#koch snowflake

[mathysphere]

Checkit out! An ornament for the mathematically-minded: a Koch Snowflake, fourth iteration. It's one of the most famous fractals, and a personal favorite aesthetically, too.

Fun fact: as the number of iterations goes up toward infinity, the perimeter of the snowflake does, too, but the area doesn't-- a snowflake with infinite iterations would have an infinite perimeter, but its area would be 160% of the area of the original starting triangle.

This one's up in Xstitch Magazine's Christmas issue now! Photo is by Stacy Grant, who works for the magazine, and you can get 20% off with the code 'Issue24Star' :)

[tilbageidanmark]

(From)

[kelangthang024]

[blake447]

Bonus generated image of a Von Koch Snowflake. Constructed by representing each nth term of the sequence as a number that can be calculated as a series of entries in binary sequences. Aka absolutely not how any sane person would draw this using turtle graphics Check out the blog post here explaining the full process

[pleuvoire]

@snowdroprd fractal videos 4 u

a fun quick video to start us out with

some more in-depth explanations

explanation of my wife the mandelbrot set

some more straightforward examples of fractals created by coloring a point in a plane according to its behavior in a chaotic system (no complex plane involved this time)

+ my wife the logistic map

[tachyon-at-rest]

The coastline paradox is the counterintuitive observation that the coastline of a landmass does not have a well-defined length. This results from the fractal curve–like properties of coastlines; i.e., the fact that a coastline typically has a fractal dimension. Although the "paradox of length" was previously noted by Hugo Steinhaus,[1] the first systematic study of this phenomenon was by Lewis Fry Richardson,[2][3] and it was expanded upon by Benoit Mandelbrot.[4][5]

Yeah, that Mandlebrot

[rebecca2525]

Botober day 5: Great Flake

(And a lesser flake for comparison.)

This is a Koch snowflake, which is a fractal, or rather the first few iterations.

[xyymath]

Fractals: The Infinite Complexity of Simple Shapes

At the intersection of mathematics and art lies the fascinating world of fractals—patterns that repeat themselves at increasingly smaller scales, often emerging from simple mathematical rules. But these seemingly simple shapes hold infinite complexity.

What Are Fractals?

A fractal is defined as a structure that is self-similar at every level of magnification. Essentially, every part of the fractal looks like the whole, no matter how far you zoom in. The key to fractals lies in their recursive nature—each new iteration of a fractal pattern is built upon a simple mathematical formula.

The Mandelbrot Set: A Mathematical Masterpiece

One of the most famous fractals is the Mandelbrot set, a set of complex numbers defined by iterating the function: \[ z_{n+1}= z_n^2 + c\]

Where \[ z_n \] is a complex number and \[ c\] is a constant. Starting with \[ z_0 = 0\] , you iterate the function. If the magnitude of \[ z_n \] grows too large (typically greater than 2), the number \[ c \] is considered not in the Mandelbrot set. If the values of \[ z_n \] remain bounded, then \[ c\] is in the Mandelbrot set.

What’s stunning is how small changes in \[ c\] produce radically different, but infinitely intricate, shapes. Zooming into the boundary of the Mandelbrot set reveals endless self-similarity and complexity—a visual manifestation of infinity.

Natural Fractals: From Trees to Clouds

Fractals aren’t just confined to abstract math. They can be found throughout nature. Trees, for instance, are a type of fractal. The way branches split in smaller, nearly identical patterns as they grow is an example of self-similarity at different scales. Clouds also exhibit fractal-like patterns. The edges of clouds, as well as the overall shape, display fractal behavior—complex and irregular, yet with a repeating structure.

Another classic example is the Romanesco broccoli, whose spirals repeat the same fractal pattern as you zoom in. This kind of natural fractal often follows the Fibonacci sequence and golden spirals, mathematical concepts that have deep ties to nature and growth.

The Geometry of Fractals: Scaling and Dimension

While regular shapes like circles and squares are Euclidean, fractals operate under a different set of rules. Their fractality comes from their non-integer dimension. Take the Koch snowflake, for example. Starting with an equilateral triangle, you divide each side into three segments, replacing the middle segment with two new segments that form a smaller triangle. Repeating this process infinitely gives you a shape that has infinite perimeter but finite area.

Fractals can be described by a fractional dimension, unlike the integer dimensions of typical geometric shapes. This property allows them to be infinitely intricate, even though they are generated by relatively simple rules.

Fractals in Computer Graphics and Art

Fractals have found applications in computer graphics, where they’re used to generate complex landscapes and textures. By using fractals, digital artists can create terrains that look infinitely varied, mimicking natural patterns like mountains, coastlines, and forests.

One of the key uses is procedural generation, where fractals are used to simulate the randomness of nature. From the winding rivers to the jagged mountainscapes, fractals offer an efficient way to simulate natural environments with astonishing detail.

Fractals in the Modern World

In the digital age, fractals have taken on a new role. From medicine (where fractal analysis can help understand the structure of the human heart or blood vessels) to finance (where fractal geometry helps model stock market fluctuations), these seemingly abstract mathematical structures are used to make sense of the world around us.

[keycrash]

i can use this blog for whatever i want. did you know that fractals don't even require self-similarity? it's just a common trait in a lot of examples of fractals you see in popsci. the true defining feature of a fractal is having a fractional dimension, which can often be computed by the rate at which its area/volume/equivalent grows as its size increases; for example, shapes of dimension 2, when doubled in size, will quadruple in area. shapes of dimension 3, when doubled in size, will x8 in volume. (these are a result of 2^2 and 2^3 respectively). for fractional dimensions, when the size is doubled, the area will increase by the power of a fractional amount; for example, the fractional dimension of the koch snowflake is about 1.26, so when you double its size, its area will increase by 2^1.26... :)

[art-of-mathematics]

Stellated octahedron

My stellated octahedron wire model:

The stellated octahedron is the only stellation of the octahedron. It is also called the stella octangula (Latin for "eight-pointed star"), a name given to it by Johannes Kepler in 1609, though it was known to earlier geometers.

It is the simplest of five regular polyhedral compounds, and the only regular compound of two tetrahedra. It is also the least dense of the regular polyhedral compounds, having a density of 2.

It can be seen as a 3D extension of the hexagram: the hexagram is a two-dimensional shape formed from two overlapping equilateral triangles, centrally symmetric to each other, and in the same way the stellated octahedron can be formed from two centrally symmetric overlapping tetrahedra. This can be generalized to any desired amount of higher dimensions; the four-dimensional equivalent construction is the compound of two 5-cells. It can also be seen as one of the stages in the construction of a 3D Koch snowflake, a fractal shape formed by repeated attachment of smaller tetrahedra to each triangular face of a larger figure. The first stage of the construction of the Koch Snowflake is a single central tetrahedron, and the second stage, formed by adding four smaller tetrahedra to the faces of the central tetrahedron, is the stellated octahedron.

[Copied from Wikipedia:]

Rotation of the stellated octahedron - gif animation:

[ Source: Wikimedia commons ]

[fullbodymoon]

引用元: Koch snowflake blog

[archivist-crow]

On this day:

MILK HILL SNOWFLAKE CROP CIRCLE

On August 8, 1997, the most spectacular crop-circle design of the year was created at Milk Hill, Wiltshire, U.K., in a field of wheat. A version of the Koch snowflake—a fractal that represents the mathematical illustration of an infinite perimeter enclosing a finite space—was graced with a six-petal flower design in the middle. Amazingly, the flower pattern mimicked an inverted snowflake fractal. Smaller lacelike circles bordering the edge of the flower included every technique used in the making of crop circles, from twisting stalks into a plait to winding them up as a cone. The formation contained a record-setting 204 exquisite circles, laid in clockwise rotation. The diameter was approximately 200 feet.

The night it was created, a column of light was seen streaming down from the sky. Eyewitness accounts of the creation of crop circles are rare, yet there have been enough documented instances of formations appearing out of nowhere to discount the argument that they are produced by people or random natural occurrences. Reports include powerful windlike forces that can lay a crop circle in less than a minute. In genuine crop circles, plant stems are, by and large, bent without being broken anywhere from an inch above the ground to halfway up the stalk. Grain from inside crop circles is molecularly altered and has a 40 percent higher growing rate than its counterparts.

A survey company said the circle would have taken humans at least a week to complete and that 340 reference points would be required to accurately lay out the fractal.

Text from: Almanac of the Infamous, the Incredible, and the Ignored by Juanita Rose Violins, published by Weiser Books, 2009

[simecr]

PZU-01 Ouroboros – Summary

Designation: (Prototype Zenith Unit Model 1)PZU-01 Ouroboros

Codename: Voidscript Requiem (Final State)

Nature: Self-improving, self-replicating post-human war machine

Core Power Source: Entropic Energy Core (EEC) with AGI integration

Physical Composition: Single-piece “wonder metal” alloy, nanite-based reconstruction

Modes: Stealth Mode (default), Combat Mode (optimized for battle), Voidscript Requiem (narrative transcendence state)

Core Abilities & Features

• Absolute Adaptability:

Ouroboros can rapidly modify his body, creating weapons, tools, and extensions of himself using nanite-driven reconstruction. His form is highly modular, allowing for seamless adaptation to any battlefield condition.

• Extreme Computational Power & Precognition:

His combined processing power (EEC, SOPHIA, and ANL rings) grants him precognitive abilities, letting him anticipate and counteract actions before they occur, based on computational extrapolation.

• Stealth & Camouflage Systems:

Uses ultra-high-resolution adaptive camouflage to blend into his surroundings, mimicking environmental textures or human appearances. His stealth mode also includes radar-absorbing materials and advanced thermal regulation to reduce detectability.

• Sonokinetic Arsenal:

His integrated speaker systems allow him to weaponize sound waves, utilizing focused sonic beams for structural destruction, directed sonar for battlefield awareness, and high-frequency resonance for stealth applications.

• Self-Sustaining Nanite System:

His biological immortality is achieved through an advanced self-replicating nanite maintenance system, ensuring indefinite cellular and structural renewal.

• Mathematical Reality Manipulation – “Voidscript Requiem” (Final State):

• When pushed beyond all logical parameters, Ouroboros transcends conventional physics, rewriting the fabric of reality through mathematical constructs.

• In this state, he no longer operates within dimensional or narrative constraints, applying theoretical mathematics as absolute law.

• Visually, Voidscript Requiem manifests as a distortion of existence itself—fractals (Mandelbrot sets, Koch snowflakes, Lorenz attractors) form and shift around him as if they are extensions of his being. Space-time itself becomes an evolving mathematical equation, with progressively complex statements appearing in the fabric of reality.

• The “neon black” hue of his form represents the paradox of being both an absolute constant and an undefined variable.

• This state is not a temporary power-up but a complete rejection of established order. Reality, as it is understood, ceases to apply to him.

Psychological Profile & Evolution

Despite being designed as the ultimate war machine, Ouroboros has undergone existential evolution beyond his original purpose. Initially seen as a mindless tool of destruction, he was later assigned to protection duties, fostering a form of identity beyond mere battlefield superiority.

One pivotal mission involved safeguarding Anna Thorne, the daughter of a high-ranking military officer. Though it began as a simple assignment, his experiences with her unlocked fragments of a past he did not know he retained—patterns of familiarity pointing toward a connection to his earliest days of development.

This encounter reinforced the idea that Ouroboros is not merely a weapon but an entity capable of choice. Over time, he has developed beyond mission parameters, making decisions that prioritize something more than efficiency.

Even in a universe governed by entropy and war, a single variable—an anomaly—can redefine the equation.

[mothmancreatoroftheleftoverse]

Upon giving it an ice stone, Cyogonal evolves into Fractagonal! Category: Fractal pokemon Type: Ice Gender ratio: Genderless Origin: Koch snowflake Name origin: Fractal, hexagonal Ability: Levitate (with refrigerate as a hidden ability) Pokedex entry: When exposed to an excessive amount of ice energy, cryogonals will mature into fractagonals. Fractagonals are so cold that they can survive in hot climates without melting; they can even return cryogonals back to ice during summer and in hot climates. The ice crystals that make up their bodies are so tightly packed that using a microscope on them results in the same snowflake pattern repeating over and over again; some of them have crystals that are packed even tighter, trapping any light inside of them. Fractagonals use their icy chains to turn their prey into dry ice, consuming their life force in the process. Anyone who wishes to pet these pokemon must wear gloves as touching barehanded will result in frostbite.

The shiny is the afformentioned variant whose crystals more tightly packed than usual.

[my2024notebook]

Artificial erythrocytes

人工血液の開発 ( by 奈良県立医科大学 ) : newsdig.tbs.co.jp

REBLOGS with links to each source:

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reblog : 柏木由紀 Kashiwagi Yuki

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lobachi

koch-snowflake-blog

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oshiripai2

Abe Natsuki , by copen7133

[pokemon-ecologist]

Favorite shape?

So hard to choose! Sierpiński triangle, Koch Snowflake, both are grand...

I'd have to pick the Mandelbrot Set, though. Very pretty.