrect and somewhat unstable). /p> !-- p stylemin-height:90px>a hreffractal?f9&r-.4&i0&s3.5&1-2&c17&l3&resettrue>img --> p stylemin-height:90px> a hreffractal?f8&c17&l3&rm1&resettrue> img classleft srcimages/icon_negabrot_periodic.png width80 height80 border1 altNegabrot (Periodic) /> /a> strong>Negabrot (Periodic)/strong> is the Negabrot variant above rendered “correctly” using em>period mapping/em>. /p> !--p stylemin-height:90px> a hreffractal?f2&c17&l3&resettrue> img classleft srcimages/icon_ripples.png width80 height80 border1 altRipples /> /a> strong>Ripples/strong>* inserts a sine function into the formula, creating an underwater effect. This is not part of the multibrot set. /p>--> !--p stylemin-height:90px> a hreffractal?f5&c17&l3&resettrue> img classleft srcimages/icon_feathered.png width80 height80 border1 altFeathered /> /a> strong>Feathered/strong>* uses an arctangent function to create a feathered, or windy, effect. This is not part of the multibrot set. /p>--> p stylemin-height:90px> a hreffractal?f11&c17&l3&resettrue> img classleft srcimages/icon_powers.png width80 height80 border1 altTower of Powers /> /a> strong>Tower of Powers*/strong> is a little different. It calculates the initial value em>c/em> raised to the power of the result em>z/em> over and over again, em>z csup>z/sup>/em>, starting with em>z 1/em>. You would think that this would quickly escape to infinity, but there are some stable areas that produce interesting results. This is very similar to what appears when you render the Multibrot with a very high exponent. /p> p stylemin-height:90px> a hreffractal?f12&c11&l1&resettrue> img classleft srcimages/icon_newton.png width80 height80 border1 altNewton /> /a> strong>Newton*/strong> fractal illustrates the chaotic nature of the Newton method for iteratively discovering the roots of a complex equation. By default, it renders in the Julia set, but things get interesting using other rendering options. This fractal is essentially a Multibrot formula divided by its derivative. /p> p classcenter> em> Select an image to go directly to the live rendering. /em> /p> p> * Some of the formulas are more calculation-intensive than others, and will take more time to render. /p> h3>More Details/h3> p> Here are some more details on this application. First, it is designed for functionality, not to be pretty. If youre using a tablet or mobile device, touch support is rudimentary. The primary goals are flexibility, quality of output, and calculation speed (more below). I am always tweaking it with these goals in mind. /p> svg height90 width90 classinset drawing right> line x115 y115 x215 y275 /> circle cx15 cy30 r5 classfill /> line x130 y115 x230 y275 /> circle cx30 cy45 r5 classfill /> line x145 y115 x245 y275 /> circle cx45 cy60 r5 classfill /> line x160 y115 x260 y275 /> circle cx60 cy45 r5 classfill /> line x175 y115 x275 y275 /> circle cx75 cy60 r5 classfill /> /svg> p> strong>Flexibility/strong> A strong, customizable theming engine is built in for theoretically unlimited color theming of the fractals. It supports traditional gradient (ramp) themes, sinewave- based (wave) themes, and more complicated (and hard to manage) interference matrices. Many presets are provided to get you started. You can also build your own, which are saved in your browsers local storage. Note that if you clear your browsers storage, you will lose your themes! The theme can be applied as a linear, logarithmic, or exponential progression (see the em>Render/em> menu) to control complexity. For finished work, the resolution can be changed to create print-quality results. There are many other parameters that can be fine-tuned. /p> p> The em>depth/em> controls the maximum iterations allowed before the point is considered “escaped.” Without such a maximum, the calculation would take forever. This is automatically determined based upon the magnification. Several presets are provided, em>Moderate/em> being the typical setting. You can also specify this number manually in the em>Details/em> pane. /p> svg height90 width90 classinset right> rect x0 y0 width18 height18 stylefill:#000 /> rect x18 y0 width18 height18 stylefill:#000 /> rect x36 y0 width18 height18 stylefill:#333 /> rect x54 y0 width18 height18 stylefill:#666 /> rect x72 y0 width18 height18 stylefill:#999 /> rect x0 y18 width18 height18 stylefill:#000 /> rect x18 y18 width18 height18 stylefill:#333 /> rect x36 y18 width18 height18 stylefill:#666 /> rect x54 y18 width18 height18 stylefill:#999 /> rect x72 y18 width18 height18 stylefill:#FFF /> rect x0 y36 width18 height18 stylefill:#333 /> rect x18 y36 width18 height18 stylefill:#666 /> rect x36 y36 width18 height18 stylefill:#999 /> rect x54 y36 width18 height18 stylefill:#FFF /> rect x72 y36 width18 height18 stylefill:#FFF /> rect x0 y54 width18 height18 stylefill:#666 /> rect x18 y54 width18 height18 stylefill:#999 /> rect x36 y54 width18 height18 stylefill:#FFF /> rect x54 y54 width18 height18 stylefill:#FFF /> rect x72 y54 width18 height18 stylefill:#FFF /> rect x0 y72 width18 height18 stylefill:#999 /> rect x18 y72 width18 height18 stylefill:#FFF /> rect x36 y72 width18 height18 stylefill:#FFF /> rect x54 y72 width18 height18 stylefill:#FFF /> rect x72 y72 width18 height18 stylefill:#FFF /> /svg> p> strong>Quality/strong>. em>Oversampling/em> performs multiple calculations per pixel to create a high-quality image. You can choose from em>Fast/em> calculation with no oversampling to em>Fine/em> 8x8 oversampling. For most exploration, you will probably stay in em>Good/em> 2x2 oversampling for a balance of speed and quality. The calculation overruns by three extra iterations to smooth out banding artifacts that are typically created by an iterated process. em>Preview/em> mode temporarily turns off calculation-intensive enhancements for quicker rendering. /p> svg height90 width90 classinset left> rect x0 y0 width30 height90 stylefill:#F00 /> rect x30 y0 width30 height90 stylefill:#0F0 /> rect x60 y0 width30 height90 stylefill:#00F /> /svg> p> For extra detail on your screen, you can enable em> Subpixel Rendering /em>. Most LCD screens are set up with red, green, and blue elements arranged side-by-side on each pixel. This option takes the positions of these elements into consideration when rendering, effectively tripling the resolution of your display on the em>Detailed/em> and em>Fine/em> quality settings. The oversampling adjusts slightly to 6x4 and 9x8, respectively. Note that you should turn this off when rendering for print or web. /p> div classinset right stylefont-size:14pt;width:90;height:90;line-height:67pt;> ASM.JS /div> p> strong>Speed/strong>. The calculation engine is fully-optimised to run in the browsers em>asm.js/em> compiler. This means that the browser distills the core calculation into native machine code (really fast)! The application contains its own benchmark, which Ive used to fine-tune the performance. You can adjust certain performance parameters in the options. This application is multi- threading, and will automatically adjust to your devices capabilities, even if youre using a tablet. /p> p> Many factors will affect the overall speed of the rendering, CPU power being the primary constraint. The deeper you go into a portion of the fractal set, the more iterations are usually required to render a result. Oversampling quality and smoothing also increase demand. You can temporarily turn off all calculation-intensive enhancements with the em>Preview/em> option under the em>Quality/em> menu. /p> p> The em>Performace/em> option under the em>Options/em> menu will attempt to utilize the full capability of your CPU. Note that some browsers (ahem, Internet Explorer) will be extra aggressive when allocating your systems resources. By default, this option is not turned on to give your system some breathing room to operate other tasks. /p> h3>Extended Precision for Deep Zoom/h3> p> This application supports em>double-double-precision math/em> for zooming in beyond the CPUs double-precision limit. Normally, the image will degrade into blocks if you zoom in too far; youve hit the precision limit of your computer when this happens. This application will automatically switch to em>extended precision/em> when you hit this limit. This requires extra computation, however, and results in the calculation slowing down dramatically. You will see a brief notification when this occurs, and there is a “Precision” indicator in the “Details” panel that will switch from “Standard” to “Extended.” Id recommend lowering the quality setting to “Fast” when exploring this deep. Expect long render times for finished fractals. /p> p> At this time, the formulas that require trigonometry dont yet support extended precision. The application will notify you if it cant do it. You can currently zoom deep into the Mandelbrot, Mandelbar, Burning Ship, and Cubed variations. Eventually it will have support for all variations. /p> h3>Various Tips/h3> p> You can save the browsers URL at any time to a bookmark in your web browser to remember the current fractal displayed. The displayed coordinates are compatible with any other Mandelbrot set application, however, the options in the URL are unique to this application. The strong> em>Save Location/em> option under the em>Options/em> menu /strong> will automatically remember where you were when you come back later. Note that a saved URL (bookmark) will override this option. /p> svg height90 width80 classdrawing right> polygon points17,25,27,15,27,25,17,25,17,60,62,60,62,15,27,15 /> rect x0 y60 width80 height20 /> /svg> p> strong>Printing a rendering/strong> usually requires higher resolution output than your screen. You can manually set the pixel width and height through the em>Resolution/em> setting on the em>Details/em> pane. You will want to calculate the target width and height by multiplying the paper size by the desired DPI. strong>Gamma/strong> is usually set to 1. If you are rendering for a print, you may wish to change this to around 1.2. /p> p> strong>Resolution/strong> defaults to your web browser window size, or the screen size of your mobile device. With em>Auto Render/em> enabled under the em>Options/em>, your fractal will automatically re-render if your window size changes. This includes any bars that appear along the bottom of the window. To prevent this, click on the em>Set/em> or em>Screen/em> link under the em>Resolution/em> setting to fix the resolution, ignoring resize events. /p> p> strong>Saving your rendering/strong> is best achieved by right-clicking on the image and selecting the option to save the image from the menu. If your browser responds to the right-click by sending a regular click to the application, click em>Lock Coords/em> above the coordinates first. A em>Save Image/em> link appears on the em>Details/em> pane when rendering completes, but this is memory-intensive and has limited support by the browsers. Note that some browsers will show a status bar at the bottom once you save an image, triggering a resize (and re-render) on the window. On mobile devices, use the em>Full Screen/em> and em>Hide Controls/em> options to clear all user interface elements, then take a screenshot. /p> p> strong>Settings and custom themes/strong> are saved in your browsers local storage. em> If you clear this, all of your customization will disappear. /em> There is no export function. However, custom themes are copied into the URL on the em>Coordinates/em> link, so if you have saved a bookmark, your theme will be preserved. I take pains during development to honor older bookmarks so they always work. /p> /div> a nameapp>/a> div classcontent idapp> h3>Fractal App for Android/h3> p> em> Note: The latest version of Android Web View has a display issue on some devices. The display will not update properly and you may see a blank screen, or a partial rendering. We have to wait for Google to fix the issue. /em> /p> p> Ive created a simple em>container/em> app for Android that runs my fractal application on your smartphone or tablet. The advantage of this is that it is available offline at any time, and it remembers your place. The app version is simplified, as it is not running in a full web browser. There are buttons along the bottom to launch your work in the devices web browser, copy the URL to the clipboard, copy the clipboard back into the app, and to enter fullscreen mode on the device. It does not have capability to save the image at this time (use the devices screenshot capability). /p> p aligncenter> a hrefapk/com.fractalartdesign.fractal.apk download> img srcapk/appicon.png width50 height50 border0 alignabsmiddle hspace8 altApp />Fractal Everywhere /a> /p> p> This app is not “official” and is not available in any app store. It is available “as-is” from this web site only. You must enable your devices ability to install apps from outside sources. I do not have an Apple development environment, so I dont plan on developing an iDevice version. This is a work in progress. /p> p> em>Tip:/em> Access the latest updates by clearing the apps data on the device, then launching the app while connected to the Internet. Note that this will lose your place. Do this through the devices application management settings. Alternatively, you can uninstall and reinstall the app. /p> /div> a namemath>/a> div classcontent idmath> h3>Math & Art/h3> p> Wait, “Math” em>and/em> “Art?” If youve come this far, you may be thinking twice about these apparent opposites. Yes, its possible for the two to coexist and even complement each other. /p> p> This is a space to discuss various topics that dont fit elsewhere regarding art, math, programmatic details, etc. I hope you find this interesting. Chances are, you arrived here through a search result. If so, please explore the rest of the site! I plan to add more to this page as my exploration continues in this area. /p> a namemath_contents>/a> h3>Contents/h3> ol> li>a href#math_how>How is This Art?/a>/li> li>a href#math_motion>Motion/a>/li> li>a href#math_perturb>Perturbation Theory/a>/li> li>a href#math_smooth>Smoothing/a>/li> li>a href#math_period>Period-Mapping/a>/li> /ol> a namemath_how>/a> h3>How Is This Art?/h3> p> img classright srcwallpaper/Solar.jpg width240 height240 border0 altFractal Art /> When creating fractal art, the computer does all the work. Really, You press a button and out comes an image. You may be asking, em>how is this art?/em> /p> p> The art is in the painstaking em>preparation/em>. For example, an exquisite bronze sculpture is created when a metal foundry pours molten metal into a cast. The metal cools, the cast is opened, and the sculpture is there! It isnt that simple, though. An artist had to create the cast. /p> p> Computer-generated art requires an artist to set up the parameters. This requires careful exploration of the fractal space and fine adjustment of form and color. Like photography, framing and lighting are critical. It can take several hours (or days!) to create something thats worth rendering to print. /p> p> To continue the bronze sulpture metaphor, multiple copies of the sculpture can be created simply by reusing the mold. This reduces the value of each copy, unless em>the mold is broken/em> after the first successful cast. This creates a one-of-a-kind piece of work. /p> p> The same can be done with fractal art. If only one print is created from the work, and the parameters are never shared, then the work becomes unique. Its possible to “break the mold” by securing or even em>deleting/em> the parameters file and the rendered image file after printing. These become em>single-edition/em> prints. /p> p> I have not had the courage to hit the em>delete/em> button after finishing a work. The prints are one-of-a-kind, yet the original fractal provides opportunity for continued study. Unlike a mold, it is still a “living” work of art that can be explored further. /p> a namemath_motion>/a> h3>Motion/h3> p> img classleft srcimages/motion.gif width240 height240 border0 altMoving Fractal /> Creating moving art with fractals involves gradually changing one or more of the values that affects the underlying fractal formula, as well as the method used to render the fractal such as the approach to color the fractal form. /p> h4>Possible Parameters/h4> ul> li>em>X/em> and em>Y/em> coordinates in the complex plane/li> li>The exponent em>e/em> in the formula, em>z/em> em>z/em>sup>e/sup> + em>c/em>/li> li>When slicing through multibrot space, the slice angle and position/li> li>Coloration using the red, green, and blue channels of each rendered pixel/li> /ul> p> There are two methods of rendering a moving fractal: rendering to video or real-time generation. With a modern graphics processing unit (GPU) that is built-in to todays computers and devices, it is possible to generate motion in real-time on nearly any device. Higher complexity, such as varying the exponent in a trigonometric fractal, requires more capable devices such as the a hrefhttps://layer.com>Layer Frame/a>. /p> p> This is the next natural step for development of my fractal program, as all the components are in place. I have a little development to do... /p> a namemath_perturb>/a> h3>Perturbation Theory/h3> p> em>Perturbation Theory/em> has been around for a while, and it is a method for figuring out the results of an unknown relationship by calculating known relationships around it, then using the differences (or perturbations) as a means to get to the unknown value. See a hrefhttps://en.m.wikipedia.org/wiki/Perturbation_theory>Wikipedia - Perturbation Theory/a> for more information. /p> p> How does this apply to fractals? In March of 2013, a hrefhttp://www.superfractalthing.co.nf/sft_maths.pdf>Kevin Martin published a paper/a> explaining a theoretical method to apply perturbation theory to the calculation of the Mandelbrot set. Application of the theory promised a couple of significant advantages: /p> ol> li> You only have to iterate /em>once/em>, leveraging the values collected during the iteration of a single sample point to figure out all the rest of the points in the fractal. /li> li> All calculations are performed using standard double-precision floating-point processing, em>no matter how deep the zoom/em>. There is no need to exotic extended-precision math. /li> /ol> p> These two advantages would eliminate the “brute force” technique traditionally utilized to render fractals, which requires million or billions of iterations of the fractal formula. By iterating just one point, the rest of the fractal could be rendered simply by leveraging the data collected, calculating em>perturbations/em> for all the points around that reference point. In practice, more reference points are needed for complex areas of the fractal, but this usually results in only a handful of fully-iterated points being calculated. /p> p> The pertubation formulas for the Mandelbrot set em>cancel out/em> the need for ultra-precise floating-point numbers when zooming into the fractal beyone 10sup>15/sup> magnification. All calculations can be performed natively on any computers floating-point processor for any zoom level. This eliminates the breakdown or blockiness that occurs in deep zooms, as well as the need for CPU-intensive extended-precision techniques, and opens up the ability for nearly-unlimited zoom into a fractal. /p> p styletext-align:center>strong>em>The result: nearly real-time rendering with unlimited zoom!/em>/strong>/p> p> Multiple existing fractal applications have already taken advantage of this theory and have implemented it. I am late to the game, and am currently working on implementing it for my own project. At this time, my application em>does not/em> utilize perturbation methods, and relies on the brute-force iteration method. This will change. /p> p> em>Why do this if others have already done it?/em> I am doing this to learn-by-doing, to enjoy the process of figuring this out, and to build my artistic toolset. By building it myself, I gain a better understanding of the subject and can better create art. /p> a namemath_smooth>/a> h3>Smoothing/h3> p> img classright srcimages/smooth-banded.jpg width240 height240 border0 altSmoothing vs. unenhanced banding /> Calculating a Mandelbrot-type fractal requires counting the iterations required for the formula to em>escape/em> beyond a predefined value. This results in integer values, which when mapped produces a marked “banded” effect. A little calculus allows us to figure out how these bands are progressing overall and to smooth out the quantization effect of an iterating formula. This image shows the difference, the right half of the image being unenhanced. /p> h4>Smoothing Formula/h4> p classcenter> span classmath> iterations + /span> span classmath> span classmath under> ln (2) - log span classmath large> ( /span> span classmath> span classunder> log (r² + i²) /span>br /> 2 ∙ ln (2) /span> span classmath large> ) /span> /span>br /> log (exponent) /span> /p> p> In most cases em>ln (2)/em>, em>2 ∙ ln (2)/em>, and em>log (exponent)/em> can be precalculated into constants to speed iterative calculations: /p> p classcenter> span classmath> iterations + /span> span classmath> span classmath under> em>LN2/em> - log span classmath large> ( /span> span classmath> span classunder> log (r² + i²) /span>br /> em>2LN2/em> /span> span classmath large> ) /span> /span>br /> em>LOGE/em> /span> /p> svg height80 width80 classdrawing right> lineargradient idbanded x10% y10% x20% y2100%> stop offset0% stylestop-color:#FFF /> stop offset25% stylestop-color:#FFF /> stop offset25% stylestop-color:#AAA /> stop offset50% stylestop-color:#AAA /> stop offset50% stylestop-color:#555 /> stop offset75% stylestop-color:#555 /> stop offset75% stylestop-color:#000 /> stop offset100% stylestop-color:#000 /> /lineargradient> lineargradient idsmooth x10% y10% x20% y2100%> stop offset0% stylestop-color:white /> stop offset100% stylestop-color:black /> /lineargradient> rect x0 y0 width38 height80 fillurl(#smooth) /> rect x42 y0 width38 height80 fillurl(#banded) /> /svg> p> The iteration cycles must overshoot beyond the escape value a few times to collect enough samples for smoothing. This is a small price to pay for a more finished result. /p> a namemath_period>/a> h3>Period-Mapping/h3> p> img classleft srcimages/negabrot.jpg width240 height240 border0 altNegabrot with periodicity texturing /> Rendering negative exponents “breaks” traditional em>escape-time/em> rendering of the inside of the set. Instead, em>periodic/em> rendering must be utilized to render under negative exponents. Typically, this requires time-consuming sampling of many iterations for each point. However, a hrefhttps://en.wikipedia.org/wiki/Aleksandr_Lyapunov>Aleksandr Lyapunov/a> developed a methodology to determine period-calculation without such sampling. /p> p> Applying Lyapunovs work to the Multibrot universe, the calculation is fairly simple: take the natural logarithm of the average result and divide by the number of iterations run. No sampling, just apply a formula to the final figures. We can collect the sum over the iteration process without much extra CPU work. /p> p> Implementing this into a computer program, I realized that the formula can be simplified, I mean em>really/em> simplified, with the goal of a satisfying visual result. The final periodic calculation used by this application resolves the average magnitude of a set of vectors. This provides a visual texture of the periodicity inherrent to negative exponents. My piece a hrefgallery.html#p10>Sunset on Ice/a> is an example of how complex this texture can become. /p> h4>Simplified Periodic Formula/h4> p classcenter> span classmath> 1br /> span classover> (∑r² + ∑i²) ÷ iterations + 1 /span> /span> /p> p> Since we are already calculating these squares as a matter of building the fractal, collecting the sums is a trivial addition (no pun intended). Adding one to the denominator avoids a runaway result to infinity for vector lengths close to zero. On the other hand, the inversion inhibits runaway values in the other direction. The overall magnitude of the result can be controlled by changing the constant numerator from one to another value. /p> p> These calculations are leveraged for optional effect-rendering on the inside of the set for positive exponents, and the outer area for negative exponents. em>Period-rendering/em> can be applied at any time in my application through the em>Render/em> menu. Also, variations of the effect can be selected through the em>Effect/em> menu. The application can render both escape-time em>and/em> period-mapping at the same time for interesting results. Try it with the trigonometric variants of the Mandelbrot fractal. /p> p> See the Wikipedia articles linked in the em>Resources/em> section for more detail about smoothing and period-mapping. /p> /div> a nameresources>/a> div classcontent idresources> h3>Resources/h3> p>Here are a few additional resources for more information./p> ul classspaced linklist> li> a hrefhttps://wikipedia.org/wiki/Fractal>Wikipedia: Fractal/a> A comprehensive overview /li> li> a hrefhttps://wikipedia.org/wiki/Mandelbrot_set>Wikipedia: Mandelbrot Set/a> A math-heavy, detailed explanation /li> li> a hrefhttps://wikipedia.org/wiki/Multibrot_set>Wikipedia: Multibrot Set/a> More info on changing the exponent /li> li> a hrefhttps://wikipedia.org/wiki/Benoit_Mandelbrot>Wikipedia: Benoit Mandelbrot/a> The “founder of fractals” /li> li> a hrefhttp://fractalforums.org/>Fractal Forums/a> In-depth discussions of all things fractal /li> /ul> h3>Related Organizations/h3> p>Some organizations that I have worked with are listed below./p> ul classspaced linklist> li> a hrefhttp://www.lightspacetime.art>Light, Space & Time/a> Online art gallery and artist competitions /li> li> a hrefhttp://www.naaa-arthub.org>NAAA ArtHub/a> Nashua Area Artists Association /li> !-- li> a hrefhttp://fineartamerica.com/shop/prints/fractal>Fractal Art Prints for Sale/a> by many talented artists at Fine Art America /li> --> /ul> /div> a namecontact>/a> div classcontent idcontact> h3>Contact Me/h3> p>You may contact me through this web site./p> p classcenter> a hrefmailto:fractal@fractalartdesign.com stylefont-size:larger>fractal@fractalartdesign.com/a>br /> /p> p>em>Note that there is a “spam” filter in place./em>/p> p> Facebook: a hrefhttps://www.facebook.com/fractalartanddesign/>www.facebook.com/fractalartanddesign/a> /p> p> Paypal: a hrefhttps://paypal.me/GregorySearle>paypal.me/GregorySearle/a> /p> /div> div classfooter> div> Copyright © span idyear> /span> Gregory B. 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