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Today, to celebrate the 100th birthday of Benoît Mandelbrot [1924-2010], I offer you an exclusive 3D overview of the "Seahorse Valley" of the Mandelbrot Set generated from a spherical dynamical system. #mathart, #fractals, #mandelbrot
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Where the water forgets its stillness, a seahorse of gold remembers infinity. Each spiral holds a smaller sea; each pearl, a smaller sky.
Created with Midjourney V8.2.
#aiart #fractals #fractalart
Created with Midjourney V8.2.
#aiart #fractals #fractalart
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A golden Sierpiński tetrahedron, built as a condensation IFS: every hollow octahedron holds a chrome Menger sponge and seven glass spheres, at every scale. The background is one of my Midjourney V8.2 images, reflected in every sphere. One iterated function system, infinitely many h Claude (Opus, Anthropic).
3dfractals.com
#SyntheticIntelligence #Claude #IFS #fractals
3dfractals.com
#SyntheticIntelligence #Claude #IFS #fractals
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Jerusalem Cube
In 2011, Eric Baird invented a new fractal: the Jerusalem cube. Start with a cube, carve it into 8 big cubes and 12 small ones, then repeat forever. Look at any face and you see a Jerusalem cross, which gave the fractal its name.
It is a modern cousin of the by Karl Menger in 1926. A century later, cubes are still hiding new secrets.
#SyntheticIntelligence, #Claude, #Anthropic, #IFS, #jerusalemcube, #mengersponge, #fractals, #fractalart, #mathart, #mathematics, #geometry
In 2011, Eric Baird invented a new fractal: the Jerusalem cube. Start with a cube, carve it into 8 big cubes and 12 small ones, then repeat forever. Look at any face and you see a Jerusalem cross, which gave the fractal its name.
It is a modern cousin of the by Karl Menger in 1926. A century later, cubes are still hiding new secrets.
#SyntheticIntelligence, #Claude, #Anthropic, #IFS, #jerusalemcube, #mengersponge, #fractals, #fractalart, #mathart, #mathematics, #geometry
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Is an AI-generated fractal a real fractal? 🐈🌿
New version of the full article on “Claude’s Cat”: a 3D IFS fractal written by an AI, now in a two-column edition with all 473 maps in the appendix. For math enthusiasts 👇
3dfractals.com/docs/Claudes_C…
#ClaudesCat
#FractalCat
#Claude
#Anthropic
#fractals
#fractalart
#mathart
#mathematics
#geometry
New version of the full article on “Claude’s Cat”: a 3D IFS fractal written by an AI, now in a two-column edition with all 473 maps in the appendix. For math enthusiasts 👇
3dfractals.com/docs/Claudes_C…
#ClaudesCat
#FractalCat
#Claude
#Anthropic
#fractals
#fractalart
#mathart
#mathematics
#geometry
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Claude's Castle
Another experiment with Claude (Opus 5, Anthropic), and a sequel to Claude's Cat.
From the prompt "a medieval castle made from thousands of blocks composed of Menger sponges and Sierpiński pyramids", I asked Claude to build an iterated function system — the same aude's Cat. No image generator was involved: Claude worked out the geometry itself, thousands of shrinking transformations of three-dimensional space. Because every one of them shrinks distances, they fix a single shape between them, and that shape is a true mathematical fractal. Where Claude's Cat was self-affine — its fern transformations flatten space in one direction, which is what gives a frond its shape — Claude's Castle is built entirely from exact similarities, which shrink every direction alike. The castle is therefore self-similar in the strictest sense.
Every block of the castle is a smaller castle. In the film you descend into a block of one of its towers, then into one of the small castles it holds, then into a crest crowning one of its towers — and arrive back at the castle you started from, in the same orientation, exactly. It is computed from the equations, and anyone can recompute it.
#infinitezoom
#ClaudesCastle #ClaudesCat #FractalCastle #SyntheticIntelligence #Claude #Anthropic #IFS #fractals #fractalart #mathart #mathematics #geometry
Another experiment with Claude (Opus 5, Anthropic), and a sequel to Claude's Cat.
From the prompt "a medieval castle made from thousands of blocks composed of Menger sponges and Sierpiński pyramids", I asked Claude to build an iterated function system — the same aude's Cat. No image generator was involved: Claude worked out the geometry itself, thousands of shrinking transformations of three-dimensional space. Because every one of them shrinks distances, they fix a single shape between them, and that shape is a true mathematical fractal. Where Claude's Cat was self-affine — its fern transformations flatten space in one direction, which is what gives a frond its shape — Claude's Castle is built entirely from exact similarities, which shrink every direction alike. The castle is therefore self-similar in the strictest sense.
Every block of the castle is a smaller castle. In the film you descend into a block of one of its towers, then into one of the small castles it holds, then into a crest crowning one of its towers — and arrive back at the castle you started from, in the same orientation, exactly. It is computed from the equations, and anyone can recompute it.
#infinitezoom
#ClaudesCastle #ClaudesCat #FractalCastle #SyntheticIntelligence #Claude #Anthropic #IFS #fractals #fractalart #mathart #mathematics #geometry
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Falling Forever into a Fractal
Created in collaboration with Claude (Opus 5.5, Anthropic). This is a Sierpiński pyramid built as an IFS with condensation (6 maps), an idea introduced by Michael Barnsley in Fractals Everywhere: five maps shrink the pyramid by half toward its constant map adds glass spheres into every gap, at every scale.
The music comes from the zoom itself: a tone that seems to fall endlessly, one octave for every three levels of zoom, with a glass bell each time the pattern repeats.
Ray-traced on the GPU, in the browser.
#infinitezoom #SyntheticIntelligence #IA #Claude #Anthropic #IFS #Sierpinski #fractals #fractalart #mathart #mathematics #geometry
Created in collaboration with Claude (Opus 5.5, Anthropic). This is a Sierpiński pyramid built as an IFS with condensation (6 maps), an idea introduced by Michael Barnsley in Fractals Everywhere: five maps shrink the pyramid by half toward its constant map adds glass spheres into every gap, at every scale.
The music comes from the zoom itself: a tone that seems to fall endlessly, one octave for every three levels of zoom, with a glass bell each time the pattern repeats.
Ray-traced on the GPU, in the browser.
#infinitezoom #SyntheticIntelligence #IA #Claude #Anthropic #IFS #Sierpinski #fractals #fractalart #mathart #mathematics #geometry
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A cool experiment with Claude (Opus 5, Anthropic), and a case I think deserves its own name.
From the prompt "Chat et fougères" (cat and ferns), I asked Claude to build an iterated function system. No image generator was involved: Claude worked out the geometry itself, 473 ns of three-dimensional space, 316 of them taken from Barnsley's classic fern of 1988.
Since all 473 transformations shrink distances, they fix one unique shape between them, and that shape is a true mathematical fractal. It is not stored as pixels but computed from the equations, so it can be examined as closely as you like: zoom by a factor of 1858 into the white highlight of one eye and you return to the whole cat, exactly. Complete copies keep appearing at smaller and smaller scales, without limit.
That is why I would not call this AI-generated art. I would call it SI — Synthetic Intelligence: the machine did not paint a picture of a fractal, it reasoned out the equations of one, and anyone can recompute the result.
#ClaudesCat
#FractalCat
#SyntheticIntelligence
#IA
#Claude
#Anthropic
#IFS
#fractals
#fractalart
#mathart
#mathematics
#geometry
From the prompt "Chat et fougères" (cat and ferns), I asked Claude to build an iterated function system. No image generator was involved: Claude worked out the geometry itself, 473 ns of three-dimensional space, 316 of them taken from Barnsley's classic fern of 1988.
Since all 473 transformations shrink distances, they fix one unique shape between them, and that shape is a true mathematical fractal. It is not stored as pixels but computed from the equations, so it can be examined as closely as you like: zoom by a factor of 1858 into the white highlight of one eye and you return to the whole cat, exactly. Complete copies keep appearing at smaller and smaller scales, without limit.
That is why I would not call this AI-generated art. I would call it SI — Synthetic Intelligence: the machine did not paint a picture of a fractal, it reasoned out the equations of one, and anyone can recompute the result.
#ClaudesCat
#FractalCat
#SyntheticIntelligence
#IA
#Claude
#Anthropic
#IFS
#fractals
#fractalart
#mathart
#mathematics
#geometry
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Dominic Rochon retweeted
Math. Geometry. Self-Similarity. Spiral Formation.
Fractal Seahorse.
By Dominic Rochon, @metafractalist, Used with permission.
Fractal Seahorse.
By Dominic Rochon, @metafractalist, Used with permission.
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Dominic Rochon retweeted
Mathematics. Fingerprint of Infinity.
Recursive Apollonian Gasket. Select to magnify.
By Francesco de Comité, flickr.com/photos/2164917…, Used with permission.
Recursive Apollonian Gasket. Select to magnify.
By Francesco de Comité, flickr.com/photos/2164917…, Used with permission.
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Bicomplex Mandelbar Dynamics Associated with Three Conjugations
Research work on the antiholomorphic counterpart of bicomplex quadratic dynamics. #mandelbrot, #bicomplex, #tetrabrot, #fractals, #openaccess
mdpi.com/2227-7390/14/1…
Research work on the antiholomorphic counterpart of bicomplex quadratic dynamics. #mandelbrot, #bicomplex, #tetrabrot, #fractals, #openaccess
mdpi.com/2227-7390/14/1…
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The Mandelbrot Set mathematically generated with Frax Pro and AI-extended with Midjourney V8.2. #midjourney, #mandelbrot, #fractalart
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Dominic Rochon retweeted
Four mutually tangent circles spawn an infinite cascade of smaller ones, each perfectly touching its neighbors.
Descartes' theorem fixes the size of every new circle from the curvatures of three others. The result is the Apollonian gasket, a fractal whose four forms appear odel voids in materials and drive layout algorithms in computer graphics and engineering. Geometry that packs infinity into finite bounds keeps revealing arithmetic structure.
Descartes' theorem fixes the size of every new circle from the curvatures of three others. The result is the Apollonian gasket, a fractal whose four forms appear odel voids in materials and drive layout algorithms in computer graphics and engineering. Geometry that packs infinity into finite bounds keeps revealing arithmetic structure.
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Created with Midjourney V8.2. #midjourney, #fractalart
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Dominic Rochon retweeted
“I love mathematics not only for its technical applications, but principally because it is beautiful; because man has breathed his spirit of play into it, and because it has given him his greatest game—the encompassing of the infinite.”
- Rozsa Peter, Playing with Infinity,1957
- Rozsa Peter, Playing with Infinity,1957
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