- Email: [email protected]
Layering techniques in fractal art
Computers & Graphics 24 (2000) 611}616
Chaos and Graphics
Layering techniques in fractal art Alice Kelley 23 East Maynard Avenue, Columbus, OH 43202, USA
Abstract A fractal program released in 1999, Ultra Fractal, was the "rst publicly available fractal software package to include convenient layering methods previously limited to image editing programs. The artistic e!ects of layering within a fractal program are demonstrated. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Fractal art; Fractals; Computer art; Ultra fractal; Lapsing techniques
Computer-generated fractal imagery, originally in the realm of physicists and mathematicians, has been appearing with increasing frequency as popular art. Galleries display high-quality prints of fractal images, and stores o!er fractal merchandise such as posters and calendars [1]. `Fractala geometry, from the Latin `fractusa, meaning `brokena, was introduced in 1975 by mathematician Benoit Mandelbrot to describe irregular and intricate natural phenomena such as coastlines, plant branching, and mountains that cannot be described by Euclidean geometry. Fractal shapes, like coastlines, exhibit selfsimilarity * similar details at di!erent size scales. These characteristics continue for many magni"cations, both with natural phenomena and digital fractal art. Computers, with their ability to quickly perform the thousands of iterations necessary to graphically render a fractal mapping, have made it possible to create and explore abstract fractal geometric shapes. As fractal generating programs become easier to use and have more options, dazzling fractals are being created with increasing ease. Most computer-generated digital fractal images start out as a single layer. The concept of an image `layera is well known to users of image-editing tools such as Adobe Photoshop. Generally speaking, layers allow di!erent images to be superimposed, or merged, with a variety of options. A fractal generating program called Ultra Frac-
E-mail address: [email protected] (A. Kelley). http://www.KelleyArt.com
tal, written by Frederik Slijkerman (http://www.ultrafractal.com, Win 95/98/NT), allows additional fractals to be rendered within the same boundaries as the "rst fractal. With each added fractal consisting of a new layer with its own set of properties, and with a merge mode that dictates how the added fractal layer interacts visually with the previous fractal's layer. Individual layers can be manipulated without a!ecting the other layers. Users may combine as many layers as they wish in one image, either using variations of the original layer, or adding entirely new layers that use di!erent fractal generating formulas. There are 19 di!erent merge modes, including screen, overlay, di!erence, and hue, with each having an adjustable opacity from 0 to 100%. By combining di!erent layers and altering the merge modes between them, it is possible to create several di!erent fractals from one original single-layer image, which can allow for more artistic interpretation of traditional fractal forms. Moonscape (Fig. 1), a spiral derived from the wellknown Mandelbrot set, `z"z#ca [2], is an example of a single-layer fractal that was originally rendered using the DOS, 256 color program Fractint (The Stone Soup Group, http://spanky.triumf.ca/www/fractint/ fractint.html) [3]. Ultra Fractal has the ability to read Fractint parameter "les, which are the text "le `recipesa that contain all the fractal's information, allowing the fractal to be recreated when Fractint, or in this case Ultra Fractal, reads the "le. Moonscape was re-rendered in Ultra Fractal, and the color banding within the image that results from Fractint's 256 color limitation was automatically smoothed by Ultra Fractal's true color
0097-8493/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 9 7 - 8 4 9 3 ( 0 0 ) 0 0 0 6 2 - 5
612
A. Kelley / Computers & Graphics 24 (2000) 611}616
Fig. 1. Moonscape.
Fig. 2. Corona.
(2 colors) capability. A stark image such as this does not bene"t from the addition of layers. Corona (Fig. 2), a spiral created using the Phoenix formula `z(n#1)"z(n)?#cHz(n)@#p*z(n!1)a [4], is
another image that was "rst rendered in Fractint. Shown with it are two examples of how adding a single layer can alter the original fractal. In the "rst example, Etched Corona (Fig. 3), a new layer was added that was identical
A. Kelley / Computers & Graphics 24 (2000) 611}616
613
Fig. 3. Etched Corona.
Fig. 4. Beta Lyrae.
to the "rst. The coloring algorithm, a function that transforms the color index value of each point, was switched in the new layer from the original `outside color"itera algorithm, to a `hearta orbit trap; all coloring algorithms mentioned are documented and available in Ultra Frac-
tal. The `di!erencea merge method was selected, which returns the absolute di!erence between the blend color and the base color [5], and which typically is used at opacity 100%. In the second example, Beta Lyrae (Fig. 4), which the author "nds even more artistically e!ective,
614
A. Kelley / Computers & Graphics 24 (2000) 611}616
again a new layer was added that was identical to the "rst. This time a Gaussian integer coloring algorithm was applied to the new layer, and a `Hard lighta merge method, at opacity 100%, was chosen, which multiplies or screens the colors, depending on the blend color [5], and "nally the coloring gradient editor, which controls
the colors in each layer, was shifted slightly in this layer to give the "nal image a more pleasing appearance. Ultra Fractal's graphical user interface allows the user to see the e!ects these changes have as they are being made, and to quickly adjust variables in order to attain a visually pleasing image. A fractal like Corona may also be
Fig. 5. Tropical Fish.
Fig. 6. Tropical Fish/Sea.
A. Kelley / Computers & Graphics 24 (2000) 611}616
duplicated several times within the program's window, allowing exploration of di!erent variables and layers for each duplication to occur nearly simultaneously. Tropical Fish (Fig. 5), a fractal made with a formula called Gallet-5-08, `z"F(real(z), imag(z))#iHF(imag(z), real(z))a [6], was also "rst rendered in Fractint. Once its
615
parameter "le was read by Ultra Fractal, the image could be altered any number of di!erent ways by adding layers. Tropical Fish/Sea (Fig. 6) is one example out of the "ve fractals that ultimately evolved from the original, which incorporates a total of four layers, all using the Overlay merge method, which multiplies or screens the colors,
Fig. 7. Rainforest.
Fig. 8. Hatchling.
616
A. Kelley / Computers & Graphics 24 (2000) 611}616
Fig. 9. Terraria.
depending on the base color [5], at between 44 and 50% opacity. The "rst new layer uses the same formula, with the coloring algorithm change to `outside"reala, which creates a very simple fractal that provides the diagonal white swirl that runs across the image. The second new layer uses the formula used in the Corona example, the Phoenix Julia [4], to provide the `seaweeda in the image with the bene"t of a `linesa orbit trap coloring algorithm. The last new layer uses the formula `Newton's method for exp(z)"log(z)a [7] with a `Shapes 2"ellipsea coloring algorithm to provide the green and blue swirling shapes that look like water. This version of tropical "sh is also notable because two of the layers make use of Ultra Fractal's `alpha channela feature, which allows the user to specify an opacity value for each point in the gradient [5]. Each layer has its own alpha channel which may be toggled on or o!. Examples of fractals that did not start out as singlelayer Fractint images and which exist because of the complexity and interaction of multiple layers include Rainforest (Fig. 7), and Hatchling (Fig. 8), which are both three-layer images, and Terraria (Fig. 9), a fourlayer image. Though these images are complex, artists routinely mix 15 or more layers to create images that in the past would have been very di$cult to create and manipulate. The approaches described in this paper appear to further expand the artistic possibilities of fractal creation. To see more examples of both single and multi-layer fractal art, visit the author's web gallery: http://www.KelleyArt.com.
Acknowledgements The author thanks Cli!ord Pickover for his encouragement and comments.
Appendix A. Computer details All "gures were rendered on a 233 MHx Pentium II running Windows 95 with 64 megs RAM and a 1024;768 screen display.
References [1] Kelley A. Fractal Cosmos Calendar. Amber Lotus, http://www.amberlotus.com/fractal.html, P.O. Box 31538, San Francisco, CA, 94131, 2000, 2001. [2] Mandelbrot B. The fractal geometry of nature. New York: Freeman, 1982. [3] Wegner T, Tyler B. Fractal creations. 2nd ed. Corte Madera, CA: The Waite Group, Inc., 1993. [4] Jones DM. DMJ formula "le for Ultra Fractal, http://www.fractalus.com/ultrafractal/. [5] Slijkerman F. Ultra fractal help "les, http://www.ultrafractal.com. [6] Gallet S. Gallet formula "les. http://ourworld.compuserve.com/homepages/Sylvie}Gallet/linke.htm. [7] Mitchell, L. Kerry LKM formula "le for Ultra Fractal. http://www.primenet.com/&lkmitch/uf.htm.
Chaos and Graphics
Layering techniques in fractal art Alice Kelley 23 East Maynard Avenue, Columbus, OH 43202, USA
Abstract A fractal program released in 1999, Ultra Fractal, was the "rst publicly available fractal software package to include convenient layering methods previously limited to image editing programs. The artistic e!ects of layering within a fractal program are demonstrated. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Fractal art; Fractals; Computer art; Ultra fractal; Lapsing techniques
Computer-generated fractal imagery, originally in the realm of physicists and mathematicians, has been appearing with increasing frequency as popular art. Galleries display high-quality prints of fractal images, and stores o!er fractal merchandise such as posters and calendars [1]. `Fractala geometry, from the Latin `fractusa, meaning `brokena, was introduced in 1975 by mathematician Benoit Mandelbrot to describe irregular and intricate natural phenomena such as coastlines, plant branching, and mountains that cannot be described by Euclidean geometry. Fractal shapes, like coastlines, exhibit selfsimilarity * similar details at di!erent size scales. These characteristics continue for many magni"cations, both with natural phenomena and digital fractal art. Computers, with their ability to quickly perform the thousands of iterations necessary to graphically render a fractal mapping, have made it possible to create and explore abstract fractal geometric shapes. As fractal generating programs become easier to use and have more options, dazzling fractals are being created with increasing ease. Most computer-generated digital fractal images start out as a single layer. The concept of an image `layera is well known to users of image-editing tools such as Adobe Photoshop. Generally speaking, layers allow di!erent images to be superimposed, or merged, with a variety of options. A fractal generating program called Ultra Frac-
E-mail address: [email protected] (A. Kelley). http://www.KelleyArt.com
tal, written by Frederik Slijkerman (http://www.ultrafractal.com, Win 95/98/NT), allows additional fractals to be rendered within the same boundaries as the "rst fractal. With each added fractal consisting of a new layer with its own set of properties, and with a merge mode that dictates how the added fractal layer interacts visually with the previous fractal's layer. Individual layers can be manipulated without a!ecting the other layers. Users may combine as many layers as they wish in one image, either using variations of the original layer, or adding entirely new layers that use di!erent fractal generating formulas. There are 19 di!erent merge modes, including screen, overlay, di!erence, and hue, with each having an adjustable opacity from 0 to 100%. By combining di!erent layers and altering the merge modes between them, it is possible to create several di!erent fractals from one original single-layer image, which can allow for more artistic interpretation of traditional fractal forms. Moonscape (Fig. 1), a spiral derived from the wellknown Mandelbrot set, `z"z#ca [2], is an example of a single-layer fractal that was originally rendered using the DOS, 256 color program Fractint (The Stone Soup Group, http://spanky.triumf.ca/www/fractint/ fractint.html) [3]. Ultra Fractal has the ability to read Fractint parameter "les, which are the text "le `recipesa that contain all the fractal's information, allowing the fractal to be recreated when Fractint, or in this case Ultra Fractal, reads the "le. Moonscape was re-rendered in Ultra Fractal, and the color banding within the image that results from Fractint's 256 color limitation was automatically smoothed by Ultra Fractal's true color
0097-8493/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 9 7 - 8 4 9 3 ( 0 0 ) 0 0 0 6 2 - 5
612
A. Kelley / Computers & Graphics 24 (2000) 611}616
Fig. 1. Moonscape.
Fig. 2. Corona.
(2 colors) capability. A stark image such as this does not bene"t from the addition of layers. Corona (Fig. 2), a spiral created using the Phoenix formula `z(n#1)"z(n)?#cHz(n)@#p*z(n!1)a [4], is
another image that was "rst rendered in Fractint. Shown with it are two examples of how adding a single layer can alter the original fractal. In the "rst example, Etched Corona (Fig. 3), a new layer was added that was identical
A. Kelley / Computers & Graphics 24 (2000) 611}616
613
Fig. 3. Etched Corona.
Fig. 4. Beta Lyrae.
to the "rst. The coloring algorithm, a function that transforms the color index value of each point, was switched in the new layer from the original `outside color"itera algorithm, to a `hearta orbit trap; all coloring algorithms mentioned are documented and available in Ultra Frac-
tal. The `di!erencea merge method was selected, which returns the absolute di!erence between the blend color and the base color [5], and which typically is used at opacity 100%. In the second example, Beta Lyrae (Fig. 4), which the author "nds even more artistically e!ective,
614
A. Kelley / Computers & Graphics 24 (2000) 611}616
again a new layer was added that was identical to the "rst. This time a Gaussian integer coloring algorithm was applied to the new layer, and a `Hard lighta merge method, at opacity 100%, was chosen, which multiplies or screens the colors, depending on the blend color [5], and "nally the coloring gradient editor, which controls
the colors in each layer, was shifted slightly in this layer to give the "nal image a more pleasing appearance. Ultra Fractal's graphical user interface allows the user to see the e!ects these changes have as they are being made, and to quickly adjust variables in order to attain a visually pleasing image. A fractal like Corona may also be
Fig. 5. Tropical Fish.
Fig. 6. Tropical Fish/Sea.
A. Kelley / Computers & Graphics 24 (2000) 611}616
duplicated several times within the program's window, allowing exploration of di!erent variables and layers for each duplication to occur nearly simultaneously. Tropical Fish (Fig. 5), a fractal made with a formula called Gallet-5-08, `z"F(real(z), imag(z))#iHF(imag(z), real(z))a [6], was also "rst rendered in Fractint. Once its
615
parameter "le was read by Ultra Fractal, the image could be altered any number of di!erent ways by adding layers. Tropical Fish/Sea (Fig. 6) is one example out of the "ve fractals that ultimately evolved from the original, which incorporates a total of four layers, all using the Overlay merge method, which multiplies or screens the colors,
Fig. 7. Rainforest.
Fig. 8. Hatchling.
616
A. Kelley / Computers & Graphics 24 (2000) 611}616
Fig. 9. Terraria.
depending on the base color [5], at between 44 and 50% opacity. The "rst new layer uses the same formula, with the coloring algorithm change to `outside"reala, which creates a very simple fractal that provides the diagonal white swirl that runs across the image. The second new layer uses the formula used in the Corona example, the Phoenix Julia [4], to provide the `seaweeda in the image with the bene"t of a `linesa orbit trap coloring algorithm. The last new layer uses the formula `Newton's method for exp(z)"log(z)a [7] with a `Shapes 2"ellipsea coloring algorithm to provide the green and blue swirling shapes that look like water. This version of tropical "sh is also notable because two of the layers make use of Ultra Fractal's `alpha channela feature, which allows the user to specify an opacity value for each point in the gradient [5]. Each layer has its own alpha channel which may be toggled on or o!. Examples of fractals that did not start out as singlelayer Fractint images and which exist because of the complexity and interaction of multiple layers include Rainforest (Fig. 7), and Hatchling (Fig. 8), which are both three-layer images, and Terraria (Fig. 9), a fourlayer image. Though these images are complex, artists routinely mix 15 or more layers to create images that in the past would have been very di$cult to create and manipulate. The approaches described in this paper appear to further expand the artistic possibilities of fractal creation. To see more examples of both single and multi-layer fractal art, visit the author's web gallery: http://www.KelleyArt.com.
Acknowledgements The author thanks Cli!ord Pickover for his encouragement and comments.
Appendix A. Computer details All "gures were rendered on a 233 MHx Pentium II running Windows 95 with 64 megs RAM and a 1024;768 screen display.
References [1] Kelley A. Fractal Cosmos Calendar. Amber Lotus, http://www.amberlotus.com/fractal.html, P.O. Box 31538, San Francisco, CA, 94131, 2000, 2001. [2] Mandelbrot B. The fractal geometry of nature. New York: Freeman, 1982. [3] Wegner T, Tyler B. Fractal creations. 2nd ed. Corte Madera, CA: The Waite Group, Inc., 1993. [4] Jones DM. DMJ formula "le for Ultra Fractal, http://www.fractalus.com/ultrafractal/. [5] Slijkerman F. Ultra fractal help "les, http://www.ultrafractal.com. [6] Gallet S. Gallet formula "les. http://ourworld.compuserve.com/homepages/Sylvie}Gallet/linke.htm. [7] Mitchell, L. Kerry LKM formula "le for Ultra Fractal. http://www.primenet.com/&lkmitch/uf.htm.