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

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 Menger sponge, imagined 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

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_… #ClaudesCat #FractalCat #SyntheticIntelligence #Claude #Anthropic #fractals #fractalart #mathart #mathematics #geometry

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 method that produced Claude'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

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 vertices, and a sixth, 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

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 shrinking transformations 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

Bicomplex Mandelbar Dynamics Associated with Three Conjugations Research work on the antiholomorphic counterpart of bicomplex quadratic dynamics. #mandelbrot, #bicomplex, #tetrabrot, #fractals, #openaccess mdpi.com/2227-7390/14/17/320…

The Mandelbrot Set mathematically generated with Frax Pro and AI-extended with Midjourney V8.2. #midjourney, #mandelbrot, #fractalart

Created with Midjourney V8.2. #midjourney, #fractalart

Created with Midjourney V8.2. #midjourney, #fractalart

Created with Midjourney V8.2. #midjourney, #fractalart