Fractal Art - The beauty of mathematical algorithms

[orchid copper]

Fractal art is a fascinating art form that uses mathematical algorithms to create intricate and complex patterns. It combines science and creativity, showcasing the beauty of mathematics. Different mathematical formulas produce different fractal patterns. For example, the Mandelbrot set creates complex, swirling patterns, while the Julia set produces intricate, web-like structures.

Fractals are patterns that repeat themselves regardless of how much you zoom in or out. A small part of a fractal looks like the whole pattern. For example, a tree branch resembles the entire tree, with smaller branches splitting off in a similar way. Similarly, in a fractal, the same shapes repeat at different sizes.

While fractal patterns can be beautiful on their own, they also work well as a design element for a primary subject, like this cat, which just got back from an overzealous grooming session.

Returned prompts
• digital animation portrait of a long-haired white cat who just got back from the groomer, with its fur styled in intricate Julia set fractal patterns. The cat has a displeased expression, clearly not happy with the new hairstyle. The simple background highlights the cat's unique fractal-styled fur. • photo of a whiteboard in a classroom showing a Julia set fractal. The whiteboard includes the intricate details of the fractal pattern, along with mathematical formulas, set in a typical academic classroom environment. • digital animation scene of a fern in a forest. The chartreuse fern glows with life and freshness, with golden sunlight illuminating the intricate fractal patterns from the side. This composition evokes a sense of wonder and beauty, highlighting the natural elegance and magical feeling of the fern in its lush forest environment.

[quick merlin]

Fractal Patterns in Nature - Various Art Styles
There are many examples of fractal patterns in nature, from lightening and river delta, to ferns and snowflakes. Here, I took three examples of fractal patterns in nature and matched them up with different art styles to best bring out their intriguing looks. Here are the base prompts:

“White peacock, feathers in fractal patterns, against dark background. Art Nouveau style painting.”

“Fauvist painting, a closeup view of a Romanesco broccoli, vivid color fields following fractal patterns.”

“A closeup view of a Nautilus shell on beach sand, showing fractal patterns. Ink and watercolor drawing, photorealistic finish, full color.”

[dapper valve]

The art form in the images utilizes fractal mathematics to create intricate and captivating designs. Here are the key correspondences:

Self-Similarity: Patterns repeat at different scales, creating cohesive and visually appealing designs.
Iteration and Recursion: Simple shapes and designs are iterated to build complexity, mimicking natural growth patterns.
Fractal Dimension: The use of non-integer dimensions adds depth and intricacy, enhancing the mystical atmosphere.
Natural Patterns: Inspired by natural fractals, the artwork blends organic and geometric elements harmoniously.
By applying these mathematical principles, the artwork achieves a unique blend of mystical beauty and intellectual engagement.

A surreal bird with a body showcasing intricate fractal patterns in iridescent colors, perched on a tree. The bird is adorned with fractals and 3D reflective objects or shapes, adding to its mystical appearance. The tree and surroundings are complex, with natural elements fusing seamlessly with geometric fractal patterns. The environment is rich in shades of iridescent colors, complemented by black and subtle rainbow iridescence, creating a glossy, otherworldly atmosphere that captivates the viewer. Natural light photo.

[noble jasper]

DallE via CGPT: one prompt idea three styles.

Primary prompt: Create a high-resolution modern flat design abstract art image bright colors indicative of a waterfall striking down over a lake in a forest, using mathematical art forms like fractals and tessellations. The waterfall should be depicted with intricate fractal patterns, while the forest features tessellated geometric shapes. The overall composition should blend the rythm of nature with the order of mathematics, creating a visually striking and detailed widescreen scene.

Next prompt: modern flat design abstract art

Last prompt: impressionist art

[noble jasper]

High-resolution bright watercolor art of a multicolor rose bed in a garden, using mathematical art forms like fractals and tessellations.

[quick merlin]

Fractal Patterns in Nature - Various Media
Intricate fractal patterns lend themselves naturally to kiri-e (paptercut art) and lace work. Felt figurine is a good choice for chunkier items like a Romanesco. Here are the base prompts:

"Handcrafted precision cut kiri-e artwork, depicting a Nautilus shell featuring fractal patterns, metallic yellow and silver paper photographed against dark background." (Copilot)
"Handcrafted lace work, metallic color yarns, depicting a peacock, feathers featuring fractal patterns, photographed against dark background." (Copilot)
"Fuzzy felt figurine of a Romanesco broccoli, featuring fractal patterns. Every element is made of felt in matcha green and white, photographed against dark background." (ChatGPT 4o)

[jolly junco]

The Sierpinski Triangle: Using GPT4 (or 4o) to explore different types of fractals

I love making art from mathematical equations and have posted on it in the gallery. A great way to get started in exploring fractal art, specifically, can just be simply asking the gpt to recommend a list of different fractals to try when creating visually interesting ai art images. Julia and Mandelbrot will usually come up first. In this case, I selected,

The Sierpinski Triangle
• Description: The Sierpinski triangle is a fractal made by recursively removing equilateral triangles from a larger equilateral triangle. The result is a pattern of interlocking triangles with a self-similar structure.

The first Image is just a basic rendering of what the Sierpinski Triangle actually is. It was the first thing I got. I have used GPT4, and even Data Analysis, prior, but this time I was using GPT4o.

I then asked GPT4o to do an Art Nouveau version of the Sierpinski Triangle. This became the prompt:
Create an Art Nouveau-inspired version of the Sierpinski Triangle fractal. Use flowing, curvilinear lines and incorporate organic motifs like vines, flowers, and leaves. Utilize a color palette with natural tones such as greens, browns, and golds. Add intricate detailing and texture to give the design a decorative and ornate appearance typical of the Art Nouveau style. That led to the second image.

Asking about different fractals and rendering them can be a lot of fun. Often seeing the initial fractal pattern will give great ideas for rendering. But asking the ChatGPT engine for generation ideas works rather well.

[orchid copper]

Fractal elements can be used as design elements in many situations. Here we see how ChatGPT uses the entire context of your conversation to carry forward art style and content instructions across multiple prompts. The more complex your prompt is, the lower the goal weight will be on any single component. If the concept of “fractal” is important, you can repeat the term to add goal weight to that feature.

First Prompt: make a digital animation cartoon honeybee architect at work building a fractal honeycomb. whimsical illustration suitable for a children's book. the fractal patterns in the honeycomb are clearly defined examples of mathematical order. the honeybee is wearing construction worker gear. cute. chibi. fun colors

**Second Prompt: **now show the honeybee architect with a work crew of construction worker bees working on a new section of the fractal honeycomb. the fractal additional is a new wing on the fractal honeybee hive. same colors and art style to match the previous illustration

Third Prompt: same prompt but make the fractal honeycomb an asymmetrical spiral

[dapper valve]

A Mandelbulb is a three-dimensional fractal, an extension of the two-dimensional Mandelbrot set into the third dimension. It was first proposed by Daniel White and Paul Nylander in 2009. The Mandelbulb attempts to create a three-dimensional analogue of the Mandelbrot set using spherical coordinates instead of complex numbers.

Key Characteristics of the Mandelbulb:
Mathematical Foundation:

The Mandelbulb is based on iterating a mathematical formula that involves raising spherical coordinates to a power and then transforming them back to Cartesian coordinates. This iterative process creates complex and highly detailed structures.
Visual Appearance:

The Mandelbulb exhibits intricate, organic-looking shapes and surfaces that often resemble natural forms such as cauliflower, coral, or other intricate patterns found in nature.

A surreal garden filled with flowers inspired by the Mandelbulb fractal. The flowers have intricate, fractal petals that split infinitely into smaller patterns, creating an otherworldly appearance. The garden is vibrant with a variety of colors and shapes, each flower uniquely detailed with bulbous and cauliflower-like formations. The stems and leaves follow fractal patterns as well, twisting and turning in complex, recursive shapes. The background features a kaleidoscope of swirling colors, with lace-like fractal clouds drifting across the sky. Bioluminescent plants add a soft glow to the scene, illuminating the detailed fractal flowers.

[dapper valve]

Flame fractals are a type of fractal created by a specific algorithm that generates visually stunning images with intricate patterns and vibrant colors. These fractals are part of the broader category of iterated function systems, but they stand out due to their unique rendering method which emphasizes smooth transitions and glowing effects.