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The Generalized Circle Fractal: Exploration & Straightedge-Compass Construction

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Xuất bản 09/08/2016
Explore the fractal online at http://demonstrations.wolfram.com/GeneralizedStraightedgeAndCompassFractal/ I recently found this interesting fractal which could be created from a triangle in a very simple way. The basic idea is to start with an equilateral triangle, and keep the new finding midpoint triangles, and then adding the `circles' of those triangles. Here, by `the circles of a triangle' I mean the three circles centered around the vertices of the triangle, which have a radius equal to m times the edge-length of the triangle (where m=1). By repeatedly creating the new midpoint triangles, and their circles I created the interesting patterns described in my previous videos (see below for links). However, today I started to look at what happens when one changes the circle-radius multiplier m. If one creates a configuration of circles by running the original system for a given number of iterations, and then one alters the value of the `multiplier' m one sees the circles change size and position in a beautiful way. In this video I give a rough outline of the idea behind the systems, then I show how the configurations of circles change as the multiplier is changed from 0 to 3, then I describe how these patterns can be made via a straightedge and compass construction (using GeoGebra). New Compass And Straightedge Fractal Explored In Color https://www.youtube.com/watch?v=QNuZ9yXdev0 A New Fractal Created With Compass And Straight Edge https://www.youtube.com/watch?v=SGDQkncxia0 http://www.geogebra.org/cms/en/ https://sites.google.com/site/richardsouthwell254/home
Computer Compass Mathematics Programming Language Paradigm Geometry Field Of Study Circle Midpoint Vertex Compass-and-straightedge Construction Exploration Taxonomy Subject Fractal Concepts/Theories Triangle Mathematical Concept Sierpinski Triangle Namesake Pascal's Triangle Namesake Benoit Mandelbrot Academic Stephen Wolfram Computer Scientist Mechanisns Euclidean Geometry Concepts/Theories GeoGebra Software computational geometry