The Koch snowflake / Mark McClure

I'm a professor of mathematics at the University of North Carolina Asheville. I've also done a fair amount of consulting work over the years focusing on scientific and data visualization. I'm happy to be part of the first Observable Ambassador's cohort.

Workspace

Published

Edited

Mar 23, 2022

2 stars

Math

A Julia set on the Riemann sphere The Z-Curve Barnsley's fern A stochastic digraph IFS algorithm Self-affine tiles The Twindragon The Eisenstein fractions A self-affine tile with holes Self-affine tiles via polygon merge Golden rectangle fractals Bifurcation diagram with critical curves The tame twindragon Illustrations for the proof of Green's theorem Non-orientability of a Mobius strip Examples of parametric surfaces Penrose tiling The extended unit circle Penrose three coloring Newtons's method on the Riemann sphere Conic sections Divisor graphs The dance of Earth and Venus Iterating multiples of the sine function Borderline fractals Self-similar intersections Box-counting dimension examples Mandelbrot by dimension Inverse iteration for quadratic Julia sets Integer Apollonian Packings Illustrations of two-dimensonal heat flow The logistic bifurcation locus The eleven unfoldings of the cube A unimodal function with fractal level curves Green's theorem and polygonal area The geometry and numerics of first order ODEs The xxx^xxx-spindle Animated beats Rauzy Fractals Hilbert's coordinate functions Pluck Not Pi Drum strike

The Koch snowflake

Fractalized square A Taylor series about π/4\pi/4π/4 PlotX3D Hyperboloid A PlotX3D animation Modular arithmetic in 5th grade art Simple S-I-R Model The Poisson Kernel Poly-gaskets Classification of 2D linear systems via trace and determinant Julia sets and the Mandelbrot set Water waves Fourier Series Disks for a solid of revolution Orbit detection for the Mandelbrot set Tracing a path on a sphere Plot for mathematicians Functions of two variables Partial derivatives Dijkstra's algorithm on an RGG Gradient ascent Unfolding polyhedra Tangent plane to a level surface A strange discontinuity Examples of level surfaces McMullen carpets Hills and valleys The definition of ⇒Double and iterated integrals MST in an RGG Trees are bipartite Fractal typesetting d3.hierarchy and d3.tree K23 is Planar Polar Coordinates Parametric region generator Parametric Plot 2D Contour plots Greedy graph coloring Graph6 A few hundred interesting graphs The Kings Problem First order, autonomous systems of ODEs Runge-Kutta for systems of ODEs

Also listed in…

Fractals

flakeIFS = new IteratedFunctionSystem([

scale(1 / 3, [0, 1]),

scale(1 / 3, [-Math.sqrt(3) / 2, 1 / 2]),

scale(1 / 3, [-Math.sqrt(3) / 2, -1 / 2]),

scale(1 / 3, [0, -1]),

scale(1 / 3, [Math.sqrt(3) / 2, -1 / 2]),

scale(1 / 3, [Math.sqrt(3) / 2, 1 / 2]),

scale(1 / Math.sqrt(3)).compose(rotate(Math.PI / 6))

])

// The odd points like [-Math.sqrt(3) / 2, 1 / 2] arise since we'd like the

// resulting curve to correspond to one "side" of the snowflake.

curveIFS = {

let f0 = scale(1 / 3, [-Math.sqrt(3) / 2, 1 / 2]);

let f1 = shift([1 / Math.sqrt(3), 0])

.compose(scale(1 / 3, [-Math.sqrt(3) / 2, 1 / 2]))

.compose(rotate(Math.PI / 3, [-Math.sqrt(3) / 2, 1 / 2]));

let f2 = shift([Math.sqrt(3) / 2, 1 / 2])

.compose(scale(1 / 3, [-Math.sqrt(3) / 2, 1 / 2]))

.compose(rotate(-Math.PI / 3, [-Math.sqrt(3) / 2, 1 / 2]));

let f3 = scale(1 / 3, [Math.sqrt(3) / 2, 1 / 2]);

return new IteratedFunctionSystem([f0, f1, f2, f3]);

}

koch_flake = function(n) {

let [f0, f1, f2, f3] = curveIFS.affine_function_list.map((af) => af.f);

let step = (pts) =>

pts

.map(f0)

.slice(0, -1)

.concat(pts.map(f1).slice(0, -1))

.concat(pts.map(f2).slice(0, -1))

.concat(pts.map(f3));

let pts = [

[-Math.sqrt(3) / 2, 1 / 2],

[Math.sqrt(3) / 2, 1 / 2]

];

d3.range(n).map(() => (pts = step(pts)));

let pts2 = pts.map(

rotate((-2 * Math.PI) / 3, [Math.sqrt(3) / 2, 1 / 2]).compose(

shift([Math.sqrt(3), 0])

).f

);

let pts3 = pts.map(

rotate((2 * Math.PI) / 3, [0, -1]).compose(

shift([Math.sqrt(3) / 2, -3 / 2])

).f

);

return pts.slice(0, -1).concat(pts2).concat(pts3);

}

import {

shift,

scale,

rotate,

reflect,

degree

} from "@mcmcclur/affinefunction-class"

import { IteratedFunctionSystem } from "@mcmcclur/iteratedfunctionsystem-class"

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